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.
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 this paper, we propose a novel many-body approach for determining the amplitudes of generalized operators in a projection-based formalism. To implicitly account for the effects of higher-order excitations, we begin with the well-established double-exponential coupled-cluster (CC) ansatz, parametrized by both one- and two-body excitation operators, complemented by a set of vacuum-annihilating two-body generalized operators with effective excitation rank of one. A systematic formalism is developed that effectively bypasses the constraints due to the vacuum-annihilation property of the generalized operators toward a set of closed-form residual equations for their optimization. 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 onto an internally contracted lower-dimensional manifold. This many-body formalism is integrated with the near-term projective quantum eigensolver (PQE) framework that leverages the conventional CC-like residual minimization to iteratively decouple the excited manifold from the reference. With the application of several molecular systems within PQE architecture, we demonstrate that the developed methodology enables us to achieve similar accuracy to the disentangled unitary coupled cluster with singles, doubles, and triples ansatz while utilizing an order of magnitude fewer quantum resources. 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.
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.
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.
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.
In this study, we study the projective quantum eigensolver (PQE) approach, a hybrid quantum-classical algorithm that, by optimizing a unitary coupled cluster wave function, aims at computing the ground state of many-body systems. Instead of trying to minimize the energy of the system like the variational quantum eigensolver (VQE), PQE uses projections of the Schrödinger equation to efficiently bring the trial state closer to an eigenstate of the Hamiltonian. In this study, we provide a mathematical study of the algorithm. We derive a bound relating off-diagonal coefficients (residues) of the Hamiltonian to the energy error of the algorithm and the overlap achieved by the obtained wavefunction. These bounds not only give formal guarantees to PQE, but they also allow us to formulate a well-informed convergence criterion for residue-based optimizers. We then study the classical optimization itself and derive convergence guarantees under certain conditions. We propose a new residue-based optimizer, with numerical evidence of the superiority of this new approach for H4, H6, BeH2, and LiH dissociation curves over both the optimization introduced in Stair and Evangelista [PRX Quantum 2, 030301 (2021)] and VQE optimized using the Broyden-Fletcher-Goldfarb-Shanno method.
The Coupled Cluster (CC) method is used to compute the electronic correlation energy in atoms and molecules and often leads to highly accurate results. However, due to its single-reference nature, standard CC in its projected form fails to describe quantum states characterized by strong electronic correlations and multi-reference projective methods become necessary. On the other hand, quantum algorithms for the solution of many-electron problems have also emerged recently. The quantum unitary variant of CC (UCC) with singles and doubles (q-UCCSD) is a popular wavefunction Ansatz for the variational quantum eigensolver algorithm. The variational nature of this approach can lead to significant advantages compared to its classical equivalent in the projected form, in particular, for the description of strong electronic correlation. However, due to the large number of gate operations required in q-UCCSD, approximations need to be introduced in order to make this approach implementable in a state-of-the-art quantum computer. In this work, we evaluate several variants of the standard q-UCCSD Ansatz in which only a subset of excitations is included. In particular, we investigate the singlet and pair q-UCCD approaches combined with orbital optimization. We show that these approaches can capture the dissociation/distortion profiles of challenging systems, such as H4, H2O, and N2 molecules, as well as the one-dimensional periodic Fermi–Hubbard chain. These results promote the future use of q-UCC methods for the solution of challenging electronic structure problems in quantum chemistry.