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FERMILAB-SLIDES-19-040-QIS This document has been authored by Fermi Research Variational quantum simulation of Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, interacting bosons on NISQ

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SLIDE 1

Variational quantum simulation of interacting bosons on NISQ devices

Andy C. Y. Li Kavli ACP Spring Workshop: Intersections QIS/HEP 20 May 2019

This document has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.

Andy C. Y. Li Panagiotis Spentzouris Alex Macridin

FERMILAB-SLIDES-19-040-QIS

This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department

  • f Energy, Office of Science, Office of High Energy Physics
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SLIDE 2
  • Noisy Intermediate-Scale Quantum (NISQ) devices
  • Quantum-classical hybrid variational algorithms
  • Variational quantum eigensolver (VQE) of interacting bosons
  • Proof-of-principle experiment of a 3-qubit implementation
  • Open questions about scalability

Outline

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 2

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SLIDE 3
  • “Nature isn’t classical, dammit, and if you

want to make a simulation of nature, you’d better make it quantum mechanical” – Richard Feynman (1982)

  • Digital: all operations are represented by

qubit gates

  • Time evolution, quantum phase estimation,

quantum annealing, …

  • Targets: highly entangled quantum states,

non-perturbative system dynamics, …

Digital quantum simulation

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 3

Science 334.6052 (2011): 57-61

  • A. A. Houck et al, Nat Phys 8, 292 (2012)

control

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SLIDE 4
  • Decoherence: relaxation, pure dephasing,

correlated noise, …

→ device loses ‘quantumness’ after a limited coherence time

  • Control error: inaccurate gate

implementation due to imperfect calibration, qubit drift, …

→ reliable result only within a limited number

  • f gate operations
  • Only shallow circuits can be reliably

implemented in the near future

Challenges: noise and control error

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 4

Martin Savage’s group: PRA 98, 032331 (2018)

Relaxation (𝑈

")

Pure dephasing (𝑈#)

Δ𝐹(𝑢) ←noise

𝜔 𝑢 = 𝑏 0 + 𝑐 𝑓0123 4 4 1

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SLIDE 5

Superconducting qubit coherence

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 5

  • M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013)

2019

  • Phys. Rev. A 76, 042319 (2007)

Transmon: simple design with advances in fabrication and materials

=

0-pi and other more advanced designs: much more complicated circuit structure

  • Phys. Rev. A 87, 052306 (2013),
  • Phys. Rev. X 3, 011003 (2013)
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SLIDE 6

Quantum computing for NISQ devices

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 6

Fault-tolerant qubit – unclear path to realize

https://qutech.nl/ majorana-trilogy- completed/

D-wave quantum annealing – unclear quantum advantage Analog open-system simulation – very limited applications

  • Phys. Rev. X 7,

011016 (2017)

Noisy Intermediate- Scale Quantum (NISQ) devices: ~100 pre-threshold qubits capable for shallow circuits

https://ai.googleblog.com/ 2018/03/a-preview-of- bristlecone-googles- new.html

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SLIDE 7
  • Quantum Approximate Optimization Algorithm (QAOA)

– Approximated solutions for combinatorial optimization problems through a series of classically optimized gate

  • perations
  • Quantum kernel method

– Support vector machine (SVM) with kernel function evaluated by quantum devices

  • Quantum autoencoder

– encoding in Hilbert space with encoder trained classically

  • Variational quantum eigensolver (VQE)

– Variational ansatz represented by a list of quantum gate and optimized by a classical optimizer

Quantum-classical hybrid variational algorithms

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 7

Classical

  • ptimization

algorithm Cost function 𝐷( ⃗ 𝜄) Evaluate by a classical computer Evaluate by a Quantum device

Classical computing Hybrid

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SLIDE 8
  • Relatively shallow circuit
  • Tolerant to control errors (coherent rotation

angle errors )

  • Quantum advantage?

– Heuristic and most likely problem-specific

  • Possible sources of quantum advantage

– Quantum tunneling (QAOA) – Hilbert space size: 2: (Quantum machine learning) – Natural way to evaluate ⟨𝐼⟩, … (VQE)

Why hybrid variational algorithms?

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 8

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Variational quantum eigensolver (VQE)

9

Low-energy spectrum Variational ansatz: parameterized circuit to prepare the trial state Noisy intermediate Scale Quantum (NISQ) devices Trial state’s energy

Efficient measurement

Classical optimization algorithm

Update

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Encoding: system representation qubits

  • Ground-state

properties

  • Long-time scale

responses

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SLIDE 10

VQE applications in quantum chemistry

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 10

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SLIDE 11

VQE: much less developed for non-fermionic systems

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 11

  • Fermions ↔ Qubits

– Jordan-Wigner transformation

  • Many models in high-energy and condensed-

matter involve non-fermionic degrees of freedom

  • Goal: many-body systems with bosons
  • light-matter interaction
  • electron-phonon coupling

By en:User_talk:S_kliminUpgrade (New vectorial edition) : Olivier d'ALLIVY KELLY - en:File:Polaron_scheme1.jpg, CC BY-SA 4.0, https://commons.wikimedia.org/w/i ndex.php?curid=8461871

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Boson encoding by qubits

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Goal: encode a truncated boson Hilbert space in qubits

…

𝑜 = 0 = 0 … 00 @ 𝑜 = 1 = 0 … 01 @ 𝑜 = 2 = 0 … 10 @ 𝑜 = 𝑂 = 1 … 11 @

Number basis binary encoding

𝑦

…

𝑦 = Δ:0"

#

= 1 … 11 @ 𝑦 = Δ :0"

# 0"

= 1 … 10 @ 𝑦 = Δ 0:0"

#

= 0 … 00 @

Δ

Ref: Phys. Rev. Lett. 121, 110504

Position basis binary encoding

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

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Variational ansatz

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 13

|𝜔 ⃗ 𝜄 ⟩

Parameterized gates natively supported by the hardware

|𝜔 ⃗ 𝜄 ⟩

𝑉(𝛽F, 𝜄F) 𝑉(𝛽", 𝜄")

𝑉 𝛽, 𝜄 = 𝑓01 H( "0I JKLIJM)

Ground state of 𝐼N Ground state of 𝐼O

Motivated by adiabatic state transfer

Hardware efficient Model motivated

Increasing circuit depth Increasing number of

  • ptimization parameters
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SLIDE 14

Cost function for ground state & excited states

14

Ground-state cost function = trial state’s energy 𝐷F( ⃗ 𝜄) = 𝜔 ⃗ 𝜄 𝐼 𝜔 ⃗ 𝜄 Ground state: 𝜔F = argmin

|V(H)⟩

𝐷F 2nd-excited state cost function: 𝐷# = 𝜔 ⃗ 𝜄 𝐼 𝜔 ⃗ 𝜄 + 𝜗 𝜔F|𝜔 ⃗ 𝜄

#+ 𝜗 𝜔"|𝜔 ⃗

𝜄

#

…

1st-excited state cost function: 𝐷" = 𝜔 ⃗ 𝜄 𝐼 𝜔 ⃗ 𝜄 + 𝜗 𝜔F|𝜔 ⃗ 𝜄

#

1st-excited state: 𝜔" = argmin

|V(H)⟩

𝐷"

Overlap with the ground state

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

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SLIDE 15
  • Expensive to evaluate gradient of cost function

– Numerical differentiation, cost ~ Ο 𝑜 , where 𝑜 = number of parameters – In contrast, for neural network, cost ~ Ο log 𝑜 by back propagation – Preferable: gradient-free optimizer

  • Noisy cost function

– Hardware fidelities, sampling error, … – Preferable: noise insensitive

  • Local minimums

– Low energy but physically very different from the ground state (or targeted state) – Preferable: global optimizer / knowledge to make reasonably good initial guess

Optimization algorithm

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 15

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SLIDE 16

Proof-of-principle expt. – Rabi model using Rigetti’ s device

16

𝑕 𝜕 Ω TLS Photon

Rabi Hamiltonian: two-level system (TLS) coupled to a photon mode 𝐼 = 𝜕𝑏_𝑏 + 𝛻 2 𝜏b + 𝑕 𝑏_ + 𝑏 𝜏c Number-basis binary encoding: photon mode truncated to up to 3 photons

𝑜 = 0 = 00 @ 𝑜 = 1 = 01 @ 𝑜 = 2 = 10 @ 𝑜 = 3 = 11 @

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

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Hardware efficient ansatz

17

|𝜔 ⃗ 𝜄 ⟩ 1Q-gate layer Entanglement-gate layers

Ansatz consists only of native gates supported by the hardware e.g. Rf(𝜄), Rg(𝜄) and CZ 3 qubits with 1 entanglement layer

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

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SLIDE 18

Optimizers

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Optimization algorithm Simultaneous Perturbation Stochastic Approximation (SPSA) Stochastic Nelder-Mead Gradient-free Constrained Optimization BY Linear Approximations (COBYLA) Gradient-free Bound Optimization BY Quadratic Approximation (BOBYQA) Gradient-free Covariance Matrix Adaptation Evolution Strategy (CMA-ES) Evolutionary algorithm: stochastic & gradient-free

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

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SLIDE 19

Optimizers with noisy device

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Optimizer |𝑭 − 𝑭𝒇𝒚𝒃𝒅𝒖| CMA-ES 0.062 SPSA 0.099 COBYLA 0.165 BOBYQA 0.219 Nelder-Mead 0.223

  • Stochastic algorithm ✓
  • CMA-ES: slightly better

Expt: 𝑕 = 0.6Ω, Ω = 𝜕

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

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SLIDE 20

Experimental result

20

perturbative USC non-perturbative USC ⎯ non-perturbative DSC

Zero-detuning: 𝜕 = Ω coupling strength: 𝑕/Ω Error bars: sampling error of 200000 shots

Energy gap

  • Consistent trend

across multiple parameter regimes Deviation

  • Hardware

fidelities

  • Photon cutoff for

𝑕 ≥ 0.8Ω

𝐹

w/Ω

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

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  • Proof-of-principle experiment of Rabi model

– 3-qubit implementation on Rigetti’s device

  • Error mitigation techniques
  • Generalize to bigger system?

Generalizing to bigger systems?

21 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

𝑢

?

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SLIDE 22

Rabi dimer: hardware efficient ansatz

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 22

𝑢

# of parameters 204 # of 1Q gates 110 # of 2Q gates 49 Circuit depth 51

Simulation

t/g

Dimer: crossover between hopping and blockade of bosons

  • Too many # of optimization steps
  • Large # of local minima
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SLIDE 23

Rabi dimer: model motivated ansatz

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 23

|0⟩ … 9 parameters per ‘Trotter step’:

  • No-coupling ground state (𝑕 = 𝑢 = 0)
  • Boson vacuum state (discretized

Gaussian state) and spin down state

  • Gaussian state preparation: scalable

using a shallow hardware-efficient circuit

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SLIDE 24

Circuit depth of model-motivated ansatz

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 24

Per one ‘Trotter step’ # of parameters 9 # of 1Q gates 24 # of 2Q gates 39

120 1Q gates 195 2Q gates

Fidelity for 5 steps 0.999**120 * 0.995**195 = 0.33

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Overview

5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics 25

Hardware efficient Model motivated

Increasing circuit depth Increasing number of

  • ptimization parameters

?

Nature Communications 9, 4812 (2018)

  • VQE algorithms for interacting bosons
  • Demonstrate with a 3-qubit implementation
  • Scalability:

– Hardware-efficient ansatz: large # of parameters – Explore model-motivated ansatz with less demanding circuit depth – Optimizing a high-dimension cost function