On Computational Complexity of Unitary and State Design Properties

TL;DR

This paper explores the computational complexity of verifying and computing properties of unitary and state t-designs, revealing significant hardness results and the difficulty of approximation tasks in quantum information theory.

Contribution

It introduces complexity classifications for computing frame potentials and deciding design properties, highlighting the computational challenges in quantum design verification.

Findings

Exact frame potential computation is -hard and requires a -oracle.

Deciding if a set approximates a t-design is -hard.

Promise problems related to design properties are -complete or -hard depending on parameters.

Abstract

We investigate unitary and state $t$-designs from a computational complexity perspective. First, we address the problems of computing frame potentials that characterize (approximate) $t$-designs. We present a quantum algorithm for computing frame potentials and establish the following: (1) exact computation can be achieved by a single query to a $\# \textsf{P}$-oracle and is $\# \textsf{P}$-hard; (2) for state vectors, deciding whether the frame potential is larger than or smaller than certain values is $\textsf{BQP}$-complete, provided the promise gap between the two values is inverse-polynomial in the number of qubits; and (3) for both state vectors and unitaries, this promise problem is $\textsf{PP}$-complete if the promise gap is exponentially small. Second, we address the promise problem of deciding whether or not a given set is a good approximation to a design. Given a certain promise gap that could be constant, we show that this problem is $\textsf{PP}$-hard, highlighting the inherent computational difficulty of determining properties of unitary and state designs. We further identify the implications of our results across diverse areas, including variational methods for constructing designs, diagnosing quantum chaos through out-of-time-ordered correlators (OTOCs), and exploring emergent designs in Hamiltonian systems.