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Exact spectral gaps of random one-dimensional quantum circuits

Andrew E. Deneris, Pablo Bermejo, Paolo Braccia, Lukasz Cincio, M. Cerezo

TL;DR

The authors derive exact expressions for the t=2 spectral gap of the second-moment operator for one-dimensional random quantum circuits with nearest-neighbor m-qudit gates, distinguishing open and closed boundary conditions. They prove that the closed-boundary gap is the square of the open-boundary gap for unitary gates, and provide explicit eigenvectors, leading to tighter bounds on the number of layers required for the circuit to form an approximate 2-design. The work also analyzes orthogonal and symplectic ensembles numerically, showing a more complex relation between boundary conditions and gaps. Numerical results up to 70 qubits corroborate the unitary theory and reveal richer behavior for non-unitary local gates. The framework combines moment operators, Weingarten calculus, and a detailed eigenstructure analysis to sharpen convergence estimates and guide future exploration of higher moments and different gate groups.

Abstract

The spectral gap of local random quantum circuits is a fundamental property that determines how close the moments of the circuit's unitaries match those of a Haar random distribution. When studying spectral gaps, it is common to bound these quantities using tools from statistical mechanics or via quantum information-based inequalities. By focusing on the second moment of one-dimensional unitary circuits where nearest neighboring gates act on sets of qudits (with open and closed boundary conditions), we show that one can exactly compute the associated spectral gaps. Indeed, having access to their functional form allows us to prove several important results, such as the fact that the spectral gap for closed boundary condition is exactly the square of the gap for open boundaries, as well as improve on previously known bounds for approximate design convergence. Finally, we verify our theoretical results by numerically computing the spectral gap for systems of up to 70 qubits, as well as comparing them to gaps of random orthogonal and symplectic circuits.

Exact spectral gaps of random one-dimensional quantum circuits

TL;DR

The authors derive exact expressions for the t=2 spectral gap of the second-moment operator for one-dimensional random quantum circuits with nearest-neighbor m-qudit gates, distinguishing open and closed boundary conditions. They prove that the closed-boundary gap is the square of the open-boundary gap for unitary gates, and provide explicit eigenvectors, leading to tighter bounds on the number of layers required for the circuit to form an approximate 2-design. The work also analyzes orthogonal and symplectic ensembles numerically, showing a more complex relation between boundary conditions and gaps. Numerical results up to 70 qubits corroborate the unitary theory and reveal richer behavior for non-unitary local gates. The framework combines moment operators, Weingarten calculus, and a detailed eigenstructure analysis to sharpen convergence estimates and guide future exploration of higher moments and different gate groups.

Abstract

The spectral gap of local random quantum circuits is a fundamental property that determines how close the moments of the circuit's unitaries match those of a Haar random distribution. When studying spectral gaps, it is common to bound these quantities using tools from statistical mechanics or via quantum information-based inequalities. By focusing on the second moment of one-dimensional unitary circuits where nearest neighboring gates act on sets of qudits (with open and closed boundary conditions), we show that one can exactly compute the associated spectral gaps. Indeed, having access to their functional form allows us to prove several important results, such as the fact that the spectral gap for closed boundary condition is exactly the square of the gap for open boundaries, as well as improve on previously known bounds for approximate design convergence. Finally, we verify our theoretical results by numerically computing the spectral gap for systems of up to 70 qubits, as well as comparing them to gaps of random orthogonal and symplectic circuits.
Paper Structure (18 sections, 8 theorems, 65 equations, 3 figures)

This paper contains 18 sections, 8 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Framework
  3. Moment operators
  4. Weingarten calculus
  5. Spectral gap
  6. Theoretical results for unitary random circuits
  7. Proof of Theorem \ref{['th:theorem_1']} and explicit eigenvector construction
  8. Open boundary conditions
  9. Closed boundary conditions
  10. Numerical Results
  11. Discussion
  12. Acknowledgments
  13. Proof of Lemma \ref{['lemma:1']}
  14. Proof of Lemma \ref{['lemma:2']}
  15. Proof of Eq. \ref{['eq:Lambda-1-switch']}: Action of $\Lambda$ on 1-switch terms
  16. ...and 3 more sections

Key Result

Theorem 1

Let $U$ be a circuit as in Eqs. eq:circuit and eq:layer, i.e., and where all the local gates are sampled i.i.d from the Haar measure over $\mathbb{U}(d^{2m})$. Then, for open boundary conditions ($\Delta=0$) the spectral gap is while for closed boundary conditions ($\Delta=1$) it is

Figures (3)

  • Figure 1: Schematic representation of a one-dimensional random circuit. As shown, random gates act on alternating groups of $m$ neighboring qudits in a brick-like fashion. The colored gates are removed (added) with open (closed) boundary conditions.
  • Figure 2: Schematic representation of the second moment operator. After the reduction of Eq. \ref{['eq:layer-moment']}, the action of $\mathcal{T}_{\mathcal{E}_L}^{(2)}$ can be studied as a $2^\eta\times 2^\eta$ operator where $A$ matrices act on two two-dimensional spaces following the same topology as that in $U$.
  • Figure 3: Spectral gap for unitary, orthogonal and symplectic circuits. We plot the spectral gap versus the number of qubits for different groups, as well as different boundary conditions on a single layered random quantum circuit.

Theorems & Definitions (12)

  • Definition 1: Approximate design.
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 2 more