Further advancement of quantum computing (QC) is contingent on enabling many-body models that avoid deep circuits and excessive use of CNOT gates. To this end, we develop a QC approach employing finite-order connected moment expansions (CMX) and affordable procedures for initial state preparation. We demonstrate the performance of our approach employing several quantum variants of CMX through the classical emulations on the H2 molecule potential energy surface and the Anderson model with a broad range of correlation strength. The results show that our approach is robust and flexible. Good agreement with exact solutions can be maintained even at the dissociation and strong correlation limits.
Further advancement of quantum computing (QC) is contingent on enabling many-body models that avoid deep circuits and excessive use of CNOT gates. To this end, we develop a QC approach employing finite-order connected moment expansions (CMX) and affordable procedures for initial state preparation. We demonstrate the performance of our approach employing several quantum variants of CMX through the classical emulations on the H2 molecule potential energy surface and the Anderson model with a broad range of correlation strength. The results show that our approach is robust and flexible. Good agreement with exact solutions can be maintained even at the dissociation and strong correlation limits.
Abstract The still-maturing noisy intermediate-scale quantum technology faces strict limitations on the algorithms that can be implemented efficiently. In the realm of quantum chemistry, the variational quantum eigensolver (VQE) algorithm has become ubiquitous, with many variations. Alternatively, a promising new avenue has been unraveled by the quantum variants of techniques grounded on expansions of the moments of the Hamiltonian, notably the connected moments expansion (CMX) and the Peeters–Devreese–Soldatov (PDS) energy functional. Common to those approaches is that, upon preparing an approximate ground state used to compute the necessary moments, the accuracy of the estimated ground state energy depends on the degree of overlap between the prepared state and the true ground state. Thus, we use the ADAPT-VQE algorithm to test shallow circuit construction strategies for the purpose of increasing the overlap with the exact ground state, validated by the sizable accuracy improvement herein reported in the PDS and CMX ground state energies. We also show that we can take advantage of the fact that the terms to be measured are highly recurring in different moments, incurring a substantial reduction in the number of necessary measurements. By coupling this measurement caching with a threshold that determines whether a given term is to be measured based on its associated scalar coefficient, we observe a further reduction in the number of circuit implementations while allowing for tunable accuracy.
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.
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. On a superconducting quantum processor, Hamiltonian moments, ⟨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. Calculated dissociation curves indicate precision at about 10mH for this system and as low as 0.1mH for molecular hydrogen, H2, 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 performance of quantum algorithms for eigenvalue problems, such as computing Hamiltonian spectra, depends strongly on the overlap of the initial wave function and the target eigenvector. In a basis of Slater determinants, the representation of energy eigenstates of systems with N strongly correlated electrons requires a number of determinants that scales exponentially with N . On classical processors, this restricts simulations to systems where N is small. Here, we show that quantum computers can efficiently simulate strongly correlated molecular systems by directly encoding the dominant entanglement structure in the form of spin-coupled initial states. This avoids resorting to expensive classical or quantum state preparation heuristics and instead exploits symmetries in the wave function. We provide quantum circuits for deterministic preparation of a family of spin eigenfunctions with ( N N / 2 ) Slater determinants with depth O ( N ) and O ( N 2 ) local gates. Their use as highly entangled initial states in quantum algorithms reduces the total runtime of quantum phase estimation and related fault-tolerant methods by orders of magnitude. Furthermore, we assess the application of spin-coupled wave functions as initial states for a range of heuristic quantum algorithms, namely, the variational quantum eigensolver, adiabatic state preparation, and different versions of quantum subspace diagonalization (QSD) including QSD based on real-time-evolved states. We also propose a QSD algorithm that exploits states obtained through adaptive quantum eigensolvers. For all algorithms, we demonstrate that using spin-coupled initial states drastically reduces the quantum resources required to simulate strongly correlated ground and excited states. Our work provides a crucial component for enabling scalable quantum simulation of classically challenging electronic systems.
Recently a new class of quantum algorithms that are based on the quantum computation of the connected moment expansion has been reported to find the ground and excited state energies. In particular, the Peeters-Devreese-Soldatov (PDS) formulation is found variational and bearing the potential for further combining with the existing variational quantum infrastructure. Here we find that the PDS formulation can be considered as a new energy functional of which the PDS energy gradient can be employed in a conventional variational quantum solver. In comparison with the usual variational quantum eigensolver (VQE) and the original static PDS approach, this new variational quantum solver offers an effective approach to navigate the dynamics to be free from getting trapped in the local minima that refer to different states, and achieve high accuracy at finding the ground state and its energy through the rotation of the trial wave function of modest quality, thus improves the accuracy and efficiency of the quantum simulation. We demonstrate the performance of the proposed variational quantum solver for toy models, H 2 molecule, and strongly correlated planar H 4 system in some challenging situations. In all the case studies, the proposed variational quantum approach outperforms the usual VQE and static PDS calculations even at the lowest order. We also discuss the limitations of the proposed approach and its preliminary execution for model Hamiltonian on the NISQ device.
Quantum computing allows, in principle, the encoding of the exponentially scaling many-electron wave function onto a linearly scaling qubit register, offering a promising solution to overcome the limitations of traditional quantum chemistry methods. An essential requirement for ground state quantum algorithms to be practical is the initialization of the qubits to a high-quality approximation of the sought-after ground state. Quantum state preparation enables the generation of approximate eigenstates derived from classical computations but is frequently treated as an oracle in quantum information. In this study, we investigate the quantum state preparation of prototypical strongly correlated systems' ground state, up to 28 qubits, using the Hyperion-1 GPU-accelerated state-vector emulator. Various variational and nonvariational methods are compared in terms of their circuit depth and classical complexity. Our results indicate that the recently developed Overlap-ADAPT-VQE algorithm offers the most advantageous performance for near-term applications.