Quantum inverse iteration algorithm for programmable quantum simulators

Oleksandr Kyriienko

doi:10.1038/s41534-019-0239-7

Abstract

We propose a quantum inverse iteration algorithm, which can be used to estimate ground state properties of a programmable quantum device. The method relies on the inverse power iteration technique, where the sequential application of the Hamiltonian inverse to an initial state prepares the approximate ground state. To apply the inverse Hamiltonian operation, we write it as a sum of unitary evolution operators using the Fourier approximation approach. This allows to reformulate the protocol as separate measurements for the overlap of initial and propagated wavefunction. The algorithm thus crucially depends on the ability to run Hamiltonian dynamics with an available quantum device, and can be used for analog quantum simulators. We benchmark the performance using paradigmatic examples of quantum chemistry, corresponding to molecular hydrogen and beryllium hydride. Finally, we show its use for studying the ground state properties of relevant material science models, which can be simulated with existing devices, considering an example of the Bose-Hubbard atomic simulator.

Highlights

Quantum computing offers drastic speed up for certain computational problems, and has evolved as a unique direction in the theoretical information science.[1]
We have presented the algorithm for the ground state energy (GSE) estimation of a quantum Hamiltonian
It is based on the iterative application of the Hamiltonian inverse to the initial state, and can be represented as a sum of unitary evolution operators

Summary

Introduction

Quantum computing offers drastic speed up for certain computational problems, and has evolved as a unique direction in the theoretical information science.[1]. Whereas textbook examples of quantum algorithms with exponential and quadratic speed up for factoring and search serve as a great motivation,[1] the estimates of gate counts are daunting, making them distant goals for the future fault-tolerant quantum computers.[5] Recent developments in this fast evolving field call for new short depth algorithms which can solve useful problems in the era of noisy intermediate scale quantum (NISQ) devices,[6] and in future lead to quantum advantage. One of the most promising directions for quantum computation is the field of quantum chemistry and materials.[5,7] Targeting the access to ground state properties of molecules and strongly correlated matter, it can offer huge gain for various technological applications, for instance helping to find a catalyst for the nitrogen fixation.[8] To date, different quantum theoretical protocols were developed, and several proof-of-principle experiments on various platforms were performed in the simplest cases. From the material science perspective, the use of cold atom quantum simulators has shown great promise, where simulations of Fermi-Hubbard lattice dynamics,[15] large scale quantum Rydberg chain[16] and Ising model,[17] and twodimensional many-body localization[18] have been performed

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Results

Conclusion