PaperlayerView exact revision on arXiv

Source-linked AI summary

A Domain-agnostic, Noise-resistant, Hardware-efficient Evolutionary Variational Quantum Eigensolver

Arthur G. Rattew, Shaohan Hu, Marco Pistoia, Richard Chen, Steve Wood

arXiv:1910.09694v4quant-phcs.NE

TL;DR

Fixed preconstructed variational forms limit VQE’s generality and accuracy, while domain-specific forms require specialized knowledge. EVQE uses evolutionary programming to generate and optimize circuit forms dynamically. The paper reports hardware-efficient molecular circuits, improved noisy-simulation results, and a real 5-qubit IBMQ demonstration.

  • Problem

    VQE’s fixed variational forms limit general applicability and accuracy, while heuristic or domain-specific forms require specialized knowledge.

    Source: §I · pp. 1–2

  • Method

    EVQE uses evolutionary programming to adaptively search circuit forms and their parameterizations for structures customized to a given problem instance.

    Source: §I · p. 2

  • Results

    EVQE’s molecular circuits were up to 18.6× shallower and used up to 12× fewer CX gates than VQE with domain-specific forms, and EVQE was demonstrated on a real 5-qubit IBMQ processor.

    Source: §I · p. 2, §VI · p. 11

  • Takeaways & Limitations

    EVQE is presented as a domain-agnostic approach for accurate energy evaluation with hardware-efficient ansatzes across diverse optimization domains.

    Source: §I · p. 2, §VI · p. 11

Abstract

from arXiv · show

Variational quantum algorithms have shown promise in numerous fields due to their versatility in solving problems of scientific and commercial interest. However, leading algorithms for Hamiltonian simulation, such as the Variational Quantum Eigensolver (VQE), use fixed preconstructed ansatzes, limiting their general applicability and accuracy. Thus, variational forms---the quantum circuits that implement ansatzes ---are either crafted heuristically or by encoding domain-specific knowledge. In this paper, we present an Evolutionary Variational Quantum Eigensolver (EVQE), a novel variational algorithm that uses evolutionary programming techniques to minimize the expectation value of a given Hamiltonian by dynamically generating and optimizing an ansatz. The algorithm is equally applicable to optimization problems in all domains, obtaining accurate energy evaluations with hardware-efficient ansatzes. In molecular simulations, the variational forms generated by EVQE are up to $18.6\times$ shallower and use up to $12\times$ fewer CX gates than those obtained by VQE with a unitary coupled cluster ansatz. EVQE demonstrates significant noise-resistance properties, obtaining results in noisy simulation with at least $3.6\times$ less error than VQE using any tested ansatz configuration. We successfully evaluated EVQE on a real 5-qubit IBMQ quantum computer. The experimental results, which we obtained both via simulation and on real quantum hardware, demonstrate the effectiveness of EVQE for general-purpose optimization on the quantum computers of the present and near future.

I. INTRODUCTION

VQE’s fixed variational forms limit its accuracy and generality, while domain-specific forms are difficult to construct and may not suit target hardware. EVQE searches circuit structures and parameterizations together, reporting efficient molecular circuits and improved noisy-simulation results.

I. INTRODUCTION

  • NISQ devices are limited by 2-qubit-gate fidelity and qubit-coherence times, making deep circuits with many 2-qubit operations infeasible on foreseeable hardware.The paper frames hybrid quantum/classical algorithms as possible routes to applications across fields including chemistry, logistics, healthcare, and finance.Source: §I · p. 1
  • VQE’s tractably parameterized fixed circuits cover only an exponentially small subspace of an n-qubit Hilbert space.A circuit capable of mapping to any state would require exponentially many parameters, whereas practical variational forms keep their parameter counts polynomial.Source: §I · p. 1
  • Domain-specific variational forms require specialized knowledge and may fail to produce circuits optimal in accuracy, depth, or 2-qubit-gate count.Such circuits are also often designed independently of the execution hardware.Source: §I · pp. 1–2
  • EVQE adaptively searches circuit forms and their parameterizations to generate structures customized to each problem instance.The algorithm uses evolutionary programming techniques to minimize a Hamiltonian’s expectation value.Source: §I · p. 2
  • 5.0×–15.2× shallower circuits and 3.5×–5.0× fewer CX gates on LiH, and 18.7× shallower circuits and 12.0× fewer CX gates on BeH2, were reported for EVQE versus VQE/UCCSD.The introduction also reports at least 3.4× less error than the best VQE result in noisy LiH simulation and a successful real 5-qubit IBMQ demonstration.Source: §I · p. 2

II. BACKGROUND

Fixed variational forms are efficient to optimize but cannot represent arbitrary ground states, motivating methods that vary circuit structure. EVQE applies an asexual evolutionary search, with identity-initialized growth and ancestry-based speciation, to explore and optimize parameterized circuits.

II. BACKGROUND

  • Efficiently parameterized fixed variational forms cannot map to the ground state of an arbitrary Hamiltonian, motivating variation of circuit form.The background notes that finding an efficient gate set for arbitrary Ising Hamiltonians is tied to NP-Hardness or the difficulty of efficiently optimizing a fixed circuit.Source: §II · p. 2
  • Earlier adaptive approaches include ADAPT-VQE and a greedy method that grows circuits using single-qubit gates while leaving 2-qubit gates fixed.The paper notes that ADAPT-VQE may be noise-sensitive and that the greedy method retains limitations similar to VQE.Source: §II · p. 2
  • NEAT concurrently grows and optimizes neural networks, and the paper notes that some of its techniques transfer to EVQE’s changing circuit structures.This contrasts with quantum machine-learning algorithms described as using fixed variational forms.Source: §II · p. 2
  • EVQE uses one-parent mutation rather than crossover, avoiding circuit fusion that can disrupt mappings and require parameter re-optimization.Its identity-initialized gate additions preserve the circuit’s expectation value until optimization alters the new parameters.Source: §II · pp. 2–3
  • EVQE’s single-random-layer initialization and identity-initialized growth are described as mechanisms for avoiding barren-plateau regions.The paper also says users can choose a starting population outside a barren region; the concern is exponentially small probability of nonzero gradients in sufficiently deep random circuits.Source: §II · p. 3

III. EVOLUTIONARY VARIATIONAL QUANTUM EIGENSOLVER (EVQE)

  • EVQE is presented as a speciated, asexual evolutionary algorithm for general-purpose multimodal optimization.It explores both quantum-circuit forms and their parameterizations.Source: §III · p. 3

A. Genetic Representation of Quantum Circuits

EVQE represents candidate circuits as genomes composed of gene instances, with genes specifying layer gates and instances supplying their parameters. Its asexual mutation and ancestry-based speciation support circuit growth, population diversity, and noise resistance.

A. Genetic Representation of Quantum Circuits

B. Asexual Reproduction and Speciation Through Genetic Ancestry

  • EVQE explores circuits primarily through mutation of one-parent offspring rather than crossover between parent genomes.The paper argues that fusing quantum circuits can disrupt their mappings because of entanglement and superposition.Source: B. Asexual Reproduction and Speciation… · p. 3
  • New gates are initialized to perform the identity, leaving the circuit expectation unchanged until parameter optimization can reduce it.Unlike crossover, this approach preserves the existing circuit’s expectation value and avoids re-optimizing all parameters in the offspring.Source: B. Asexual Reproduction and Speciation… · pp. 3–4
  • Speciation maintains diverse candidate solutions across optima, while fit genomes can produce multiple offspring whose lineage must be harmed together for noise to extinguish it.Under a simple stochastic additive-noise model, the paper says that simultaneous loss has exponentially small probability in the number of lineage members.Source: B. Asexual Reproduction and Speciation… · p. 4
  • EVQE defines genetic distance by the total genes in two genomes minus their shared genes, corresponding to their distance from a most recent common ancestor.Each genome has one parent, allowing the population’s genetic ancestry to be represented as a tree.Source: B. Asexual Reproduction and Speciation… · p. 4

C. Fitness Evaluation

EVQE’s fitness combines energy evaluation with penalties for circuit resources. These penalties encourage shallow, low-CX circuits and reduce incentives to overfit noise, while fitness sharing maintains multiple species.

C. Fitness Evaluation

  • Genome fitness combines the circuit’s energy evaluation with penalties based on circuit depth and controlled-U3 gate count.Depth is the number of genes because each gene corresponds to one circuit layer.Source: C. Fitness Evaluation · p. 4
  • Small user-specified penalty coefficients encourage the shallowest, fewest-CX circuits that still meet the desired energy precision and can mitigate noise susceptibility.The paper compares this effect to regularization: resource penalties discourage circuits more susceptible to hardware noise without preventing naturally lower-energy circuits.Source: C. Fitness Evaluation · p. 4
  • Explicit fitness sharing divides a genome’s fitness among members of its species, preventing one species from dominating and supporting concurrent search across optima.The adjusted score is defined using the size of the genome’s species.Source: C. Fitness Evaluation · p. 5

D. Mutation Operators

EVQE mutates circuit genomes through topological search, parameter search, and removal. These operators explore circuit structures and parameterizations while allowing shallower search spaces to be revisited.

D. Mutation Operators

  • EVQE uses topological search, parameter search, and removal to mutate circuit genomes.Source: D. Mutation Operators · p. 5
  • Topological search appends a layer built from randomly assigned gates, with new-gate parameters initialized to zero so those gates act as identity operations.The zero initialization reduces the generations needed to converge to a solution with fixed precision.Source: D. Mutation Operators · p. 5
  • Parameter search optimizes each circuit layer in a randomly selected order, exploring parameterizations while holding circuit form fixed.The iteration optimization count limits the optimizer’s iterations each time it is called on a layer.Source: D. Mutation Operators · p. 5
  • Removal deletes a uniformly random number of contiguous gene instances from the genome’s end, allowing EVQE to revisit shallower search spaces.The number removed is selected from 1 through the genome’s gene count.Source: D. Mutation Operators · p. 5

E. Algorithm Summary

EVQE repeatedly optimizes circuit genomes, evaluates their fitness, selects parents, and mutates them into successive generations. It returns the best expectation value found when the generation budget is reached.

E. Algorithm Summary

  • EVQE initializes a population by applying τ to each genome, chooses one representative per species, and optimizes each genome’s last circuit layer.Source: E. Algorithm Summary · p. 5
  • The algorithm calculates genome fitness and species-adjusted fitness before parent selection.Source: E. Algorithm Summary · p. 5
  • Each generation selects parent genomes with replacement in proportion to adjusted fitness, mutates them, and assigns the resulting genomes to species using a genetic-distance threshold.Source: E. Algorithm Summary · p. 5
  • If no existing species representative is within the genetic-distance threshold, EVQE creates a new species; it returns the fittest genome’s expectation value after the generation budget.Source: E. Algorithm Summary · p. 5

IV. EXPERIMENTAL EVALUATION

The evaluation uses state-vector simulation to compare EVQE and VQE across molecular simulation and general optimization. The chemistry experiments use tapered molecular Hamiltonians and assess accuracy across molecular instances.

IV. EXPERIMENTAL EVALUATION

  • Circuit depths and CX counts are transpiled with Qiskit Terra at optimization level 3, and displayed error bars represent one standard deviation.Source: §IV · p. 5

A. State-vector Simulation

  • State-vector simulations compare EVQE and VQE under theoretically optimal, noiseless conditions, with each circuit evaluated using its output state vector.The Hamiltonian expectation value is calculated from that state vector; this is equivalent to noiseless simulation with infinite shots.Source: A. State-vector Simulation · p. 5
  • The evaluation covers molecular simulation and general optimization, with chemistry tasks targeting LiH and BeH2 ground-state energies and optimization tasks using small randomly generated NP-hard instances.Source: A. State-vector Simulation · pp. 5–6
  • Across five-trial averages, EVQE and VQE/UCCSD all achieve chemical accuracy for LiH at every tested interatomic distance.Source: Figure 3 · p. 6
  • Jordan–Wigner mapping and precision-preserving symmetry tapering reduce LiH from 12 to 6 qubits and BeH2 from 14 to 7 without approximation.Figure 3 compares LiH across interatomic distances; Figure 4 compares BeH2 at one distance.Source: A. State-vector Simulation · p. 6

B. Hardware Simulation

Hardware-simulation experiments examine EVQE’s noise resistance and compare it with VQE under device-informed noise, while a separate experiment evaluates EVQE on real quantum hardware. The supplied passages describe experiment settings and scopes but do not report the corresponding outcome comparisons.

B. Hardware Simulation

  • Hardware-simulation comparisons are designed to assess intrinsic noise resistance without applying error-mitigation techniques.Source: B. Hardware Simulation · p. 6
  • The supplied BeH2, Max-Cut, and vehicle-routing captions describe state-vector comparisons, including chemical-accuracy outcomes for BeH2 and optimal-solution markers for the optimization plots.The Max-Cut instance has 10 nodes; the vehicle-routing instance involves 32 vehicles and four destinations, with 10 trials for each plot.Source: Figure 4 · p. 6, Figure 5 · p. 7, Figure 6 · p. 8
  • The variable-noise study tests EVQE on 10 random two-qubit Hamiltonians using IBMQ Tokyo’s noise profile, 1300 shots, and five trials per Hamiltonian.It varies simulated coherence times and uses SPSA because the device noise requires a noise-resistant optimizer.Source: B. Hardware Simulation · p. 6, Table I · p. 7
  • The LiH noise-resistance experiment reduces the modeled system from six qubits to four by excluding unoccupied spin orbitals, with negligible imprecision that preserves chemical accuracy.The comparison uses comparable total circuit executions; VQE/UCCSD has a reported error of 0.602 ± 0.026 Hartree in this experiment.Source: B. Hardware Simulation · pp. 6–7

C. Execution on Quantum Hardware

  • The real-hardware experiment runs a random five-vertex Max-Cut instance on five-qubit IBMQ Vigo after exclusive device access was obtained.The paper notes that contiguous, relatively uninterrupted access is likely necessary because calibration drift can change a circuit parameterization’s mapping.Source: C. Execution on Quantum Hardware · p. 7
  • For the real-device run, EVQE uses a population of eight and a two-generation cap; EVQE and VQE each execute with 1300 shots.The hardware-time constraint motivates EVQE’s conservative configuration; VQE is tested with RyRz/linear and Ry/linear layouts at depths zero through two.Source: C. Execution on Quantum Hardware · pp. 7–8

V. DISCUSSION

The discussion weighs solution error, circuit resources, and evaluation cost, while the state-vector results show EVQE using fewer resources than VQE/UCCSD and matching or outperforming VQE on optimization tasks.

V. DISCUSSION

  • Circuit depth and CX count indicate hardware execution time and accumulated gate-fidelity or decoherence error, while total circuit evaluations matter but are not directly comparable between EVQE and VQE.For noisy execution or simulation, total solution error is the primary metric.Source: §V · p. 8

A. State-vector Simulation

  • With near-ideal parallelism, EVQE’s sequential circuit evaluations are comparable to VQE’s total evaluations at similar error, yielding similar or faster runtimes on 64-core processors.This comparison applies to the experiments in Figures 3–6.Source: A. State-vector Simulation · p. 8
  • 5.0×–15.2× shallower circuits and 3.5×–5.0× fewer CX gates let EVQE reach chemical accuracy for LiH; for BeH2, it is 18.6× shallower and uses 12.0× fewer CX gates than VQE/UCCSD.The LiH comparison spans 10 interatomic distances; EVQE uses no domain-specific variational form.Source: A. State-vector Simulation · pp. 8–9
  • EVQE always finds optimal solutions for two NP-Hard optimization problems, with shallower circuits than VQE’s shallowest two-body configurations and comparable CX counts.Some VQE configurations return sub-optimal results in up to 90% of trials.Source: A. State-vector Simulation · p. 9
  • EVQE optimizes one layer at a time, with O(n) concurrent parameters versus VQE’s O(dn), while population speciation helps distribute circuits across search-space peaks and reduce premature convergence.In one experiment, EVQE’s higher error was intentional; reducing the depth and U3-gate penalties can yield greater accuracy.Source: A. State-vector Simulation · pp. 9–10

B. Hardware Simulation

In noisy LiH simulation, EVQE outperformed VQE’s best variational form, and only EVQE produced results accounting for some molecular correlation energy. EVQE also adapted its circuits to noise constraints, while its error and resource use plateaued around noise setting 2.

B. Hardware Simulation

  • Around noise setting 2, with coherence times 5× actual, EVQE’s error and resource use plateau; rising error favors fewer CX gates and energies approach the optimal separable-state energy.At noise setting 0 only shot noise is present; settings 1–3 also include constant gate-fidelity error.Source: B. Hardware Simulation · p. 10
  • In noisy LiH simulation, EVQE errors were 3.4× to 11.6× smaller than VQE’s best variational form; only EVQE beat Hartree-Fock and captured some correlation energy.No VQE configuration obtained an energy below Hartree-Fock, while all EVQE trials did.Source: B. Hardware Simulation · p. 10
  • VQE/RyRz/linear showed no improvement from depth 0 to 1 despite three added CX gates; EVQE evolves circuits rather than repeating a fixed pattern.EVQE can find circuits with sufficient depth and CX gates to capture necessary entanglement while minimizing energy evaluations despite noise.Source: B. Hardware Simulation · p. 10
  • Optimizer choice remains a caveat: tested COBYLA, SLSQP, and Nelder-Mead configurations did not match EVQE, though optimizer effects cannot be ruled out.Tighter COBYLA termination tolerances more than tripled average circuit evaluations without meaningful error improvements; Nelder-Mead had comparable results to COBYLA with substantially more evaluations.Source: B. Hardware Simulation · p. 10
  • A depth-16 EVQE circuit with 14 CX gates achieved 0.0308 Hartree error, confirmed at 0.0391 Hartree in noiseless state-vector simulation.The noiseless test was conducted to check that random noise did not account for the low noisy-simulation energy.Source: B. Hardware Simulation · p. 10

C. Execution on Real Quantum Hardware

On real quantum hardware, EVQE’s fittest genome found the optimal solution in most shots, while the conclusion reports efficient molecular energy evaluations and reduced error in noisy simulation. The hardware results are illustrative because each algorithm configuration was tested only once.

C. Execution on Real Quantum Hardware

  • After the second generation, EVQE’s fittest genome had error 0.059055, depth 5, and 2 CX gates; it obtained the optimal solution in 97.4% of 1,300 shots.Only VQE/RyRz/linear at depth 0 among the tested configurations obtained the optimal solution as the most probable state, in 81.4% of shots.Source: C. Execution on Real Quantum Hardware · p. 10
  • The hardware comparison is illustrative rather than demonstrative because each algorithm configuration was tested only once.Exclusive hardware access limited the number of tests.Source: C. Execution on Real Quantum Hardware · p. 10

VI. CONCLUSION

  • EVQE’s molecular circuits were up to 18.6× shallower and used up to 12× fewer CX gates than VQE with domain-specific variational forms, with at least 3.6× less noisy-simulation error than VQE using any tested ansatz.The conclusion presents these results alongside EVQE’s domain-agnostic and quantum-hardware-adaptive properties.Source: §VI · p. 11

IX. APPENDIX

Identity-initialized circuit growth achieved lower error than growth using CX gates, though it tended to require more circuit evaluations per generation. In molecular simulations, EVQE generally reached chemical accuracy with fewer circuit resources than tested heuristic VQE forms, with a LiH exception at two distances.

A. Identity-initialized Growth

  • Identity-initialized growth produced average error about half an order of magnitude lower than growth with CX gates in the random 4-qubit Hermitian-matrix experiment.The comparison held other algorithm parameters constant, but did not control for differing parameter counts.Source: A. Identity-initialized Growth · p. 12, Figure 8 · p. 13
  • EVQE-Standard tended to decrease error with each added layer, while EVQE-CX trials could remain stuck at an error level for over 10 generations; Standard also tended to use more circuit evaluations per generation.The Standard method adds parameterized identity-initialized ∧1(U3) gates, while EVQE-CX adds non-parameterized CX gates.Source: A. Identity-initialized Growth · p. 13

B. State-vector Simulation of BeH2 and LiH Using Heuristic Ansatzes

  • EVQE reached chemical accuracy with shallower circuits and fewer CX gates than all tested heuristic VQE configurations except RyRz/linear at LiH distances 3.0 Å and 3.4 Å.Heuristic configuration required manual testing across depths and optimizers and many circuit evaluations; Ry/linear did not achieve chemical accuracy at all tested distances.Source: Figure 9 · p. 13, B. State-vector Simulation of BeH2 and… · p. 13
  • EVQE obtained chemical accuracy for BeH2 with fewer resources than every tested heuristic VQE configuration.The authors suggest the larger EVQE population or heuristic forms’ difficulty as qubit counts increase may explain the result.Source: Figure 10 · p. 14, B. State-vector Simulation of BeH2 and… · p. 14

C. Pruning the Space of Circuit Forms

EVQE prunes redundant circuit constructions to reduce search-space complexity and make evolution more efficient. The molecular test Hamiltonians are prepared through orbital and symmetry reductions followed by the Jordan-Wigner transformation.

C. Pruning the Space of Circuit Forms

  • The genome-construction rules prohibit consecutive U3 gates on one qubit because two can be contracted into one, reducing six parameters to three.The stated aim is to make classical optimization less challenging.Source: C. Pruning the Space of Circuit Forms · p. 14
  • Consecutive CU3 gates with the same control and target are disallowed unless control and target are reversed, because matching gates can be contracted into one.The rule avoids redundant circuit forms.Source: C. Pruning the Space of Circuit Forms · p. 14
  • Identity gates are added only when neither U3 nor CU3 can be added, reducing parameter-space dimension; the prototype applies all listed pruning rules to make evolution more efficient.This also prevents an identity gate immediately following CU3 or directly preceding U3.Source: C. Pruning the Space of Circuit Forms · p. 14

D. Qubit Hamiltonian Preparation

  • Molecular Hamiltonian preparation aligns atoms on the x-axis, freezes Li and Be 1s orbitals, reduces qubits using symmetries or unoccupied-orbital removal, then applies Jordan-Wigner mapping.The calculations use the STO-3G basis and PySCF; the alignment leaves controlled Rz gates as an unexploited optimization opportunity.Source: D. Qubit Hamiltonian Preparation · p. 14
  • Symmetry reduction converts 10-mode LiH to a precision-preserving 6-qubit Hamiltonian by identifying four symmetries.The passage attributes the symmetry approach to Bravyi et al. and Setia et al. [5] [31].Source: D. Qubit Hamiltonian Preparation · p. 14
  • For precision-preserving BeH2, five symmetries reduce 12 fermionic modes to 7 qubits.Source: D. Qubit Hamiltonian Preparation · p. 14
  • The approximated LiH Hamiltonian removes four weakly interacting modes and finds two additional symmetries, reducing the system to 4 qubits.The removed orbitals are 2py and 2pz, given the atoms’ x-axis alignment.Source: D. Qubit Hamiltonian Preparation · p. 14

Related papers

Loading 1910.09694v4…