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

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 and estimation

Phase estimation Amplitude amplification and estimation e − iHt for sparse H

Phase estimation Amplitude amplification and estimation e − iHt for sparse H Linear systems

Phase estimation Amplitude amplification and estimation e − iHt for sparse H Linear systems h ψ | φ i by swap test

Phase estimation Amplitude amplification and estimation e − iHt for sparse H Linear systems h ψ | φ i by swap test All require quantum error correction!

Quantum error correction in a nutshell

Quantum error correction in a nutshell N

Quantum error correction in a nutshell ψ ψ N

Quantum error correction in a nutshell ψ N

Quantum error correction in a nutshell ψ N N 0 N 0

Quantum error correction in a nutshell ψ N V N 0 N 0

Quantum error correction in a nutshell ψ N V N 0 W N 0

Quantum error correction in a nutshell ψ ψ N V N 0 W N 0

Quantum error correction in a nutshell ≤

State of the art Surface code Realized on 2D architecture Syndrome measurements local Gates below threshold Instruction manual for building a quantum computer!

Google’s 49 qubit processor

1 logical qubit

Google’s 49 qubit processor 1 logical qubit

Google’s 49 qubit processor 1 logical qubit 10,000 physical qubits

Is there a better way?

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)

Opportunities for improving QEC “… in the usual case in the laboratory, one works with a specific apparatus with a particular dominant quantum noise process.” “The GF(4) codes do not take advantage of this specific knowledge, and may thus be sub-optimal, in terms of transmission rate…” -Leung, Nielsen, Chuang, & Yamamoto PRA (1997)

Single qubit decoherence ρ =

Single qubit decoherence N ( ρ ) = Phase damping ( ) T 2

Single qubit decoherence N ( ρ ) = Amplitude damping ( ) T 1

T 2 = 60 µs = T 1 / 3

Five-qubit code T 2 = 60 µs = T 1 / 3

∗ Five-qubit code ∗ T 2 = 60 µs = T 1 / 3

∗ Bi-convex optimization! ∗ ∗ Kosut and Lidar (2009) ∗ Five-qubit code ∗ T 2 = 60 µs = T 1 / 3

∗ Bi-convex optimization! ∗ ∗ Kosut and Lidar (2009) Five-qubit code T 2 = 60 µs = T 1 / 3

1. Require noise model N ? =

2. Optimization unscalable Cost function evaluation = O (exp(#qubits))

3. Gate compilation needed ? ? ? ? ? ? ? V ? ? = ∗ ? ? ? ? ? ? ?

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

Previous approaches Our algorithm 1. Require noise model 2. Optimization unscalable 3. Gate compilation needed

Previous approaches Our algorithm 1. Require noise model Model free 2. Optimization unscalable 3. Gate compilation needed

Previous approaches Our algorithm 1. Require noise model Model free 2. Optimization unscalable E ffi cient evaluation 3. Gate compilation needed

Previous approaches Our algorithm 1. Require noise model Model free 2. Optimization unscalable E ffi cient evaluation Built-in gate 3. Gate compilation needed decomposition

Variational quantum algorithms… Quantum autoencoder Romero et al. (2016) Quantum Variational adiabatic quantum optimization eigensolver algorithm Peruzzo et al. (2014) Farhi et al. (2014) … training quantum circuits.

Q uantum V ariational E rror C orrec TOR Variational quantum optimization algorithm for designing quantum error correcting schemes…

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

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

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

E ffi cient evaluation 0 or 1 ~ ~ p q random samples S (from 2-design) N Fraction of -outcomes estimates 0 √ average fidelity to . O (1 / N )

Built-in gate decomposition ~ ~ p q

Input Output Classical Optimizer q ∗ , h F i h F i ∗ ~ p ∗ , ~ ~ p 0 , ~ q 0 p, ~ ~ q h F i h F i q Quantum Estimator p ~ ~ p q

QVECTOR

QVECTOR

QVECTOR

Outlook 1. Simulate extension to error corrected gates 2. Discover good circuit ansatz 3. Perform on existing hardware!

Thank you!

Back-up slides…

Input Output Classical Optimizer q ∗ , h F i h F i ∗ ~ p ∗ , ~ ~ p 0 , ~ q 0 p, ~ ~ q h F i h F i Quantum Estimator { | 0 i | 0 i logical k k S † S . . . . qubits . | 0 i | 0 i . V ~ W ~ { | 0 i | 0 i p q syndrome n - k n - k . . qubits . . . . | 0 i | 0 i

a) d) Quantum-classical interface Segment of Variational Encoding Circuit Input Output Classical Z p 1 Z p 13 X p 6 X p 18 Optimizer q ∗ , h F i h F i ∗ p 0 , ~ ~ p ∗ , ~ ~ q 0 Z p 14 Z p 2 X p 7 Z p 11 X p 19 p, ~ ~ q . . . . . . h F i h F i X p 8 Z p 15 X p 20 Z p 23 Z p 3 Quantum Z p 4 X p 9 Z p 12 Z p 16 X p 21 Estimator X p 10 Z p 17 X p 22 Z p 24 Z p 5 b) c) Average Fidelity Estimator Average Fidelity Sampling Circuit Sample from 2-design S S µ µ { | 0 i | 0 i logical k k S † . S . . . Repeat times qubits L L L . | 0 i | 0 i . V † V ~ { | 0 i | 0 i syndrome p ~ Measure Apply Initialize n - k n - k p . . W ~ qubits . . V † V † . h 0 | ⊗ n S † h 0 | ⊗ n S † S L | 0 i ⊗ n S L | 0 i ⊗ n . | 0 i | 0 i p � W ~ p � W ~ q � V ~ q � V ~ q ~ ~ p p L L { | 0 i refresh r . . Output average fidelity estimate h F i h F i qubits . . . . | 0 i

Initialize | 0 i | 0 i | 0 i | 0 i | 0 i

Run circuit | 0 i Z p 1 Z p 13 X p 18 X p 6 | 0 i Z p 14 Z p 2 X p 7 Z p 11 X p 19 | 0 i Z p 15 X p 20 Z p 23 X p 8 Z p 3 | 0 i Z p 4 X p 9 Z p 12 X p 21 Z p 16 | 0 i X p 22 Z p 24 X p 10 Z p 17 Z p 5

Measure | 0 i Z p 1 Z p 13 X p 18 X p 6 | 0 i Z p 14 Z p 2 X p 7 Z p 11 X p 19 | 0 i Z p 15 X p 20 Z p 23 X p 8 Z p 3 | 0 i Z p 4 X p 9 Z p 12 X p 21 Z p 16 | 0 i X p 22 Z p 24 X p 10 Z p 17 Z p 5

Repeat!

Classical processor

0 2 4 6 8 10 12 1 0.95 0.9 0 200 400 1 0.85 0.8 0.8 0.6 0.75 0 3.6 7.2 10.8 14.4 18 21.6

In situ quantum optimization O ptimized R andomized B enchmarking for I mmediate T une-up Kelly et al. PRL (2014) A daptive C ontrol via R andomized O ptimization N early Y ielding M aximization Ferrie & Moussa PRA (2015)