Classical algorithms for quantum mean values

• Published in: Nature physics 2021-03, Vol.17 (3), p.337-341
• Main Authors: Bravyi, Sergey, Gosset, David, Movassagh, Ramis
• Format: Article
• Language: English
• Subjects: 639/766/483/2802; 639/766/483/3926; 639/766/483/481; Algorithms; Atomic; Circuits; Classical and Continuum Physics; Complex Systems; Condensed Matter Physics; Fault tolerance; Machine learning; Mathematical and Computational Physics; Molecular; Optical and Plasma Physics; Optimization; Physics; Physics and Astronomy; Qubits (quantum computing); Run time (computers); Theoretical
• ISSN: 1745-2473; 1745-2481
• DOI: 10.1038/s41567-020-01109-8

Quantum algorithms hold the promise of solving certain computational problems dramatically faster than their classical counterparts. The latest generation of quantum processors with ~50 qubits are expected to be at the brink of outperforming classical computers. However, due to the lack of fault tolerance, the qubits can be operated for only a few time steps, making the quantum circuits shallow in depth. Variational quantum algorithms are leading candidates in the effort to find shallow-depth quantum algorithms that outperform classical computers. Here we consider the task of computing the mean values of multi-qubit observables, which is a cornerstone of variational quantum algorithms for optimization, machine learning and the simulation of quantum many-body systems. We develop sub-exponential time classical algorithms for solving the quantum mean value problem for general classes of quantum observables and constant-depth quantum circuits. In the special case of geometrically local two-dimensional quantum circuits, the runtime of our algorithm scales linearly with the number of qubits. Our results demonstrate that appropriate choices of circuit parameters such as geometric locality and depth are essential to obtain quantum speed-ups based on variational quantum algorithms. Finding expectation values is a key step in variational quantum algorithms that are hoped to provide a near-term quantum advantage. Bravyi et al. show that a classical approximation is possible when the quantum circuits are limited to constant depth.