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Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets

Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, Jay M. Gambetta

arXiv:1704.05018v2quant-phcond-mat.supr-con

TL;DR

Classical methods face costly or problematic solutions for interacting fermion systems, motivating quantum approaches to molecular structure. This paper experimentally combines a hardware-efficient VQE, compact fermion-to-qubit encodings, and stochastic optimization, reaching molecular Hamiltonians up to six qubits and extending the approach to quantum magnetism. The reported experiments agree with noisy-device simulations and clarify resources relevant to scaling.

Abstract

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Quantum computers can be used to address molecular structure, materials science and condensed matter physics problems, which currently stretch the limits of existing high-performance computing resources. Finding exact numerical solutions to these interacting fermion problems has exponential cost, while Monte Carlo methods are plagued by the fermionic sign problem. These limitations of classical computational methods have made even few-atom molecular structures problems of practical interest for medium-sized quantum computers. Yet, thus far experimental implementations have been restricted to molecules involving only Period I elements. Here, we demonstrate the experimental optimization of up to six-qubit Hamiltonian problems with over a hundred Pauli terms, determining the ground state energy for molecules of increasing size, up to BeH2. This is enabled by a hardware-efficient variational quantum eigensolver with trial states specifically tailored to the available interactions in our quantum processor, combined with a compact encoding of fermionic Hamiltonians and a robust stochastic optimization routine. We further demonstrate the flexibility of our approach by applying the technique to a problem of quantum magnetism. Across all studied problems, we find agreement between experiment and numerical simulations with a noisy model of the device. These results help elucidate the requirements for scaling the method to larger systems, and aim at bridging the gap between problems at the forefront of high-performance computing and their implementation on quantum hardware.

Supplementary Information is available in the on-line version of the paper.

The supplementary information credits experimental, theoretical, and manuscript contributions across the collaboration and acknowledges external support for the work.

Acknowledgments

  • The work acknowledges experimental, technical, and scientific discussions with numerous collaborators and support from IBM Research Frontiers Institute and IARPA.IARPA support is identified under a device-fabrication contract.Source: Supplementary Information is available… · p. 5

Author contributions

  • A.K. and A.M. contributed equally, while experimental design, device characterization, experiments, theory, simulations, and data interpretation were distributed across the team.The listed contributions also include device fabrication, manuscript composition, and experimental-control software assistance.Source: Supplementary Information is available… · p. 5

I. DEVICE AND CHARACTERIZATION

The experiments use a six-qubit superconducting processor built around transmons, CPW-bus connectivity, individual readout resonators, and calibrated microwave control. Device characterization addresses tunability, readout performance, and gate stability.

Device architecture

  • The processor uses six fixed-frequency transmons and one weakly tunable asymmetric transmon, with two CPW resonators providing quantum-bus connectivity.Six qubits labeled Q1–6 are used in the experiments, and each qubit has an independent resonator for control and readout.Source: §I · p. 7
  • The asymmetric transmon Q4 is directly coupled to the other qubits and operated at an upper flux sweet spot to reduce flux-noise sensitivity and support CR gates.At this operating point, Q4 behaves as a fixed-frequency transmon with coherence times comparable to the other qubits.Source: §I · p. 7

Control and readout

  • The schematic combines microwave qubit control, CR entangling gates, resonator-based measurement, and JPC-then-HEMT amplification for readout.Single-qubit drives address each qubit frequency, while CR gates drive a control qubit at the target frequency.Source: Figure S1 · p. 7
  • Readout characterization reports joint assignment errors below 0.06 for Q2, Q4, and Q6 and below 0.03 for Q1, Q3, and Q5 at 1.5 µs measurement time.Each readout line uses a JPC followed by a HEMT amplifier.Source: §I · p. 8

Gate stability

  • Repeated Xπ calibrations over 18 hours show angle deviations below 1.5°, establishing the measured scale of single-qubit pulse-amplitude drift.The drift is expressed relative to the initial 180° rotation.Source: §I · pp. 8–9, Figure S3 · p. 9

II. HARDWARE-EFFICIENT OPTIMIZATION OF QUANTUM HAMILTONIAN PROBLEMS

The optimization algorithm maps a Hamiltonian to qubits, prepares hardware-efficient trial states, estimates their energies, and iteratively updates controls until convergence. Circuit depth, update budget, and sampling are increased when needed, subject to hardware limits.

Algorithm

  • The method is presented as a compact optimization framework whose subroutines separately handle encoding, entangling-gate preparation, energy evaluation, and classical optimization.Resource requirements are numerically estimated for circuit depth, control updates, and samples.Source: §II · pp. 9–10
  • The workflow maps the Hamiltonian to qubits, chooses circuit depth and variational controls, samples trial states, updates controls, and returns a converged energy estimate.The final estimate uses a separate sample count and the best controls found during the feedback loop.Source: §II · p. 9
  • The algorithm increases circuit depth, control updates, and sampling until the result converges or available resources such as decoherence and sampling time are saturated.Increasing these parameters after convergence does not improve the final energy estimate.Source: §II · p. 9

III. MOLECULAR HAMILTONIANS

The molecular Hamiltonians are constructed in the STO-3G basis, reduced using filling and symmetry assumptions, and mapped to compact qubit Hamiltonians for H2, LiH, and BeH2. The resulting systems contain 4, 99, and 164 Pauli terms on 2, 4, and 6 qubits.

Molecular Hamiltonians

  • The Hamiltonians are computed in the STO-3G basis using one- and two-electron integrals, with selected orbitals retained for LiH and BeH2.The constructions use 4 spin-orbitals for H2, 8 for LiH, and 10 for BeH2 before subsequent reductions.Source: §III · p. 10

Fermionic formulation

  • The fermionic formulation uses creation and annihilation operators obeying anticommutation relations before mapping the Hamiltonian onto qubits.The number of spin-orbitals is M = 4, 8, and 10 for H2, LiH, and BeH2.Source: §III · pp. 10–11

Molecular Hamiltonians

  • Filled inner 1s orbitals in LiH and BeH2 are treated through a Bogoliubov transformation and contribute effective one-body terms or energy shifts.The approximation is stated to apply when the filled orbitals are sufficiently low energy and weakly interacting with the retained orbitals.Source: §III · p. 10

Qubit mappings

  • Binary-tree mapping is used for H2, while parity mapping is used for LiH and BeH2, placing particle-number and spin symmetries in designated qubits for tapering.The parity values are fixed from the total electron number and spin-up electron parity.Source: §III · p. 11

Molecular Hamiltonians

  • The final qubit-tapered Hamiltonians contain 4, 99, and 164 Pauli terms on 2, 4, and 6 qubits for H2, LiH, and BeH2, respectively.They require 2, 25, and 44 tensor-product basis sets, respectively.Source: §III · p. 11

IV. CHARACTERIZATION OF THE ENTANGLERS

The experiment uses collective cross-resonance entanglers, characterized through Hamiltonian tomography and entanglement-sensitive diagnostics. Fixed 150 ns gates are chosen within broad low-error phase plateaus near maximal concurrence while limiting decoherence.

IV. CHARACTERIZATION OF THE ENTANGLERS

  • Cross-resonance drives form the collective entanglers, and high-fidelity CNOT construction is unnecessary provided the drive supplies sufficient entanglement.The gates are implemented by driving a control qubit at the target-qubit frequency.Source: §IV · p. 11
  • Hamiltonian tomography reveals multiple CR interaction terms, while ||R⃗|| = 0 identifies gate times of maximal entanglement.For CR2−4, the measured terms include ZX: 1.04 MHz, ZY: 0.07 MHz, and ZZ: 0.05 MHz.Source: §IV · p. 11
  • Chemical accuracy is reached across a range of fixed gate phases around maximal-concurrence points, producing minimum-energy-error plateaus.The simulations use ZX gates with the experimental connectivity and omit decoherence and stochastic fluctuations.Source: §IV · p. 11
  • The chosen 150 ns gate time places CR phases near the beginning of minimum-error plateaus, balancing sufficient entanglement against decoherence.CR drives on different buses are applied simultaneously to reduce state-preparation time.Source: §IV · pp. 11–12
  • At depth d = 6, the simulated energy error reaches chemical accuracy of 0.0016 Hartree.The supplementary figure compares energy error across depths and entangler phases.Source: Figure S4 · p. 12

V. ENERGY ESTIMATION

Energy estimation reduces molecular Hamiltonians to measurements of weighted Pauli expectations and variances on prepared trial states. These estimates drive the classical angle updates, but finite sampling makes their accuracy central to optimization outcomes.

V. ENERGY ESTIMATION

  • The optimization updates variational angles using measured Hamiltonian expectations to approximate the energy landscape gradient.Every molecular Hamiltonian is represented as a weighted sum of Pauli terms on N qubits.Source: §V · p. 12
  • The mapped Hamiltonian has the form H = Σα hαPα, with each Pα a tensor product of single-qubit Pauli operators and identity.The coefficients hα are real, and the desired quantity is the trial-state expectation ⟨H⟩k.Source: §V · p. 12
  • The mean energy is estimated by sampling each Pauli operator after preparing the same parameterized trial state and applying required post-rotations.The experiment separately estimates Pauli means and variances rather than directly measuring H or its variance.Source: §V · p. 13
  • Sampling variance of the mean energy differs from the quantum Hamiltonian variance: an H eigenstate can have ⟨∆H²⟩ = 0 but finite Var[H].This distinction arises because the experiment samples individual Pauli terms separately.Source: §V · p. 13
  • Individual Pauli measurements use simultaneous dispersive readout in the Z basis, with single-qubit post-rotations for non-diagonal Pauli factors.The largest Pauli coefficient hmax contributes to the sampling-error bound.Source: §V · p. 13

A. Grouping Pauli Operators

Grouping Pauli terms into tensor-product-basis sets reuses state preparations and post-rotations, reducing sampling overhead. Numerical sampling shows this grouping lowers mean-energy variance for all studied molecular Hamiltonians despite covariance trade-offs.

A. Grouping Pauli Operators

  • Grouping Pauli terms into tensor-product-basis sets lets one state preparation sample all terms sharing the same measurement basis.The method reduces repeated post-rotations and state preparations.Source: A. Grouping Pauli Operators · p. 13
  • The grouping trade-off combines fewer samples per Pauli term with covariance contributions within each set, which can be positive or negative.The largest set size smax enters the grouped sampling-error expression.Source: A. Grouping Pauli Operators · p. 13
  • Grouping into TPB sets lowers the variance of the mean energy for H2, LiH, and BeH2 at fixed total sample count.The comparison uses S = 10^3 samples per TPB set and matched total measurements for ungrouped sampling.Source: A. Grouping Pauli Operators · p. 14, Figure S5 · p. 14
  • The grouped and ungrouped procedures estimate Pauli means and covariances from repeated measurements of random states before evaluating total-energy variance.The TPB simulations use A·S total measurements, while ungrouped simulations allocate S_A/T samples per Pauli term.Source: A. Grouping Pauli Operators · p. 14

B. Assignement Errors

Readout assignment errors are modeled by deformed single-qubit projectors and corrected through calibration. For higher-weight Pauli operators, the resulting measurement contrast decreases exponentially with operator weight.

B. Assignement Errors

  • Independent readout assignment errors deform the ideal projectors through two parameters, η0 and η1, while preserving ˆΠ0 + ˆΠ1 = I.The model distinguishes errors when reading |0⟩ as |1⟩ and |1⟩ as |0⟩.Source: B. Assignement Errors · pp. 14–15
  • The measured Pauli signal is shifted and attenuated, with contrast factor 2η1 and shift 1 − 2η0.For a Pauli operator of weight w, the contrast loss becomes exponential in w.Source: B. Assignement Errors · p. 15
  • The experiment calibrates readout before every angle update and corrects sampling outcomes using those calibrations.The stated model includes independent readout errors, while correlated errors may also occur.Source: B. Assignement Errors · p. 15
  • Binary-tree encoding is identified as important for larger systems because its locality scaling can counter the exponential contrast loss with Pauli weight.This is presented as a scaling consideration rather than a demonstrated correction in the reported experiments.Source: B. Assignement Errors · p. 15

VI. OPTIMIZATION USING A SIMULTANEOUS PERTURBATION METHOD

SPSA makes VQE optimization practical under noisy energy estimates by approximating gradients with only two energy evaluations, while numerical simulations assess scaling and resource requirements.

VI. OPTIMIZATION USING A SIMULTANEOUS PERTURBATION METHOD

  • The optimizer’s parameter count p = N(3d−1) grows linearly with qubit number and circuit depth, increasing classical and measurement overheads.SPSA is selected because it remains comparable in accuracy to finite differences while saving O(p) cost-function evaluations.Source: §VI · p. 15
  • SPSA approximates each p-parameter gradient using energy evaluations at two simultaneous perturbations, reducing cost-function evaluations from finite-difference scaling to a constant two.The controls are then updated from the approximate gradient, with convergence possible under stochastic fluctuations when initialization lies in the attraction domain.Source: §VI · p. 15
  • The experiment calibrates perturbation scales from measured energy differences and averages 25 random gradient directions to estimate the local landscape slope.The parameter c is chosen for robustness against stochastic energy fluctuations, while a is calibrated to produce a target first update of 2π/10.Source: §VI · p. 16
  • The final energy is measured only after optimization by averaging the last 25 control settings, using 10^5 samples to reduce stochastic fluctuations.During optimization, perturbed energies generate new gradient estimates, while the underlying optimized angles are evaluated at the end.Source: §VI · p. 16

VII. NUMERICAL SIMULATIONS AND SCALING OF RESOURCES

  • The numerical-simulation section evaluates decoherence, finite-sampling fluctuations, circuit-depth scaling, and resources needed for chemical accuracy across three molecules.The simulations also examine how circuit depth and noise jointly affect optimization outcomes.Source: §VII · p. 17

A. Numerical model of the experiment

The numerical model reproduces the experimental processor’s entangling topology, gate characteristics, decoherence, and finite-sampling fluctuations to simulate optimization outcomes.

A. Numerical model of the experiment

  • Simulations use ZX entanglers with experimentally motivated additional Pauli terms and the same qubit connectivity as the four- and six-qubit experiments.Initial Z angles are sampled from N(0,1), while X angles are initialized at π/2.Source: A. Numerical model of the experiment · p. 17
  • Amplitude damping and dephasing channels are applied after single-qubit rotations and entanglers, with strengths set by measured coherence times and gate durations.The model uses alternating channel durations for Euler-gate sequences and entangler steps.Source: A. Numerical model of the experiment · p. 17
  • For H2 the simulations use T1 = T2* = 40 µs and 150 ns entanglers, whereas four- and six-qubit simulations use T1 = 30 µs, T2* = 20 µs, and 450 ns entanglers.Parallel two-qubit gates make the four- and six-qubit entangler durations equal.Source: A. Numerical model of the experiment · p. 17
  • Finite-sampling noise is modeled by adding normally distributed energy errors calibrated from 10^3 samples on 100 random states, with 250 angle updates and final averaging over 25 updates.Each interatomic distance or J/B ratio is represented by 100 numerical optimization runs.Source: A. Numerical model of the experiment · p. 17

B. Scaling of resources: depth, function calls, sampling

Resource simulations quantify circuit depth, function calls, and sampling needed for chemical accuracy under experimental and all-connected qubit topologies.

B. Scaling of resources: depth, function calls, sampling

  • Chemical accuracy requires depths d = 1, 8, 28 for H2, LiH, and BeH2 with experimental connectivity, versus d = 1, 6, 16 with all-connected connectivity.The target energy error is approximately 0.0016 Hartree.Source: B. Scaling of resources: depth,… · pp. 17–18, Figure S7 · p. 18
  • Approximately 2×10^3 function calls suffice for H2, about 2×10^4 for LiH, and about 2×10^4–3×10^4 for BeH2 depending on connectivity.These estimates average ten optimizations at the shortest depth meeting chemical accuracy.Source: B. Scaling of resources: depth,… · p. 18
  • All three molecules reach chemical accuracy when each optimization energy uses approximately S ≈ 10^6 samples.The estimate extrapolates energy standard deviations from S = 10^3 samples using ϵA → ϵA√(10^3/S).Source: B. Scaling of resources: depth,… · p. 18
  • The simulations indicate that resource scaling with problem size is not very dramatic when decoherence is set aside.Rapid qubit-reset protocols could increase function-call and sampling rates in the near future.Source: B. Scaling of resources: depth,… · p. 18

C. Scaling of resources: decoherence

Noise simulations show that the useful circuit depth depends on depolarizing noise, while molecular and spin Hamiltonians respond differently to additional entanglers.

C. Scaling of resources: decoherence

  • Chemical accuracy for LiH is reached at depolarizing noise rates of approximately 10^-5 with 6 and 8 entanglers.At low noise rates, deeper circuits perform better; as noise increases, shallower circuits become preferable.Source: C. Scaling of resources: decoherence · p. 19, Figure S8 · p. 19
  • For molecular Hamiltonians, LiH optimization results at depths d = 0, 1, 2 show substantial overlap under combined decoherence, sampling noise, and limited trial states.The corresponding numerical simulations include 100 outcomes for each depth.Source: C. Scaling of resources: decoherence · p. 19, Figure S9 · p. 20
  • Interacting spin Hamiltonians benefit from entanglement: d = 1, 2, 3 circuits produce significantly better estimates than d = 0 circuits.This contrasts with the weak depth dependence observed for the molecular Hamiltonians discussed in the same analysis.Source: C. Scaling of resources: decoherence · p. 19, Figure S9 · p. 20
  • The Hamiltonian resource tables contain 4, 99, and 164 Pauli terms for H2, LiH, and BeH2, grouped into 2, 25, and 44 tensor-product basis sets.The listed coefficients exclude energy shifts from inner-orbital filling and nuclear Coulomb repulsion.Source: Table S2 · p. 21

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