Quantum variational error correction by Peter Johnson, Jonathan - - PowerPoint PPT Presentation

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quantum variational error correction

Quantum variational error correction by Peter Johnson, Jonathan - - PowerPoint PPT Presentation

Nov 30, 2023 383 likes •1.03k views

Quantum Techniques in Machine Learning Workshop Verona, Italy, November 7, 2017 Quantum variational error correction by Peter Johnson, Jonathan Romero, Jonathan Olson, Yudong Cao, and Alan Aspuru-Guzik Phase estimation Amplitude amplification

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Quantum variational error correction

Quantum Techniques in Machine Learning Workshop Verona, Italy, November 7, 2017 by Peter Johnson, Jonathan Romero, Jonathan Olson, Yudong Cao, and Alan Aspuru-Guzik

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SLIDE 2

Amplitude amplification and estimation Phase estimation

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Amplitude amplification and estimation Phase estimation for sparse

e−iHt H

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Amplitude amplification and estimation Phase estimation for sparse

e−iHt H

Linear systems

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Amplitude amplification and estimation Phase estimation for sparse

e−iHt H hψ|φi by swap test

Linear systems

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Amplitude amplification and estimation Phase estimation for sparse

e−iHt H hψ|φi by swap test

All require quantum error correction! Linear systems

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SLIDE 7

Quantum error correction in a nutshell

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Quantum error correction in a nutshell

N

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Quantum error correction in a nutshell

N ψ ψ

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N

Quantum error correction in a nutshell

ψ

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N N N

Quantum error correction in a nutshell

ψ

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N N N V

Quantum error correction in a nutshell

ψ

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N N N V W

Quantum error correction in a nutshell

ψ

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N N N V W

Quantum error correction in a nutshell

ψ ψ

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Quantum error correction in a nutshell

≤

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SLIDE 16

State of the art

Instruction manual for building a quantum computer!

Surface code

Syndrome measurements local Gates below threshold Realized on 2D architecture

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SLIDE 17

Google’s 49 qubit processor

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1 logical qubit

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1 logical qubit

Google’s 49 qubit processor

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10,000 physical qubits

Google’s 49 qubit processor

1 logical qubit

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SLIDE 21

Is there a better way?

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Opportunities for improving QEC

“… in the usual case in the laboratory, one works with a specific apparatus with a particular dominant quantum noise process.”

Leung, Nielsen, Chuang,

& Yamamoto PRA (1997)

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SLIDE 23

Opportunities for improving QEC

“… in the usual case in the laboratory, one works with a specific apparatus with a particular dominant quantum noise process.”

Leung, Nielsen, Chuang,

& Yamamoto PRA (1997)

“The GF(4) codes do not take advantage of this specific knowledge, and may thus be sub-optimal, in terms of transmission rate…”

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Single qubit decoherence

ρ =

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Single qubit decoherence Phase damping ( )

N(ρ) = T2

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Amplitude damping ( ) Single qubit decoherence

N(ρ) = T1

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T2 = 60µs = T1/3

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Five-qubit code

T2 = 60µs = T1/3

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∗ ∗

Five-qubit code

T2 = 60µs = T1/3

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∗ ∗

Five-qubit code

∗

Bi-convex optimization! Kosut and Lidar (2009)

∗ ∗

T2 = 60µs = T1/3

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Kosut and Lidar (2009)

∗

Bi-convex optimization!

∗ ∗

Five-qubit code

T2 = 60µs = T1/3

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1. Require noise model

N ? =

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2. Optimization unscalable

O(exp(#qubits))

Cost function evaluation

=

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SLIDE 34

3. Gate compilation needed

V

∗ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

=

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1. Require noise model
2. Optimization unscalable
3. Gate compilation needed

Previous approaches

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SLIDE 36

1. Require noise model
2. Optimization unscalable
3. Gate compilation needed

Previous approaches Our algorithm

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1. Require noise model
2. Optimization unscalable
3. Gate compilation needed

Previous approaches Our algorithm

Model free

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1. Require noise model
2. Optimization unscalable
3. Gate compilation needed

Previous approaches Our algorithm

Model free Efficient evaluation

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1. Require noise model
2. Optimization unscalable
3. Gate compilation needed

Model free Efficient evaluation Built-in gate decomposition

Previous approaches Our algorithm

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Variational quantum algorithms… … training quantum circuits.

Variational quantum eigensolver

Peruzzo et al. (2014)

Quantum autoencoder

Romero et al. (2016)

Quantum adiabatic

ptimization

algorithm

Farhi et al. (2014)

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Variational quantum optimization algorithm for designing quantum error correcting schemes… Quantum Variational Error CorrecTOR

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Variational quantum optimization algorithm for designing quantum error correcting schemes… QVECTOR

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Variational quantum optimization algorithm for designing quantum error correcting schemes… QVECTOR Objective: maximize average fidelity

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Model free In situ optimization… … noise “perfectly” simulates itself.

~ p ~ q

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~ p ~ q

Efficient evaluation Fraction of -outcomes estimates average fidelity to .

N O(1/ √ N)

r

1

random samples

S (from 2-design)

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~ p ~ q

Built-in gate decomposition

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Input Output

~ p0, ~ q0 ~ p∗, ~ q∗, hFi

Classical Optimizer

hFi hFi ~ p, ~ q

Quantum Estimator

hFi∗

~ p ~ q p q

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SLIDE 48

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QVECTOR

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QVECTOR

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QVECTOR

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1. Simulate extension to error corrected gates

Outlook

3. Perform on existing hardware!
2. Discover good circuit ansatz

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SLIDE 53

Thank you!

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SLIDE 54

Back-up slides…

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Input Output

~ p0, ~ q0 ~ p∗, ~ q∗, hFi

Classical Optimizer

hFi

hFi ~ p, ~ q

Quantum Estimator

hFi∗

S

V~

p

W~

q

S†

logical qubits syndrome qubits

{

k n-k

{

|0i |0i |0i |0i

. . . . . .

. . . . . .

k n-k

|0i |0i |0i |0i

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Input Output

~ p0, ~ q0 ~ p∗, ~ q∗, hFi

Classical Optimizer

hFi hFi ~ p, ~ q

Quantum Estimator

hFi∗

L L L S S µ µ hFi hFi

Measure Repeat times

Average Fidelity Estimator

Sample from 2-design Output average fidelity estimate Apply Initialize

a) b) d) c)

S

S†

logical qubits syndrome qubits

{

k n-k

{

|0i |0i |0i |0i

. . . . . .

. . . . . .

k n-k

|0i |0i |0i |0i

Average Fidelity Sampling Circuit Segment of Variational Encoding Circuit Quantum-classical interface

refresh qubits

{

. . .

. . .

|0i |0i

W~

q

SL|0i⊗n SL|0i⊗n h0|⊗nS†

L

h0|⊗nS†

L

V~

p

V†

~ p

r

. . . . . .

V†

~ p W~ q V~ p

V†

~ p W~ q V~ p

Xp6 Xp7 Xp8 Xp9 Xp10 Xp18 Xp19 Xp20 Xp21 Xp22 Zp1 Zp2 Zp3 Zp4 Zp5 Zp11 Zp12 Zp13 Zp14 Zp15 Zp16 Zp17 Zp23 Zp24

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SLIDE 57

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|0i |0i |0i |0i |0i

Initialize

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Xp6 Xp7 Xp8 Xp9 Xp10 Xp18 Xp19 Xp20 Xp21 Xp22 Zp1 Zp2 Zp3 Zp4 Zp5 Zp11 Zp12 Zp13 Zp14 Zp15 Zp16 Zp17 Zp23 Zp24

|0i |0i |0i |0i |0i

Run circuit

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Xp6 Xp7 Xp8 Xp9 Xp10 Xp18 Xp19 Xp20 Xp21 Xp22 Zp1 Zp2 Zp3 Zp4 Zp5 Zp11 Zp12 Zp13 Zp14 Zp15 Zp16 Zp17 Zp23 Zp24

|0i |0i |0i |0i |0i

Measure

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Repeat!

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Classical processor

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3.6 7.2 10.8 14.4 18 21.6 0.75 0.8 0.85 0.9 0.95 1 2 4 6 8 10 12

200 400 0.6 0.8 1

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Adaptive Control via Randomized Optimization Nearly Yielding Maximization

In situ quantum optimization

Optimized Randomized Benchmarking for Immediate Tune-up

Kelly et al. PRL (2014) Ferrie & Moussa PRA (2015)