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Optimizing quantum heuristics with meta-learning

Max Wilson, Sam Stromswold, Filip Wudarski, Stuart Hadfield, Norm M. Tubman, Eleanor Rieffel

arXiv:1908.03185v1quant-phcs.NE

TL;DR

The paper asks how to optimize parameters of variational quantum heuristics reliably despite noisy quantum evaluations. It compares an LSTM meta-learner with four established optimizers across QAOA and VQE problems and three simulation environments. The meta-learner outperforms L-BFGS-B in most cases by average gain and finds the most near-optimal solutions in the Noisy environment, while comparisons remain scope-limited and larger problem sizes are untested.

  • Problem

    Parameter optimization in variational quantum algorithms is difficult because of hardware noise, stochastic measurements, and theoretical limitations such as barren plateaus.

    Source: §I · p. 2

  • Method

    The study compares an LSTM gradient-based meta-learner with L-BFGS-B, Nelder-Mead, Bayesian optimization, and evolutionary strategies across QAOA and VQE problems.

    Source: §I · p. 1, §1 · pp. 4–5

  • Results

    The meta-learner outperforms L-BFGS-B in most cases by average percent gain and performs better than all optimizers in the Noisy environment by near-optimal-solution count.

    Source: §V · p. 9, §VI · p. 10, B. Noise · p. 10

  • Takeaways & Limitations

    The results suggest that meta-learning may be useful for optimizing quantum heuristics on noisy near-term quantum hardware.

    Source: §I · p. 2, §VI · p. 10

    The study does not investigate scaling to larger problem sizes, and its optimizer comparison has limited scope because gradient-based methods require auxiliary quantum-circuit evaluations.

    Source: §V · p. 9, §VI · p. 11

Abstract

from arXiv · show

Variational quantum algorithms, a class of quantum heuristics, are promising candidates for the demonstration of useful quantum computation. Finding the best way to amplify the performance of these methods on hardware is an important task. Here, we evaluate the optimization of quantum heuristics with an existing class of techniques called `meta-learners'. We compare the performance of a meta-learner to Bayesian optimization, evolutionary strategies, L-BFGS-B and Nelder-Mead approaches, for two quantum heuristics (quantum alternating operator ansatz and variational quantum eigensolver), on three problems, in three simulation environments. We show that the meta-learner comes near to the global optima more frequently than all other optimizers we tested in a noisy parameter setting environment. We also find that the meta-learner is generally more resistant to noise, for example seeing a smaller reduction in performance in Noisy and Sampling environments and performs better on average by a `gain' metric than its closest comparable competitor L-BFGS-B. These results are an important indication that meta-learning and associated machine learning methods will be integral to the useful application of noisy near-term quantum computers.

I. INTRODUCTION

The paper proposes meta-learning for parameter optimization in noisy variational quantum algorithms and compares it with established optimizers across problems and simulation environments.

  • Meta-learning is proposed as a robust classical optimization framework for variational quantum algorithms facing hardware noise, stochastic measurements, and barren plateaus.Source: §I · pp. 1–2
  • The study compares an LSTM meta-learner with L-BFGS-B, Nelder-Mead, Bayesian optimization, and evolutionary strategies.Source: §I · p. 2
  • The experiments use Wave Function, Sampling, and Noisy simulation environments to assess optimizer performance under increasingly realistic conditions.Source: §I · p. 2
  • The comparison covers QAOA for MAX-2-SAT and Graph Bisection and VQE for Fermi-Hubbard models.Source: §I · p. 2
  • The meta-learner generally performs as well as or better than competing optimizers and is especially resistant to noise, finding more near-optimal solutions in the Noisy environment.Source: §I · p. 2

II. BACKGROUND

The paper studies two variational quantum heuristics: QAOA for combinatorial optimization and VQE for estimating Hamiltonian ground-state energies.

A. Quantum Alternating Operator Ans¨atz

B. Variational Quantum Eigensolver

C. Meta-learning

The meta-learner is an LSTM optimizer that maps cost gradients to parameter updates, while experiments model exact, sampling-based, and parameter-noisy quantum evaluations.

C. Meta-learning

  • The meta-learner uses an LSTM that receives preprocessed cost gradients and outputs parameter updates iteratively.Source: C. Meta-learning · p. 3
  • Meta-learners can optimize general functions, and their ability to learn optimization procedures motivates their application to quantum-circuit parameters.Source: C. Meta-learning · p. 3

A. Simulation Environments

  • The study compares exact Wave Function simulation, finite-shot Sampling simulation, and Noisy simulation with parameter-setting perturbations.Source: A. Simulation Environments · p. 4
  • In the Noisy environment, each single-qubit rotation receives normally distributed parameter noise with σ = 0.1 at every optimization step.Source: A. Simulation Environments · p. 4

B. Optimizers

The benchmark includes local gradient-based and gradient-free methods, Bayesian optimization, and evolutionary strategies, spanning distinct search mechanisms and computational trade-offs.

1. Local optimizers

  • L-BFGS-B and Nelder-Mead are standard local optimizers, with L-BFGS-B serving as the meta-learner’s closest comparator because both use gradients.Source: §1 · p. 4

2. Bayesian Optimization

  • Bayesian optimization uses Gaussian-process posterior updates over candidate functions and is designed to find global optima with few steps.Source: §2 · p. 4

C. Evolutionary Strategies

  • Evolutionary strategies maintain populations of candidate solutions, select lower-cost parents, and generate children through crossover and mutation.Source: C. Evolutionary Strategies · p. 5
  • Evolutionary strategies are presented as robust, parallelizable, and relatively inexpensive candidates for quantum-heuristic optimization.Source: C. Evolutionary Strategies · p. 5

1. Meta-learning on quantum circuits

The meta-learner is a coordinatewise LSTM local optimizer that converts cost-function gradients into parameter updates. Its loss is chosen for rapid convergence and fewer quantum-device calls, although this can lead to local minima.

Meta-learner architecture

  • The LSTM receives preprocessed Hamiltonian gradients and outputs parameter updates, iteratively moving through the optimization landscape.Gradient preprocessing preserves sign information while handling potentially exponentially large values; the update is φ_t+1 = φ_t + Δφ_t.Source: §1 · p. 5
  • A single shared LSTM model is applied coordinatewise, with separate cell and hidden states maintained for each parameter.This makes model size only indirectly dependent on the number of optimization parameters.Source: §1 · p. 5

Training objective

  • The training loss accumulates weighted losses across an unrolled time horizon, with ω_t = 1 in these runs.The horizon T determines how many optimization steps are unrolled before losses are accumulated and backpropagated.Source: §1 · pp. 5–6
  • The loss was selected to converge rapidly and reduce quantum-device calls, but it can converge to local minima in some cases.The authors report that it performed better than L-BFGS-B on these problems.Source: §1 · p. 6

D. Problems

The VQE problem uses a simplified spinless three-site Fermi-Hubbard model mapped to three qubits. Its variational circuit applies repeated single-qubit rotations followed by nearest-neighbor entangling gates.

1. Fermi-Hubbard Model

  • The Hamiltonian includes hopping, many-body interaction, and onsite chemical-potential terms represented by t, U, and μ.The model is difficult to solve at large lattice sizes and large U/t, motivating continued method development.Source: §1 · p. 6
  • The VQE targets the ground state of a spinless three-site Fermi-Hubbard model with unequal couplings in [−2, 2] and U = μ = 0.The model is mapped to a qubit Hamiltonian using the Jordan-Wigner transformation.Source: §1 · p. 6, Figure 6 · p. 6
  • The ansatz contains three blocks of RZ(α)RY(β)RZ(γ) rotations applied to every qubit, followed by CNOT gates on qubit pairs (1,2) and (2,3).The construction follows prior results cited by the paper.Source: §1 · p. 6

2. MAX-2-SAT

MAX-2-SAT is encoded as an n-qubit cost-minimization problem whose Hamiltonian preserves the solutions that maximize satisfied clauses. The reported instances use eight qubits, eight clauses, and QAOA depth three.

2. MAX-2-SAT

  • MAX-2-SAT asks for the maximum number of simultaneously satisfied two-literal clauses and is NP-hard.The best known efficient classical algorithm achieves only a constant-factor worst-case approximation.Source: §2 · p. 6
  • The encoding maps Boolean variables to qubit computational-basis states, then transforms them into Ising variables and Pauli-Z cost terms.The resulting cost Hamiltonian is minimized when the number of satisfied clauses is maximized.Source: §2 · p. 6
  • The evaluated MAX-2-SAT instances contain n = 8 qubits and 8 clauses, with QAOA circuit depth p = 3.The implementation uses the original QAOA initial state.Source: §2 · pp. 6–7

3. Graph Bisection

Graph Bisection partitions an even-sized graph into equal sets while minimizing crossing edges. The QAOA construction encodes feasible partitions in fixed-Hamming-weight states and uses an XY mixer that preserves feasibility.

3. Graph Bisection

  • Graph Bisection seeks an equal-size partition minimizing edges crossing between the two sets and is NP-hard.The best known efficient classical algorithm has a logarithmic-factor worst-case approximation ratio.Source: §3 · p. 7
  • Feasible partitions are encoded as n-bit strings of Hamming weight n/2, with the cost function minimized over that constrained subspace.The encoding is transformed into an Ising cost Hamiltonian.Source: §3 · p. 7
  • Each instance starts from a uniformly random feasible computational-basis state with Hamming weight n/2.This initial state is selected from the valid partition space.Source: §3 · p. 7
  • The XY partial mixers preserve Hamming weight, so the ordered QAOA mixer keeps evolution and output samples within the feasible subspace.The evaluated instances use n = 8 qubits, 8 edges, and QAOA depth p = 3.Source: §3 · p. 7

IV. METHODS

The study evaluates optimizers using gain to the global minimum and distance to the global minima, with gain measuring average progress from initialization.

A. Metrics

  • Gain G measures average progress from initialized parameters toward the global minimum across problem instances.The metric uses the optimizer’s converged cost, initial cost, and ground-state energy.Source: A. Metrics · p. 7
  • The ground-state reference is obtained by exhaustive evaluation for MAX-2-SAT and Graph Bisection and exact diagonalization for Fermi-Hubbard models.Source: A. Metrics · p. 7
  • Distance D evaluates how close the final solution is to the global minimum as a percentage of the possible energy range.Source: A. Metrics · p. 7

B. Configuring Optimizers

The experiments configure multiple optimizers across quantum problems and simulation environments, train separate meta-learners, and compare performance using gain and near-optimal-solution frequency.

B. Configuring Optimizers

  • The evaluation uses 20 problems with five initializations, shared local-optimizer starts, and optimizer-specific convergence settings.QAOA uses m = 60 parameters, VQE uses m = 297, and evolutionary-strategy populations contain 20 individuals.Source: B. Configuring Optimizers · p. 7

V. DISCUSSION & RESULTS

  • Figure 7 compares optimizer gain across Fermi-Hubbard, Graph Bisection, and MAX-2-SAT problems in Wave Function, Sampling, and Noisy simulations.Gain represents average progress toward the minimum from the initial expectation-value evaluation.Source: Figure 7 · p. 8, §V · p. 9

C. Training the meta-learner

  • Nine meta-learner models are trained across three simulation environments and three problem classes, with training truncated at 100 problems for Fermi-Hubbard models.MAX-2-SAT and Graph Bisection models use 200 training problems.Source: C. Training the meta-learner · pp. 8–9

V. DISCUSSION & RESULTS

  • Near-optimal solutions are counted when an optimizer finishes within 2% of the global optimum, complementing the gain metric.L-BFGS-B is the closest comparable competitor because both it and the meta-learner use gradients.Source: §V · p. 9, Figure 8 · p. 9
  • The Figure 7 comparison is limited because QPU iterations count expectation-value evaluations while gradient-based methods also use auxiliary quantum-circuit calls.The authors therefore draw conservative conclusions about optimizer comparisons.Source: Figure 7 · p. 8, §V · p. 9

A. General Performance

The meta-learner performs competitively across problem classes, shows greater resistance to increasing noise, and often finds near-optimal solutions more frequently than evolutionary strategies.

A. General Performance

  • The meta-learner achieves competitive gain across multiple problem classes at small instance size, using G to compare optimizers.Source: A. General Performance · p. 9

B. Noise

  • The meta-learner’s performance falls less under noise than L-BFGS-B, decreasing by 0.2 versus 0.35 in Fermi-Hubbard models.Performance worsens from Wave Function to Sampling and from Sampling to Noisy simulations for all optimizers.Source: B. Noise · p. 10, A. General Performance · p. 10
  • In Noisy simulation, the meta-learner finds 80 near-optimal solutions compared with 17 for evolutionary strategies.Near-optimal means within 2% of the ground state, and all optimizers show fewer such solutions as noise increases.Source: A. General Performance · p. 9, B. Noise · p. 10

C. Evolutionary Strategies

  • Evolutionary strategies produce gain spikes every 21 evaluations because each generation evaluates 20 individuals plus the previous generation’s fittest individual.Source: C. Evolutionary Strategies · p. 10
  • Long-timescale results suggest evolutionary strategies can perform well on particularly hard problems, whereas Figure 8 favors the meta-learner for near-optimal searches.The paper states that different comparison metrics are needed to determine their respective use cases.Source: C. Evolutionary Strategies · p. 10

D. Problems and algorithms

Across quantum-heuristic problems and simulation settings, the meta-learner generally outperforms L-BFGS-B and all optimizers in the Noisy environment. The conclusions also identify unresolved boundaries involving scaling and QAOA design choices.

D. Problems and algorithms

  • Fermi-Hubbard models were easiest to solve, while their VQE implementations used 24 parameters compared with 6 for QAOA.The passage attributes the global optimizers’ underperformance to the size of the parameter space despite the smaller problem size.Source: D. Problems and algorithms · p. 10
  • Graph Bisection was harder than MAX-2-SAT by classical approximability, and every optimizer generally achieved worse gain on it except Bayesian optimization.The passage identifies Bayesian optimization as significantly more robust on Graph Bisection.Source: D. Problems and algorithms · p. 10

VI. CONCLUSION

  • The meta-learner outperforms L-BFGS-B in most cases by average percent gain and performs best among all optimizers in the Noisy environment by near-optimal-solution count.These results span MAX-2-SAT, Graph Bisection, and Fermi-Hubbard models using QAOA and VQE across Wave Function, Sampling, and Noisy environments.Source: §VI · p. 10
  • The study did not investigate how meta-learner performance scales to larger problem sizes because larger quantum simulations and gradient computation are burdensome.Performance on larger systems therefore remains an open question for future hardware-based evaluation.Source: §VI · p. 11
  • Differences between MAX-2-SAT and Graph Bisection may reflect both problem structure and QAOA implementation choices, because the implementations used different mixers and initial states.Additional mixer variants and initial states could further affect QAOA and parameter-optimization performance.Source: §VI · p. 11

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