Unitary designs in nearly optimal depth

TL;DR

This paper presents nearly optimal depth constructions for approximate unitary $k$-designs on $n$ qubits, significantly improving efficiency and establishing optimal parameter dependencies, with applications to quantum pseudorandomness.

Contribution

It introduces the first nearly optimal depth constructions for approximate unitary $k$-designs, using structured random ensembles and a new analytical framework for error bounding.

Findings

Constructed $ ilde{O}(nk)$ ancilla-based unitary $k$-designs with $O( ext{log} k ext{ log log } n k / ext{ε})$ depth.

Achieved an alternative design with $ ilde{O}(n)$ ancillas and $O(k ext{ log log } n k / ext{ε})$ depth.

Provided a new framework for analyzing errors in quantum experiments with many random unitaries.

Abstract

We construct $\varepsilon$-approximate unitary $k$-designs on $n$ qubits in circuit depth $O(\log k \log \log n k / \varepsilon)$. The depth is exponentially improved over all known results in all three parameters $n$, $k$, $\varepsilon$. We further show that each dependence is optimal up to exponentially smaller factors. Our construction uses $\tilde{{O}}(nk)$ ancilla qubits and ${O}(nk)$ bits of randomness, which are also optimal up to $\log(n k)$ factors. An alternative construction achieves a smaller ancilla count $\tilde{{O}}(n)$ with circuit depth ${O}(k \log \log nk/\varepsilon)$. To achieve these efficient unitary designs, we introduce a highly-structured random unitary ensemble that leverages long-range two-qubit gates and low-depth implementations of random classical hash functions. We also develop a new analytical framework for bounding errors in quantum experiments involving many queries to random unitaries. As an illustration of this framework's versatility, we provide a succinct alternative proof of the existence of pseudorandom unitaries.