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A Non-Orthogonal Variational Quantum Eigensolver

William J. Huggins, Joonho Lee, Unpil Baek, Bryan O'Gorman, K. Birgitta Whaley

arXiv:1909.09114v2quant-phphysics.chem-ph

TL;DR

Strongly correlated quantum chemistry motivates variational methods that can improve wavefunction flexibility despite near-term hardware constraints. The paper introduces NOVQE, combining parametrized quantum states through generalized eigenvalue problems, efficient matrix-element measurements, and Monte Carlo-guided allocation. Applied to H4 and hexatriene, NOVQE improves the logical ansatz without additional coherent quantum resources, but its measurement cost remains substantially above standard VQE.

  • Problem

    Strongly correlated quantum chemistry requires methods suited to noisy near-term devices, where precision demands and circuit errors limit straightforward advantages over classical variational approaches.

  • Method

    NOVQE approximates the ground state in a subspace spanned by independently parametrized quantum states, using pairwise Hamiltonian and overlap measurements, generalized eigenvalue problems, and adaptive Monte Carlo measurement allocation.

  • Results

    NOVQE increases ansatz flexibility without additional coherent quantum resources, with continued marginal utility from adding states in H4 and hexatriene experiments.

  • Takeaways & Limitations

    The approach combines individually limited ansätze into a more powerful logical ansatz while retaining systematic subspace growth.

  • Takeaways & Limitations

    Adaptive measurement uses fewer measurements than a non-adaptive strategy but dramatically more than standard VQE energy measurement.

Abstract

from arXiv · show

Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the ground state of a system by solving a generalized eigenvalue problem in a subspace spanned by a collection of parametrized quantum states. This allows for the systematic improvement of a logical wavefunction ansatz without a significant increase in circuit complexity. To minimize the circuit complexity of this approach, we propose a strategy for efficiently measuring the Hamiltonian and overlap matrix elements between states parametrized by circuits that commute with the total particle number operator. We also propose a classical Monte Carlo scheme to estimate the uncertainty in the ground state energy caused by a finite number of measurements of the matrix elements. We explain how this Monte Carlo procedure can be extended to adaptively schedule the required measurements, reducing the number of circuit executions necessary for a given accuracy. We apply these ideas to two model strongly correlated systems, a square configuration of H$_4$ and the $π$-system of Hexatriene (C$_6$H$_8$).

I. INTRODUCTION

NOVQE extends VQE by combining independently parametrized quantum states in a non-orthogonal subspace and solving a generalized eigenvalue problem. It increases ansatz flexibility while controlling circuit complexity and measurement cost.

  • I. INTRODUCTION: The method measures pairwise Hamiltonian and overlap matrix elements, then classically solves a generalized eigenvalue problem to optimize the subspace coefficients.The basis-state parameters can be optimized in an outer classical loop, with a new generalized eigenvalue problem at each step.
  • I. INTRODUCTION: NOVQE represents the logical wavefunction as a linear combination of independently parametrized quantum-circuit states.The component circuits need efficient implementations but are not restricted to a particular wavefunction form.
  • I. INTRODUCTION: Matrix-element circuits require the gate counts for preparing the two states plus a factor scaling linearly with system size.The required quantum volume is independent of the number of wavefunctions in the logical ansatz.
  • I. INTRODUCTION: Monte Carlo uncertainty estimation samples noisy Hamiltonian and overlap matrices, while adaptive allocation targets measurements that most reduce ground-state-energy uncertainty.This addresses the increased measurement burden introduced by the larger logical ansatz.
  • I. INTRODUCTION: The approach is applied to a square H4 configuration and the π-system of hexatriene, systems exhibiting strong and dynamical correlation effects.The examples focus on strongly entangled electrons alongside dynamical correlation from the remaining electrons.

II. THEORY

The theory measures off-diagonal Hamiltonian and overlap elements through ancilla-based pairwise superpositions rather than direct observables. Particle-number-preserving circuits enable efficient controlled-unitary constructions, while standard Hamiltonian averaging handles commuting or grouped Pauli terms.

  • A. Matrix Element Measurement: Off-diagonal Hamiltonian elements are not physical observables, so the method estimates them using ancilla-assisted circuits and conditional Hamiltonian measurements.The construction combines superposition preparation with ancilla measurements to recover real and imaginary matrix elements.
  • A. Matrix Element Measurement: The presentation assumes commuting Hamiltonian terms, while noncommuting terms can be handled by grouping them into separately measurable sets.This preserves the usual Hamiltonian-averaging procedure.
  • A. Matrix Element Measurement: Particle-number-preserving unitaries that act trivially on the vacuum bypass the impossibility of controlling an arbitrary unitary with one black-box use.The construction uses an added vacuum register, controlled-SWAP operations, and unmodified executions of the variational circuits.
  • A. Matrix Element Measurement: The measurement circuit prepares the two variational states, applies controlled-SWAP operations, measures the ancilla, and then measures the Hamiltonian on the system register.The resulting conditional expectation values are combined with the ancilla eigenvalues.
  • A. Matrix Element Measurement: The protocol prepares pairwise superpositions of |φi⟩ and |φj⟩, whose ancilla observables encode overlap and Hamiltonian matrix elements.For example, the real overlap is obtained from the ancilla Z expectation value, and the Hamiltonian analogue uses HZanc.
  • A. Matrix Element Measurement: For a Pauli-sum Hamiltonian, repeated measurements estimate real off-diagonal Hamiltonian elements with approximately the same measurement scaling as diagonal elements at fixed precision.Imaginary overlap and Hamiltonian elements are obtained with an analogous phase-modified preparation.

B. Diagonalization With Uncertainty

Finite-sample estimates of Hamiltonian and overlap elements induce uncertainty in the lowest generalized eigenvalue. The method models matrix elements probabilistically, repeatedly solves sampled eigenproblems, and uses the resulting energy distribution to quantify uncertainty.

  • B. Diagonalization With Uncertainty: Monte Carlo sampling draws Hamiltonian and overlap matrices from these distributions and solves a generalized eigenvalue problem for each draw.The sample variance of the lowest eigenvalues estimates uncertainty in the NOVQE ground-state energy.
  • B. Diagonalization With Uncertainty: The uncertainty procedure models experimentally estimated matrix elements with normal distributions whose variances scale with the number of measurements.Hamiltonian variances use established upper bounds, while overlap variance is bounded by 1/m.
  • B. Diagonalization With Uncertainty: Canonical orthogonalization discards sampled overlap-matrix eigenvalues below a numerical cutoff when noise destroys positive semidefiniteness.The associated eigenvectors are discarded with those eigenvalues.
  • B. Diagonalization With Uncertainty: The Monte Carlo energy distribution is non-Gaussian, and its mean is not generally an unbiased estimate because eigenvalue corrections depend quadratically on off-diagonal noise.Convergence with respect to Monte Carlo sample count should be checked, and standard errors should be used cautiously.

1. Experiment Design Heuristic

The adaptive heuristic estimates how additional measurements affect Monte Carlo energy uncertainty and allocates measurements to the matrix element with the greatest predicted reduction. Measurements are added in batches until the target accuracy or budget is reached.

  • 1. Experiment Design Heuristic: The heuristic estimates the impact of additional Hamiltonian and overlap measurements by differentiating the Monte Carlo energy standard deviation with respect to measurement counts.These derivatives approximate how much uncertainty would shrink after measuring each matrix element further.
  • 1. Experiment Design Heuristic: After an initial measurement of every matrix element, the procedure measures the element whose estimated derivative is most negative.Additional measurements are performed in small batches to limit classical-processing overhead.
  • 1. Experiment Design Heuristic: The allocation loop repeats until the desired accuracy is reached or a predefined measurement budget is exhausted.The goal is an approximate optimization of measurement allocation across matrix elements.

C. Implementation

The implementation supports varied ansätze, provided their circuits share a reference state and conserve fermionic excitation number. Numerical experiments use k-UpCCGSD, whose depth scales linearly with system size and improves with k.

  • The matrix-element measurement tools apply to varied ansätze whose circuits act on a common reference state and conserve fermionic excitation number.
  • The numerical experiments focus on k-UpCCGSD wavefunctions.
  • k-UpCCGSD circuit depth scales linearly with system size and can be systematically improved by increasing k.

1. The k-UpCCGSD Ansatz

The k-UpCCGSD ansatz applies k independently parametrized approximations to unitary coupled-cluster circuits on a Hartree–Fock reference. It uses sparse single and paired-double excitations, with implementation based on FSIM and FSWAP gates.

  • k-UpCCGSD acts on a Hartree–Fock reference with k independently parametrized blocks approximating unitary coupled-cluster circuits.
  • Each block uses a sparse cluster operator containing only single and paired double excitations.
  • Unlike UCCSD, k-UpCCGSD restricts double excitations to electron pairs moving between spatial orbitals, while generalized excitations and repetition provide additional flexibility.
  • The circuit implementation includes FSIM2, FSIM4, and FSWAP gates.
  • Compiling each FSWAP with the immediately preceding FSIM2 yields an overall depth of kN(τ2 + τ4/2).The passage defines τ2 and τ4 as effective gate durations.

2. Computational Details

The computational study compares NOVQE results with FCI and Hartree–Fock references, optimizing the newest ansatz state while repeatedly diagonalizing Hamiltonian and overlap matrices. Random parameter kicks help escape local minima.

  • FCI ground states and Hartree–Fock reference wavefunctions were computed with Psi4 and OpenFermion.
  • NOVQE optimization varies the newest ansatz wavefunction's parameters and diagonalizes the Hamiltonian and overlap matrices at each step.
  • Random parameter kicks are applied after gradient-evaluation intervals, with variances σ2 = 1, 10^-1, and 10^-2 at 500, 1000, and 1500 steps.

III. RESULTS

The results study NOVQE representations for strongly correlated H4 and hexatriene systems, varying ansatz complexity to assess how well linear combinations of parametrized wavefunctions represent ground states.

  • Square H4 at D4h geometry has two exactly degenerate determinants and combines strong and weak correlation effects.
  • Hexatriene is studied as a hydrocarbon whose torsional potential-energy surface includes a strongly correlated θ = 90° geometry.
  • The hexatriene active space contains six electrons in six orbitals, including the system's π electrons and both Π and Π* orbitals.
  • The experiments vary ansatz complexity to assess how effectively optimized linear combinations of parametrized wavefunctions represent ground states.

1. A Hydrogen Complex, H4

NOVQE improves ground-state estimates by combining parametrized states in a generalized subspace, with flexibility increased through either more states or more circuit blocks. H4 and Hexatriene calculations show chemical-accuracy performance, while convergence and circuit-cost trade-offs depend on the ansatz and system.

  • H4: For H4, k ≥ 2 UpCCGSD is essentially exact with one state, equivalent to regular VQE.One k = 2 optimization failed to find the global optimum, and optimization challenges are expected to worsen with system size and stochastic measurements.
  • H4: For k = 1 H4, increasing the NOVQE subspace systematically improves accuracy, with M = 3 independent states representing the ground state almost exactly.The result supports combining individually limited ansatz states into a stronger logical ansatz.
  • Hexatriene: For Hexatriene, increasing either circuit blocks k or subspace size M drives the ground-state error below chemical accuracy.The study considers equilibrium and 90° twisted configurations, with circuits up to k = 5 and subspaces up to M = 10.
  • Hexatriene: For Hexatriene, multiple k = 1 states require almost four times less circuit depth and half as many gates than VQE with k = 4.This comparison concerns the NOVQE procedure for k = 1 states.
  • Hexatriene: Hexatriene errors decrease exponentially with M for k = 1 in both configurations and k = 2 in the twisted configuration.The authors contrast this with slow convergence from dynamic-correlation tails in classical non-orthogonal calculations and identify possible breakdown for more complicated systems as future work.
  • Measurement analysis: NOVQE matrix-element measurements use optimized states and compare adaptive scheduling with uniform non-adaptive allocation under finite measurement noise.The adaptive protocol selects matrix elements iteratively, whereas the non-adaptive protocol measures each element equally often.

1. A Hydrogen Complex, H4

Adaptive measurement scheduling converges faster than uniform allocation for NOVQE energy estimation in H4 and Hexatriene. It yields few-fold savings relative to non-adaptive scheduling, but the total measurement cost remains substantially above regular VQE and requires further reduction.

  • H4: In H4, adaptive scheduling converges more quickly than non-adaptive scheduling for two M = 4 subspaces of optimized k = 1 states.The comparison uses estimated ground-state trajectories with error bars; the variance model is based on upper bounds.
  • Limitations: NOVQE measurement costs remain orders of magnitude larger than ordinary VQE energy measurement, despite adaptive scheduling gains.The authors identify adapting regular-VQE measurement strategies and modifying NOVQE optimization as possible routes to reducing circuit executions.
  • Hexatriene: For Hexatriene, adaptive scheduling has κ ≈ 3.7·10^5 and 6.5·10^5, compared with κ ≈ 2.9·10^6 for non-adaptive panels A and B.The adaptive scheme provides a slightly larger gain than in H4 but remains far from experimentally plausible measurement counts.

IV. DISCUSSION AND OUTLOOK

NOVQE extends VQE by combining parametrized quantum states through a generalized eigenvalue problem, increasing ansatz flexibility through pairwise matrix measurements. The authors demonstrate adaptive measurement scheduling in model strongly correlated systems while identifying measurement cost and conditioning challenges.

  • NOVQE approach: NOVQE solves a generalized eigenvalue problem in a subspace spanned by M parametrized wavefunctions, forming a more flexible logical ansatz from pairwise Hamiltonian and overlap measurements.The approach increases ansatz flexibility without requiring additional coherent quantum resources.
  • Measurement strategy: The measurement protocol uses modified Hadamard tests and exploits common reference states with conserved fermionic excitation number to avoid controlled ansatz circuits.This design targets lower circuit complexity for off-diagonal matrix-element measurements.
  • Measurement scheduling: The uncertainty framework estimates ground-state energy errors from finite matrix-element sampling and adaptively selects further measurements through Monte Carlo sampling.The adaptive procedure is designed to reduce circuit executions for a specified accuracy.
  • Numerical demonstrations: For trans-Hexatriene with M = 8 optimized k = 1 k-UpCCGSD states, adaptive scheduling converged significantly faster than non-adaptive scheduling in two measurement-record realizations.The plotted estimates include 2σ error bars, while the actual subspace energies are shown as dashed reference lines.
  • Numerical demonstrations: The experiments found that adding NOVQE states retained substantial marginal utility, unlike classical nonorthogonal methods that can develop slow convergence from dynamic-correlation tails.The authors attribute the potential benefit to using wavefunctions more sophisticated than Slater determinants.
  • Numerical demonstrations: In H4 and trans-Hexatriene experiments, adaptive scheduling used somewhat fewer measurements than equal-allocation scheduling but substantially more than standard VQE energy estimation.The comparisons used optimized NOVQE states with frozen parameters in fixed subspaces.
  • Limitations and outlook: Nearly linearly dependent NOVQE states can dramatically increase measurement cost, motivating regularized optimization, improved initialization, and gradient-based extensions.The authors also identify adapting tensor-factorization measurement strategies as a possible route to reduce separately measured terms.

Appendix A: Hexatriene Geometries

The appendix specifies two Hexatriene geometries used in the study: the equilibrium trans configuration and a configuration with the central double bond twisted by 90°.

  • Equilibrium configuration: The equilibrium trans-Hexatriene geometry was obtained through geometry optimization using ωB97X-D and ccpVTZ.The calculation used a development version of Q-Chem.
  • Twisted configuration: The 90° twisted Hexatriene geometry was generated by rotating the middle C-C double bond out of plane.This configuration is listed as the twisted geometry in Table II.
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