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Efficient approximate unitary designs from random Pauli rotations

Jeongwan Haah, Yunchao Liu, Xinyu Tan

TL;DR

This paper demonstrates that random Pauli rotations generate efficient approximate unitary t-designs by analyzing the t-th moment channel through the lens of Lie-algebra representations and quadratic Casimir operators. It proves a spectral gap of Ω(1/t) for the associated random-walk on SU(2^n) and establishes an ε-approximate unitary t-design in depth roughly O(n t^2 + t log 1/ε) with polylogarithmic circuit depth to implement each Pauli rotation. The approach yields strong implications for circuit complexity, seed-length efficiency via discrete or derandomized sampling, and extends to orthogonal designs (SO(N)) with analogous guarantees. Central to the argument are the kernel-projection bounds obtained from averaging over random Pauli rotations and the Casimir-based bounds that bound sums of Pauli-squared representations across irreps. Overall, the work provides a conceptually simple, scalable method to achieve high-quality random designs with explicit, near-optimal t-dependence and practical circuit-depth metrics.

Abstract

We construct random walks on simple Lie groups that quickly converge to the Haar measure for all moments up to order $t$. Specifically, a step of the walk on the unitary or orthognoal group of dimension $2^{\mathsf n}$ is a random Pauli rotation $e^{\mathrm i θP /2}$. The spectral gap of this random walk is shown to be $Ω(1/t)$, which coincides with the best previously known bound for a random walk on the permutation group on $\{0,1\}^{\mathsf n}$. This implies that the walk gives an $\varepsilon$-approximate unitary $t$-design in depth $O(\mathsf n t^2 + t \log 1/\varepsilon)d$ where $d=O(\log \mathsf n)$ is the circuit depth to implement $e^{\mathrm i θP /2}$. Our simple proof uses quadratic Casimir operators of Lie algebras.

Efficient approximate unitary designs from random Pauli rotations

TL;DR

This paper demonstrates that random Pauli rotations generate efficient approximate unitary t-designs by analyzing the t-th moment channel through the lens of Lie-algebra representations and quadratic Casimir operators. It proves a spectral gap of Ω(1/t) for the associated random-walk on SU(2^n) and establishes an ε-approximate unitary t-design in depth roughly O(n t^2 + t log 1/ε) with polylogarithmic circuit depth to implement each Pauli rotation. The approach yields strong implications for circuit complexity, seed-length efficiency via discrete or derandomized sampling, and extends to orthogonal designs (SO(N)) with analogous guarantees. Central to the argument are the kernel-projection bounds obtained from averaging over random Pauli rotations and the Casimir-based bounds that bound sums of Pauli-squared representations across irreps. Overall, the work provides a conceptually simple, scalable method to achieve high-quality random designs with explicit, near-optimal t-dependence and practical circuit-depth metrics.

Abstract

We construct random walks on simple Lie groups that quickly converge to the Haar measure for all moments up to order . Specifically, a step of the walk on the unitary or orthognoal group of dimension is a random Pauli rotation . The spectral gap of this random walk is shown to be , which coincides with the best previously known bound for a random walk on the permutation group on . This implies that the walk gives an -approximate unitary -design in depth where is the circuit depth to implement . Our simple proof uses quadratic Casimir operators of Lie algebras.
Paper Structure (17 sections, 19 theorems, 67 equations, 1 figure)

This paper contains 17 sections, 19 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous spectral gap bounds
  3. Implications
  4. Circuit complexity.
  5. Seed length.
  6. Orthogonal designs and more.
  7. Overview of the argument
  8. Note added.
  9. Lie algebras and probability distributions
  10. Random Pauli rotations
  11. The special case of su(2)
  12. A spectral gap bound by quadratic Casimir invariants
  13. Orthogonal designs
  14. Skew-symmetric Pauli operators
  15. Skew-symmetric elementary matrix basis
  16. ...and 2 more sections

Key Result

theorem 1.1

For any integers ${\mathsf{n}},t\geq 1$, it holds that Here, ${\mathbf{P}}_{\mathsf{n}} = \{ {\mathbf{1}}_2, \sigma^x, \sigma^y, \sigma^z\}^{\otimes {\mathsf{n}}} \setminus \{ {\mathbf{1}}_{2^{\mathsf{n}}} \}$ is the set of all nonidentity ${\mathsf{n}}$-qubit Pauli operators, the norm denotes the greatest singular value, $\bar{U}$ denotes the complex c

Figures (1)

  • Figure 1: Implementation of $e^{\mathrm{i} \frac{\theta}{2} P}$ ($P\in {\mathbf{P}}_{{\mathsf{n}}}$) by an $\mathcal{O}\left( \log {\mathsf{n}} \right)$ depth circuit. The example corresponds to the Pauli string XZXZXXXZ. Gates between two dashed lines are implemented in parallel.

Theorems & Definitions (42)

  • theorem 1.1
  • corollary 1.2
  • lemma 2.1
  • proof
  • lemma 2.2
  • proof
  • lemma 2.3
  • proof
  • proposition 2.4
  • proof
  • ...and 32 more