Unitary property testing lower bounds by polynomials

TL;DR

This paper introduces a generalized polynomial method for quantum unitary property testing, establishing new lower bounds and exploring implications for quantum complexity class separations.

Contribution

It develops a novel polynomial technique for unitary property testing and applies it to key problems, advancing understanding of quantum query complexity and complexity class separations.

Findings

Lower bounds for recurrence time determination

Bounds on approximating subspace dimension

Results on entanglement entropy approximation

Abstract

We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between $\mathsf{QMA}$ and $\mathsf{QMA(2)}$, a long standing question in quantum complexity theory.