Symmetric Pruning in Quantum Neural Networks

Abstract

Many fundamental properties of a quantum system are captured by its Hamiltonian and ground state. Despite the significance, ground states preparation (GSP) is classically intractable for most large-scale Hamiltonians. Quantum neural networks (QNNs), which exert the power of modern quantum machines, have emerged as a leading protocol to conquer this issue. As such, the performance enhancement of QNNs becomes the core in GSP. Empirical evidence showed that QNNs with handcraft symmetric ansätze generally experience better trainability than those with asymmetric ansätze, while theoretical explanations remain vague. To fill this knowledge gap, here we propose the effective quantum neural tangent kernel (EQNTK) and connect this concept with over-parameterization theory to quantify the convergence of QNNs towards the global optima. We uncover that the advance of symmetric ansätze attributes to their large EQNTK value with low effective dimension, which requests few parameters and quantum circuit depth to reach the over-parameterization regime permitting a benign loss landscape and fast convergence. Guided by EQNTK, we further devise a symmetric pruning (SP) scheme to automatically tailor a symmetric ansatz from an over-parameterized and asymmetric one to greatly improve the performance of QNNs when the explicit symmetry information of Hamiltonian is unavailable. Extensive numerical simulations are conducted to validate the analytical results of EQNTK and the effectiveness of SP.

1. Introduction

The law of quantum mechanics advocates that any quantum system can be described by a Hamiltonian, and many important physical properties are reflected by its ground state. For this reason, the ground state preparation (GSP) of Hamiltonians is the key to understanding and fabricating novel quantum matters. Due to the intrinsic hardness of GSP (Poulin & Wocjan, 2009; Carleo et al., 2019) , the required computational resources of classical methods are unaffordable when the size of Hamiltonian becomes large. Quantum computers, whose operations can harness the strength of quantum mechanics, promise to tackle this problem with potential computational merits. In the noisy intermediate-scale quantum (NISQ) era (Preskill, 2018) , quantum neural networks (QNNs) (Farhi & Neven, 2018; Cong et al., 2019; Cerezo et al., 2021a) are leading candidates toward this goal. The building blocks of QNNs, analogous to deep neural networks, consist of variational ansätze (also called parameterized quantum circuits) and classical optimizers. In order to enhance the power of QNNs in GSP, great efforts have been made to design advanced ansätze with varied circuit structures (Peruzzo et al., 2014; Wecker et al., 2015; Kandala et al., 2017) . Despite the achievements aforementioned, recent progress has shown that QNNs may suffer from severe trainability issues when the circuit depth of ansätze is either shallow or deep. Namely, for the deep ansätze, the magnitude of the gradients exponentially decays with the increased system size (McClean et al., 2018; Cerezo et al., 2021b) . This phenomenon, dubbed the barren plateau, hints at the difficulty of optimizing deep QNNs, where an exponential runtime is necessitated for convergence. The wisdom to alleviate barren plateaus is exploiting shallow ansätze to accomplish learning tasks (Grant et al., 2019; Skolik et al., 2021; Zhang et al., 2020; Pesah et al., 2021) , while the price to pay is incurring another serious trainability issue-convergence (Boyd & Vandenberghe, 2004; Du et al., 2021) . The trainable parameters may get stuck into sub-optimal local minima or saddle points with high probability because of the unfavorable loss landscape (Anschuetz, 2021; Anschuetz & Kiani, 2022) . Orthogonal to these negative results, several studies pointed out that when the depth of ansätze becomes overwhelmingly deep and surpasses a critical point, the overparameterized QNNs embrace a benign landscape and permit fast convergence towards good local minima (Kiani et al., 2020; Wiersema et al., 2020; Larocca et al., 2021b) . Nevertheless, the criteria to reach such a critical point is stringent, i.e., the number of parameterized gates or the circuit depth scales exponentially with the problem size, which hurdles the application of over-parameterized QNNs in practice. Empirical evidence sheds new light on exploiting overparameterized QNNs to tackle GSP. QNNs with symmetric ansätze only demand a polynomial number of trainable parameters and the circuit depth with the problem size to reach the over-parameterized region and achieve a fast convergence rate (Herasymenko & O'Brien, 2021; Gard et al., 2020; Zheng et al., 2021; 2022; Shaydulin & Wild, 2021; Mernyei et al., 2022; Marvian, 2022; Meyer et al., 2022; Larocca et al., 2022; Sauvage et al., 2022) . A common feature of these symmetric ansätze is capitalizing on the symmetric properties underlying the problem Hamiltonian to shrink the solution space and facilitate seeking near-optimal solutions. Unfortunately, current symmetric ansätze are inapplicable to a broad class of Hamiltonians whose symmetry is implicit, since their constructions rely on the explicit information for the symmetry of Hamiltonians. Besides, it is unknown whether the symmetry contributes to lowering the critical point to reach the over-parameterization regime. Here we fill the above knowledge gap from both theoretical and practical aspects. Concretely, we develop a novel notion-effective quantum neural tangent kernel (EQNTK) to capture the training dynamic of various ansätze via their effective dimension. In doing so, we expose that compared with the asymmetric ansätze, the symmetric ansätze possess dramatically lower effective dimensions and the required number of parameters and circuit depth to reach the over-parameterization may polynomially scale with the problem size (see Fig. 1 for an intuition). By leveraging EQNTK, we next prove that when the condition of over-parameterization is satisfied, the trainable parameters of QNNs with symmetric ansätze can exponentially converge to the global optima with the increased iterations. Taken together, our analysis recognizes that overparameterized QNNs with symmetric ansätze is a possible solution toward large-scale GSP tasks. Envisioned by EQNTK and pruning techniques in deep neural networks (Han et al., 2015; Blalock et al., 2020; Frankle et al., 2020; Wang et al., 2022) , we further devise a symmetric pruning scheme (SP) to automatically tailor a symmetric ansatz from an over-parameterized and asymmetric one with the enhanced trainability and applicability. Conceptually, SP continuously eliminates the redundant quantum gates from the given asymmetric ansatz and correlates parameters to assign different types of symmetries on the slimmed ansatz. In this way, SP generates a modest-depth symmetric ansatz with a fast convergence guarantee and thus improves the hardware efficiency. Extensive simulations on many-body physics and combinatorial problems validate the theory of EQNTK and the



Figure 1: The critical point of the over-parameterized regime. When the number of parameters is beyond the critical point (the red circle), the training error exponentially converges to a nearly global minimum. Symmetric ansätze (the blue curve) require few parameters to reach the critical point over the asymmetric ansätze.

