Classical algorithms for quantum mean values

Quantum algorithms hold the promise of solving certain computational problems dramatically faster than their classical counterparts. The latest generation of quantum processors with similar to 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.

Automated summary

Quantum algorithms show potential in solving computational problems faster than classical ones, with the latest quantum processors on the brink of outperforming classical computers. Variational quantum algorithms are leading contenders in this quest for shallow-depth algorithms that surpass classical computers. Proper selection of circuit parameters such as geometric locality and depth is crucial for achieving quantum speed-ups with variational quantum algorithms.