Variational quantum eigensolver of interacting bosons with NISQ - PowerPoint PPT Presentation

FERMILAB-SLIDES-19-003-QIS Variational quantum eigensolver of interacting bosons with NISQ devices Andy C. Y. Li This document has been authored by Fermi Research Alliance, LLC under APS March Meeting 2019 Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. 6 March 2019

Variational quantum eigensolver (VQE) Parameterized trial Noisy intermediate Scale state Quantum (NISQ) devices Quantum-classical hybrid algorithm Update Efficient measurement Classical optimization algorithm Trial state’s energy Goal: many-body systems with bosons Eigenstates and • light-matter interaction eigenenergies • electron-phonon coupling 2 3/6/2019 Andy C. Y. Li | APS March Meeting 2019

Boson encoding by qubits Goal: encode a truncated boson Hilbert space in qubits Position basis binary encoding Number basis binary encoding Ref: Phys. Rev. Lett. 121, 110504 ) ! = ( = 1 … 11 % ) = Δ +,- … = 1 … 11 % . Δ ! = 2 = 0 … 10 % ) = Δ +,- = 1 … 10 % . ,- ! = 1 = 0 … 01 % … ! = 0 = 0 … 00 % ) = Δ ,+,- = 0 … 00 % . 3 3/6/2019 Andy C. Y. Li | APS March Meeting 2019

Hardware efficient trial state’s ansatz Entanglement-gate layers 1Q-gate layer 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 4 3/6/2019 Andy C. Y. Li | APS March Meeting 2019

Cost function for ground state & excited states Ground-state cost function = trial state’s energy ! " = $ ⃗ & ' $ ⃗ & Ground state: $ " = argmin ! " |/(1)⟩ 4 1st-excited state cost function: ! 7 = $ ⃗ & ' $ ⃗ + 6 $ " |$ ⃗ & & Overlap with the ground state 1st-excited state: $ 7 = argmin ! 7 |/(1)⟩ 4 + 6 $ 7 |$ ⃗ 4 2nd-excited state cost function: ! 4 = $ ⃗ & ' $ ⃗ + 6 $ " |$ ⃗ & & & … 5 3/6/2019 Andy C. Y. Li | APS March Meeting 2019

Proof-of-principle expt. – Rabi model using Rigetti’s device TLS Rabi Hamiltonian: two-level system (TLS) coupled to a photon mode Ω $ = "& ' & + ) 2 + , + ! & ' + & + - ! " Number-basis binary encoding: . = 0 = 00 0 . = 1 = 01 0 photon mode truncated to up to Photon . = 2 = 10 0 . = 3 = 11 0 3 photons 6 3/6/2019 Andy C. Y. Li | APS March Meeting 2019

Optimizers Optimization algorithm Simultaneous Perturbation Stochastic Stochastic Approximation (SPSA) Nelder-Mead Gradient-free Constrained Optimization BY Linear Gradient-free Approximations (COBYLA) Bound Optimization BY Quadratic Gradient-free Approximation (BOBYQA) Covariance Matrix Adaptation Evolutionary algorithm: Evolution Strategy (CMA-ES) stochastic & gradient-free 7 3/6/2019 Andy C. Y. Li | APS March Meeting 2019

Optimizer with noisy device Expt: ) = 0.6Ω, Ω = 0 Optimizer |" − " $%&'( | CMA-ES 0.062 0.099 SPSA 0.165 COBYLA 0.219 BOBYQA 0.223 Nelder-Mead • Stochastic algorithm ✓ • CMA-ES: slightly better 8 3/6/2019 Andy C. Y. Li | APS March Meeting 2019

Experimental result Error bars: sampling error of 200000 shots Energy gap • Consistent trend across multiple parameter , /Ω regimes + Deviation • Hardware fidelities • Photon cutoff for Zero-detuning: " = Ω % ≥ 0.8Ω coupling strength: %/Ω perturbative USC non-perturbative USC ⎯ non-perturbative DSC 9 3/6/2019 Andy C. Y. Li | APS March Meeting 2019

Summary • 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… Andy C. Y. Li Alex Macridin Panagiotis Spentzouris 10 3/6/2019 Andy C. Y. Li | APS March Meeting 2019