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Learning to learn with quantum neural networks via classical neural networks

Guillaume Verdon, Michael Broughton, Jarrod R. McClean, Kevin J. Sung, Ryan Babbush, Zhang Jiang, Hartmut Neven, Masoud Mohseni

arXiv:1907.05415v1quant-phcs.LG

TL;DR

QNN optimization needs efficient parameter initialization because local optimization and quantum cost evaluation can require many expensive iterations. The paper trains classical RNNs as meta-learned few-shot optimizers for QAOA and VQE tasks, then uses their parameter suggestions to initialize local optimizers. The learned strategies outperform alternative initialization approaches, generalize across problem sizes, and may reduce quantum-device optimization runs by an order of magnitude or more.

  • Problem

    QNN optimization faces costly local methods and a need for more efficient, flexible parameter-initialization heuristics.

  • Method

    The paper trains classical RNNs on random instances from QAOA MaxCut, QAOA Sherrington-Kirkpatrick Ising, and VQE Hubbard-model tasks to propose approximate optima that initialize local optimizers.

  • Results

    RNN-generated initializations produce higher-quality optima with fewer quantum-classical optimization iterations and generalize from small training systems to larger problem instances.

  • Takeaways & Limitations

    Classically pre-trained RNN optimizers could initialize larger QNN instances on quantum processors and reduce required QPU runs by an order of magnitude or more.

Abstract

from arXiv · show

Quantum Neural Networks (QNNs) are a promising variational learning paradigm with applications to near-term quantum processors, however they still face some significant challenges. One such challenge is finding good parameter initialization heuristics that ensure rapid and consistent convergence to local minima of the parameterized quantum circuit landscape. In this work, we train classical neural networks to assist in the quantum learning process, also know as meta-learning, to rapidly find approximate optima in the parameter landscape for several classes of quantum variational algorithms. Specifically, we train classical recurrent neural networks to find approximately optimal parameters within a small number of queries of the cost function for the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut, QAOA for Sherrington-Kirkpatrick Ising model, and for a Variational Quantum Eigensolver for the Hubbard model. By initializing other optimizers at parameter values suggested by the classical neural network, we demonstrate a significant improvement in the total number of optimization iterations required to reach a given accuracy. We further demonstrate that the optimization strategies learned by the neural network generalize well across a range of problem instance sizes. This opens up the possibility of training on small, classically simulatable problem instances, in order to initialize larger, classically intractably simulatable problem instances on quantum devices, thereby significantly reducing the number of required quantum-classical optimization iterations.

I. INTRODUCTION

QNN optimization is promising for near-term quantum computing but faces costly local optimization and difficult parameter initialization. The paper uses meta-learned classical RNNs to produce useful initial parameters across several variational algorithm problem classes.

  • QNNs optimize parameterized quantum circuits using classical optimization techniques, paralleling classical deep-learning parameter training.
  • Finite-difference, backpropagation-based, and sampled-gradient methods remain costly because they require additional resources or many quantum circuit evaluations.
  • Good parameter initialization places optimization near a local minimum, promoting more consistent convergence, fewer iterations, and better local minima.
  • The paper trains classical RNNs as black-box controllers for QAOA MaxCut, QAOA Sherrington-Kirkpatrick Ising models, and VQE Hubbard-model ansatze.
  • Numerical simulations show that RNNs trained on small QNNs generalize update heuristics to larger problem sizes and instances.
  • The RNN acts as a few-shot global approximate optimizer that initializes local optimizers, improving over standard initialization methods across the studied cases.

A. Variational Quantum Algorithms

Variational quantum algorithms iteratively exchange parameters and cost estimates between a CPU and QPU. The QPU estimates Hamiltonian expectation values from repeated circuit executions, while the CPU updates parameters for black-box optimization.

  • Variational quantum algorithms use an iterative quantum-classical optimization loop between a classical processing unit and a quantum processing unit.
  • The CPU sends candidate parameters θ to a parameterized circuit, whose output state |ψθ⟩ defines the cost f(θ)=⟨ψθ|H|ψθ⟩.
  • Repeated QPU runs estimate the cost Hamiltonian expectation, which is returned to the CPU for proposing parameters in the next iteration.
  • The CPU treats the objective as a stochastic black-box function f:R^m→R and seeks θ*=argminθ∈R^m f(θ), often accepting an approximate minimum.
  • Finite measurement samples create inherent noise even with perfect quantum gates, while Pauli decompositions support expectation estimation through repeated measurements and classical combination.
  • Unrolling the CPU-QPU loop produces a temporal hybrid computational graph through which gradients can be propagated by reverse-mode autodifferentiation.

B. Meta-learning with Neural Optimizers

The paper frames quantum optimization as a meta-learning problem in which an RNN learns parameter-update heuristics from sequential cost observations. Its hidden state carries optimization memory while each update proposes new QNN parameters for quantum evaluation.

  • Meta-learning learns how to modify learning-algorithm parameters or hyperparameters for purposes such as faster fitting, generalization, or transfer learning.
  • The optimizer neural network is trained to learn parameter-update heuristics for quantum neural networks using noisy unbiased cost estimates at test time.
  • Because test-time gradients are expensive on NISQ devices, the learner is designed to use black-box function queries, while gradients may assist training.
  • The RNN processes QNN parameters and cost evaluations as a sequence-to-sequence problem, retaining information in memory across iterations.
  • An LSTM-based architecture recursively proposes QNN parameter updates, receives estimated costs and hidden state, and sends each new proposal to the QPU.
  • The update mapping produces both a new QNN parameter suggestion and a new hidden state for the next iteration.
  • The selected QAOA-related ansatze use variational parameters as dynamical control parameters that guide wavefunctions toward lower-energy regions.

1. Meta-Training & Loss functions

The paper trains specialized RNN optimizers to learn parameter-update heuristics for distributions of QNN cost functions. A short RNN optimization phase initializes local optimizers, while an improvement-based loss encourages rapid progress and exploration.

  • Meta-learning objective: Meta-learning trains an optimizer to efficiently find sufficiently optimal approximate local minima across a distribution of QNN cost functions.Efficiency is measured by the number of black-box function queries required.
  • Specialized optimizers: Specialized RNNs are trained on targeted QAOA and VQE problem classes, then tested on larger unseen instances from those classes.This differs from general optimizer networks that make few assumptions about the optimizee.
  • Loss functions: The final-cost loss is sparse because its optimization signal must backpropagate through much of the unrolled RNN graph.It averages the final cost over sampled functions.
  • Loss functions: Cumulative regret provides a denser signal and proxies the minimum cost achieved over the optimization history, but can prioritize finding and retaining an early approximate optimum.It uniformly averages the cost-function history over the time horizon.
  • Loss functions: The experiments use summed observed improvement, rewarding progress over the historical best and allowing temporary cost increases when they lead to larger later improvements.The per-step improvement compares the proposed cost with the best previous cost and contributes nothing without improvement.
  • Optimization pipeline: Because training uses backpropagation through time, the RNN runs for a short fixed horizon and initializes a separate local optimizer.Nelder-Mead is the main example, although other local optimizers could also be initialized.

III. NUMERICAL EXPERIMENTS

The numerical experiments train and test specialist RNN optimizers separately for three quantum variational problem classes: MaxCut QAOA, Ising QAOA, and a Hubbard-model VQE ansatz.

  • III. NUMERICAL EXPERIMENTS: Three specialist RNN optimizers are trained and tested, one for each of the paper’s quantum variational problem classes.The classes are MaxCut QAOA, Sherrington-Kirkpatrick Ising QAOA, and a Trotter-based Hubbard-model VQE.
  • III. NUMERICAL EXPERIMENTS: The first class is quantum approximate optimization for MaxCut.
  • III. NUMERICAL EXPERIMENTS: The other classes are quantum approximate optimization for Sherrington-Kirkpatrick Ising models and a Trotter-based variational quantum eigensolver ansatz for the Hubbard model.

A. Quantum Approximate Optimization Algorithms

QAOA prepares low-energy states by alternating cost- and mixer-Hamiltonian evolutions whose variational parameters are optimized. The experiments specialize this framework to MaxCut and SK Ising models.

  • General QAOA: QAOA alternates exponentials of noncommuting cost and mixer Hamiltonians, with variational parameters optimized through the resulting circuit.The cost Hamiltonian is typically diagonal, while the mixer does not commute with it.
  • General QAOA: QAOA is suited to NISQ processors because its gate count scales linearly with circuit depth and its Hamiltonian exponentials are usually easy to compile.The cost expectation can be estimated directly from each repetition for typical cost Hamiltonians.
  • MaxCut QAOA: For MaxCut, qubits encode graph-vertex partitions, and measuring the optimized state yields bitstrings corresponding to partitions with large cut sizes.The mixer is the sum of Pauli X operators and the initial state is the uniform superposition.
  • MaxCut QAOA: The MaxCut experiments use depth P = 2 QAOA circuits built from random graphs sampled from a broader random-graph ensemble.The graph generation fixes n, samples k in [3, n − 1], and uses G_n,p with p = k/n.
  • MaxCut QAOA: MaxCut QAOA exhibits narrower initial 95% confidence intervals across instances than SK QAOA, consistent with concentration of the fixed-parameter cost expectation.The paper states that the numerical results corroborate this concentration finding.

2. Ising QAOA

The SK Ising experiments apply QAOA to finding low-energy configurations of a fully connected spin-glass model. The model is relevant to combinatorial optimization but its ground-state problem is NP-Hard.

  • 2. Ising QAOA: The Sherrington-Kirkpatrick model is an Ising spin glass whose low-energy states are targeted with QAOA.Many combinatorial optimization problems can be mapped to such models.
  • 2. Ising QAOA: The SK QAOA uses a transverse-field mixer, a uniform-superposition initial state, and a regular QAOA ansatz with the SK Hamiltonian as its cost Hamiltonian.Optimized measurements yield bitstrings representing relatively low-energy spin configurations.
  • 2. Ising QAOA: The experiments use the fully connected SK model with random coupling and bias coefficients sampled from independent zero-mean, unit-variance Gaussian distributions.The system size n is sampled before drawing the coefficients.

1. Hubbard Model VQE

The Hubbard-model VQE prepares approximate ground states of lattice fermions using a Trotterized variational circuit with parameters for horizontal hopping, vertical hopping, and interactions.

  • The Hubbard Hamiltonian separates into horizontal hopping, vertical hopping, and spin-interaction terms.
  • The studied VQE problems variationally prepare ground states, or approximations thereof, for Hubbard-model lattices.
  • The variational ansatz is based on Trotterizing time evolution under the Hubbard Hamiltonian.
  • Each of the P Trotter steps uses three variational parameters, θ = {θh, θv, θU}, and a single fermionic swap network.
  • The experiments use P = 5 steps, half-filling, t = 1.0, and initial states matching the ground state's particle number and total spin.

C. Meta-learning Methods & Results

The study trains LSTM recurrent networks for three QNN problem classes and uses their ten-step outputs to initialize local optimization. Testing evaluates relative error, parameter distance, noise robustness, and size generalization against standard strategies.

  • Meta-learning Methods: LSTMs are trained separately for MaxCut QAOA, Ising QAOA, and Hubbard VQE to learn parameter updates across problem instances.
  • Meta-learning Methods: The quantum circuits use Cirq simulation, while neural-network training and inference use TensorFlow with OpenFermion-Cirq for VQE ansätze.
  • Meta-learning Methods: The RNN receives a cost function normalized by the Pauli coefficient norm, constraining the squashed loss to [−1, 1].
  • Results: Figure 3 compares average relative error across 50 test instances under noiseless and Gaussian-noisy readout, including LSTM initialization and alternative strategies.
  • Meta-learning Methods: Relative error is measured against the globally optimal squashed cost found by basin hopping for QAOA and exact diagonalization for VQE.
  • Meta-learning Methods: Testing uses larger systems than training while keeping the ansatz parameter count fixed, enabling evaluation of size generalization.
  • Meta-learning Methods: The LSTM is trained for T = 10 quantum-classical iterations because longer horizons increase simulation and backpropagation-through-time overhead.
  • Meta-learning Methods: The LSTM's ten-step output initializes Nelder-Mead, while comparisons include Gaussian-process optimization, random seeds, and application-specific heuristic seeds.

IV. DISCUSSION

Across the evaluated settings, the neural optimizer reaches strong approximate optima quickly and provides useful initializations for local optimization. Its learned parameter patterns generalize across sizes, supporting a hybrid strategy that shifts training cost to classical simulation.

  • Results: In both noisy and noiseless readout, the neural optimizer reaches a higher-quality approximate optimum in 10 iterations than other optimizers manage in hundreds.
  • Results: The LSTM places parameters in a basin of attraction that helps local optimizers converge more consistently with fewer iterations.
  • Figures: Figure 4 compares post-initialization Euclidean distance to the true optimum for LSTM and alternative heuristics across the three problem classes.
  • Results: Nelder-Mead initialized by the LSTM tends to achieve the highest-quality optimum with lower performance variance than alternative initializations.
  • Results: The LSTM's initialization advantage is especially notable for noisy VQE, where Nelder-Mead struggles to improve upon its starting point.
  • Parameter structure: MaxCut QAOA has the most concentrated optimal parameters, Ising QAOA shows some concentration, and Hubbard VQE shows the least but still clusters.
  • Interpretation: The learned optimizer adjusts problem-class-specific parameters case by case, resembling a heuristic for rapidly selecting useful hyperparameters.
  • Implications: Training on small, classically simulatable instances could initialize larger quantum-device instances and reduce required QPU runs by an order of magnitude or more.

V. CONCLUSION & OUTLOOK

The paper applies meta-learning with classical neural network optimizers to several variational quantum algorithm tasks, finding higher-quality optima with fewer optimization iterations and generalization across problem sizes. The authors identify further extensions while noting that good initialization remains an open challenge.

  • The study applies meta-learning with a classical neural network optimizer to QAOA and VQE optimization tasks.The evaluated tasks include QAOA for MaxCut and Sherrington-Kirkpatrick Ising spin glasses, plus VQE ansätze for Hubbard models.
  • The neural network rapidly finds approximate global parameter optima that initialize local search heuristics.This initialization strategy produced higher-quality optima than alternatives using orders of magnitude more quantum-classical optimization iterations.
  • The learned parameter-update heuristics generalize from small quantum neural networks to larger system sizes and problem instances.This suggests classically pre-training neural optimizers on simulatable instances before applying them to larger instances requiring quantum processors.
  • The authors conclude that quantum-classical meta-learning mitigates fine-tuning challenges but leaves parameter initialization as an open problem.Their conclusion distinguishes the readout-noise-related fine-tuning challenge from the unresolved initialization challenge.
  • Future work could extend the approach to arbitrary QNN architectures and variable parameter counts using newer meta-learning optimizer networks.The authors also suggest meta-learning optimizers for Quantum Dynamical Descent hyperparameters.
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