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IEEE International Conference on Quantum Computing and Engineering (QCE)

Efficient Training of Layerwise-Commuting PQCs with Parallel Gradient Estimation

Marko Brnović, Dmitri Iouchtchenko, Maciej Koch-Janusz

Published Sep 15, 2024• 0 citations
Publisher

Abstract

Variational quantum algorithms, such as the variational quantum eigensolver (VQE), have a very high cost due to the large number of measurement outcomes that must be acquired from the quantum device during training. This is due in large part to the use of the parameter shift rule for estimating the gradient at each step, which requires two circuit evaluations per parameter. It was recently shown that if all the gate generators in the ansatz commute with each other, then the complete gradient vector can be found from a single evaluation of an augmented circuit (Bowles et al., 2023), and this has the potential to significantly reduce the quantum cost. Unfortunately, this is quite a severe restriction, as it prevents the ansatz from having a nontrivial dynamical Lie algebra (DLA). In order to obtain a more expressive ansatz than permitted by the commutation requirement, we construct it iteratively and train it in a layerwise fashion using the parallel gradient method. For VQE training of an 8-qubit transverse field Ising model (TFIM) on an ideal simulator, we show that this approach is able to reach a lower energy using fewer circuit evaluations than the parameter shift rule with a more conventional ansatz.

DOI
10.1109/qce60285.2024.10410

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Variational quantum algorithms, such as the variational quantum eigensolver (VQE), have a very high cost due to the large number of measurement outcomes that must be acquired from the quantum device during training. This is due in large part to the use of the parameter shift rule for estimating the gradient at each step, which requires two circuit evaluations per parameter. It was recently shown that if all the gate generators in the ansatz commute with each other, then the complete gradient vector can be found from a single evaluation of an augmented circuit (Bowles et al., 2023), and this has the potential to significantly reduce the quantum cost. Unfortunately, this is quite a severe restriction, as it prevents the ansatz from having a nontrivial dynamical Lie algebra (DLA). In order to obtain a more expressive ansatz than permitted by the commutation requirement, we construct it iteratively and train it in a layerwise fashion using the parallel gradient method. For VQE training of an 8-qubit transverse field Ising model (TFIM) on an ideal simulator, we show that this approach is able to reach a lower energy using fewer circuit evaluations than the parameter shift rule with a more conventional ansatz.
Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation, which under the parameter-shift rule scales linearly in the number of trainable parameters and dominates the total shot budget of training at scale. In this work, we propose a framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation, that yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives and recovers SPSA, random coordinate descent, and the parameter-shift rule as limiting cases, with no ancilla qubits or controlled-gate overhead. We prove that stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the single-direction extreme of SPSA and the full-gradient extreme of parameter-shift. Within this framework we derive QUIVER (Quantum Iterative V-adaptive Estimator Rule), an adaptive optimiser for parameterised circuits whose update rule follows from a closed-form minimum measurement-cost allocation. We show numerically that forward gradients train Hamming-weight-preserving orthogonal quantum neural networks with up to 60 qubits and 1770 parameters on the ECG5000 and MNIST datasets orders of magnitude more efficiently than the parameter-shift rule. We also demonstrate that our proposed QUIVER optimiser can outperform iCANS and gCANS measurement-frugal optimisers on optimisation problems using the quantum approximate optimisation algorithm and quantum simulation with the variational quantum eigensolver.
Variational quantum algorithms (VQAs) are a promising approach for leveraging near-term quantum computers, but trainability and quantum-resource costs remain major challenges. Here we address these challenges with the \textbf{H}ardware \textbf{E}fficient and dynamical \textbf{LI}e algebra supported \textbf{A}nsatz (HELIA) and two training methods combining the classical-enhanced \g method with the quantum-based Parameter-Shift Rule (PSR). This distributes gradient estimation and training across classical and quantum hardware. We evaluate ground-state estimation of 6- to 18-qubit Hamiltonians using the Variational Quantum Eigensolver (VQE) and quantum phase classification for Hamiltonians up to 12 qubits. For VQE, our methods improve accuracy and success rates, with a mean reduction in estimated logical quantum evaluations of up to 60%60\%60% relative to Full-PSR training of the same ansatz. For classification, the largest improvement in mean peak test accuracy relative to Full-PSR is 2.82.82.8 percentage points. We also use these methods as a warm-start pretraining strategy for Hamiltonians with exponential-dimensional dynamical Lie algebras. Gradient diagnostics for the studied Hamiltonian configurations show slower gradient-variance decay than the selected deep hardware-efficient reference over the simulated system sizes, potentially paving the way for training large-qubit quantum models.
Many optimization methods for training variational quantum algorithms are based on estimating gradients of the cost function. Due to the statistical nature of quantum measurements, this estimation requires many circuit evaluations, which is a crucial bottleneck of the whole approach. We propose a new gradient estimation method to mitigate this measurement challenge and reduce the required measurement rounds. Within a Bayesian framework and based on the generalized parameter shift rule, we use prior information about the circuit to find an estimation strategy that minimizes expected statistical and systematic errors simultaneously. We demonstrate that this approach can significantly outperform traditional gradient estimation methods, reducing the required measurement rounds by up to an order of magnitude for a common QAOA setup. Our analysis also shows that an estimation via finite differences can outperform the parameter shift rule in terms of gradient accuracy for small and moderate measurement budgets.
A central challenge in quantum machine learning is the design and training of parameterized quantum circuits (PQCs). Similar to deep learning, vanishing gradients pose immense problems in the trainability of PQCs, which have been shown to arise from a multitude of sources. One such cause are non-local loss functions, that demand the measurement of a large subset of involved qubits. To facilitate the parameter training for quantum applications using global loss functions, we propose a Sequential Hamiltonian Assembly, which iteratively approximates the loss function using local components. Aiming for a prove of principle, we evaluate our approach using Graph Coloring problem with a Varational Quantum Eigensolver (VQE). Simulation results show, that our approach outperforms conventional parameter training by 29.99% and the empirical state of the art, Layerwise Learning, by 5.12% in the mean accuracy. This paves the way towards locality-aware learning techniques, allowing to evade vanishing gradients for a large class of practically relevant problems.
Variational Quantum algorithms, especially Quantum Approximate Optimization and Variational Quantum Eigensolver (VQE) have established their potential to provide computational advantage in the realm of combinatorial optimization. However, these algorithms suffer from classically intractable gradients limiting the scalability. This work addresses the scalability challenge for VQE by proposing a classical gradient computation method which utilizes the parameter shift rule but computes the expected values from the circuits using a tensor ring approximation. The parametrized gates from the circuit transform the tensor ring by contracting the matrix along the free edges of the tensor ring. While the single qubit gates do not alter the ring structure, the state transformations from the two qubit rotations are evaluated by truncating the singular values thereby preserving the structure of the tensor ring and reducing the computational complexity. This variation of the Matrix product state approximation grows linearly in number of qubits and the number of two qubit gates as opposed to the exponential growth in the classical simulations, allowing for a faster evaluation of the gradients on classical simulators.
Variational quantum algorithms, especially quantum approximate optimization and the variational quantum eigensolver (VQE), have demonstrated their potential to provide computational advantage in the realm of combinatorial optimization. However, these algorithms suffer from classically intractable gradients limiting the scalability. This work addresses the scalability challenge for the VQE by proposing a classical gradient computation method which utilizes the parameter-shift rule but computes the expected values from the circuits using a tensor-ring approximation. The parametrized gates from the circuit transform the tensor ring by contracting the matrix along the free edges of the tensor ring. While the single-qubit gates do not alter the ring structure, the state transformations from the two-qubit rotations are evaluated by truncating the singular values, thereby preserving the structure of the tensor ring and reducing the computational complexity. This variation of the matrix-product-state approximation grows linearly in number of qubits and the number of two-qubit gates as opposed to the exponential growth in the classical simulations, allowing for a faster evaluation of the gradients on classical simulators.