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The theory of variational hybrid quantum-classical algorithms

Jarrod R. McClean, Jonathan Romero, Ryan Babbush, Alán Aspuru-Guzik

arXiv:1509.04279v1quant-phphysics.chem-ph

TL;DR

Near-term quantum algorithms face resource requirements beyond available hardware, motivating hybrid methods that combine quantum and classical resources. This paper develops a general theory of VQE, introduces variational ansätze and error suppression, and adds measurement and optimization improvements, including computational savings of up to three orders of magnitude over earlier optimization techniques.

  • Problem

    Many quantum algorithms require hardware resources beyond near-term quantum devices, motivating hybrid quantum-classical approaches that use limited quantum resources with classical routines.

  • Method

    The paper extends VQE theory with variational adiabatic and unitary coupled-cluster ansätze, variational error suppression, Hamiltonian-averaging reductions, and derivative-free optimization techniques.

  • Results

    Up to three orders of magnitude in computational savings are reported for modern derivative-free optimization techniques compared with previously used optimization techniques.

  • Takeaways & Limitations

    VQE offers a framework for combining quantum and classical resources on pre-threshold devices while reducing measurement and optimization costs.

Abstract

from arXiv · show

Many quantum algorithms have daunting resource requirements when compared to what is available today. To address this discrepancy, a quantum-classical hybrid optimization scheme known as "the quantum variational eigensolver" was developed with the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines. In this work we extend the general theory of this algorithm and suggest algorithmic improvements for practical implementations. Specifically, we develop a variational adiabatic ansatz and explore unitary coupled cluster where we establish a connection from second order unitary coupled cluster to universal gate sets through relaxation of exponential splitting. We introduce the concept of quantum variational error suppression that allows some errors to be suppressed naturally in this algorithm on a pre-threshold quantum device. Additionally, we analyze truncation and correlated sampling in Hamiltonian averaging as ways to reduce the cost of this procedure. Finally, we show how the use of modern derivative free optimization techniques can offer dramatic computational savings of up to three orders of magnitude over previously used optimization techniques.

I. INTRODUCTION

The variational quantum eigensolver addresses the resource demands of quantum algorithms by combining quantum state preparation and measurement with classical optimization. Its variational principle supports ground- and excited-state estimation, including for mixed states and operators decomposed into measurable terms.

  • Near-term quantum hardware cannot support many quantum algorithms because their hardware requirements and overhead exceed available capabilities.
  • The VQE combines parameterized quantum state preparation, expectation-value measurement, and classical optimization to estimate eigenvalues and eigenvectors.
  • The paper extends VQE theory and practice with hardware-adaptable ansätze, error suppression, measurement-cost reductions, and improved optimization methods.
  • Hamiltonian expectation values can be measured by decomposing operators into polynomially many simple terms, including weighted tensor products of Pauli operators.
  • The variational principle identifies parameters minimizing the Hamiltonian expectation as an approximation to the ground state, and orthogonality constraints extend the approach to excited states.
  • The same variational principle applies to mixed states, making energy minimization relevant when state preparation includes errors or environmental influence.

B. Fermionic Hamiltonians and Quantum Chemistry

Quantum chemistry maps electronic-structure problems into finite fermionic Hamiltonians and then onto qubits. The resulting operators provide the form needed for quantum-computer treatment of molecular eigenvalue problems.

  • Electronic-structure calculations seek eigenvectors and eigenvalues of fermionic Hamiltonians defined by nuclear charges, electron coordinates, and nuclear positions.
  • The Born-Oppenheimer approximation treats nuclei as classical point charges with fixed positions because nuclear and electronic masses are widely separated.
  • Second quantization projects the problem into a finite orthogonal spin-orbital basis and represents fermionic antisymmetry with creation and annihilation operators.
  • Fermionic Hamiltonians must be mapped to qubits, with Jordan-Wigner and Bravyi-Kitaev transformations identified as common mappings.

C. Reference States

Reference states provide efficient starting points for quantum chemistry ansätze, but mean-field references can fail under strong correlation. The VQE workflow then prepares, measures, and iteratively optimizes a parameterized state.

  • Reference states are product states used as starting points for defining more general electronic-structure wavefunctions.
  • Local basis transformations convert product references into simple computational-basis forms, trading classical Hamiltonian transformation effort for reduced quantum preparation resources.
  • When mean-field theory is adequate, excitation-based corrections motivate MP2, configuration interaction, and coupled-cluster methods.
  • Strong correlation can make mean-field descriptions poor starting points, motivating multi-reference methods that begin with entangled states.
  • The VQE prepares a parameterized state, evaluates the objective, uses a classical nonlinear optimizer, and iterates until energy convergence.
  • The paper develops additional algorithmic and conceptual improvements to these preparation, measurement, and optimization steps.

III. STATE PARAMETERIZATION AND PREPARATION

State parameterization determines which quantum states VQE can prepare and how adiabatic evolution is optimized under finite-time and noisy conditions. Variationally optimizing the evolution path can improve preparation when fixed schedules are inadequate.

  • A useful VQE ansatz should describe the target solution while remaining difficult to prepare or sample classically with known methods.
  • Energy and variance measurements provide bounds for estimating eigenvalue accuracy and assessing whether an ansatz should be changed.
  • Even a linear adiabatic path may require only one parameter when the evolution is sufficiently slow under ideal conditions.
  • Finite evolution time, technological constraints, and decoherence can make fixed or slower adiabatic evolution unsuitable for ground-state preparation.
  • The schedule Hamiltonian has an avoided crossing at A(s) = 1/2 whose gap is controlled by ϵ, illustrated with ϵ = 0.1.
  • Intermediate Hamiltonians and boundary-constrained paths provide additional variational flexibility for adiabatic state preparation.
  • Adiabatic state preparation becomes a VQE ansatz when parameters define the evolution path and classical optimization minimizes the final Hamiltonian expectation.

1. Variational Adiabatic Path Example

A variational spline schedule prepares a ground state under limited evolution time by optimizing the final Hamiltonian expectation. It slows near the avoided crossing and achieves comparable performance to a linear schedule using roughly one-tenth the evolution time.

  • Variational Adiabatic Path Example: The example optimizes a two-parameter cubic B-spline schedule for ground-state preparation when the maximum evolution time is limited.The parameters are chosen by nonlinear minimization of the final-state Hamiltonian expectation.
  • Variational Adiabatic Path Example: The optimized path slows evolution near the closing gap without prior spectral knowledge, using only endpoint measurements.Compared with the standard linear path, the schedule is otherwise only slightly distorted.
  • Variational Adiabatic Path Example: 10 times less evolution time produces similar results to the linear path, reducing the quantum evolution-time requirement by a factor of 10.The reduction is obtained at the cost of classical minimization and relies on final-state measurements in a black-box procedure.

2. Pontryagin’s Principle and Non-Adiabatic Bang-Bang

The paper extends variational state preparation beyond adiabatic schedules through non-adiabatic control and unitary coupled-cluster constructions. Relaxing exponential-splitting parameters connects second-order unitary coupled cluster to universal two-qubit gates.

  • Pontryagin’s Principle and Non-Adiabatic Bang-Bang: Non-adiabatic state preparation can use variationally optimized schedules, including bang-bang controls from optimal-control theory.The schedule can be adapted to engineer a desired state or perform more general quantum computation.
  • Unitary Coupled Cluster: Unitary coupled cluster explores Hilbert space through a unitary generator, unlike projective coupled cluster, whose truncation is well defined classically.The unitary form is naturally suited to quantum state preparation but lacks the projective form’s well-defined truncation.
  • Unitary Coupled Cluster: Second-order unitary coupled cluster has O((3N)^2) parameters and can represent arbitrary quantum operations at sufficient order and implementation precision.The practically common k = 2 case has a parameter count that grows quadratically with system size.
  • Unitary Coupled Cluster: Relaxing parameter sharing across Trotter steps allows second-order unitary coupled cluster to express arbitrary two-qubit gates and therefore universal gate sets.The construction can generate arbitrary SU(4) operations on any pair of qubits, including gate sets such as Clifford+T.

D. Fermionic UCC

The fermionic unitary coupled-cluster construction generalizes the variational ansatz to interacting two-level systems while preserving fermion-number structure. A state-universal form broadens reference-state coverage beyond conventional reference-specific operators.

  • D. Fermionic UCC: The fermionic formulation specializes the approach to quantum chemistry and generalizes it to generic interacting two-level systems.The antisymmetric electronic case is included as a specialization.
  • D. Fermionic UCC: Fermionic cluster operators conserve particle number and use O(M^(2k)) real parameters at order k.The operators are defined from occupied and unoccupied orbitals relative to a reference.
  • D. Fermionic UCC: Mapping fermionic operators to qubits through Jordan-Wigner produces allowed operations forming a non-trivial subgroup of SU(2^k).The mapping can produce spin flips involving all qubits at every fermionic order k.
  • D. Fermionic UCC: Constructing the ansatz in the fermionic framework avoids irrelevant or symmetry-broken states that spin-first construction might explore.The cited examples include mixtures of different particle-number states.
  • D. Fermionic UCC: The state-universal quantum unitary coupled ansatz is state agnostic, fermion-number conserving, and optimized variationally against the target Hamiltonian.Its parameters are selected by minimizing the Hamiltonian expectation value.

E. Quantum Error Suppression and Symmetries

Variational quantum algorithms can suppress some implementation errors on pre-threshold devices by adjusting ansatz parameters, provided the operator manifold can represent the needed correction. Symmetry penalties extend this capability and can target selected excited states.

  • E. Quantum Error Suppression and Symmetries: For fermionic coupled cluster, particle-number conservation is a built-in symmetry, while partial error correction makes symmetry choices practically consequential.The ansatz and Hamiltonian commute with the number operator, but pre-threshold devices require additional consideration.
  • E. Quantum Error Suppression and Symmetries: Variational error suppression treats an implementation error as suppressible when a parameter correction can compensate for it.The framework distinguishes the formally specified ansatz from the imperfect operation sequence actually executed.
  • E. Quantum Error Suppression and Symmetries: An error is suppressible only when the ansatz manifold contains a correction direction returning the state to the variational minimum.Adding operators can enlarge the manifold and make previously unsuppressible errors suppressible.
  • E. Quantum Error Suppression and Symmetries: A random single-qubit bit-flip error is not suppressible under the original number-conserving operator set.Extending the generators with spin-flip or fermionic number-nonconserving operators can make this error suppressible.
  • E. Quantum Error Suppression and Symmetries: Symmetry-penalty optimization can preserve desired expectation values while enabling variational error suppression, and can access symmetry-defined excited-state minima.The construction is described for states such as molecular triplets or ionic states after photodissociation.

IV. OPERATOR AVERAGING

VQE evaluates the energy objective by averaging measurements of decomposed Hamiltonian terms rather than requiring fully coherent phase estimation. Frequentist and Bayesian variance estimates provide practical stopping rules, though small-sample and numerical-convolution issues remain.

  • Operator averaging: Hamiltonian averaging replaces prohibitive coherent phase estimation with weighted measurements of simple operator terms.The Hamiltonian is decomposed into measurable Hermitian components, whose expectation values are combined linearly.
  • Operator averaging: The total estimator variance is obtained by summing variances of independent term estimators.Independent state preparations make covariances between distinct term estimators zero.
  • Frequentist estimation: Measurements continue term by term until the estimated variance falls below ϵ2/M.This frequentist procedure uses the sample mean and unbiased sample variance, but small samples and zero-variance eigenstates create ambiguities.
  • Bayesian estimation: Bayesian updating uses a Beta posterior, Beta(α+r, β+N−r), to estimate term means and variances from binary measurement outcomes.Starting from Beta(1,1), measurements update α and β until the variance threshold is met; informative priors may reduce cost but can bias results for poor reference states.
  • Convergence: After convergence checks, the normal approximation yields an energy estimate precise to the desired ϵ.Exact convolution of Beta distributions is unavailable analytically and must be performed numerically, although convergence to normality is described as rapid.

B. Cost Reduction

Hamiltonian-averaging cost can be reduced by truncating terms whose bounded contributions are negligible, while reallocating the variance budget to control total mean-square error.

  • Truncation: Truncation removes the k* lowest-contribution terms when their maximum cumulative bias remains below Cϵ.Weighted Pauli terms satisfy |⟨Hγ⟩| ≤ |hγ|, enabling ordering by maximum possible contribution.
  • Truncation: The remaining terms must satisfy C2ϵ2 + Σ Var[⟨Hγ⟩] < ϵ2 to preserve the target mean-square error.The per-term variance threshold becomes (1−C2)ϵ2/(M−k*) after truncation.
  • Truncation: Locality-aware truncation can potentially reduce quantum-chemistry costs dramatically.The parameter C may be selected according to experimental constraints and the operator distribution.

2. Commuting Groups and Correlated Sampling

Grouping commuting Hamiltonian terms can reduce state preparations, but correlations within groups affect variance and therefore the optimal grouping. The two-spin example shows that fewer groups alone does not guarantee the lowest cost.

  • Commuting groups: Commuting operators can be measured sequentially on one state preparation without biasing expectation values.This can save resources because state preparation is expected to cost more than projective measurement.
  • Example: The two-spin example compares five-term, three-group, and two-group measurement prescriptions.The Hamiltonian contains −X1X2, −Y1Y2, Z1Z2, Z1, and Z2, with the displayed groupings determining preparation cost and covariance.
  • Example: 10/ϵ2, 6/ϵ2, and 8/ϵ2 expected state preparations result from the first, second, and third groupings, respectively.The second grouping is best because it reduces preparations while grouping terms with zero covariance.
  • Example: The second grouping reduces cost by almost a factor of 2 relative to measuring all terms individually.The grouping with the fewest commuting sets performs worse because covariance between X1X2 and Y1Y2 adds variance.
  • Correlated sampling: Grouping is state- and operator-dependent, so covariance should guide heuristic choices without biasing the final result.Covariance can be estimated classically from an approximate state or on the quantum device before the main experiment.

C. Beyond Energy to General Observables

The variational framework provides access to observables beyond energy and supports practical optimization improvements. Derivative-free TOMLAB methods achieved higher energy accuracy with substantially fewer function evaluations, while larger problem dimensions and stochastic objectives remain open concerns.

  • General observables: Reduced one- and two-electron density matrices provide access to observables such as dipole moments and charge density without additional measurements.Measured density-matrix elements can be combined classically to compute these expectation values; the approach is also viewed as scalable partial tomography.
  • Optimization results: TOMLAB methods provided superior final-energy accuracy at essentially all measurement precisions above ϵ = .1.Figure 5 averaged results over 20 repetitions and reported error bars as one standard deviation.
  • Optimization challenges: The objective function is generally nonlinear and non-convex, so global optimization and solution verification may be infeasible.Local optima can nevertheless be sufficient when prior knowledge provides high-quality starting points.
  • Optimization results: TOMLAB methods converged to higher energy accuracy and sometimes used 1000 times fewer function evaluations than Nelder-Mead in noisy H2 benchmarks.The benchmark optimized a unitary coupled cluster wavefunction for H2 in a minimal STO-3G basis, repeating each setting 20 times.
  • Optimization results: Because evaluation cost scales roughly as 1/ϵ2 while convergence evaluations remain approximately constant, variable-precision optimization may yield additional savings.The proposed savings concern the cost of evaluating the objective at different measurement precisions.
  • Limitations: The TOMLAB methods require further numerical testing as problem dimension grows, and none were specifically designed for stochastic objectives.The authors identify stochastic-objective optimization as important for reducing overall runtime and as ongoing research.

VII. APPENDIX

The appendix derives bounds connecting the variance of a Hamiltonian to the quality of an estimated eigenvector. Ground-state analysis uses the spectral gap to obtain an overlap bound that approaches one as the variance vanishes.

  • Eigenvector-quality bound: The appendix derives a bound on eigenvector quality determined by the variance of the operator.The derivation treats ground states separately because their spectral structure permits a simpler argument than for general eigenstates.
  • Ground-state derivation: For the ground state, a lower bound on the gap between the ground and first excited eigenvalues is used to bound overlap with the ground state.The derivation rearranges the energy decomposition to obtain the desired overlap bound.
  • Ground-state result: The overlap estimate converges to 1 as Var(⃗θ) is reduced to 0.This conclusion is stated under the appendix’s assumptions, including the requirement that the error be less than the gap.
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