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MotivationBackground Theory
Project DetailsSummary
Topological Quantum Error Correction in SiliconPhotonics
The University of British Columbia
November 20, 2018
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MotivationBackground Theory
Project DetailsSummary
Overview
1 MotivationWhat is Quantum Computing?
2 Background TheoryQuantum DotsPhotonic Crystal CavitiesQuantum Error Correction
3 Project DetailsTopological Quantum Error CorrectionEntangling GateLosses in Photonic CircuitsResources and Schedule
4 Summary
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Origins and Applications of Quantum Computing
Feynman’s idea: Simulate quantum systems using otherquantum systems
Quantum algorithms:
Prime Factorization: Exponential speedup
Discrete Logarithm: Exponential speedup
Unstructured Database Search: Square root speedup
Solutions to Linear Systems: Exponential speedup
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Origins and Applications of Quantum Computing
Feynman’s idea: Simulate quantum systems using otherquantum systems
Quantum algorithms:
Prime Factorization: Exponential speedup
Discrete Logarithm: Exponential speedup
Unstructured Database Search: Square root speedup
Solutions to Linear Systems: Exponential speedup
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Origins and Applications of Quantum Computing
Feynman’s idea: Simulate quantum systems using otherquantum systems
Quantum algorithms:
Prime Factorization: Exponential speedup
Discrete Logarithm: Exponential speedup
Unstructured Database Search: Square root speedup
Solutions to Linear Systems: Exponential speedup
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Origins and Applications of Quantum Computing
Feynman’s idea: Simulate quantum systems using otherquantum systems
Quantum algorithms:
Prime Factorization: Exponential speedup
Discrete Logarithm: Exponential speedup
Unstructured Database Search: Square root speedup
Solutions to Linear Systems: Exponential speedup
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Origins and Applications of Quantum Computing
Feynman’s idea: Simulate quantum systems using otherquantum systems
Quantum algorithms:
Prime Factorization: Exponential speedup
Discrete Logarithm: Exponential speedup
Unstructured Database Search: Square root speedup
Solutions to Linear Systems: Exponential speedup
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Origins and Applications of Quantum Computing
Feynman’s idea: Simulate quantum systems using otherquantum systems
Quantum algorithms:
Prime Factorization: Exponential speedup
Discrete Logarithm: Exponential speedup
Unstructured Database Search: Square root speedup
Solutions to Linear Systems: Exponential speedup
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Classical and Quantum Computation
Classical: bits can beeither 0 or 1, invertibleand non-invertible gatesapplied to bit strings like1011
Quantum: bits are |0〉 or|1〉, unitary operationsapplied to bit strings|1011〉
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Classical and Quantum Computation
Classical: bits can beeither 0 or 1, invertibleand non-invertible gatesapplied to bit strings like1011
Quantum: bits are |0〉 or|1〉, unitary operationsapplied to bit strings|1011〉
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Experimental Realizations
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Experimental Realizations
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Experimental Realizations
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MotivationBackground Theory
Project DetailsSummary
What is Quantum Computing?
Experimental Difficulties
Pure quantum states are extremely fragile
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MotivationBackground Theory
Project DetailsSummary
Quantum DotsPhotonic Crystal CavitiesQuantum Error Correction
Quantum Dots
V (x , y , z) =
{12m
∗ω20(x2 + y2), |z | < L
2
∞, |z | > L2
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MotivationBackground Theory
Project DetailsSummary
Quantum DotsPhotonic Crystal CavitiesQuantum Error Correction
Photonic Crystal Cavities
Permitivity of material varies periodically: D(r) = ε(r)E(r),ε(r) = ε(r + R).
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MotivationBackground Theory
Project DetailsSummary
Quantum DotsPhotonic Crystal CavitiesQuantum Error Correction
Quantum Error Correction
Encode information using redundancy
|0〉 → |000〉
|1〉 → |111〉
If 1 qubit gets flipped, we can measure and flip it back:
|000〉 −−−→Error
|100〉 Measure−−−−−→Error
σx ⊗ I2 ⊗ I2|100〉 → |000〉
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MotivationBackground Theory
Project DetailsSummary
Quantum DotsPhotonic Crystal CavitiesQuantum Error Correction
Quantum Error Correction
Encode information using redundancy
|0〉 → |000〉
|1〉 → |111〉
If 1 qubit gets flipped, we can measure and flip it back:
|000〉 −−−→Error
|100〉 Measure−−−−−→Error
σx ⊗ I2 ⊗ I2|100〉 → |000〉
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MotivationBackground Theory
Project DetailsSummary
Quantum DotsPhotonic Crystal CavitiesQuantum Error Correction
Stabilizers
Pauli Group Πn: Operators of the form σ(i1)1 ⊗ · · · ⊗ σ(in)n ,
ij = 0, 1, 2, 3.
σ(0) = I2 =
(1 00 1
), σ(1) = σx =
(0 11 0
),
σ(2) = σy =
(0 −ii 0
), σ(3) = σz =
(1 00 −1
)Stabilizer group: A subgroup of Πn which acts trivially on aset of states, g |ψ〉 = |ψ〉 for g ∈ Πn.
Example: σz ⊗ σz ⊗ σz |000〉 = |000〉 → |000〉 is stabilized byσz ⊗ σz ⊗ σz . Same for |110〉, |101〉, |011〉.
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MotivationBackground Theory
Project DetailsSummary
Quantum DotsPhotonic Crystal CavitiesQuantum Error Correction
Stabilizers
Pauli Group Πn: Operators of the form σ(i1)1 ⊗ · · · ⊗ σ(in)n ,
ij = 0, 1, 2, 3.
σ(0) = I2 =
(1 00 1
), σ(1) = σx =
(0 11 0
),
σ(2) = σy =
(0 −ii 0
), σ(3) = σz =
(1 00 −1
)
Stabilizer group: A subgroup of Πn which acts trivially on aset of states, g |ψ〉 = |ψ〉 for g ∈ Πn.
Example: σz ⊗ σz ⊗ σz |000〉 = |000〉 → |000〉 is stabilized byσz ⊗ σz ⊗ σz . Same for |110〉, |101〉, |011〉.
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MotivationBackground Theory
Project DetailsSummary
Quantum DotsPhotonic Crystal CavitiesQuantum Error Correction
Stabilizers
Pauli Group Πn: Operators of the form σ(i1)1 ⊗ · · · ⊗ σ(in)n ,
ij = 0, 1, 2, 3.
σ(0) = I2 =
(1 00 1
), σ(1) = σx =
(0 11 0
),
σ(2) = σy =
(0 −ii 0
), σ(3) = σz =
(1 00 −1
)Stabilizer group: A subgroup of Πn which acts trivially on aset of states, g |ψ〉 = |ψ〉 for g ∈ Πn.
Example: σz ⊗ σz ⊗ σz |000〉 = |000〉 → |000〉 is stabilized byσz ⊗ σz ⊗ σz . Same for |110〉, |101〉, |011〉.
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MotivationBackground Theory
Project DetailsSummary
Quantum DotsPhotonic Crystal CavitiesQuantum Error Correction
Stabilizers
Pauli Group Πn: Operators of the form σ(i1)1 ⊗ · · · ⊗ σ(in)n ,
ij = 0, 1, 2, 3.
σ(0) = I2 =
(1 00 1
), σ(1) = σx =
(0 11 0
),
σ(2) = σy =
(0 −ii 0
), σ(3) = σz =
(1 00 −1
)Stabilizer group: A subgroup of Πn which acts trivially on aset of states, g |ψ〉 = |ψ〉 for g ∈ Πn.
Example: σz ⊗ σz ⊗ σz |000〉 = |000〉 → |000〉 is stabilized byσz ⊗ σz ⊗ σz . Same for |110〉, |101〉, |011〉.
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MotivationBackground Theory
Project DetailsSummary
Topological Quantum Error CorrectionEntangling GateLosses in Photonic CircuitsResources and Schedule
Topological Quantum Error Correction
The number of encoded qubits is 2, regardless of the size ofthe lattice.
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MotivationBackground Theory
Project DetailsSummary
Topological Quantum Error CorrectionEntangling GateLosses in Photonic CircuitsResources and Schedule
Topological Quantum Error Correction
The number of encoded qubits is 2, regardless of the size ofthe lattice.
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MotivationBackground Theory
Project DetailsSummary
Topological Quantum Error CorrectionEntangling GateLosses in Photonic CircuitsResources and Schedule
Entangling Gate
Entangling gate proposed by Deepak Sridharan and Edo Waksfrom the University of Maryland.
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MotivationBackground Theory
Project DetailsSummary
Topological Quantum Error CorrectionEntangling GateLosses in Photonic CircuitsResources and Schedule
Losses in Photonic Circuits
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MotivationBackground Theory
Project DetailsSummary
Topological Quantum Error CorrectionEntangling GateLosses in Photonic CircuitsResources and Schedule
Resources and Schedule
Photon loss and QEC calculations are being done analyticallyand using Mathematica.
Simulations of toric code will be done in Python.
Task Date
Literature Review May-AugStudy effects of photon loss Sept-NovImplement toric code with entangling gate DecAnalyze fault tolerance of code Jan-FebRepeat analysis with Colour Code. MarSubmit thesis Apr
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MotivationBackground Theory
Project DetailsSummary
Acknowledgements
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MotivationBackground Theory
Project DetailsSummary
Summary
Quantum error correction will be fundamental in anyimplementation of quantum information processing.
Quantum dots in silicon photonic circuits is a promisingscalable platform. I will investigate this area further.
I will develop a scalable architecture using the 2-qubitentangling gate and toric codes as building blocks.
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