Random Unitaries in Constant (Quantum) Time

TL;DR

This paper demonstrates that unitary designs and pseudorandom unitaries can be efficiently constructed in constant-time quantum models with nonlocal gates, challenging previous assumptions about circuit complexity and opening new avenues in quantum complexity theory.

Contribution

It shows that unitary designs and PRUs are achievable in constant-depth quantum circuits with nonlocal operations, expanding the understanding of quantum circuit complexity.

Findings

Unitary designs and PRUs can be constructed in constant-time models.

Constructed PRUs imply no polynomial-time learning algorithm for certain circuit classes.

Results suggest new approaches to longstanding quantum complexity questions.

Abstract

Random unitaries are a central object of study in quantum information, with applications to quantum computation, quantum many-body physics, and quantum cryptography. Recent work has constructed unitary designs and pseudorandom unitaries (PRUs) using $\Theta(\log \log n)$-depth unitary circuits with two-qubit gates. In this work, we show that unitary designs and PRUs can be efficiently constructed in several well-studied models of $\textit{constant-time}$ quantum computation (i.e., the time complexity on the quantum computer is independent of the system size). These models are constant-depth circuits augmented with certain nonlocal operations, such as (a) many-qubit TOFFOLI gates, (b) many-qubit FANOUT gates, or (c) mid-circuit measurements with classical feedforward control. Recent advances in quantum computing hardware suggest experimental feasibility of these models in the near future. Our results demonstrate that unitary designs and PRUs can be constructed in much weaker circuit models than previously thought. Furthermore, our construction of PRUs in constant-depth with many-qubit TOFFOLI gates shows that, under cryptographic assumptions, there is no polynomial-time learning algorithm for the circuit class $\mathsf{QAC}^0$. Finally, our results suggest a new approach towards proving that PARITY is not computable in $\mathsf{QAC}^0$, a long-standing question in quantum complexity theory.