How to find expressible and trainable parameterized quantum circuits?

TL;DR

This paper investigates the construction of parameterized quantum circuits that balance expressibility and trainability, providing theoretical guarantees and a practical framework for designing efficient quantum algorithms.

Contribution

It introduces a dimension-independent concentration bound for PQC cost functions and proposes a property-based ansatz-search framework to find circuits that are both trainable and expressive.

Findings

Anticorrelation between trainability and expressibility observed across ans"atze.

Proposed framework successfully identifies circuits with improved properties on real quantum hardware.

Achieved VQE results with fewer parameters and comparable accuracy to traditional methods.

Abstract

Whether parameterized quantum circuits (PQCs) can be systematically constructed to be both trainable and expressive remains an open question. Highly expressive PQCs often exhibit barren plateaus, while several trainable alternatives admit efficient classical simulation. We address this question by deriving a finite-sample, dimension-independent concentration bound for estimating the variance of a PQC cost function, yielding explicit trainability guarantees. Across commonly used ans\"atze, we observe an anticorrelation between trainability and expressibility, consistent with theoretical insights. Building on this observation, we propose a property-based ansatz-search framework for identifying circuits that combine trainability and expressibility. We demonstrate its practical viability on a real quantum computer and apply it to variational quantum algorithms. We identify quantum neural network ans\"atze with improved effective dimension using over $6 \times$ fewer parameters, and for VQE on $\mathrm{H}_2$ we achieve UCCSD-like accuracy at substantially reduced circuit complexity.