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Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz

Jonathan Romero, Ryan Babbush, Jarrod R. McClean, Cornelius Hempel, Peter Love, Alán Aspuru-Guzik

arXiv:1701.02691v2quant-ph

TL;DR

Classical molecular-energy simulation is difficult for strongly correlated systems, while exact methods require exponential resources. The paper studies VQE with a UCC ansatz and introduces strategies for more efficient state preparation and optimization, including analytical gradients. The proposed approach reduces computational resources while preserving solution quality, and analytical gradients substantially reduce sampling cost compared with numerical gradients.

  • Problem

    Strongly correlated molecular systems challenge approximate classical quantum-chemistry methods, while exact diagonalization requires exponential resources.

  • Method

    The paper implements VQE with a UCC ansatz and uses improved initial guesses, cluster-amplitude pre-screening, active spaces, and analytical energy gradients.

  • Results

    The proposed analytical gradients achieve solutions of the same quality as derivative-free and numerical-gradient approaches while having much smaller sampling costs.

  • Takeaways & Limitations

    The paper shows that UCC-based VQE efficiency can be improved through reduced computational resources and more effective gradient-based optimization.

Abstract

from arXiv · show

The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectation values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose an analytical method to compute the energy gradient that reduces the sampling cost for gradient estimation by several orders of magnitude compared to numerical gradients. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the circuit depth of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.

INTRODUCTION

Molecular-energy simulation remains difficult for strongly correlated systems because exact classical methods require exponential resources, motivating quantum approaches such as VQE. The paper develops efficiency strategies for VQE with a UCC ansatz, including improved initialization, resource reduction, and analytical gradients.

  • Exact diagonalization requires exponential classical resources, limiting exact simulation of molecular energies.
  • VQE combines quantum expectation-value measurements with classical optimization to minimize the energy of a parameterized wavefunction.
  • UCC provides a polynomial-gate hierarchy of wavefunctions, but its parameter count may still be too large for practical calculations on large molecules.
  • The paper introduces initial guesses, cluster-amplitude pre-screening, active-space approaches, and analytical energy gradients to improve UCC-VQE efficiency.
  • Accurate molecular energies matter because chemical rates are exponentially sensitive to changes in the potential energy surface.

B. Classical ab initio approaches to quantum chemistry

Classical quantum-chemistry methods approximate many-electron wavefunctions through determinant expansions or exponential coupled-cluster forms, balancing accuracy, correlation treatment, and computational tractability. Their limitations become pronounced for strongly correlated systems and non-equilibrium molecular geometries.

  • Hartree-Fock provides a mean-field wavefunction but cannot capture electron-correlation effects needed for near-chemical-accuracy energies.
  • FCI is exact within a basis but becomes rapidly intractable because the number of determinants grows factorially with the number of spin orbitals.
  • Truncated CI reduces cost by retaining excitations up to a fixed order, but converges slowly for highly correlated systems and is not size-extensive.
  • Coupled cluster uses an exponential wavefunction ansatz, producing size-extensive energies and multiplicatively separable wavefunctions.
  • Single-reference truncated coupled cluster generally performs poorly for strongly correlated systems and may fail near transition states or multiple-bond dissociation limits.
  • Projected coupled cluster is classically tractable, but its non-unitary operator yields a non-variational energy and relies on a single-reference assumption.

C. Unitary coupled cluster

Unitary coupled cluster restores a variational formulation that can be implemented on a quantum computer, although its energy equations are not classically tractable because the BCH expansion is infinite. VQE prepares the UCC state, measures Hamiltonian expectations, and iteratively updates amplitudes until convergence.

  • Unitary coupled cluster redefines the excitation operator to be unitary, producing a variational ansatz that spans the same Hilbert space as traditional coupled cluster.
  • The UCC total energy is obtained from the variational principle.
  • The UCC BCH expansion contains commutators between T and T† with no natural termination point, making the series infinite.
  • Because of the infinite BCH series, no known method efficiently evaluates UCC energy and amplitude equations classically without further approximation.
  • A quantum computer can readily apply eT−T†, enabling preparation of UCC wavefunctions with truncated cluster expansions.
  • VQE iterates state preparation, Hamiltonian-term expectation measurements, and classical parameter updates, avoiding phase-estimation overhead.
  • For UCC, the algorithm prepares a reference state, applies U(t), sums Hamiltonian-term expectations, and updates amplitudes until energy and amplitude changes meet thresholds.

A. Implementation of UCC on a quantum computer

The UCC ansatz is mapped to quantum-gate operations through fermion-to-qubit transformations, trotterization, and Pauli-string decompositions. Gate choices and mappings determine implementation cost, with truncation and hardware-specific entangling gates reducing resources.

  • UCC decomposition: The UCC unitary is approximated by trotterization because excitation operators do not necessarily commute.The paper uses trotter numbers ρ = 1 and ρ = 2 in its state-preparation unitaries.
  • Fermion-to-qubit mapping: Jordan-Wigner and Bravyi-Kitaev mappings convert fermionic UCC operators into products of Pauli matrices executable on quantum computers.Each excitation produces multiple Pauli subterms; a double excitation contains eight subterms.
  • Gate implementation: Pauli-string exponentials use single-qubit basis rotations, parity-computing CNOTs, and an inverse basis-change stage when X or Y operators occur.Hadamard gates rotate to the X basis, while Rx(π/2)-type gates rotate to the Y basis.
  • Truncation: UCCSD retains only single and double excitations, preserving a polynomial implementation and providing an exact description for two-electron systems.The number of parameters grows as O(N^2η^2).
  • Resource scaling: Bravyi-Kitaev implementation scales as O(N^2η^2) gates up to logarithmic factors, compared with O(N^3η^2) for Jordan-Wigner.Limited connectivity can require SWAP operations that may eliminate the Bravyi-Kitaev advantage, while Jordan-Wigner may be more robust to noise.
  • Hardware-specific implementation: Mølmer-Sørensen gates can require two entangling operations per parameter, reducing entangling-gate counts relative to CNOT-based implementations.This matters because entangling operations remain a limiting factor in leading architectures.

B. Choice and preparation of the reference state

Reference-state quality affects UCC convergence, especially for strongly correlated molecular wavefunctions. Hartree-Fock provides a simple product-state reference, while multireference states can improve the starting point when Hartree-Fock is inadequate.

  • Reference-state requirements: A high-overlap reference state facilitates convergence of the UCC ansatz toward the many-body solution.The Hartree-Fock solution generally provides such a reference.
  • Hartree-Fock reference: After Jordan-Wigner or Bravyi-Kitaev mapping, the Hartree-Fock state is a single computational-basis product state that can be initialized directly.In Jordan-Wigner ordering it is represented by occupied and unoccupied qubit strings.
  • Strong correlation: Strongly correlated wavefunctions can make Hartree-Fock a poor starting guess, motivating entangled multireference states from MCSCF or DMRG calculations.Such states are useful when they contain only a polynomial number of computational-basis states.

C. Energy measurement

VQE evaluates molecular energy by preparing a parameterized UCC state and measuring Hamiltonian terms individually. The section also introduces classical amplitude-based parameter reduction and an analytical gradient strategy intended to lower gradient-estimation sampling costs.

  • Energy evaluation: The VQE objective is the molecular energy obtained by measuring each Pauli-term expectation value and summing the Hamiltonian contributions.Hamiltonian averaging avoids phase estimation, whose circuit depth is prohibitively large for current and near-future devices.
  • Measurement procedure: Each Hamiltonian Pauli term is measured with projective measurements after applying basis changes appropriate to its Pauli operators.The total measurement precision depends on the variances and coefficients of the individual terms.
  • Parameter optimization: Classical MP2 or coupled-cluster amplitudes can initialize VQE parameters and screen excitation operators with small estimated amplitudes.Additional operators may be restored after optimization until the desired convergence threshold is reached.
  • Analytical gradients: The paper proposes evaluating the energy gradient analytically on the quantum computer for a product of parametrized unitaries.The circuit uses an ancilla-controlled Pauli operator and measures an ancilla observable to recover gradient contributions.
  • Gradient cost: Numerical-gradient sampling cost increases quadratically as the finite-difference step decreases, and is expected to exceed analytical-gradient cost for δ < 0.5.Finite differences also impose a step-size-dependent accuracy limit.

F. VQE-UCC with an active space approximation

The CAS-UCC strategy restricts excitations to an active orbital space and progressively enlarges that space. This reduces qubit and parameter requirements while using smaller-space solutions as initial guesses for larger calculations.

  • Active-space construction: The complete active space approximation divides orbitals into inactive and active sets, restricting UCC excitations to active orbitals.The active space is denoted CAS(ηA, NA), with ηA electrons distributed among NA spatial orbitals.
  • Resource reduction: CAS-UCC reduces the number of qubits required by a factor of NA/N.The reduction follows from separating active and inactive orbital spaces and forming an effective active-space Hamiltonian.
  • Parameter reduction: The UCCSD parameter count is reduced by a factor of (ηA NA)^2/(Nη)^2 relative to full-UCCSD.The active-space scaling is O(ηA^2N^2).
  • Active-space selection: Natural-orbital occupations can select an initial active space by retaining orbitals with fractional occupations and discarding generally integer-occupied orbitals.This selection targets static-correlation effects.
  • Progressive enlargement: A CAS-UCC solution can initialize a calculation with a larger active space, repeating the process until the full basis is reached.The paper expects faster convergence because each stage supplies a better approximation to the exact UCC wavefunction.

A. Classical simulation of VQE-UCC

The simulations implement VQE-UCC for small molecules using quantum-chemistry integrals, Jordan–Wigner mapping, and classical optimization. The setup uses several optimizers, numerical gradients for L-BFGS-B, fixed convergence thresholds, and double-precision evaluations.

  • Molecular integrals were obtained with PSI4 and mapped to qubits using the Jordan–Wigner transformation.
  • The UCC unitary used a truncated cluster operator whose symbolic form was converted into Pauli-string unitaries.
  • Nelder-Mead, Powell, COBYLA, and L-BFGS-B were used for classical optimization, with numerical gradients for L-BFGS-B.
  • All numerical experiments used double-precision arithmetic unless otherwise indicated.

B. VQE-UCC results for H4 molecular systems

The H4 benchmark scans potential-energy surfaces along rectangular, trapezoidal, and linear geometrical paths. These paths vary the wavefunction’s entanglement, providing a model for assessing VQE-UCC across changing correlation regimes.

  • Scanning geometries along a potential-energy surface changes the wavefunction’s entanglement and affects ansatz performance.
  • The study examines H4 potential-energy surfaces along rectangular, trapezoidal, and linear paths.
  • The H4 model was selected because its wavefunction entanglement can be varied continuously while remaining suitable for simulation.
  • The rectangular, trapezoidal, and linear geometries are used as benchmark paths for multireference methods.

1. Influence of the optimization method in the VQE performance

The H4 experiments compare optimizers, initialization strategies, trotterization, parameter screening, and gradient estimation. They show that optimizer choice and classical amplitude guesses strongly affect convergence, while circuit reductions can preserve accuracy under the tested conditions.

  • Evaluation criteria: The evaluation compares energy error, wavefunction infidelity, and function evaluations using random, zero, and MP2 initial amplitudes.Random initialization uses values uniformly sampled from −0.25 to 0.25, while zero initialization corresponds to the Hartree–Fock reference.
  • Optimization methods: Nelder-Mead performs poorly from random or zero initialization, while COBYLA and L-BFGS-B usually reach similar minima across the potential-energy surface.Nelder-Mead can exceed 10 kcal/mol energy error and fail to converge within 20,000 evaluations.
  • Initialization strategies: MP2 amplitude guesses significantly reduce function evaluations for all optimization methods and improve Nelder-Mead and Powell energy accuracy.For strongly correlated systems, MP2 may be a poor starting point, while DMRG with a small active space can provide more reliable guesses at higher classical cost.
  • Trotterization: Trotterized unitaries produce nearly identical quality to the exact UCC implementation with L-BFGS-B, while two trotter steps converge faster on average than one.
  • Trotterization: Along the strongly multireference linear path, COBYLA yields overlaps below 0.95 and as low as 0.78, whereas L-BFGS-B performs better.Increasing the number of trotter steps partially alleviates COBYLA’s difficulty in this region.
  • Parameter screening: Only 10 of 34 H4 double-excitation operators significantly affect the total energy, and discarded-parameter errors remain below chemical accuracy.Pre-screening reduces circuit depth and the number of optimization parameters.
  • Gradient estimation: Figure 8 compares analytical and numerical gradient errors against measurement count for 100 random amplitudes in linear H4 at r = 1.2 ˚A.
  • Parameter screening: A threshold of 10−5 reduces the number of UCC parameters and circuit depth by almost one order of magnitude across the tested molecules.Thresholds above 10−5 can produce subquadratic scaling in Nη instead of the formal quadratic scaling of full UCCSD.

4. Gradient based optimization

The paper compares analytical and numerical gradients for VQE optimization, finding that analytical gradients retain similar accuracy while reducing sampling and improving convergence under control errors.

  • Sampling cost: Analytical gradients have a much smaller sampling cost than numerical gradients.Numerical gradients can stop improving beyond 10^8 samples because of an accuracy limit, while smaller step sizes require exceedingly larger sampling costs.
  • Control errors: The gradient error scales linearly with control-error magnitude ∆Θ, whereas its numerical-gradient contribution scales quadratically with the step size δ.Control errors impose a practical lower bound on δ and can dominate the numerical-gradient error.
  • Optimization under noise: Analytical gradients required 20% fewer gradient evaluations on average than numerical gradients under control errors with ∆Θ = 0.01.The two approaches produced final-energy accuracies in similar ranges across 150 VQE optimization runs.
  • Hybrid resource reduction: Classical cluster-amplitude approximations provide effective initial guesses and prescreening that remove operators with negligible contributions to the optimal wavefunction.These strategies reduce VQE cost by combining classical approximations with quantum optimization.
  • Classical optimization: Derivative-free Nelder–Mead and Powell methods performed poorly with many parameters unless supplied with physically meaningful initial guesses, while COBYLA performed better.The comparison identifies optimizer choice as an important factor in UCC wavefunction optimization.
  • Gradient comparison: Analytical gradients achieve similar solution quality to derivative-free and numerical gradient approaches for UCC optimization.The comparison includes numerical simulations of UCC wavefunction optimization.

Appendix A: Commutativity of subterms in excitation operators

The appendix shows that the Pauli-string subterms produced by UCC excitation operators commute, allowing the operator structure to be rearranged without changing the result.

  • Operator structure: A UCC operator of order n contains 2^(2n−1) Pauli-string terms sharing a common Z-string structure.The remaining factors span possible X and Y strings on 2n qubits with odd X and Y counts.
  • Commutator analysis: The commutator between the X/Y strings determines commutativity because the common Z-string factors do not alter the relevant comparison.The appendix reduces the analysis to strings of X and Y operators acting on the same qubits.
  • Pauli products: Multiplying corresponding X and Y operators yields 1, iσ_z, or −iσ_z depending on whether the symbols match or differ in order.The three cases are specified by the local operator pair at each qubit.
  • Commutativity result: For UCC operators, the relevant X/Y parity conditions make the commutator zero, so the subterms of a single UCC operator commute.This conclusion follows after rearranging the commutator expression using the parity of the X and Y counts.
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