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arXiv:2306.14560 · The Journal of Chemical Physics

Development of zero-noise extrapolated projective quantum algorithm for accurate evaluation of molecular energetics in noisy quantum devices

Sonaldeep Halder, Chinmay Shrikhande, Rahul Maitra

Published Sep 19, 2023• 13 citations
arXivHTMLPDF

Abstract

The recently developed Projective Quantum Eigensolver (PQE) offers an elegant procedure to evaluate the ground state energies of molecular systems in quantum computers. However, the noise in available quantum hardware can result in significant errors in computed outcomes, limiting the realization of quantum advantage. Although PQE comes equipped with some degree of inherent noise resilience, any practical implementation with apposite accuracy would require additional routines to eliminate or mitigate the errors further. In this work, we propose a way to enhance the efficiency of PQE by developing an optimal framework for introducing Zero Noise Extrapolation (ZNE) in the nonlinear iterative procedure that outlines the PQE, leading to the formulation of ZNE-PQE. Moreover, we perform a detailed analysis of how various components involved in it affect the accuracy and efficiency of the reciprocated energy convergence trajectory. Additionally, we investigate the underlying mechanism that leads to the improvements observed in ZNE-PQE over conventional PQE by performing a comparative analysis of their residue norm landscape. This approach is expected to facilitate practical applications of quantum computing in fields related to molecular sciences, where it is essential to determine molecular energies accurately.

DOI
10.1063/5.0166433

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The recently developed Projective Quantum Eigensolver (PQE) offers an elegant procedure to evaluate the ground state energies of molecular systems in quantum computers. However, the noise in available quantum hardware can result in significant errors in computed outcomes, limiting the realization of quantum advantage. Although PQE comes equipped with some degree of inherent noise resilience, any practical implementation with apposite accuracy would require additional routines to eliminate or mitigate the errors further. In this work, we propose a way to enhance the efficiency of PQE by developing an optimal framework for introducing Zero Noise Extrapolation (ZNE) in the nonlinear iterative procedure that outlines the PQE, leading to the formulation of ZNE-PQE. Moreover, we perform a detailed analysis of how various components involved in it affect the accuracy and efficiency of the reciprocated energy convergence trajectory. Additionally, we investigate the underlying mechanism that leads to the improvements observed in ZNE-PQE over conventional PQE by performing a comparative analysis of their residue norm landscape. This approach is expected to facilitate practical applications of quantum computing in fields related to molecular sciences, where it is essential to determine molecular energies accurately.
The recently developed Projective Quantum Eigensolver (PQE) has been demonstrated as an elegant methodology to compute the ground state energy of molecular systems in Noisy Intermdiate Scale Quantum (NISQ) devices. The iterative optimization of the ansatz parameters involves repeated construction of residues on a quantum device. The quintessential pattern of the iteration dynamics, when projected as a time discrete map, suggests a hierarchical structure in the timescale of convergence, effectively partitioning the parameters into two distinct classes. In this work, we have exploited the collective interplay of these two sets of parameters via machine learning techniques to bring out the synergistic inter-relationship among them that triggers a drastic reduction in the number of quantum measurements necessary for the parameter updates while maintaining the characteristic accuracy of PQE. Furthermore the machine learning model may be tuned to capture the noisy data of NISQ devices and thus the predicted energy is shown to be resilient under a given noise model.
Determination of molecular energetics and properties is one of the core challenges in the near-term quantum computing. To this end, hybrid quantum-classical algorithms are preferred for Noisy Intermediate Scale Quantum (NISQ) architectures. The Projective Quantum Eigensolver (PQE) is one such algorithms that optimizes the parameters of the chemistry-inspired unitary coupled cluster (UCC) ansatz using a conventional coupled cluster-like residual minimization. Such a strategy involves the projection of the Schrodinger equation on to linearly independent basis towards the parameter optimization, restricting the ansatz is solely defined in terms of the excitation operators. This warrants the inclusion of high-rank operators for strongly correlated systems, leading to increased utilization of quantum resources. In this manuscript, we develop a methodology for determining the generalized operators in terms of a closed form residual equations in the PQE framework that can be efficiently implemented in a quantum computer with manageable quantum resources. Such a strategy requires the removal of the underlying redundancy in high-rank excited determinants, generated due to the presence of the generalized operators in the ansatz, by projecting them on to an internally contracted lower dimensional manifold. With the application on several molecular systems, we have demonstrated our ansatz achieves similar accuracy to the (disentangled) UCC with singles, doubles and triples (SDT) ansatz, while utilizing an order of magnitude fewer quantum gates. Furthermore, when simulated under stochastic Gaussian noise or depolarizing hardware noise, our method shows significantly improved noise resilience compared to the other members of PQE family and the state-of-the-art variational quantum eigensolver.
In recent times, a variety of hybrid quantum-classical algorithms have been developed that aim to calculate the ground state energies of molecular systems on Noisy Intermediate-Scale Quantum (NISQ) devices. Albeit the utilization of shallow depth circuits in these algorithms, the optimization of ansatz parameters necessitates a substantial number of quantum measurements, leading to prolonged runtimes on the scantly available quantum hardware. Through our work, we lay the general foundation for an interdisciplinary approach that can be used to drastically reduce the dependency of these algorithms on quantum infrastructure. We showcase these pertinent concepts within the framework of the recently developed Projective Quantum Eigensolver (PQE) that involves iterative optimization of the nonlinearly coupled parameters through repeated residue measurements on quantum hardware. We demonstrate that one may perceive such a nonlinear optimization problem as a collective dynamic interplay of fast and slow relaxing modes. As such, the synergy among the parameters is exploited using an on-the-fly supervised machine learning protocol that dynamically casts the PQE optimization into a smaller subspace by reducing the effective degrees of freedom. We demonstrate analytically and numerically that our proposed methodology ensures a drastic reduction in the number of quantum measurements necessary for the parameter updates without compromising the characteristic accuracy. Furthermore, the machine learning model may be tuned to capture the noisy data of NISQ devices, and thus the predicted energy is shown to be resilient under a given noise model.
As a hybrid quantum-classical algorithm, the variational quantum eigensolver is widely applied in quantum chemistry simulations, especially in computing the electronic structure of complex molecular systems. However, on existing noisy intermediate-scale quantum devices, some factors such as quantum decoherence, measurement errors, and gate operation imprecisions are unavoidable. To overcome these challenges, this study proposes an efficient noise-mitigating variational quantum eigensolver for accurate computation of molecular ground state energies in noisy environments. We design the quantum circuit with reference to the structure of matrix product states and utilize it to pre-train the circuit parameters, which ensures circuit stability and mitigates fluctuations caused by initialization. We also employ zero-noise extrapolation to mitigate quantum noise and combine it with neural networks to improve the accuracy of the noise-fitting function, which significantly eliminates noise interference. Furthermore, we implement an intelligent grouping strategy for measuring Hamiltonian Pauli strings, which not only reduces measurement errors but also improves sampling efficiency. We perform numerical simulations to solve the ground state energy of the H4molecule by using MindSpore Quantum framework, and the results demonstrate that our algorithm can constrain noise errors within the range of O(10−2)∼O(10−1)\mathcal{O}\left( {{{10}^{ - 2}}} \right)\sim \mathcal{O}\left( {{{10}^{ - 1}}} \right)O(10−2)∼O(10−1), outperforming mainstream variational quantum eigensolvers. This work provides a new strategy for high-precision quantum chemistry calculations on near-term noisy quantum hardware.
Quantum computers hold immense potential in the field of chemistry, ushering new frontiers to solve complex many-body problems that are beyond the reach of classical computers. However, noise in the current quantum hardware limits their applicability to large chemical systems. This work encompasses the development of a projective formalism that aims to compute ground-state energies of molecular systems accurately using noisy intermediate scale quantum (NISQ) hardware in a resource-efficient manner. Our approach is reliant upon the formulation of a bipartitely decoupled parameterized ansatz within the disentangled unitary coupled cluster framework based on the principles of nonlinear dynamics and synergetics. Such decoupling emulates total parameter optimization in a lower dimensional manifold, while a mutual synergistic relationship among the parameters is exploited to ensure characteristic accuracy via a non-iterative energy correction. Without any pre-circuit measurements, our method leads to a highly compact fixed-depth ansatz with shallower circuits and fewer expectation value evaluations. Through analytical and numerical demonstrations, we establish the method's superior performance under noise while concurrently ensuring requisite accuracy in future fault-tolerant systems. This approach enables rapid exploration of emerging chemical spaces by the efficient utilization of near-term quantum hardware resources.
, we find that we are able to obtain energies within 4 millihartree (2.5 kcal/mol) of the exact energy along the entire potential energy curve, with the accuracy limited by both the stochastic error and the inconsistent performance of the IBM devices. We find that an optimization algorithm using direct inversion of the iterative subspace can converge swiftly, even to excited states, but stochastic noise can prompt large parameter updates. For the 4-site transverse-field Ising model at its critical point, PQE with an appropriate application of qubit tapering can recover 99% of the correlation energy, even after discarding several parameters. The large number of CNOT gates needed for the additional parameters introduces a concomitant error that, on the IBM devices, results in a loss of accuracy despite the increased expressivity of the trial state. Error extrapolation techniques and tapering or postselection are recommended to mitigate errors in PQE hardware experiments.