Incompressibility and spectral gaps of random circuits
Chi-Fang Chen, Jeongwan Haah, Jonas Haferkamp, Yunchao Liu, Tony Metger, Xinyu Tan
TL;DR
<3-5 sentence high-level summary> This work proves t-independent spectral gaps of order Ω(n^{-3}) for random reversible circuits and Ω(n^{-3}) (up to t ≈ Θ(2^{n/2})) for random quantum circuits, with refinements to Ω(n^{-1}/polylog(n,t)) for larger t. These gaps yield near-optimal, multiplicative-error t-designs after ~O(n^2 t) gates for permutations and ~O(n t) depth for unitaries, and they imply linear growth of robust circuit complexity for random circuits over exponential timescales. The authors achieve this by reducing the random-circuit walks to a structured PFC/CPFPC-like ensemble featuring Kassabov’s Alt(2^n) expander, then decomposing these structured steps into local gates and applying tools from frustration-free Hamiltonians to preserve gaps under aggregation. The results connect spectral-gap methods to concrete circuit-design and complexity consequences, resolving conjectures about randomness amplification and the computational hardness growth in random circuits, and opening new directions for 1D architectures and broader design constructions.</paper_summary>
Abstract
Random reversible and quantum circuits form random walks on the alternating group $\mathrm{Alt}(2^n)$ and unitary group $\mathrm{SU}(2^n)$, respectively. Known bounds on the spectral gap for the $t$-th moment of these random walks have inverse-polynomial dependence in both $n$ and $t$. We prove that the gap for random reversible circuits is $Ω(n^{-3})$ for all $t\geq 1$, and the gap for random quantum circuits is $Ω(n^{-3})$ for $t \leq Θ(2^{n/2})$. These gaps are independent of $t$ in the respective regimes. We can further improve both gaps to $n^{-1}/\mathrm{polylog}(n, t)$ for $t\leq 2^{Θ(n)}$, which is tight up to polylog factors. Our spectral gap results have a number of consequences: 1) Random reversible circuits with $\mathcal{O}(n^4 t)$ gates form multiplicative-error $t$-wise independent (even) permutations for all $t\geq 1$; for $t \leq Θ(2^{n/6.1})$, we show that $\tilde{\mathcal{O}}(n^2 t)$ gates suffice. 2) Random quantum circuits with $\mathcal{O}(n^4 t)$ gates form multiplicative-error unitary $t$-designs for $t \leq Θ(2^{n/2})$; for $t\leq Θ(2^{2n/5})$, we show that $\tilde{\mathcal{O}}(n^2t)$ gates suffice. 3) The robust quantum circuit complexity of random circuits grows linearly for an exponentially long time, proving the robust Brown--Susskind conjecture [BS18,BCHJ+21]. Our spectral gap bounds are proven by reducing random quantum circuits to a more structured walk: a modification of the ``$\mathrm{PFC}$ ensemble'' from [MPSY24] together with an expander on the alternating group due to Kassabov [Kas07a], for which we give an efficient implementation using reversible circuits. In our reduction, we approximate the structured walk with local random circuits without losing the gap, which uses tools from the study of frustration-free Hamiltonians.