Class D Audio Amplifier With Ferroxcube Gapped Toroid Output Filter
Universal Quantum Computation with Gapped Boundaries
Transcript of Universal Quantum Computation with Gapped Boundaries
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Universal Quantum Computation with GappedBoundaries
Iris Cong, Meng Cheng, Zhenghan [email protected]
Department of PhysicsHarvard University, Cambridge, MA
April 23rd, 2018
Refs.: C, Cheng, Wang, Comm. Math. Phys. (2017) 355: 645.
C, Cheng, Wang, Phys. Rev. Lett. 119, 170504 (2017).
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Introduction: Quantum Computation
Quantum computing:
Feynman (1982): Use a quantum-mechanical computer to simulatequantum physics (classically intractable)Encode information in qubits, gates = unitary operationsApplications: Factoring/breaking RSA (Shor, 1994), quantummachine learning, ...
Major challenge: local decoherence of qubits
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Introduction: Quantum Computation
Quantum computing:
Feynman (1982): Use a quantum-mechanical computer to simulatequantum physics (classically intractable)Encode information in qubits, gates = unitary operationsApplications: Factoring/breaking RSA (Shor, 1994), quantummachine learning, ...
Major challenge: local decoherence of qubits
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Introduction: Topological Quantum Computation
Topological quantum computing (TQC) (Kitaev, 1997; Freedmanet al., 2003):
Encode information in topological degrees of freedomPerform topologically protected operations
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Topological Quantum Computation
Traditional realization of TQC: anyons in topological phases ofmatter (e.g. Fractional Quantum Hall - FQH)
Elementary (quasi-)particles in 2 dimensions s.t. |ψ1ψ2〉 = e iφ|ψ2ψ1〉
Qubit encoding: degeneracy arising from the fusion rules
e.g. Toric code: e imj ⊗ ekml = e(i+k)(mod 2)m(j+l)(mod 2) (i , j = 0, 1)e.g. Ising: σ ⊗ σ = 1⊕ ψ, ψ ⊕ ψ = 1e.g. Fibonacci: τ ⊗ τ = 1⊕ τ
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Topological Quantum Computation
Traditional realization of TQC: anyons in topological phases ofmatter (e.g. Fractional Quantum Hall - FQH)
Elementary (quasi-)particles in 2 dimensions s.t. |ψ1ψ2〉 = e iφ|ψ2ψ1〉Qubit encoding: degeneracy arising from the fusion rules
e.g. Toric code: e imj ⊗ ekml = e(i+k)(mod 2)m(j+l)(mod 2) (i , j = 0, 1)e.g. Ising: σ ⊗ σ = 1⊕ ψ, ψ ⊕ ψ = 1e.g. Fibonacci: τ ⊗ τ = 1⊕ τ
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Topological Quantum Computation
Topologically protected operations: Braiding of anyons
Move one anyon around another → pick up phase (due toAharonov-Bohm)
Figures: (1) Z. Wang, Topological Quantum Computation.
(2) C. Nayak et al., Non-abelian anyons and topological quantum computation.
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Problem
Problem: Abelian anyons have no degeneracy, so no computationpower → need non-abelian anyons for TQC (e.g. ν = 5/2, 12/5)
These are difficult to realize, existence is still uncertain
Question: Given a top. phase that supports only abelian anyons, is itpossible “engineer” other non-abelian objects?
Answer: Yes! We consider boundaries of the topological phase →gapped boundariesWe’ll even get a universal gate set from gapped boundaries of anabelian phase (specifically, bilayer ν = 1/3 FQH)
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Problem
Problem: Abelian anyons have no degeneracy, so no computationpower → need non-abelian anyons for TQC (e.g. ν = 5/2, 12/5)
These are difficult to realize, existence is still uncertain
Question: Given a top. phase that supports only abelian anyons, is itpossible “engineer” other non-abelian objects?
Answer: Yes! We consider boundaries of the topological phase →gapped boundariesWe’ll even get a universal gate set from gapped boundaries of anabelian phase (specifically, bilayer ν = 1/3 FQH)
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Problem
Problem: Abelian anyons have no degeneracy, so no computationpower → need non-abelian anyons for TQC (e.g. ν = 5/2, 12/5)
These are difficult to realize, existence is still uncertain
Question: Given a top. phase that supports only abelian anyons, is itpossible “engineer” other non-abelian objects?
Answer: Yes! We consider boundaries of the topological phase →gapped boundariesWe’ll even get a universal gate set from gapped boundaries of anabelian phase (specifically, bilayer ν = 1/3 FQH)
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Problem
Problem: Abelian anyons have no degeneracy, so no computationpower → need non-abelian anyons for TQC (e.g. ν = 5/2, 12/5)
These are difficult to realize, existence is still uncertain
Question: Given a top. phase that supports only abelian anyons, is itpossible “engineer” other non-abelian objects?
Answer: Yes! We consider boundaries of the topological phase →gapped boundariesWe’ll even get a universal gate set from gapped boundaries of anabelian phase (specifically, bilayer ν = 1/3 FQH)
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 12: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/12.jpg)
Overview
Framework of TQC
Introduce gapped boundaries and their framework
Gapped boundaries for TQC
Universal TQC with gapped boundaries in bilayer ν = 1/3 FQH
Summary and Outlook
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Overview
Framework of TQC
Introduce gapped boundaries and their framework
Gapped boundaries for TQC
Universal TQC with gapped boundaries in bilayer ν = 1/3 FQH
Summary and Outlook
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Framework of TQC
Formally, an anyon model B consists of the following data:
Set of anyon types/labels: {a, b, c ...}, one of whichshould represent the vacuum 1
Each anyon type has a topological twist θi ∈ U(1):
For each pair of anyon types, a set of fusion rules: a⊗ b = ⊕cNcabc .
The fusion space of a⊗ b is a vector space Vab with basis V cab.1
The fusion space of a1 ⊗ a2 ⊗ ...⊗ an to b is a vector space V ba1a2...an
with basis given by anyon labels in intermediate segments, e.g.
1More general cases exist, but are not used in this talk.
Figures: Z. Wang, Topological Quantum Computation (2010)
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Framework of TQC
Formally, an anyon model B consists of the following data:
Set of anyon types/labels: {a, b, c ...}, one of whichshould represent the vacuum 1
Each anyon type has a topological twist θi ∈ U(1):
For each pair of anyon types, a set of fusion rules: a⊗ b = ⊕cNcabc .
The fusion space of a⊗ b is a vector space Vab with basis V cab.1
The fusion space of a1 ⊗ a2 ⊗ ...⊗ an to b is a vector space V ba1a2...an
with basis given by anyon labels in intermediate segments, e.g.
1More general cases exist, but are not used in this talk.
Figures: Z. Wang, Topological Quantum Computation (2010)
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Framework of TQC
Formally, an anyon model B consists of the following data: [Cont’d]
Associativity: for each (a, b, c , d), a set of(unitary) linear transformations{F abc
d ;ef : V abcd → V abc
d } satisfying“pentagons”
Figures: Z. Wang, Topological Quantum Computation (2010)
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Framework of TQC
Formally, an anyon model B consists of the following data: [Cont’d]
(Non-degenerate) braiding: Foreach (a, b, c) s.t. Nc
a,b 6= 0, a set
of phases1 Rcab ∈ U(1) compatible
with associativity (“hexagons”):
Mathematically, this is captured bya (unitary) modular tensor category(UMTC)
1More general cases exist, but are not used in this talk.
Figures: Z. Wang, Topological Quantum Computation (2010)
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Framework of TQC
Formally, an anyon model B consists of the following data: [Cont’d]
(Non-degenerate) braiding: Foreach (a, b, c) s.t. Nc
a,b 6= 0, a set
of phases1 Rcab ∈ U(1) compatible
with associativity (“hexagons”):
Mathematically, this is captured bya (unitary) modular tensor category(UMTC)
1More general cases exist, but are not used in this talk.
Figures: Z. Wang, Topological Quantum Computation (2010)
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Framework of TQC
In this framework, a 2D topological phase of matter is anequivalence class of gapped Hamiltonians H = {H} whoselow-energy excitations form the same anyon model BExamples (on the lattice) include Kitaev’s quantum double models,Levin-Wen string-net models, ...
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Framework of TQC: Example
Physical system: Bilayer FQH system, 1/3 Laughlin state of oppositechirality in each layer
Equivalent to Z3 toric code (Kitaev, 2003)
Anyon types: eamb, a, b = 0, 1, 2
Twist: θ(eamb) = ωab where ω = e2πi/3
Fusion rules: eamb ⊗ ecmd → e(a+c)(mod 3)m(b+d)(mod 3)
F symbols all trivial (0 or 1)
Reamb,ecmd = e2πibc/3
UMTC: SU(3)1 × SU(3)1 ∼= D(Z3) = Z(Rep(Z3)) = Z(VecZ3)
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Overview
Framework of TQC
Introduce gapped boundaries and their framework
Gapped boundaries for TQC
Universal gate set with gapped boundaries in bilayer ν = 1/3 FQH
Summary and Outlook
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Gapped Boundaries: Framework
A gapped boundary is an equivalence class of gapped local(commuting) extensions of H ∈ H to the boundary
In the anyon model: Collection of bulk bosonic (θ = 1) anyons whichcondense to vacuum on the boundary (think Bose condensation)
All other bulk anyons condense to confined “boundary excitations”α, β, γ...
Mathematically, Lagrangian algebra A ∈ B
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Gapped Boundaries: Framework
A gapped boundary is an equivalence class of gapped local(commuting) extensions of H ∈ H to the boundary
In the anyon model: Collection of bulk bosonic (θ = 1) anyons whichcondense to vacuum on the boundary (think Bose condensation)
All other bulk anyons condense to confined “boundary excitations”α, β, γ...
Mathematically, Lagrangian algebra A ∈ B
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Gapped Boundaries: Framework
More rigorously, gapped boundaries come with M symbols (like Fsymbols for the bulk):
(More generally, we also define these with boundary excitations, but thatis unnecessary for this talk.)M symbols must be compatible with F symbols (“mixed pentagons”):
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Gapped Boundaries: Framework
More rigorously, gapped boundaries come with M symbols (like Fsymbols for the bulk):
(More generally, we also define these with boundary excitations, but thatis unnecessary for this talk.)
M symbols must be compatible with F symbols (“mixed pentagons”):
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Gapped Boundaries: Framework
More rigorously, gapped boundaries come with M symbols (like Fsymbols for the bulk):
(More generally, we also define these with boundary excitations, but thatis unnecessary for this talk.)M symbols must be compatible with F symbols (“mixed pentagons”):
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Gapped Boundaries: Example
Physical system: Bilayer ν = 1/3 FQH, equiv. to Z3 toric code
Anyon types: eamb, a, b = 0, 1, 2
Two gapped boundary types:
Electric charge condensate: A1 = 1⊕ e ⊕ e2
Magnetic flux condensate: A2 = 1⊕m ⊕m2
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Gapped Boundaries: Example
Physical system: Bilayer ν = 1/3 FQH, equiv. to Z3 toric code
Anyon types: eamb, a, b = 0, 1, 2
Two gapped boundary types:
Electric charge condensate: A1 = 1⊕ e ⊕ e2
Magnetic flux condensate: A2 = 1⊕m ⊕m2
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Gapped Boundaries: Example
Physical system: Bilayer ν = 1/3 FQH, equiv. Z3 toric code
We will work mainly with A1 = 1⊕ e ⊕ e2:
Algebraically, A1, A2 are equivalent by electric-magnetic dualityEasier to work with charge condensate - read-out can be done bymeasuring electric charge (Barkeshli, 2016)It is interesting to consider both A1 and A2 at the same time - we dothis in a separate paper3, will briefly mention in our Outlook
M symbols for this theory are all 0 or 1
3C, Cheng, Wang, Phys. Rev. B 96, 195129 (2017)Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Gapped Boundaries: Example
Physical system: Bilayer ν = 1/3 FQH, equiv. Z3 toric code
We will work mainly with A1 = 1⊕ e ⊕ e2:
Algebraically, A1, A2 are equivalent by electric-magnetic dualityEasier to work with charge condensate - read-out can be done bymeasuring electric charge (Barkeshli, 2016)It is interesting to consider both A1 and A2 at the same time - we dothis in a separate paper3, will briefly mention in our Outlook
M symbols for this theory are all 0 or 1
3C, Cheng, Wang, Phys. Rev. B 96, 195129 (2017)Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Overview
Framework of TQC
Introduce gapped boundaries and their framework
Gapped boundaries for TQC
Universal gate set with gapped boundaries in bilayer ν = 1/3 FQH
Summary and Outlook
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Gapped Boundaries for TQC
Qudit encoding:
Gapped boundaries give rise to a natural ground state degeneracy: ngapped boundaries on a plane, with total charge vacuum
For qudit encoding: use n = 2
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Gapped Boundaries for TQC
Qudit encoding:
Gapped boundaries give rise to a natural ground state degeneracy: ngapped boundaries on a plane, with total charge vacuum
For qudit encoding: use n = 2
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Gapped Boundaries for TQC: Example
For our bilayer ν = 1/3 FQH system, we have:
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Gapped Boundaries for TQC
Topologically protected operations:
Tunnel-a operations
Loop-a operations
Braiding gapped boundaries
Topological charge measurement∗
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Tunnel-a Operations
Starting from state |b〉, tunnel an a anyon from A1 to A2:
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Tunnel-a Operations
Compute using M-symbols:
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Tunnel-a Operations
Result:
Wa(γ) =∑c
Mabc (A1)[Mab
c ]†(A2)
√dadbdc
(1)
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Loop-a Operations
Starting from state |b〉, loop an a anyon around one of the boundaries:
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Loop-a Operations
Similar computation methods lead to the formula:
where sab = s̃ab/db is given by the modular S matrix of the theory:
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Braiding Gapped Boundaries
Braid gapped boundaries around each other:
(Mathematically, this gives a representation of the (spherical) 2n-strandpure braid group.)
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Braiding Gapped Boundaries
Simplify with (bulk) R and F moves to get:
σ22 =∑c,c ′
F a2a1b1b2;c ′c
Rb1a1c ′ Ra1b1
c (F a2a1b1b2
)−1c ′′c ′ (2)
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Gapped Boundaries for TQC: Example
For our bilayer ν = 1/3 FQH case: (A1 = A2 = 1⊕ e ⊕ e2)
Tunnel an e anyon from A1 to A2: Wa(γ)|b〉 = |a⊗ b〉→We(γ) = σx3 , where σx3 |i〉 = |(i + 1)(mod 3)〉Loop an m anyon around A2: Wm(α2)|e j〉 = ωj |e j〉→Wm(α2) = σz3Braid gapped boundaries: get ∧σz3
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Topological Charge Projection/Measurement
Motivation:
Property F conjecture (Naidu and Rowell, 2011): Braidings alonecannot be universal for TQC for most physically plausible systems
Topological charge projection (TCP) (Barkeshli and Freedman,2016):
Doubled theories: Wilson line lifts to a loop → measure topologicalcharge through the loop
Resulting projection operator: (Ox(β) = Wxx(γi ) or Wx(αi ))
P(a)β =
∑x∈C
S0aS∗xaOx(β). (3)
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Topological Charge Projection/Measurement
Motivation:
Property F conjecture (Naidu and Rowell, 2011): Braidings alonecannot be universal for TQC for most physically plausible systems
Topological charge projection (TCP) (Barkeshli and Freedman,2016):
Doubled theories: Wilson line lifts to a loop → measure topologicalcharge through the loop
Resulting projection operator: (Ox(β) = Wxx(γi ) or Wx(αi ))
P(a)β =
∑x∈C
S0aS∗xaOx(β). (3)
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Topological Charge Projection/Measurement
Topological charge projection (TCP): [Cont’d]
Given an anyon theory C, its S, T matrices
S =
{ }, T = diag(θi )
give mapping class group representations VC(Y ) for surfaces Y .Barkeshli and Freedman showed that topological charge projectionsgenerate all matrices in VC(Y )
General topological charge measurements (TCMs):
Projection operators P(a)β =
∑x∈C S0aS
∗xaOx(β) → topological charge
measurements perform the complement of P(a)β
1
Not always physical, but special cases are symmetry protected – weexamine this
1More general cases exist, but are not used in this talk.
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 47: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/47.jpg)
Topological Charge Projection/Measurement
Topological charge projection (TCP): [Cont’d]
Given an anyon theory C, its S, T matrices
S =
{ }, T = diag(θi )
give mapping class group representations VC(Y ) for surfaces Y .Barkeshli and Freedman showed that topological charge projectionsgenerate all matrices in VC(Y )
General topological charge measurements (TCMs):
Projection operators P(a)β =
∑x∈C S0aS
∗xaOx(β) → topological charge
measurements perform the complement of P(a)β
1
Not always physical, but special cases are symmetry protected – weexamine this
1More general cases exist, but are not used in this talk.
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 48: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/48.jpg)
Overview
Framework of TQC
Introduce gapped boundaries and their framework
Gapped boundaries for TQC
Universal gate set with gapped boundaries in bilayer ν = 1/3FQH
Summary and Outlook
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Universal Gate Set
Universal (metaplectic) gate set for the qutrit model (Cui and Wang,2015):
1 The single-qutrit Hadamard gate H3, defined asH3|j〉 = 1√
3
∑2i=0 ω
ij |i〉, j = 0, 1, 2, ω = e2πi/3
2 The two-qutrit entangling gate SUM3, defined asSUM3|i〉|j〉 = |i〉|(i + j) mod 3〉, i , j = 0, 1, 2.
3 The single-qutrit generalized phase gate Q3 = diag(1, 1, ω).
4 Any nontrivial single-qutrit classical (i.e. Clifford) gate not equal toH23 .
5 A projection M of a state in the qutrit space C3 to Span{|0〉} andits orthogonal complement Span{|1〉, |2〉}, so that the resulting stateis coherent if projected into Span{|1〉, |2〉}.
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Universal Gate Set - Bilayer ν = 1/3 FQH
Universal (metaplectic) gate set for the qutrit model (Cui and Wang,2015):
1 H3 = S for C = SU(3)1 (single layer ν = 1/3 FQH), so it can beimplemented by TCP.
2 ∧σz3 can be implemented by gapped boundary braiding. Conjugatesecond qutrit by H3 to get SUM3.
3 TCP can implement diag(1, ω, ω) (Dehn twist of SU(3)1). Follow byσx3 for Q3.
4 Any nontrivial single-qutrit classical (i.e. Clifford) gate not equal toH23 - we have a Pauli-X from tunneling e.
5 Projective measurement - we use the TCM which is the complementof
P(1)γ =
1
3
1 1 11 1 11 1 1
. (4)
Conjugating 1− P(1)γ with the Hadamard gives the result.
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 51: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/51.jpg)
Universal Gate Set - Bilayer ν = 1/3 FQH
Universal (metaplectic) gate set for the qutrit model (Cui and Wang,2015):
1 H3 = S for C = SU(3)1 (single layer ν = 1/3 FQH), so it can beimplemented by TCP.
2 ∧σz3 can be implemented by gapped boundary braiding. Conjugatesecond qutrit by H3 to get SUM3.
3 TCP can implement diag(1, ω, ω) (Dehn twist of SU(3)1). Follow byσx3 for Q3.
4 Any nontrivial single-qutrit classical (i.e. Clifford) gate not equal toH23 - we have a Pauli-X from tunneling e.
5 Projective measurement - we use the TCM which is the complementof
P(1)γ =
1
3
1 1 11 1 11 1 1
. (4)
Conjugating 1− P(1)γ with the Hadamard gives the result.
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 52: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/52.jpg)
Universal Gate Set - Bilayer ν = 1/3 FQH
Universal (metaplectic) gate set for the qutrit model (Cui and Wang,2015):
1 H3 = S for C = SU(3)1 (single layer ν = 1/3 FQH), so it can beimplemented by TCP.
2 ∧σz3 can be implemented by gapped boundary braiding. Conjugatesecond qutrit by H3 to get SUM3.
3 TCP can implement diag(1, ω, ω) (Dehn twist of SU(3)1). Follow byσx3 for Q3.
4 Any nontrivial single-qutrit classical (i.e. Clifford) gate not equal toH23 - we have a Pauli-X from tunneling e.
5 Projective measurement - we use the TCM which is the complementof
P(1)γ =
1
3
1 1 11 1 11 1 1
. (4)
Conjugating 1− P(1)γ with the Hadamard gives the result.
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 53: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/53.jpg)
Universal Gate Set - Bilayer ν = 1/3 FQH
Universal (metaplectic) gate set for the qutrit model (Cui and Wang,2015):
1 H3 = S for C = SU(3)1 (single layer ν = 1/3 FQH), so it can beimplemented by TCP.
2 ∧σz3 can be implemented by gapped boundary braiding. Conjugatesecond qutrit by H3 to get SUM3.
3 TCP can implement diag(1, ω, ω) (Dehn twist of SU(3)1). Follow byσx3 for Q3.
4 Any nontrivial single-qutrit classical (i.e. Clifford) gate not equal toH23 - we have a Pauli-X from tunneling e.
5 Projective measurement - we use the TCM which is the complementof
P(1)γ =
1
3
1 1 11 1 11 1 1
. (4)
Conjugating 1− P(1)γ with the Hadamard gives the result.
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 54: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/54.jpg)
Universal Gate Set - Bilayer ν = 1/3 FQH
Universal (metaplectic) gate set for the qutrit model (Cui and Wang,2015):
1 H3 = S for C = SU(3)1 (single layer ν = 1/3 FQH), so it can beimplemented by TCP.
2 ∧σz3 can be implemented by gapped boundary braiding. Conjugatesecond qutrit by H3 to get SUM3.
3 TCP can implement diag(1, ω, ω) (Dehn twist of SU(3)1). Follow byσx3 for Q3.
4 Any nontrivial single-qutrit classical (i.e. Clifford) gate not equal toH23 - we have a Pauli-X from tunneling e.
5 Projective measurement - we use the TCM which is the complementof
P(1)γ =
1
3
1 1 11 1 11 1 1
. (4)
Conjugating 1− P(1)γ with the Hadamard gives the result.
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 55: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/55.jpg)
Universal Gate Set - Bilayer ν = 1/3 FQH
To get the projective measurement, we introduce a symmetry-protectedtopological charge measurement:
Want to tune system s.t. quasiparticle tunneling along γ is enhanced→ implement H ′ = −tWγ(e) + h.c.
t = (complex) tunneling amplitude, Wγ(e) = tunnel-e operator
Implementing M ↔ ground state of H ′ is doubly degenerate for|e〉, |e2〉 ↔ t is real (beyond topological protection)
Physically, could realize in fractional quantum spin Hall state –quantum spin Hall + time-reversal symmetry (exchange two layers)
Topologically equiv. to ν = 1/3 Laughlin, e = bound state of spinup/down quasiholese is time-reversal invariant → tunneling amplitude of e must be real→ symmetry-protected TCM
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 56: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/56.jpg)
Universal Gate Set - Bilayer ν = 1/3 FQH
To get the projective measurement, we introduce a symmetry-protectedtopological charge measurement:
Want to tune system s.t. quasiparticle tunneling along γ is enhanced→ implement H ′ = −tWγ(e) + h.c.
t = (complex) tunneling amplitude, Wγ(e) = tunnel-e operator
Implementing M ↔ ground state of H ′ is doubly degenerate for|e〉, |e2〉 ↔ t is real (beyond topological protection)
Physically, could realize in fractional quantum spin Hall state –quantum spin Hall + time-reversal symmetry (exchange two layers)
Topologically equiv. to ν = 1/3 Laughlin, e = bound state of spinup/down quasiholese is time-reversal invariant → tunneling amplitude of e must be real→ symmetry-protected TCM
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 57: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/57.jpg)
Universal Gate Set - Bilayer ν = 1/3 FQH
To get the projective measurement, we introduce a symmetry-protectedtopological charge measurement:
Want to tune system s.t. quasiparticle tunneling along γ is enhanced→ implement H ′ = −tWγ(e) + h.c.
t = (complex) tunneling amplitude, Wγ(e) = tunnel-e operator
Implementing M ↔ ground state of H ′ is doubly degenerate for|e〉, |e2〉 ↔ t is real (beyond topological protection)
Physically, could realize in fractional quantum spin Hall state –quantum spin Hall + time-reversal symmetry (exchange two layers)
Topologically equiv. to ν = 1/3 Laughlin, e = bound state of spinup/down quasiholese is time-reversal invariant → tunneling amplitude of e must be real→ symmetry-protected TCM
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 58: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/58.jpg)
Overview
Framework of TQC
Introduce gapped boundaries and their framework
Gapped boundaries for TQC
Universal gate set with gapped boundaries in bilayer ν = 1/3 FQH
Summary and Outlook
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
![Page 59: Universal Quantum Computation with Gapped Boundaries](https://reader034.fdocuments.in/reader034/viewer/2022051323/627bb6e7a2641e7aa824d0e2/html5/thumbnails/59.jpg)
Summary
Decoherence is a major challenge to quantum computing →topological quantum computing (TQC)
TQC with anyons requires non-abelian topological phases (difficultto implement) → engineer non-abelian objects (e.g. gappedboundaries) from abelian phases
We can get a universal quantum computing gate set from apurely abelian theory (bilayer ν = 1/3 FQH), which is trivial foranyonic TQC
Topologically protected qudit encoding and Clifford gatesSymmetry-protected implementation for non-Clifford projection
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Outlook
Practical implementation of the symmetry-protected TCM
More thorough study of symmetry-protected quantum computation
Amount of protection offered and computation power
Other routes to engineer non-abelian objects
Boundary defects/parafermion zero modes from gapped boundaries ofν = 1/3 FQH (Lindner et al., 2014)Genons and symmetry defects (Barkeshli et al., 2014; C, Cheng,Wang, 2017)How would these look when combined with gapped boundaries?
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Acknowledgments
Special thanks to Cesar Galindo, Shawn Cui, Maissam Barkeshli foranswering many questions
Many thanks to Prof. Freedman and everyone at Station Q for agreat summer
None of this would have been possible without the guidance anddedication of Prof. Zhenghan Wang
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018
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Thanks!
Iris Cong, Harvard TQC with Gapped Boundaries Harvard, April 2018