Increasing the Representation Accuracy of Quantum Simulations of Chemistry without Extra Quantum Resources

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Increasing the Representation Accuracy of Quantum Simulations of Chemistry without Extra Quantum Resources

Tyler Takeshita, Nicholas C. Rubin, Zhang Jiang, Eunseok Lee, Ryan Babbush, and Jarrod R. McClean

Phys. Rev. X 10, 011004 – Published 7 January 2020

See Synopsis: A Few Qubits Go a Long Way

Abstract

Proposals for experiments in quantum chemistry on quantum computers leverage the ability to target a subset of degrees of freedom containing the essential quantum behavior, sometimes called the active space. This approximation allows one to treat more difficult problems using fewer qubits and lower gate depths than would otherwise be possible. However, while this approximation captures many important qualitative features, it may leave the results wanting in terms of absolute accuracy (basis error) of the representation. In traditional approaches, increasing this accuracy requires increasing the number of qubits and an appropriate increase in circuit depth as well. Here we explore two techniques requiring no additional qubits or circuit depth that are able to remove much of this approximation in favor of additional measurements. The techniques are constructed and analyzed theoretically, and some numerical proof-of-concept calculations are shown. As an example, we show how to achieve the accuracy of a 20-qubit representation using only four qubits and a modest number of additional measurements for a hydrogen molecule. We close with an outlook on the impact such techniques may have on both near-term and fault-tolerant quantum simulations.

Received 11 April 2019
Revised 26 September 2019

DOI:https://doi.org/10.1103/PhysRevX.10.011004

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)

Electronic structure of atoms & molecules First-principles calculations Quantum algorithms Quantum simulation

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Synopsis

A Few Qubits Go a Long Way

Published 7 January 2020

The right combination of quantum and classical computations allows for accurate quantum chemistry simulations using surprisingly few qubits.

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Authors & Affiliations

Tyler Takeshita^1,*, Nicholas C. Rubin^2,†, Zhang Jiang², Eunseok Lee¹, Ryan Babbush², and Jarrod R. McClean ^2,‡

¹Mercedes-Benz Research and Development North America, Sunnyvale, California 94085, USA
²Google LLC, Venice, California 90291, USA

^*tyler.takeshita@daimler.com
^†nickrubin@google.com
^‡jmcclean@google.com

Popular Summary

Quantum computers promise to revolutionize the way we understand the basic physics and chemistry of materials through simulation. In order to make accurate predictions, the system of interest must be accurately represented. Previously, this accuracy was fundamentally tied to the number of available qubits and number of operations one can do on a near-term quantum computer, which are currently limited in real devices. In this work, we show how to produce high-accuracy representations without additional qubits or quantum operations. This boosts the prospects for predictive near-term simulations of chemistry on a quantum computer and furthers our understanding as to how quantum and classical resources may be combined to maximize the power of these emerging devices.

Using theoretical analyses and numerical calculations, we explore two techniques for increasing accuracy in quantum simulations. One technique leverages quantum subspace expansions—mathematical descriptions of how a system behaves around a state of interest—in a novel way. The other technique relies on orbital relaxations, where a change in the molecular orbitals offers a more compact and correct description of an effect. Using these techniques, we show how to achieve the accuracy of a 20-qubit device using only four qubits and a modest number of additional measurements to describe where electrons sit in a simulated hydrogen molecule and how their energies change when shifted by a small amount.

We hope to understand how these techniques can be applied to even larger systems, with an additional focus on excited states and dynamics. It is hoped that this can reduce the representation cost for simulating physical systems that are out of reach for current computers.

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Vol. 10, Iss. 1 — January - March 2020

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Figure 1
Schematic virtual quantum subspace expansion to increase accuracy without additional qubits. The method separates the orbitals of a fermionic system into their core, active, and virtual components. The quantum device solves the active-space problem, and additional quantum measurements are taken within this space. These additional measurements are combined with classical postprocessing from data on the virtual space to correct the solutions using no additional qubits or circuit depth. The resulting corrected wave function(s) are stored in a mixed quantum-classical representation that may be used to derive any desired properties of these wave functions.
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Figure 2
Ground-state energy as a function of the internuclear separation for the $H_{2}$ molecule at different levels of theory. The notation $(n = n_{q})$ after each line indicates that $n_{q}$ qubits are needed for that representation. The use of the virtual quantum subspace expansion (VQSE) technique using only four qubits attains the same accuracy in this case as a representation using 20 qubits, with an error much smaller than chemical accuracy (approximately $10^{- 5} E_{h}$ ). The minimal basis here denotes a STO-3G calculation and 6-31G is an intermediate level of accuracy between the minimal basis and a double-zeta basis cc-pVDZ (here, VDZ).
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Figure 3
Ground-state energy error with respect to the exact solution in a cc-pVDZ (abbreviated here as VDZ) as a function of internuclear separation for the $H_{2}$ molecule for different levels of theory. The notation $(n = n_{q})$ after each line indicates that $n_{q}$ qubits are needed for that representation. We see that the VQSE technique using only four qubits attains the same accuracy as an exact solution in a 20-qubit basis, up to an error of $10^{- 5} E_{h}$ which is far below chemical accuracy.
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Figure 4
Ground-state energy as a function of internuclear separation for the $H_{2}$ molecule at different levels of theory. The notation $(n = n_{q})$ after each line indicates that $n_{q}$ qubits are needed for that representation. The orbital optimization postprocessing is labeled by “OO.” We compare active-space calculations on the full space with the 6-31G basis set (requiring eight qubits) to active-space calculations on the same model involving only four and six qubits.
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Figure 5
Ground-state energy error with respect to the exact solution in a 6-31G basis as a function of internuclear separation for $H_{2}$ . The notation $(n = n_{q})$ after each line indicates that $n_{q}$ qubits are needed for that representation. The orbital optimization postprocessing is labeled by OO. We see that orbital optimization greatly reduces the need for perturbative corrections in the stretched bonding region.
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Figure 6
Ground-state energy as a function of the OH—H bond distance for the water molecule. Using an active space within the large cc-pVTZ (VTZ) basis, we show that the OO step improves upon the active-space solution with ( $n = 14$ ) qubits (CAS) achieving the accuracy of a larger active space ( $n = 20$ ) in the bonding region, and superior accuracy in the dissociation region due to better treatment of dynamical correlation. The solution in a minimal basis exact solution, which requires the same number of qubits, is not shown here, due to being in error by over a hartree.
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