Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin...
Transcript of Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin...
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Two dimensional quantum memories
David Poulin
Département de PhysiqueUniversité de Sherbrooke
Collaborators H. Bombin, S. Bravyi, G. Duclos-Cianci, O. Landon-Cardinal, and B. Terhal
Institut transdisciplinaire d’informatique quantique, Bromont, April 2013
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 1 / 31
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Outline
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 2 / 31
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Check operators & local codes
Outline
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 3 / 31
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Check operators & local codes
Classical codes
Noisy bitAt each time interval, the bit has a probability p of being flipped.
0→ 1 & 1→ 0
Encoding :0→ 0001→ 111
Receive 001→ 000
Error probability p → 3p2 improvement provided p < 13 .
Quantum encoding :|0〉 → |000〉|1〉 → |111〉 ?
But we can’t look at the bits to see if there was an error!
α|000〉+ β|111〉 →{|000〉 with prob. |α|2|111〉 with prob. |β|2
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 4 / 31
![Page 5: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/5.jpg)
Check operators & local codes
Classical codes
Noisy bitAt each time interval, the bit has a probability p of being flipped.
0→ 1 & 1→ 0
Encoding :0→ 0001→ 111
Receive 001→ 000
Error probability p → 3p2 improvement provided p < 13 .
Quantum encoding :|0〉 → |000〉|1〉 → |111〉 ?
But we can’t look at the bits to see if there was an error!
α|000〉+ β|111〉 →{|000〉 with prob. |α|2|111〉 with prob. |β|2
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 4 / 31
![Page 6: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/6.jpg)
Check operators & local codes
Classical codes
Noisy bitAt each time interval, the bit has a probability p of being flipped.
0→ 1 & 1→ 0
Encoding :0→ 0001→ 111
Receive 001→ 000
Error probability p → 3p2 improvement provided p < 13 .
Quantum encoding :|0〉 → |000〉|1〉 → |111〉 ?
But we can’t look at the bits to see if there was an error!
α|000〉+ β|111〉 →{|000〉 with prob. |α|2|111〉 with prob. |β|2
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 4 / 31
![Page 7: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/7.jpg)
Check operators & local codes
Classical codes
Noisy bitAt each time interval, the bit has a probability p of being flipped.
0→ 1 & 1→ 0
Encoding :0→ 0001→ 111
Receive 001→ 000
Error probability p → 3p2 improvement provided p < 13 .
Quantum encoding :|0〉 → |000〉|1〉 → |111〉 ?
But we can’t look at the bits to see if there was an error!
α|000〉+ β|111〉 →{|000〉 with prob. |α|2|111〉 with prob. |β|2
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 4 / 31
![Page 8: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/8.jpg)
Check operators & local codes
Classical codes
Noisy bitAt each time interval, the bit has a probability p of being flipped.
0→ 1 & 1→ 0
Encoding :0→ 0001→ 111
Receive 001→ 000
Error probability p → 3p2 improvement provided p < 13 .
Quantum encoding :|0〉 → |000〉|1〉 → |111〉 ?
But we can’t look at the bits to see if there was an error!
α|000〉+ β|111〉 →{|000〉 with prob. |α|2|111〉 with prob. |β|2
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 4 / 31
![Page 9: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/9.jpg)
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
![Page 10: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/10.jpg)
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
![Page 11: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/11.jpg)
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
![Page 12: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/12.jpg)
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
![Page 13: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/13.jpg)
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
![Page 14: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/14.jpg)
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
![Page 15: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/15.jpg)
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
![Page 16: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/16.jpg)
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
![Page 17: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/17.jpg)
Check operators & local codes
Syndrome measurement
We do not need to know the bit values for the classical code, onlythe parities.The first two bits are the same, and the last two bits are different.⇒ Flip the last one.These are degenerate measurements: {00,11} vs {01,10}.Quantum mechanics
PE = |00〉〈00|+ |11〉〈11| PO = |01〉〈01|+ |10〉〈10|
⇔ Observable σz ⊗ σz
Measure σzσz = −1 on first two qubits and −1 on last two qubits⇒ apply σx to middle qubit.
This type of measurement requires interactions between qubits
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 5 / 31
![Page 18: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/18.jpg)
Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
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Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
![Page 20: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/20.jpg)
Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
![Page 21: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/21.jpg)
Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
![Page 22: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/22.jpg)
Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
![Page 23: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/23.jpg)
Check operators & local codes
Quantum codes
Set of states that obey a bunch of check conditionsC = {|ψ〉 : Pj |ψ〉 = |ψ〉,∀j}There must be more than one state in C for the code to beinteresting.We measure the check operators, eigenvalue 6= +1 indicates anerror.
LocalityBecause coherent measurement of checks requires coupling thequbits, we restrict the Pj to couple only neighbouring qubits insome geometry.In 2D, this leads to topological codes.
C = degenerate ground space of Hamiltonian H = −∑j Pj .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 6 / 31
![Page 24: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/24.jpg)
Check operators & local codes
Definitions
Λ is a 2D lattice.Each vertex occupied by d-level quantum particle.Hamiltonian H = −∑X⊂Λ PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX acts on 2× 2 cell.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
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Check operators & local codes
Definitions
Λ is a 2D lattice.Each vertex occupied by d-level quantum particle.Hamiltonian H = −∑X⊂Λ PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX acts on 2× 2 cell.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
![Page 26: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/26.jpg)
Check operators & local codes
Definitions
Λ is a 2D lattice.Each vertex occupied by d-level quantum particle.Hamiltonian H = −∑X⊂Λ PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX acts on 2× 2 cell.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
![Page 27: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/27.jpg)
Check operators & local codes
Definitions
Λ is a 2D lattice.Each vertex occupied by d-level quantum particle.Hamiltonian H = −∑X⊂Λ PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX acts on 2× 2 cell.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
![Page 28: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/28.jpg)
Check operators & local codes
Definitions
Λ is a 2D lattice.Each vertex occupied by d-level quantum particle.Hamiltonian H = −∑X⊂Λ PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX acts on 2× 2 cell.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
![Page 29: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/29.jpg)
Check operators & local codes
Definitions
Λ is a 2D lattice.Each vertex occupied by d-level quantum particle.Hamiltonian H = −∑X⊂Λ PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX acts on 2× 2 cell.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
![Page 30: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/30.jpg)
Check operators & local codes
Definitions
Λ is a 2D lattice.Each vertex occupied by d-level quantum particle.Hamiltonian H = −∑X⊂Λ PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX acts on 2× 2 cell.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
![Page 31: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/31.jpg)
Check operators & local codes
Definitions
Λ is a 2D lattice.Each vertex occupied by d-level quantum particle.Hamiltonian H = −∑X⊂Λ PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX acts on 2× 2 cell.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
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Check operators & local codes
Definitions
Λ is a 2D lattice.Each vertex occupied by d-level quantum particle.Hamiltonian H = −∑X⊂Λ PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX acts on 2× 2 cell.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
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Check operators & local codes
Definitions
Λ is a 2D lattice.Each vertex occupied by d-level quantum particle.Hamiltonian H = −∑X⊂Λ PX with
PX = 0 if radius(X )≥ w .[PX ,PY ] = 0.PX are projectors (optional).
Code C = {ψ : PX |ψ〉 = |ψ〉}= ground space of H= image of code projector Π =
∏X PX
With proper coarse graining, we can assume thatΛ is a regular square lattice.Each PX acts on 2× 2 cell.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 7 / 31
![Page 34: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/34.jpg)
Check operators & local codes
Well known examples
Kitaev’s toric codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost known models with topological quantum order
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
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Check operators & local codes
Well known examples
Kitaev’s toric codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost known models with topological quantum order
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
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Check operators & local codes
Well known examples
Kitaev’s toric codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost known models with topological quantum order
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
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Check operators & local codes
Well known examples
Kitaev’s toric codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost known models with topological quantum order
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
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Check operators & local codes
Well known examples
Kitaev’s toric codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost known models with topological quantum order
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
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Check operators & local codes
Well known examples
Kitaev’s toric codeBombin’s topological color codesLevin & Wen’s string-net modelsTuraev-Viro modelsKitaev’s quantum double modelsMost known models with topological quantum order
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 8 / 31
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Check operators & local codes
Lattice
l
l
Two-dimensional square latticePeriodic boundary conditions
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Check operators & local codes
Kitaev’s code
X
XX
ZZ
ZZ
X
As : Bp : H = !!
s
As !!
p
Bp
Site operator:As =
∏i∈v(s) σ
ix
Plaquette operator:Bp =
∏i∈v(p) σ
iz
H = −(∑
s As +∑
p Bp)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 10 / 31
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Check operators & local codes
Kitaev’s code
X
XX
ZZ
ZZ
X
As : Bp : H = !!
s
As !!
p
Bp
Site operator:As =
∏i∈v(s) σ
ix
Plaquette operator:Bp =
∏i∈v(p) σ
iz
H = −(∑
s As +∑
p Bp)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 10 / 31
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Check operators & local codes
Kitaev’s code
X
XX
ZZ
ZZ
X
As : Bp : H = !!
s
As !!
p
Bp
Site operator:As =
∏i∈v(s) σ
ix
Plaquette operator:Bp =
∏i∈v(p) σ
iz
H = −(∑
s As +∑
p Bp)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 10 / 31
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Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
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Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
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Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
![Page 47: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/47.jpg)
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
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Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
![Page 49: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/49.jpg)
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
![Page 50: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/50.jpg)
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
![Page 51: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/51.jpg)
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
![Page 52: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/52.jpg)
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
![Page 53: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/53.jpg)
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
![Page 54: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/54.jpg)
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
![Page 55: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/55.jpg)
Check operators & local codes
Other codes
MotivationAharonov & Eldar ’11: Topological order requires 4-qubitcommuting checks.
Low-weight non-commuting checks possible?Less error-prone
Bombin ’10, Topological subsystemcolour codes
Weight=2.Low threshold.
Bravyi, Duclos-Cianci, DP, SucharaWeight = 3.High threshold.Surface with boundaries.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 11 / 31
![Page 56: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/56.jpg)
Check operators & local codes
Desirable features
Let |ψ1〉 and |ψ2〉 be two code states (ground states).Suppose there exists a local (e.g. single spin) measurement σ thatdistinguishes them.Then the environment can also learn which state is encoded by“looking" at a single spin.
α|ψ1〉+ β|ψ2〉 →{|ψ1〉 with prob. |α|2|ψ2〉 with prob. |β|2
So a code should not have such local “order parameter" :all codes states should look identical locally.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 12 / 31
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Check operators & local codes
Desirable features
Let |ψ1〉 and |ψ2〉 be two code states (ground states).Suppose there exists a local (e.g. single spin) measurement σ thatdistinguishes them.Then the environment can also learn which state is encoded by“looking" at a single spin.
α|ψ1〉+ β|ψ2〉 →{|ψ1〉 with prob. |α|2|ψ2〉 with prob. |β|2
So a code should not have such local “order parameter" :all codes states should look identical locally.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 12 / 31
![Page 58: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/58.jpg)
Check operators & local codes
Desirable features
Let |ψ1〉 and |ψ2〉 be two code states (ground states).Suppose there exists a local (e.g. single spin) measurement σ thatdistinguishes them.Then the environment can also learn which state is encoded by“looking" at a single spin.
α|ψ1〉+ β|ψ2〉 →{|ψ1〉 with prob. |α|2|ψ2〉 with prob. |β|2
So a code should not have such local “order parameter" :all codes states should look identical locally.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 12 / 31
![Page 59: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/59.jpg)
Check operators & local codes
Desirable features
Let |ψ1〉 and |ψ2〉 be two code states (ground states).Suppose there exists a local (e.g. single spin) measurement σ thatdistinguishes them.Then the environment can also learn which state is encoded by“looking" at a single spin.
α|ψ1〉+ β|ψ2〉 →{|ψ1〉 with prob. |α|2|ψ2〉 with prob. |β|2
So a code should not have such local “order parameter" :all codes states should look identical locally.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 12 / 31
![Page 60: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/60.jpg)
Check operators & local codes
Standard definitions
Correctable region
A region M ⊂ Λ is correctable if there exists a recovery operation Rsuch that R(TrMρ) = ρ for all code states ρ.M correctable⇔ No order parameter on M ⇔ ΠOMΠ ∝ Π.
Minimum distanceThe minimum distance d is the size of the smallest non-correctableregion.
Logical operator
Operator L such that L|ψ〉 is a code state for any code state |ψ〉.
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Check operators & local codes
Standard definitions
Correctable region
A region M ⊂ Λ is correctable if there exists a recovery operation Rsuch that R(TrMρ) = ρ for all code states ρ.M correctable⇔ No order parameter on M ⇔ ΠOMΠ ∝ Π.
Minimum distanceThe minimum distance d is the size of the smallest non-correctableregion.
Logical operator
Operator L such that L|ψ〉 is a code state for any code state |ψ〉.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 13 / 31
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Check operators & local codes
Standard definitions
Correctable region
A region M ⊂ Λ is correctable if there exists a recovery operation Rsuch that R(TrMρ) = ρ for all code states ρ.M correctable⇔ No order parameter on M ⇔ ΠOMΠ ∝ Π.
Minimum distanceThe minimum distance d is the size of the smallest non-correctableregion.
Logical operator
Operator L such that L|ψ〉 is a code state for any code state |ψ〉.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 13 / 31
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Holographic Disentangling Lemma
Outline
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 14 / 31
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Holographic Disentangling Lemma
Statement of the lemma
Holographic disentangling lemma (Bravyi, DP, Terhal)Let M ⊂ Λ be a correctable region and suppose that its boundary ∂Mis also correctable. Then, there exists a unitary operator U∂M actingonly on the boundary of M such that, for any code state |ψ〉,
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉
for some fixed state |φM〉 on M.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 15 / 31
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Holographic Disentangling Lemma
With pictures
Let M be correctable.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe area law S(M) ≤ |∂M| for commuting Hamiltonians of Wolf,Verstraete, Hastings, and Cirac.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
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Holographic Disentangling Lemma
With pictures
Let M be correctable.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe area law S(M) ≤ |∂M| for commuting Hamiltonians of Wolf,Verstraete, Hastings, and Cirac.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
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Holographic Disentangling Lemma
With pictures
Let M be correctable.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
ABCD
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe area law S(M) ≤ |∂M| for commuting Hamiltonians of Wolf,Verstraete, Hastings, and Cirac.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
![Page 68: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/68.jpg)
Holographic Disentangling Lemma
With pictures
Let M be correctable.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
ABCD
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe area law S(M) ≤ |∂M| for commuting Hamiltonians of Wolf,Verstraete, Hastings, and Cirac.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
![Page 69: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/69.jpg)
Holographic Disentangling Lemma
With pictures
Let M be correctable.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
ABCD
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe area law S(M) ≤ |∂M| for commuting Hamiltonians of Wolf,Verstraete, Hastings, and Cirac.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
![Page 70: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/70.jpg)
Holographic Disentangling Lemma
With pictures
Let M be correctable.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
ABCD
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe area law S(M) ≤ |∂M| for commuting Hamiltonians of Wolf,Verstraete, Hastings, and Cirac.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
![Page 71: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/71.jpg)
Holographic Disentangling Lemma
With pictures
Let M be correctable.Assume ∂M is correctable.Let M = A ∪ B, M = C ∪ D, and ∂M = B ∪ C.
M
M = Λ\M
ABCD
There exists a unitary transformation U∂M such that, for any|ψ〉 ∈ C
U∂M |ψ〉 = |φM〉 ⊗ |ψ′M〉where |φM〉 is the same for all |ψ〉.
RemarkFor a trivial code TrΠ = 1, every region is correctable, so we recoverthe area law S(M) ≤ |∂M| for commuting Hamiltonians of Wolf,Verstraete, Hastings, and Cirac.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 16 / 31
![Page 72: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/72.jpg)
Holographic Minimum Distance
Outline
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 17 / 31
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Holographic Minimum Distance
Statement of the result
Holographic minimum distance (Bravyi, DP, Terhal)
Region M ⊂ Λ is correctable if its boundary is smaller than theminimum distance |∂M| ≤ cd .
Bulky errors are not problematic: it’s the skinny ones we need toworry about.This hints at our next result: string-like logical operators.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 18 / 31
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Holographic Minimum Distance
Statement of the result
Holographic minimum distance (Bravyi, DP, Terhal)
Region M ⊂ Λ is correctable if its boundary is smaller than theminimum distance |∂M| ≤ cd .
Bulky errors are not problematic: it’s the skinny ones we need toworry about.This hints at our next result: string-like logical operators.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 18 / 31
![Page 75: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/75.jpg)
Holographic Minimum Distance
Statement of the result
Holographic minimum distance (Bravyi, DP, Terhal)
Region M ⊂ Λ is correctable if its boundary is smaller than theminimum distance |∂M| ≤ cd .
Bulky errors are not problematic: it’s the skinny ones we need toworry about.This hints at our next result: string-like logical operators.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 18 / 31
![Page 76: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/76.jpg)
Holographic Minimum Distance
Proof
M
M = Λ\M
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.We can continue to grow M this way until |∂M| ≥ d .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
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Holographic Minimum Distance
Proof
M
M = Λ\M
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.We can continue to grow M this way until |∂M| ≥ d .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
![Page 78: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/78.jpg)
Holographic Minimum Distance
Proof
M
M = Λ\M
ABCD
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.We can continue to grow M this way until |∂M| ≥ d .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
![Page 79: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/79.jpg)
Holographic Minimum Distance
Proof
M
M = Λ\M
ABCD
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.We can continue to grow M this way until |∂M| ≥ d .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
![Page 80: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/80.jpg)
Holographic Minimum Distance
Proof
M
M = Λ\M
ABCD
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.We can continue to grow M this way until |∂M| ≥ d .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
![Page 81: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/81.jpg)
Holographic Minimum Distance
Proof
M
M = Λ\M
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.We can continue to grow M this way until |∂M| ≥ d .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
![Page 82: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/82.jpg)
Holographic Minimum Distance
Proof
M
M = Λ\M
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.We can continue to grow M this way until |∂M| ≥ d .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
![Page 83: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/83.jpg)
Holographic Minimum Distance
Proof
M
M = Λ\M
Let M ⊂ Λ be a correctable region.If |∂M| ≤ d , then ∂M is also correctable.Thus, we can reconstruct any code state ρ from ρAD = Tr∂Mρ.But from the Holographic disentangling lemma, ρAD = ηA ⊗ ρDwith ηA independent of the encoded state ρ.Thus, we can reconstruct ρ from ρD = TrM∪∂Mρ, so M ∪ ∂M iscorrectable.We can continue to grow M this way until |∂M| ≥ d .
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 19 / 31
![Page 84: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/84.jpg)
Capacity-Stability Tradeoff
Outline
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 20 / 31
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Capacity-Stability Tradeoff
Statement of the result
n = number of qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
Kitaev’s codes (with punctures) saturate this bound, so it is tight.No “good codes" in 2D, i.e. k ∝ n and d ∝ n.For 2D classical codes, k ≤ c n√
d.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
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Capacity-Stability Tradeoff
Statement of the result
n = number of qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
Kitaev’s codes (with punctures) saturate this bound, so it is tight.No “good codes" in 2D, i.e. k ∝ n and d ∝ n.For 2D classical codes, k ≤ c n√
d.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
![Page 87: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/87.jpg)
Capacity-Stability Tradeoff
Statement of the result
n = number of qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
Kitaev’s codes (with punctures) saturate this bound, so it is tight.No “good codes" in 2D, i.e. k ∝ n and d ∝ n.For 2D classical codes, k ≤ c n√
d.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
![Page 88: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/88.jpg)
Capacity-Stability Tradeoff
Statement of the result
n = number of qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
Kitaev’s codes (with punctures) saturate this bound, so it is tight.No “good codes" in 2D, i.e. k ∝ n and d ∝ n.For 2D classical codes, k ≤ c n√
d.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
![Page 89: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/89.jpg)
Capacity-Stability Tradeoff
Statement of the result
n = number of qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
Kitaev’s codes (with punctures) saturate this bound, so it is tight.No “good codes" in 2D, i.e. k ∝ n and d ∝ n.For 2D classical codes, k ≤ c n√
d.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
![Page 90: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/90.jpg)
Capacity-Stability Tradeoff
Statement of the result
n = number of qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
Kitaev’s codes (with punctures) saturate this bound, so it is tight.No “good codes" in 2D, i.e. k ∝ n and d ∝ n.For 2D classical codes, k ≤ c n√
d.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
![Page 91: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/91.jpg)
Capacity-Stability Tradeoff
Statement of the result
n = number of qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
Kitaev’s codes (with punctures) saturate this bound, so it is tight.No “good codes" in 2D, i.e. k ∝ n and d ∝ n.For 2D classical codes, k ≤ c n√
d.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
![Page 92: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/92.jpg)
Capacity-Stability Tradeoff
Statement of the result
n = number of qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
Kitaev’s codes (with punctures) saturate this bound, so it is tight.No “good codes" in 2D, i.e. k ∝ n and d ∝ n.For 2D classical codes, k ≤ c n√
d.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
![Page 93: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/93.jpg)
Capacity-Stability Tradeoff
Statement of the result
n = number of qubitsk = number of encoded qubitsd = minimum distance
Capacity-Stability Tradeoff
k ≤ c nd2
Singleton’s bound: k ≤ n − 2(d − 1).
Hamming bound: k ≤ n[1− d
2n log 3− H( d2n )].
Kitaev’s codes (with punctures) saturate this bound, so it is tight.No “good codes" in 2D, i.e. k ∝ n and d ∝ n.For 2D classical codes, k ≤ c n√
d.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 21 / 31
![Page 94: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/94.jpg)
String-Like Logical Operators
Outline
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 22 / 31
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String-Like Logical Operators
Statement of the result
String-like logical operators (Haah, Preskill)There exists a non-trivial logical operator supported on a string-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
Well known for Kitaev’s toric code.Intuitive for known models that support anyons:
The ground state can be changed by dragging an anyon around atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
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String-Like Logical Operators
Statement of the result
String-like logical operators (Haah, Preskill)There exists a non-trivial logical operator supported on a string-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
Well known for Kitaev’s toric code.Intuitive for known models that support anyons:
The ground state can be changed by dragging an anyon around atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
![Page 97: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/97.jpg)
String-Like Logical Operators
Statement of the result
String-like logical operators (Haah, Preskill)There exists a non-trivial logical operator supported on a string-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
Well known for Kitaev’s toric code.Intuitive for known models that support anyons:
The ground state can be changed by dragging an anyon around atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
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String-Like Logical Operators
Statement of the result
String-like logical operators (Haah, Preskill)There exists a non-trivial logical operator supported on a string-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
Well known for Kitaev’s toric code.Intuitive for known models that support anyons:
The ground state can be changed by dragging an anyon around atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
![Page 99: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/99.jpg)
String-Like Logical Operators
Statement of the result
String-like logical operators (Haah, Preskill)There exists a non-trivial logical operator supported on a string-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
Well known for Kitaev’s toric code.Intuitive for known models that support anyons:
The ground state can be changed by dragging an anyon around atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
![Page 100: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/100.jpg)
String-Like Logical Operators
Statement of the result
String-like logical operators (Haah, Preskill)There exists a non-trivial logical operator supported on a string-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
Well known for Kitaev’s toric code.Intuitive for known models that support anyons:
The ground state can be changed by dragging an anyon around atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
![Page 101: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/101.jpg)
String-Like Logical Operators
Statement of the result
String-like logical operators (Haah, Preskill)There exists a non-trivial logical operator supported on a string-likeregion.
Exists UM such that UM |ψ〉 = |ψ′〉.|ψ〉 6= |ψ′〉.|ψ〉, |ψ′〉 ∈ C.
Λ
M
Well known for Kitaev’s toric code.Intuitive for known models that support anyons:
The ground state can be changed by dragging an anyon around atopologically non-trivial loop.This process is realized on a string, and generated a logicaloperation.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 23 / 31
![Page 102: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/102.jpg)
Thermal instability
Outline
1 Check operators & local codes
2 Holographic Disentangling Lemma
3 Holographic Minimum Distance
4 Capacity-Stability Tradeoff
5 String-Like Logical Operators
6 Thermal instability
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 24 / 31
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Thermal instability
Classical memories are robust
0=
1=
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 104: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/104.jpg)
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 105: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/105.jpg)
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 106: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/106.jpg)
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 107: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/107.jpg)
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 108: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/108.jpg)
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 109: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/109.jpg)
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 110: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/110.jpg)
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 111: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/111.jpg)
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 112: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/112.jpg)
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 113: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/113.jpg)
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 114: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/114.jpg)
Thermal instability
Classical memories are robust
Energy barrier ∝ √n between logical states through local moves.Boltzmann: configuration x has probability ∝ exp(−E(x)/T ).Probability of flipping the whole configuration by local movesdecreases with n.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 25 / 31
![Page 115: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/115.jpg)
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
![Page 116: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/116.jpg)
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
![Page 117: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/117.jpg)
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
![Page 118: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/118.jpg)
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
![Page 119: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/119.jpg)
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
| "" . . . "i
| ## . . . #i2B
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
![Page 120: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/120.jpg)
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
| "" . . . "i
| ## . . . #i2B
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
![Page 121: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/121.jpg)
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
| "" . . . "i
| ## . . . #i2B
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
![Page 122: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/122.jpg)
Thermal instability
Local order parameter & decoherence
System has two ground states | ↑↑ . . . ↑〉 and | ↓↓ . . . ↓〉.α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 does not evolve in time.
Local observable σzi distinguishes them.
Local order parameter σz .Local perturbation Bσz lifts degeneracy:
| "" . . . "i
| ## . . . #i2B
α| ↑↑ . . . ↑〉+ β| ↓↓ . . . ↓〉 t−→ e−iBtα| ↑↑ . . . ↑〉+ eiBtβ| ↓↓ . . . ↓〉
Unknown B:(
|α|2 e−i2Btαβ∗
ei2Btα∗β |β|2) ∫
dB−−−→(|α|2 00 |β|2
)
Quantum superposition→ Statistical mixture.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 26 / 31
![Page 123: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/123.jpg)
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
![Page 124: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/124.jpg)
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
![Page 125: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/125.jpg)
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
![Page 126: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/126.jpg)
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
![Page 127: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/127.jpg)
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
![Page 128: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/128.jpg)
Thermal instability
Topological quantum order
Bravyi, Hastings, & Michalakis
(TQO1) System has no local order parameter.(TQO2) System is locally consistent.
The system has a stable spectrum.Long lived memory at zero temperature.
H = −∑
i
σzi σ
zi+1 + σz
23
The ground state manifold changes abruptly when including site 23.Can we combine this spectral stability with the thermal stability ofthe 2D Ising model?
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 27 / 31
![Page 129: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/129.jpg)
Thermal instability
Thermal stability vs spectral stabilisy
Main result (Landon-Cardinal & DP)The minimum set of conditions required to prove spectral stability implythe existence of a sequence of local maps that corrupt the system atan energy cost bounded by a constant.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 28 / 31
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Thermal instability
Noise model
1 2 k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 131: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/131.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 132: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/132.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 133: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/133.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 134: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/134.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 135: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/135.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 136: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/136.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 137: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/137.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 138: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/138.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 139: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/139.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 140: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/140.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 141: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/141.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 142: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/142.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 143: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/143.jpg)
Thermal instability
Noise model
1 2 k
Pk�1,k
1 Apply random unitary on sites 1 & 2.2 Measure P12
If P12 = 0 go to 1.3 Apply random unitary on site 3.4 Measure P23
If P23 = 0 go to 3.
Only a constant amount of energy at any given time.No need to backtrack.Number of steps ∝ lattice linear size.If successful, final state is corrupted. (not trivial)
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 29 / 31
![Page 144: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/144.jpg)
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
![Page 145: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/145.jpg)
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
![Page 146: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/146.jpg)
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
![Page 147: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/147.jpg)
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
![Page 148: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/148.jpg)
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
![Page 149: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/149.jpg)
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
![Page 150: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/150.jpg)
Conclusion
Take home messages
Quantum error correction requires joint qubit measurements.Local check operators in 2D⇒ topological codes.
Natural relation between codes and quantum many-body physics.Large minimum distance⇔ Topological quantum order(order with no local order parameter).Disentangling lemma⇔ Area law.Fault tolerant threshold⇔ phase transition.
Impossible to combine spectral and thermal stability with existingtools.
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 30 / 31
![Page 151: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/151.jpg)
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
![Page 152: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/152.jpg)
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
![Page 153: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/153.jpg)
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
![Page 154: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/154.jpg)
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
![Page 155: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/155.jpg)
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
![Page 156: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/156.jpg)
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
![Page 157: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/157.jpg)
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
![Page 158: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/158.jpg)
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
![Page 159: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/159.jpg)
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
![Page 160: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/160.jpg)
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
![Page 161: Two dimensional quantum memories - usherbrooke.ca · Two dimensional quantum memories David Poulin Département de Physique Université de Sherbrooke Collaborators H. Bombin, S. Bravyi,](https://reader036.fdocuments.in/reader036/viewer/2022062505/5ec5ff7235716311ae10928e/html5/thumbnails/161.jpg)
Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31
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Conclusion
Open questions
String-like logical operators +TQO⇒ constant energy barrier.This is not directly related to thermal instability.2D Ising model has an energy barrier ∝ √n, but an energy ∝ n atfinite temperature.What matters is entropy (for a given energy, many moreconfigurations many with small error droplets than with a large one).Can we characterize all string-like logical operators?We have shown information corruption in time ∝ √n. Can it beparallelized? (Percolation)Relation between commuting projector codes and anyon models.
Can we engineer dead ends?Memory that is stabilized by complexity.
Extension to subsystem codes?With local stabilizer (Bombin) and without (Bacon-Shor).
Extend to frustration-free Hamiltonians (and therefore to allgapped Hamiltonians, i.e. Hastings).
David Poulin (Sherbrooke) 2D quantum memories INTRIQ13 31 / 31