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Correlation-Informed Permutation of Qubits for Reducing Ansatz Depth in the Variational Quantum Eigensolver

Nikolay V. Tkachenko, James Sud, Yu Zhang, Sergei Tretiak, Petr M. Anisimov, Andrew T. Arrasmith, Patrick J. Coles, Lukasz Cincio, and Pavel A. Dub
PRX Quantum 2, 020337 – Published 9 June 2021
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Abstract

The variational quantum eigensolver (VQE) is a method of choice to solve the electronic structure problem for molecules on near-term gate-based quantum computers. However, the circuit depth is expected to grow significantly with the problem size. Increased depth can both degrade the accuracy of the results and reduce trainability. In this work, we propose an approach to reduce ansatz circuit depth. Our approach, called “PermVQE,” adds an additional optimization loop to the VQE that permutes qubits in order to solve for the qubit Hamiltonian that maximally localizes correlations in the ground state. The choice of permutations is based on mutual information, which is a measure of interaction between electrons and/or holes in spin-orbitals. Encoding strongly entangled spin-orbitals into proximal qubits on a quantum chip naturally reduces the circuit depth needed to prepare the ground state. For representative molecular systems, LiH, H2, (H2)2, H4, H3+, and N2, we demonstrate that placing entangled qubits in close proximity leads to shallower depth circuits required to reach a given eigenvalue-eigenvector accuracy. The approach is designed for hardware-efficient ansatz of any qubit connectivity, and examples are demonstrated for linear and two-dimensional grid architectures. The main ideas can also be applied to simulate molecules with other ansatz as well as variational quantum algorithms beyond the VQE. In particular, we demonstrate the beneficial effect of qubit permutations to build fermionic–adaptive derivative assembled pseudo-Trotter ansatz on a linear qubit connectivity architecture with nearly a twofold reduction of the number of controlled not gates.

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  • Received 14 September 2020
  • Revised 22 January 2021
  • Accepted 19 April 2021

DOI:https://doi.org/10.1103/PRXQuantum.2.020337

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & Technology

Authors & Affiliations

Nikolay V. Tkachenko1,§, James Sud2,3,§, Yu Zhang3,*, Sergei Tretiak3, Petr M. Anisimov4, Andrew T. Arrasmith3, Patrick J. Coles3, Lukasz Cincio3,†, and Pavel A. Dub1,‡

  • 1Chemistry Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
  • 2Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA
  • 3Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
  • 4Accelerators and Electrodynamics Group, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

  • *zhy@lanl.gov
  • lcincio@lanl.gov
  • pdub@lanl.gov
  • §These authors contributed equally.

Popular Summary

Quantum chemistry is regarded to be one of the first disciplines that will be revolutionized by quantum computing. It promises an exponential speedup over classical computing, as it reduces the scaling in the number of quantum particles as well as addresses the problem of memory storage required to represent the quantum wave function. The variational quantum eigensolver (VQE) is currently the leading approach to solve the molecular electronic Hamiltonian eigenvalue-eigenvector problem on near-term quantum computers, producing the ground and excited states of molecules and generating potential energy surfaces. Although successful implementations of the VQE on noisy intermediate-scale gate-based quantum computers have been presented across several “toy” molecular systems, demonstrating quantum chemistry advantage is challenging. The difficulty is due to the limited number of qubits and/or massive quantum circuits coupled with the large number of variational parameters to optimize.

Here we present a novel procedure to perform VQE calculations with lower depth circuits on quantum chips with limited connectivity, which is present on most quantum computing architectures. Our new algorithm abbreviated as permutation VQE (PermVQE) allows for efficient reordering of qubits, so that those qubits that are highly correlated are mapped to qubits that are closer on a given chip. This operation reduces the depth of the quantum circuit as one does not need to create long-distance correlations. Our approach makes limited connectivity quantum computing architectures more useful for quantum chemistry simulations. We believe that the proposed method will facilitate the simulation of larger molecular systems with near-term devices and ultimately contribute to the demonstration of quantum chemistry advantage

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Vol. 2, Iss. 2 — June - August 2021

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