Post-variational quantum neural networks

TL;DR

This work targets the training bottlenecks of variational quantum algorithms in the NISQ era by proposing post-variational quantum neural networks that fix quantum circuits and learn a classical convex combination of their outputs. It develops three design principles— Ansatz expansion, observable construction, and a hybrid approach—grounded in CQO to retain expressive power while ensuring a convex optimization landscape. The authors provide theoretical error bounds for measurement and propagation of estimation errors, and demonstrate empirically that post-variational models can outperform standard variational methods and match small two-layer neural networks on a Fashion-MNIST task. The study highlights practical benefits, including potentially reduced circuit depth and terminable optimization, while clearly stating that a quantum advantage is not claimed and that fixed-Ansatz selection remains a critical open factor.

Abstract

Hybrid quantum-classical computing in the noisy intermediate-scale quantum (NISQ) era with variational algorithms can exhibit barren plateau issues, causing difficult convergence of gradient-based optimization techniques. In this paper, we discuss "post-variational strategies", which shift tunable parameters from the quantum computer to the classical computer, opting for ensemble strategies when optimizing quantum models. We discuss various strategies and design principles for constructing individual quantum circuits, where the resulting ensembles can be optimized with convex programming. Further, we discuss architectural designs of post-variational quantum neural networks and analyze the propagation of estimation errors throughout such neural networks. Finally, we show that empirically, post-variational quantum neural networks using our architectural designs can potentially provide better results than variational algorithms and performance comparable to that of two-layer neural networks.

Figures (8)

• Figure 1: High-level sketch of post-variational strategies for near-term quantum computing. The variational method uses parameterized quantum circuits to transform embedded input data before suitable measurements, see panel a). Post-variational strategies, see panel b), use multiple fixed circuits, which may share similar circuit structure as the variational circuits, and multiple measurements of these circuits. The goal is to achieve approximately similar accuracy as the variational methods, while only performing optimization of classical parameters and retaining the power of quantum embeddings.
• Figure 2: Here we provide an overview of the various strategies introduced in the later text. Using the variational circuit (a) as a baseline, the Ansatz expansion approach (b) does model approximation by directly expanding the parameterized Ansatz into an ensemble of fixed Ansätze. On the other hand, the observable construction approach (c) foregoes all usage of an Ansatz and directly constructs a measurement observable by ensembling and taking the classical combination of various predefined trial observables. The hybrid approach (d) does both the expansion of the Ansatz, albeit only on the shallower layer of the model, and replaces the deeper layers of the model as well as the measurement observable with a series of measurement trial observables that can be used to retrieve a classical combination.
• Figure 3: Ansatz expansion approach to post-variational algorithm construction. Starting from a variational Ansatz, multiple non-parameterized quantum circuits are constructed by Taylor expansion of the Ansatz around a suitably chosen initial setting of the parameters $\boldsymbol{\theta^{(0)}}$. Gradients and higher-order derivatives of circuits can be obtained by parameter-shift rule. The different circuits are linearly combined with classical coefficients that are optimized via convex optimization.
• Figure 4: Observable construction approach to post-variational algorithm construction. A variational observable can be directly constructed by an ensemble of Pauli observables, and may serve as a potential approximation under locality restrictions.
• Figure 5: Hybrid approach to post-variational algorithm construction. This approach is a combination of the approaches shown in Figures \ref{['fig:ansatz_expand']} and \ref{['fig:observable_construct']}.

Theorems & Definitions (15)

• Definition 1
• Proposition 1: Measurements needed for direct estimation of all quantum neurons
• Proposition 2: Measurements needed for shadow estimation of all quantum neurons
• Theorem 3: Linear regression theoretical guarantee
• Theorem 4: Constrained linear regression theoretical guarantee
• Theorem 5: Stone's theorem on one-parameter unitary groups stone1930linearstone1932one
• Theorem 6: Baker–Campbell–Hausdorff identity campbell1897law
• proof : Proof of Proposition \ref{['proposition:measure']}
• proof : Proof of Proposition \ref{['proposition:shadow']}
• Theorem 7: Perturbation inequality for concave functions of singular values oymak2011simplifiedyue2016perturbation