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arXiv:2009.13140 · Bulletin of the American Physical Society

Quantum computed moments correction to variational estimates

Harish J. Vallury, Jones, Michael A., Charles D. Hill, Lloyd C. L. Hollenberg

Published Sep 28, 2020• 0 citations
arXivHTMLPDF

Abstract

The variational principle of quantum mechanics is the backbone of hybrid quantum computing for a range of applications. However, as the problem size grows, quantum logic errors and the effect of barren plateaus overwhelm the quality of the results. There is now a clear focus on strategies that require fewer quantum circuit steps and are robust to device errors. Here we present an approach in which problem complexity is transferred to dynamic quantities computed on the quantum processor - Hamiltonian moments, $\langle H^n\rangle$. From these quantum computed moments, estimates of the ground-state energy are obtained using the "infinum" theorem from Lanczos cumulant expansions which manifestly correct the associated variational calculation. With system dynamics encoded in the moments the burden on the trial-state quantum circuit depth is eased. The method is introduced and demonstrated on 2D quantum magnetism models on lattices up to 5 $\times$ 5 (25 qubits) implemented on IBM Quantum superconducting qubit devices. Moments were quantum computed to fourth order with respect to a parameterised antiferromagnetic trial-state. A comprehensive comparison with benchmark variational calculations was performed, including over an ensemble of random coupling instances. The results showed that the infinum estimate consistently outperformed the benchmark variational approach for the same trial-state. These initial investigations suggest that the quantum computed moments approach has a high degree of stability against trial-state variation, quantum gate errors and shot noise, all of which bodes well for further investigation and applications of the approach.

DOI
10.48550/arxiv.2009.13140

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The variational principle of quantum mechanics is the backbone of hybrid quantum computing for a range of applications. However, as the problem size grows, quantum logic errors and the effect of barren plateaus overwhelm the quality of the results. There is now a clear focus on strategies that require fewer quantum circuit steps and are robust to device errors. Here we present an approach in which problem complexity is transferred to dynamic quantities computed on the quantum processor - Hamiltonian moments, ⟨Hn⟩\langle H^n\rangle⟨Hn⟩. From these quantum computed moments, estimates of the ground-state energy are obtained using the "infinum" theorem from Lanczos cumulant expansions which manifestly correct the associated variational calculation. With system dynamics encoded in the moments the burden on the trial-state quantum circuit depth is eased. The method is introduced and demonstrated on 2D quantum magnetism models on lattices up to 5 ×\times× 5 (25 qubits) implemented on IBM Quantum superconducting qubit devices. Moments were quantum computed to fourth order with respect to a parameterised antiferromagnetic trial-state. A comprehensive comparison with benchmark variational calculations was performed, including over an ensemble of random coupling instances. The results showed that the infinum estimate consistently outperformed the benchmark variational approach for the same trial-state. These initial investigations suggest that the quantum computed moments approach has a high degree of stability against trial-state variation, quantum gate errors and shot noise, all of which bodes well for further investigation and applications of the approach.
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Quantum computers hold promise to circumvent the limitations of conventional computing for difficult molecular problems. However, the accumulation of quantum logic errors on real devices represents a major challenge, particularly in the pursuit of chemical accuracy requiring the inclusion of dynamical effects. In this work we implement the quantum computed moments (QCM) approach for hydrogen chain molecular systems up to H6_66​. On a superconducting quantum processor, Hamiltonian moments, ⟨Hp⟩\langle \mathcal{H}^p\rangle⟨Hp⟩ are computed with respect to the Hartree-Fock state, which are then employed in Lanczos expansion theory to determine an estimate for the ground-state energy which incorporates electronic correlations and manifestly improves on the variational result. Post-processing purification of the raw QCM data takes the estimate through the Hartree-Fock variational limit to within 99.9% of the exact electronic ground-state energy for the largest system studied, H6_66​. Calculated dissociation curves indicate precision at about 10mH for this system and as low as 0.1mH for molecular hydrogen, H2_22​, over a range of bond lengths. In the context of stringent precision requirements for chemical problems, these results provide strong evidence for the error suppression capability of the QCM method, particularly when coupled with post-processing error mitigation. Greater emphasis on more efficient representations of the Hamiltonian and classical preprocessing steps may enable the solution of larger systems on near-term quantum processors.
The accurate computation of properties of large molecular systems is classically infeasible and is one of the applications in which it is hoped that quantum computers will demonstrate an advantage over classical devices. However, due to the limitations of present-day quantum hardware, variational-hybrid algorithms introduced to tackle these problems struggle to meet the accuracy and precision requirements of chemical applications. Here, we apply the quantum computed moments (QCM) approach combined with a variety of noise-mitigation techniques to an eight qubit and spin-orbital representation of the water molecule (H2O{\text{H}}_{2}\text{O}H2​O). A noise-stable improvement on the variational result for a four-excitation trial state (circuit depth 25, 22 cz gates) was obtained, with the ground-state energy computed to be within $1.0\ifmmode\pm\else\textpm\fi{}0.8$ mHa of exact diagonalization in the 14 spin-orbital basis. Thus, the QCM approach, despite an increased number of measurements and noisy quantum hardware (cz error rates c.0.3% corresponding to expected error rates on the trial-state circuit of order 7%), is able to determine the ground-state energy of a nontrivial molecular system at the required accuracy (c.0.05%). To the best of our knowledge, these results are the largest calculations performed on a physical quantum computer to date to achieve millihartree agreement with exact diagonalization, and a promising indicator of how such hybrid approaches might scale to problems of interest in the low-error and fault-tolerant regimes as quantum computers develop.
Quantum computers hold promise to circumvent the limitations of conventional computing for difficult molecular problems. However, the accumulation of quantum logic errors on real devices represents a major challenge, particularly in the pursuit of chemical accuracy requiring the inclusion of electronic correlation effects. In this work we implement the quantum computed moments (QCM) approach for hydrogen chain molecular systems up to H[Formula: see text]. On a superconducting quantum processor, Hamiltonian moments, [Formula: see text] are computed with respect to the Hartree-Fock state, which are then employed in Lanczos expansion theory to determine an estimate for the ground-state energy which incorporates electronic correlations and manifestly improves on the direct energy measurement. Post-processing purification of the raw QCM data takes the estimate below the Hartree-Fock energy to within 99.9% of the exact electronic ground-state energy for the largest system studied, H[Formula: see text]. Calculated dissociation curves indicate precision at about 10mH for this system and as low as 0.1mH for molecular hydrogen, H[Formula: see text], over a range of bond lengths. In the context of stringent precision requirements for chemical problems, these results provide strong evidence for the error suppression capability of the QCM method, particularly when coupled with post-processing error mitigation. While calculations based on the Hartree-Fock state are tractable to classical computation, these results represent a first step towards implementing the QCM method in a quantum chemical trial circuit. Greater emphasis on more efficient representations of the Hamiltonian and classical preprocessing steps may enable the solution of larger systems on near-term quantum processors.
In this paper, we present a quantum computational method to calculate the many-body Green's function matrix in a spin orbital basis. We apply our approach to finite-sized fermionic Hubbard models and related impurity models within Dynamical Mean Field Theory, and demonstrate the calculation of Green's functions on Quantinuum's H1-1 trapped-ion quantum computer. Our approach involves a cumulant expansion of the Lanczos method, using Hamiltonian moments as measurable expectation values. This bypasses the need for a large overhead in the number of measurements due to repeated applications of the variational quantum eigensolver (VQE), and instead measures the expectation value of the moments with one set of measurement circuits. From the measured moments, the tridiagonalised Hamiltonian matrix can be computed, which in turn yields the Green's function via continued fractions. While we use a variational algorithm to prepare the ground state in this work, we note that the modularity of our implementation allows for other (non-variational) approaches to be used for the ground state.
A recently proposed variational quantum algorithm has expanded the horizon of variational quantum computing to nonlinear physics and fluid dynamics. In this work, we probe the ability of such approach to capture the ground state of the nonlinear Schr\"odinger equation for a range of parameters on real superconducting quantum processors. Specifically, we study the expressivity of real-amplitude, hardware-efficient Ansatz to capture the ground state of this nonlinear system across various interaction regimes and implement different noise scenarios in both simulators and cloud processors. Our investigation reveals that although quantum hardware noise impairs the evaluation of the energy cost function, certain small instances of the problem consistently converge to the ground state. We test and demonstrate a variety of cases on IBM Q superconducting devices and analyze the discrepancies in the energy cost function evaluation due to quantum hardware noise. These discrepancies are absent in the state fidelity estimation because of the shallow state preparation circuit. Our comprehensive analysis offers valuable insights into the practical implementation and advancement of the variational algorithms for nonlinear problems.