Quantum Gowers conjecture for random quantum circuits

Determine whether sufficiently deep random quantum circuits form pseudorandom unitary ensembles that are computationally indistinguishable from Haar-random unitaries by every efficient quantum distinguisher.

Background

The quantum Gowers conjecture is presented as the quantum analogue of the classical conjecture about random reversible circuits. Random local quantum circuits are known to form approximate unitary designs at polynomial depth when the design order is fixed or otherwise suitably bounded, but unitary-design guarantees alone do not establish computational pseudorandomness.

The paper’s counterexamples show that certain local ensembles can achieve strong constant-order design properties while remaining efficiently distinguishable from Haar measure. Those results do not resolve whether ordinary sufficiently deep random quantum circuits themselves become pseudorandom, which the paper identifies as a longstanding open problem.

References

Despite substantial recent progress on constructing pseudorandom unitaries from standard cryptographic assumptions, the quantum analog of the Gowers conjecture still remains wide open---just like its classical counterpart.

— On the pseudorandomness of simple quantum processes  (2610.02100 - Dujmovic et al., 1 Oct 2026) in Introduction, paragraph “Quantum pseudorandomness”

We therefore put forward the following new conjecture.

\begin{conjecture}[Pseudorandomness in the Maximal Scrambling Regime] Let ${\nu_n}_n$ be an efficiently samplable ensemble of local quantum gates on $n$ qubits, and let $\nu_n{*T}$ denote the ensemble obtained by independently composing $T=poly(n)$ gates drawn from $\nu_n$. If ${\nu_n{*T}}_n$ forms an approximate unitary $t$-design for $t=\Theta(n)$ with negligible error, then ${\nu_n{*T}}_n$ is a pseudorandom unitary ensemble. \end{conjecture}

— On the pseudorandomness of simple quantum processes  (2610.02100 - Dujmovic et al., 1 Oct 2026) in Section 3, “Pseudorandomness and Maximal Scrambling,” conjecture “Pseudorandomness in the Maximal Scrambling Regime”