Maximising Quantum-Computing Expressive Power through Randomised Circuits

TL;DR

The paper addresses the limited expressive power of fixed-gate variational quantum circuits on NISQ devices and proposes a randomised-circuit framework (VQCMC) where the variational state is an ANN-guided average over random circuit states. It develops a general formalism, derives an operator estimator via Hadamard tests, and analyzes statistical errors and a sign problem, offering two time-cost control strategies. Theoretical results establish that the random-circuit approach can match or surpass deterministic circuits in the low-cost limit, monotonically increase expressive power with time cost, and achieve universal approximation in the high-cost limit, with concrete bounds for Hamiltonian-ansatz-based VQE and scaling with circuit depth. A numerical demonstration on a random Heisenberg graph demonstrates the practical trade-off: increasing sampling time improves energy accuracy despite a fixed gate budget. Overall, the work provides a principled pathway to enhance the expressive power of VQAs and outlines how to manage the computational cost in the NISQ era and beyond.

Abstract

In the noisy intermediate-scale quantum era, variational quantum algorithms (VQAs) have emerged as a promising avenue to obtain quantum advantage. However, the success of VQAs depends on the expressive power of parameterised quantum circuits, which is constrained by the limited gate number and the presence of barren plateaus. In this work, we propose and numerically demonstrate a novel approach for VQAs, utilizing randomised quantum circuits to generate the variational wavefunction. We parameterize the distribution function of these random circuits using artificial neural networks and optimize it to find the solution. This random-circuit approach presents a trade-off between the expressive power of the variational wavefunction and time cost, in terms of the sampling cost of quantum circuits. Given a fixed gate number, we can systematically increase the expressive power by extending the quantum-computing time. With a sufficiently large permissible time cost, the variational wavefunction can approximate any quantum state with arbitrary accuracy. Furthermore, we establish explicit relationships between expressive power, time cost, and gate number for variational quantum eigensolvers. These results highlight the promising potential of the random-circuit approach in achieving a high expressive power in quantum computing.

Figures (10)

• Figure 1: (a) The schematic diagram of variational quantum algorithms using deterministic circuits. The problem is solved by finding the optimal circuit parameters $\theta$ to minimize loss function $L\left( \theta\right)$. (b) The schematic diagram of variational quantum algorithms using random circuits. Circuits are sampled according to a parameterised guiding function $\alpha\left(\theta; \lambda\right)$, which determines the possibility distribution. Instead of $\theta$, we optimise the parameterised guiding function $\alpha\left(\theta; \lambda\right)$ to solve a problem. See Section \ref{['sec:VQCMC']} for details.
• Figure 2: The subset of variational wavefunctions. In variational quantum algorithms (VQAs), the solution to a problem is represented by a quantum state. If the solution state is in the subset (e.g. state $a$) or close to the subset (e.g. state $b$) with a tolerable error $\epsilon$, VQA can successfully solve the problem (up to a proper optimiser and other practical issues). If the solution state is far from the subset (e.g. state $c$), VQA fails.
• Figure 3: The rectifier neural network with one hidden layer for parameterising the guiding function $\alpha(\theta; \lambda)$, where $\lambda=\left( w,b,w_A,w_B,b_A,b_B \right)$.
• Figure 4: The barrier term $L_b(\lambda) = -z\tanh\left[\frac{\langle \openone \rangle(\lambda)-x}{y}\right]$ in the modified loss function, where the parameter $x$ specifies the barrier's position. We illustrate this with two examples: for $y = 0.4$ and $y = 0.1$, while holding $z = 5$ and $x = 0.8$ constant.
• Figure 5: Graph of the Heisenberg model.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• Theorem 4
• Lemma 1
• Lemma 2
• Theorem 5
• Theorem 6
• proof