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Variational quantum eigensolver of interacting bosons with NISQ devices

Andy C. Y. Li APS March Meeting 2019 6 March 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.

FERMILAB-SLIDES-19-003-QIS

Variational quantum eigensolver (VQE)

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Eigenstates and eigenenergies Parameterized trial state Noisy intermediate Scale Quantum (NISQ) devices Trial state’s energy

Efficient measurement

Classical optimization algorithm

Update

Quantum-classical hybrid algorithm Goal: many-body systems with bosons

• light-matter interaction
• electron-phonon coupling

Boson encoding by qubits

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Goal: encode a truncated boson Hilbert space in qubits

…

! = 0 = 0 … 00 % ! = 1 = 0 … 01 % ! = 2 = 0 … 10 % ! = ( = 1 … 11 %

Number basis binary encoding

)

…

) = Δ+,-

.

= 1 … 11 % ) = Δ +,-

. ,-

= 1 … 10 % ) = Δ ,+,-

.

= 0 … 00 %

Δ

Ref: Phys. Rev. Lett. 121, 110504

Position basis binary encoding

Hardware efficient trial state’s ansatz

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|" ⃗ $ ⟩ 1Q-gate layer Entanglement-gate layers

Ansatz consists only of native gates supported by the hardware e.g. R'($), R*($) and CZ for Rigetti’s devices Example: 3 qubits with 1 entanglement layer

Cost function for ground state & excited states

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Ground-state cost function = trial state’s energy !" = $ ⃗ & ' $ ⃗ & Ground state: $" = argmin

|/(1)⟩

!" 2nd-excited state cost function: !4 = $ ⃗ & ' $ ⃗ & + 6 $"|$ ⃗ &

4+ 6 $7|$ ⃗

&

4

…

1st-excited state cost function: !7 = $ ⃗ & ' $ ⃗ & + 6 $"|$ ⃗ &

4

1st-excited state: $7 = argmin

|/(1)⟩

!7

Overlap with the ground state

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

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! " Ω TLS Photon

Rabi Hamiltonian: two-level system (TLS) coupled to a photon mode $ = "&'& + ) 2 +, + ! &' + & +- Number-basis binary encoding: photon mode truncated to up to 3 photons

. = 0 = 00 0 . = 1 = 01 0 . = 2 = 10 0 . = 3 = 11 0

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

Optimizer with noisy device

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Optimizer |" − "$%&'(| 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 = 0

Experimental result

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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

+

,/Ω

• Variational quantum eigensolver for bosons

– Low-energy spectrum

• Proof-of-principle experiment of Rabi model

– 3-qubit implementation on Rigetti’s device – Ground state and 1st excited state

• Future works

– Trial state’s ansatz – Error mitigation techniques – Lattice models: Rabi lattice, Holstein model…

Summary

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Andy C. Y. Li Panagiotis Spentzouris Alex Macridin