Topological Quantum Error Correcting Codes · •Homological codes Determine a more general...
Transcript of Topological Quantum Error Correcting Codes · •Homological codes Determine a more general...
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Topological Quantum Error Correcting Codes
Kasper Duivenvoorden
JARA IQI, RWTH Aachen
Topological Phases of Quantum Matter – 4 September 2014
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Topology Physical errors are local Store information globally
Definition: locally ground states are the same
Gapped topological phase with ground space degeneracy:
Use ground space manifold of a topological phase to encode quantum information
A
B
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Overview
• Homological codes
A formalism using emphasizing topology
• Fractal codes
Possibly thermally stable codes
• Chamon code
Work in progress
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Example: Toric Code
Logicals: preserves the code space
Ground states:
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Homological codes Lattice CW-complex Point 0-cells Line 1-cells Surface 2-cells ... ` k-cells
Boundary map
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Co-Boundary map
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Homological codes Boundary map
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Co-Boundary map
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Properties
1: 2:
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Homological codes
Step 1: CW-complex
Step 2: Spins on every k-cells Example: k=1
Step 3: Hamiltonian = z
z
z
z
= x
x
x
x
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Homological codes =
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Logicals
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Homological codes =
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Logicals
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Toric Codes D – dimensional lattice, qubits on k cells k and D-k dimensional logicals
2D 3D 4D
Dimensionality of logicals
Z X 1 1 1 2 2 2
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Thermal Stability
Movement of anyons
At T=0 Tunneling p = exp(-L)
At T>0 Thermal excitation p = exp(-E/kT) No stability!
E
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Thermal Stability
At T>0 Thermal excitation p = exp(-E/kT) Stability!
E
Chesi, Loss, Bravyi, Terhal, 2010
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Toric Codes D – dimensional lattice, qubits on k cells k and D-k dimensional logicals
2D 3D 4D
Dim. of logicals Stable Memory
Z X 1 1 No 1 2 2 2 Quantum
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Toric Codes
Drawbacks - Constant information - We only have 3 dimensions
Fractal codes
D – dimensional lattice, qubits on k cells k and D-k dimensional logicals
2D 3D 4D
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Overview
• Homological codes
A formalism using emphasizing topology
• Fractal codes
Possibly thermally stable codes
• Chamon code
Work in progress
Yoshida, 2013
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Fractal Codes Algebraic representation of Z-type operators
• Generalize to higher dimensions: • More spins per site:
y z x
Yoshida, 2013
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Fractal Codes
Commutation relations
Interpretation: shift X[g] by 0 , Z[f] and X[g] anticommute shift X[g] by 2 , Z[f] and X[g] anticommute
Yoshida, 2013
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Fractal Codes Hamiltonian
Example:
Z Z Z
Logical X[g] g(1+fy) = 0 modulo 2 Assumptions:
Z
y x
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Fractal Codes Logical: X[g]
X X X X X X X X X X X X X X
Z Z Z Z
X
Hamiltonian terms:
Set of Logicals: fractal like point like
y x
X X X X X X X X X 1 2 3 2 1
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Fractal Codes Excitations
Z
Z Z Z
X X X X X
X X X X X X X
Compare with Toric code: Possible energy barrier
Yoshida, 2013
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Quantum Fractal Codes Now in 3 dimensions:
Terms commute since:
Fractal logicals in x-y plane and in x-z plane
Yoshida, 2013
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First spin propagation in y direction Second spin propagation in z direction
Quantum Fractal Codes Excitations:
Example:
X
X X X
X X X
Yoshida, 2013
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X
X X X
X X X
Quantum Fractal Codes
X
X y
z
first
second cancellation
Excitations can propagate freely in the z-y direction Due to algebraic dependence of g and f
X X X
X
X
X second
first
Yoshida, 2013
Excitations:
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Quantum Fractal Codes
X
X y
z
first
second cancellation X X X
X
X
X
No algebraic dependence (at least) logarithmic energy barrier
E ≈ Log(L) p ≈ exp(-E/kT) Polynomial rate
Optimal system size
Memory time
System size
Bravyi, Haah, 2011
Bravyi, Haah, 2013
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Overview
• Homological codes
A formalism using emphasizing topology
• Fractal codes
Possibly thermally stable codes
• Chamon code
Work in progress
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Chamon Code
Chamon, 2005
X ↔ Z
z
z
z
z x
x
x
x
Z
X X
Z
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Chamon Code
Nog
Bravyi, Leemhuis, Terhal, 2011
x-X-X-X-X-x
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Chamon Code
pfailure
p
pfailure
p
Increased
system size
Ben-Or, Aharonov, 1999
Noise (p) Encode Decode pfailure
Error threshold
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Chamon Code
Noise (p) Encode Decode pfailure
Error threshold
Can be related to percolation
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Chamon Code
Bath (T) Time (t)
Encode
Memory Time
Decode psucces
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Future Work
• Homological codes Determine a more general condition for thermal
stability in terms of Hamiltonian properties
• Fractal codes Understand relation between fractal codes and
topological order
• Chamon code Consider better (but computational more
demanding) decoders
Conclustion: Existence of a quantum memory in 3D is still open
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Fractal Codes
Logicals: Z
Logicals: X
Similarly:
Yoshida, 2013
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Fractal Codes Logicals: Z
Logicals: X
Commutation relations
Both fractal like!
Yoshida, 2013
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Error Correction
noise
Communication
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Error Correction
no
ise
Storage
time
space
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Error Correction
noise
Solution: Build in Redundancy
1 11111 1 11001
Encoding Decoding
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Stabilizer Codes
Trade off Information k Stability d (number of qubits) (weight of a logical)
Logicals / Symmetries:
Gottesman, PRA 1996
• Allow for fault tollerant computations: errors do not accumalate when correcting • Overhead independent of computational time
Memory time
Gottesman, PRA 1998 Gottesman, 1310.2984
Stabilizers:
Ground states:
Error:
Exitated states:
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Relation to topological order
Topological order at T > 0 ↔ Stable quantum memory at T > 0
Mazac, Hamma, 2012 Caselnovo, Chamon , 2007/2008
Topological Entropy
2D 3D 4D
Adiabatic Evolution
Hastings, 2011
No quantum memory in 2D
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Homological codes
What can we learn k: stored information, related to genus d: distance, related to systole n: number of qubits, related to volume Intuitively... (sys)2 ≤ volume ... or better genus x (sys)2 ≤ volume
In general genus / log2 (genus) x (sys)2 ≤ volume
k / log2 (k) x (d)2 ≤ n
Gromov, 1992 Delfosse, 1301.6588
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Example: Toric Code
Ground states:
Error:
Excitations
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Plaquette:
Star:
Example: Toric Code
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Fractal Codes Algebraic representation of Hamiltonian Example: toric code
Plaquette:
Star:
Yoshida, 2013
y x