Topological Quantum Error Correction in Silicon Photonicsphys449/ProposalExample4.pdf · What is Quantum Computing? Origins and Applications of Quantum Computing Feynman’s idea:

MotivationBackground Theory

Project DetailsSummary

Topological Quantum Error Correction in SiliconPhotonics

The University of British Columbia

November 20, 2018

1 / 29



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

2 / 29



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

3 / 29






Quantum algorithms:





4 / 29






Quantum algorithms:





5 / 29






Quantum algorithms:





6 / 29






Quantum algorithms:





7 / 29






Quantum algorithms:





8 / 29




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〉

9 / 29




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〉

10 / 29




Experimental Realizations

11 / 29





12 / 29





13 / 29




Experimental Difficulties

Pure quantum states are extremely fragile

14 / 29



Quantum DotsPhotonic Crystal CavitiesQuantum Error Correction

Quantum Dots

V (x , y , z) =

{12m

∗ω20(x2 + y2), |z | < L

2

∞, |z | > L2

15 / 29




Photonic Crystal Cavities

Permitivity of material varies periodically: D(r) = ε(r)E(r),ε(r) = ε(r + R).

16 / 29




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〉

17 / 29




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〉

18 / 29




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〉.

19 / 29




Stabilizers


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.


20 / 29




Stabilizers


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



21 / 29




Stabilizers


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



22 / 29



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.

23 / 29




Topological Quantum Error Correction

The number of encoded qubits is 2, regardless of the size ofthe lattice.

24 / 29




Entangling Gate

Entangling gate proposed by Deepak Sridharan and Edo Waksfrom the University of Maryland.

25 / 29




Losses in Photonic Circuits

26 / 29




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

27 / 29



Acknowledgements

28 / 29



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.

29 / 29

Documents

Topological Quantum Error Correction in Silicon Photonicsphys449/ProposalExample4.pdf · What is Quantum Computing? Origins and Applications of Quantum Computing Feynman’s idea: