Frontiers of quantum physics

Abstract:

We formally represent the quantum interference of a single qubit embodied by a photon in the Mach–Zehnder interferometer using the classical Hamiltonian framework but with complex canonical variables. Although all operations on a single qubit can be formally expressed using complex classical Hamiltonian dynamics, we show that the resulting system is still not a proper qubit. The reason for this is that it is not capable of getting entangled to another bona fide qubit and hence it does not have the information-processing capacity of a fully-fledged quantum system. This simple example powerfully illustrates the difficulties faced by hybrid quantum–classical models in accounting for the full range of behaviour of quantum systems.

Abstract:

Quantum computing (QC) and quantum Monte Carlo (QMC) represent state-of-the-art quantum and classical computing methods, respectively, for understanding many-body quantum systems. However, straightforward integration of the two methods may encounter significant challenges, such as exponential sampling cost and inefficient walker propagation. Here, we propose an efficient hybrid quantum-classical algorithm that integrates the two methods, overcoming these limitations while leveraging their strengths in representing and manipulating quantum states. To measure the effectiveness of the hybrid approach, we first introduce nonstoquasticity indicators (NSIs) and their theoretical upper bounds, which quantify the severity of the sign problem, a major limitation of QMC. Next, we present a hybrid QC-QMC method where the walkers are represented by quantum states prepared by a shallow quantum circuit. Although the Hamiltonian in the quantum state walker basis is not sparse, we offer an efficient and scalable approach to implement walker propagation using a quantum computer. From the QMC perspective, our algorithm significantly mitigates the sign problem in the quantum state walker basis. From the QC perspective, integrating QMC increases the expressivity of shallow quantum circuits, enabling more accurate computations that are traditionally achievable only with much deeper quantum circuits. Our method has immediate applications in tackling complex quantum many-body problems. We numerically test and verify it for the N2 molecule (12 qubits) and the Hubbard model (16 qubits), observing a significant suppression of the sign problem (which exponentially decreases with circuit depth) and a notable improvement in calculation accuracy (which is about two to three orders compared to variational quantum algorithms). Our work paves the way to solving practical problems with intermediate-scale and early fault-tolerant quantum computers, with broad applications in chemistry, condensed matter physics, and materials.