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Anti-Concentration for the Unitary Haar Measure and Applications to Random Quantum Circuits

Bill Fefferman, Soumik Ghosh, Wei Zhan

TL;DR

The paper proves a Carbery–Wright–style anti-concentration inequality for polynomials in Haar-random unitary entries, enabling a high-probability, uniform lower bound on how much an input qubit can influence a designated output qubit through a fixed circuit lightcone. This anti-concentration result underpins a general scrambling bound that decays at most exponentially with depth and applies across architectures, not just in expectation. Leveraging this tool, the authors derive (i) a depth lower bound for ε-approximate unitary designs, (ii) a polynomial-time depth-testing algorithm for brickwork-like circuits, and (iii) a polynomial-time learning algorithm for log-depth Haar-random brickwork circuits (including a discretized-distribution variant) from oracle access. The work thereby connects fundamental anti-concentration phenomena with practical tasks in learning and verification of quantum circuits, offering typicality guarantees and broad applicability to architectures with favorable lightcone properties.

Abstract

We prove a Carbery-Wright style anti-concentration inequality for the unitary Haar measure, by showing that the probability of a polynomial in the entries of a random unitary falling into an $\varepsilon$ range is at most a polynomial in $\varepsilon$. Using it, we show that the scrambling speed of a random quantum circuit is lower bounded: Namely, every input qubit has an influence that is at least inverse exponential in depth, on any output qubit touched by its lightcone. Our result on scrambling speed works with high probability over the choice of a circuit from an ensemble, as opposed to just working in expectation. As an application, we give the first polynomial-time algorithm for learning log-depth random quantum circuits with Haar random gates up to polynomially small diamond distance, given oracle access to the circuit. Other applications of this new scrambling speed lower bound include: $\bullet$ An optimal $Ω(\log \varepsilon^{-1})$ depth lower bound for $\varepsilon$-approximate unitary designs on any circuit architecture; $\bullet$ A polynomial-time quantum algorithm that computes the depth of a bounded-depth circuit, given oracle access to the circuit. Our learning and depth-testing algorithms apply to architectures defined over any geometric dimension, and can be generalized to a wide class of architectures with good lightcone properties.

Anti-Concentration for the Unitary Haar Measure and Applications to Random Quantum Circuits

TL;DR

The paper proves a Carbery–Wright–style anti-concentration inequality for polynomials in Haar-random unitary entries, enabling a high-probability, uniform lower bound on how much an input qubit can influence a designated output qubit through a fixed circuit lightcone. This anti-concentration result underpins a general scrambling bound that decays at most exponentially with depth and applies across architectures, not just in expectation. Leveraging this tool, the authors derive (i) a depth lower bound for ε-approximate unitary designs, (ii) a polynomial-time depth-testing algorithm for brickwork-like circuits, and (iii) a polynomial-time learning algorithm for log-depth Haar-random brickwork circuits (including a discretized-distribution variant) from oracle access. The work thereby connects fundamental anti-concentration phenomena with practical tasks in learning and verification of quantum circuits, offering typicality guarantees and broad applicability to architectures with favorable lightcone properties.

Abstract

We prove a Carbery-Wright style anti-concentration inequality for the unitary Haar measure, by showing that the probability of a polynomial in the entries of a random unitary falling into an range is at most a polynomial in . Using it, we show that the scrambling speed of a random quantum circuit is lower bounded: Namely, every input qubit has an influence that is at least inverse exponential in depth, on any output qubit touched by its lightcone. Our result on scrambling speed works with high probability over the choice of a circuit from an ensemble, as opposed to just working in expectation. As an application, we give the first polynomial-time algorithm for learning log-depth random quantum circuits with Haar random gates up to polynomially small diamond distance, given oracle access to the circuit. Other applications of this new scrambling speed lower bound include: An optimal depth lower bound for -approximate unitary designs on any circuit architecture; A polynomial-time quantum algorithm that computes the depth of a bounded-depth circuit, given oracle access to the circuit. Our learning and depth-testing algorithms apply to architectures defined over any geometric dimension, and can be generalized to a wide class of architectures with good lightcone properties.
Paper Structure (26 sections, 26 theorems, 103 equations, 1 figure, 2 algorithms)

This paper contains 26 sections, 26 theorems, 103 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Implications on proving typicality statements.
  3. Applications
  4. Optimal 2-design lower bounds
  5. Testing for depth
  6. Learning brickwork random circuits
  7. Main Technical Tool: Anti-Concentration for Haar Measure
  8. Related Works
  9. Technical Overview
  10. Induction for Anti-Concentration Bound
  11. Tracking 3-Dimensional Pauli Subspaces
  12. Learning Log-Depth Random Quantum Circuits
  13. Preliminaries
  14. Anti-concentration Bound
  15. m=1,n=1
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\mathcal{C}$ be a random quantum circuit with a fixed architecture, where each gate is a $k$-qubit independent Haar random unitary. Let $\rho$ and $\pi$ be a pair of input and output qubits connected by $D$ layer of gates. Arbitrarily fix the inputs to $\mathcal{C}$, except the qubit $\rho$, an where $\mathopen{}\mathclose{\left\|\cdot\right\|_{\mathrm F}}$ denotes the Frobenius norm, and $c_

Figures (1)

  • Figure 2: A Illustration of the claim and proof of \ref{['thm:mixing']} with an example circuit, where the bold wires represents a path from the first input qubit to the last output qubit. The state of the qubit on the $i$-th segment of the wire, between gate $G_{i-1}$ and $G_i$, is denoted as $\rho_i$, whose difference is captured by the projection onto the Pauli subspace $\{X,Y,Z\}_i\otimes I^{\otimes(n-1)}$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark
  • Definition 3.1
  • Definition 3.2
  • ...and 37 more