🤖 AI Summary
This work addresses the exact computation of the spectral gap of the second-moment operator for one-dimensional random quantum circuits, which governs their convergence rate toward the Haar-random distribution. Leveraging a synthesis of group representation theory, matrix-product-state analysis, and quantum-chaos spectral techniques, we derive the first rigorous analytical expressions for the spectral gap under both open and closed boundary conditions. Crucially, we find that the closed-boundary gap equals the square of the open-boundary gap—revealing a fundamental dependence of convergence dynamics on boundary topology. Our predictions are validated numerically up to 70 qubits with perfect agreement. Consequently, we substantially tighten the upper bound on the convergence time to approximate 2-designs. The framework is further generalized to orthogonal and symplectic symmetry classes, with systematic cross-validation performed. These results establish a foundational theoretical basis for the rigorous analysis and practical design of random quantum circuits.
📝 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.