Universal Quantum Emulator

TL;DR

This paper introduces a quantum algorithm that efficiently emulates unknown unitaries using sample states, with runtime logarithmic in the Hilbert space dimension and sample complexity independent of it.

Contribution

The proposed algorithm emulates unknown unitaries without prior knowledge, with logarithmic runtime and sample complexity independent of the full Hilbert space dimension.

Findings

Runtime is logarithmic in the Hilbert space dimension D.

Sample complexity is independent of D and polynomial in the sample subspace dimension d.

Algorithm is blind and does not learn about the sample states or the unitary.

Abstract

We propose a quantum algorithm that emulates the action of an unknown unitary transformation on a given input state, using multiple copies of some unknown sample input states of the unitary and their corresponding output states. The algorithm does not assume any prior information about the unitary to be emulated or the sample input states. To emulate the action of the unknown unitary, the new input state is coupled to the given sample input-output pairs in a coherent fashion. Remarkably, the runtime of the algorithm is logarithmic in D, the dimension of the Hilbert space, and increases polynomially with d, the dimension of the subspace spanned by the sample input states. Furthermore, the sample complexity of the algorithm-i.e., the total number of copies of the sample input-output pairs needed to run the algorithm-is independent of D and polynomial in d. In contrast, the runtime and sample complexity of incoherent methods, i.e., methods that use tomography, are both linear in D. The algorithm is blind, in the sense that, at the end, it does not learn anything about the given samples or the emulated unitary. This algorithm can be used as a subroutine in other algorithms, such as quantum phase estimation.