[Paper Review] Incompressibility and spectral gaps of random circuits

Chi-Fang Chen, Jeongwan Haah+4 more|arXiv (Cornell University)|Jun 11, 2024

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper establishes tight spectral gap bounds for random reversible and quantum circuits, proving that their mixing rates are independent of circuit depth up to exponential time. It shows that random quantum circuits with O(n⁴t) gates form multiplicative-error t-designs for t ≤ Θ(2ⁿ/²), and that circuit complexity grows linearly for exponentially long times—resolving the robust Brown–Susskind conjecture and establishing incompressibility of random circuits.

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 $ ilde{\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 $ ilde{\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.

Motivation & Objective

• To establish tight, t-independent spectral gap bounds for random reversible and quantum circuits on n qubits or bits.
• To prove that random circuits form approximate t-designs with multiplicative error, quantifying their convergence to uniform distributions.
• To resolve the robust Brown–Susskind conjecture by showing that the quantum circuit complexity of random circuits grows linearly for exponentially long times.
• To demonstrate that random circuits are incompressible, meaning their complexity scales linearly with the number of gates.

Proposed method

• Reduces random quantum circuits to a structured walk using a modified PFC ensemble and Kassabov’s expander on the alternating group.
• Uses tools from frustration-free Hamiltonians to approximate the structured walk with local random circuits without losing spectral gap.
• Employs Kazhdan constants and moment operator analysis to bound the essential norm and derive spectral gaps.
• Constructs efficient reversible circuits that generate the alternating group Alt(2ⁿ) using Kassabov’s generators and bounded generation techniques.
• Applies polynomial interpolation and overlap theorems to bound the probability of high-fidelity state preparation.
• Uses net arguments and trace norm estimates to derive circuit complexity lower bounds from approximate t-design properties.

Experimental results

Research questions

• RQ1 What is the spectral gap of random reversible circuits, and does it remain bounded independently of circuit depth t for all t ≥ 1?
• RQ2 Can random quantum circuits form multiplicative-error unitary t-designs for t up to exponential in n, and what gate count is required?
• RQ3 Does the robust quantum circuit complexity of random circuits grow linearly with the number of gates for exponentially long times, confirming the robust Brown–Susskind conjecture?
• RQ4 Can the spectral gap of random circuits be bounded independently of t, and how does this relate to their ability to form t-designs?
• RQ5 Is a random quantum circuit with O(n⁴t) gates sufficient to form a multiplicative-error t-design for t ≤ Θ(2ⁿ/²), and can this be improved to ˜O(n²t)?

Key findings

• The spectral gap for random reversible circuits is Ω(n⁻³) for all t ≥ 1, independent of t.
• For random quantum circuits, the spectral gap is Ω(n⁻³) for t ≤ Θ(2ⁿ/²), also independent of t.
• Random reversible circuits with O(n⁴t) gates form multiplicative-error t-wise independent permutations for all t ≥ 1; for t ≤ Θ(2ⁿ/⁶.¹), ˜O(n²t) gates suffice.
• Random quantum circuits with O(n⁴t) gates form multiplicative-error unitary t-designs for t ≤ Θ(2ⁿ/²); for t ≤ Θ(2²ⁿ/⁵), ˜O(n²t) gates suffice.
• The robust quantum circuit complexity of random quantum circuits grows linearly with the number of gates L, satisfying CQ,𝛿(U|0ⁿ⟩) ≥ Ω(L/n⁴) with high probability for L ≤ O(2ⁿ/²).
• The robust Brown–Susskind conjecture is resolved: random quantum circuits are incompressible, with complexity growing linearly for exponentially long times.