More global randomness from less random local gates

TL;DR

The paper demonstrates that one-dimensional structured random circuits using non-Haar local gates can produce greater global randomness than Haar-based circuits with the same architecture, as diagnosed by second-moment operators. It achieves exact spectra and eigenvectors of the second-moment operators under a solvable condition that maps the circuit’s moment operator to free-fermion chains (Kitaev-like), yielding spectral gaps that can exceed those of Haar circuits. For local architectures, the spectral gap Δ^{(2)} scales as Δ^{(2)}_L,u = rac{2 e_u}{n e_H} ig(1 - rac{2 d}{d^2+1} rac{ ext{cos}(rac{ ext{π}}{n})}{1}ig), and brick-wall circuits admit a product-structure diagonalization with λ_k = ig(a_k ilde e_u + ig(a_k^2 ilde e_u^2 + 1 - ilde e_uig)^{1/2}ig)^2, leading to analogous gap expressions. These results show that increasing the entangling power above Haar levels, or operating on the solvable line, enhances randomness relative to Haar benchmarks, with concrete implications for faster convergence to unitary 2-designs and tighter RB-depth bounds. The work also connects randomness in these structured circuits to quantum-chaos indicators and frame potentials, and provides a framework extendable to higher moments and higher-dimensional architectures. Overall, the combination of non-Haar local gates and solvable mappings reveals a route to achieve stronger global randomness with potentially shallower circuits.

Abstract

Random circuits giving rise to unitary designs are key tools in quantum information science and many-body physics. In this work, we investigate a class of random quantum circuits with a specific gate structure. Within this framework, we prove that one-dimensional structured random circuits with non-Haar random local gates can exhibit substantially more global randomness compared to Haar random circuits with the same underlying circuit architecture. In particular, we derive all the exact eigenvalues and eigenvectors of the second-moment operators for these structured random circuits under a solvable condition, by establishing a link to the Kitaev chain, and show that their spectral gaps can exceed those of Haar random circuits. Our findings have applications in improving circuit depth bounds for randomized benchmarking and the generation of approximate unitary 2-designs from shallow random circuits.

Figures (5)

• Figure 1: Setup and main results. Structured random gates consist of a fixed two-qudit gate $u$ and four Haar random single-qudit gates sandwiching it. We consider two models of random circuits, where structured random gates are applied in the (a) local random circuit architecture and (b) brick-wall circuit architecture, where the boxes are two-qudit gates and the balls are single-qudit Haar random unitaries. As stated in Theorem \ref{['thm-more-randomness_local_random_circuits']} and \ref{['thm-more-randomness_brick-wall_random circuits']}, we show that if the entangling power of a structured random gate is greater than that of Haar random gates, the structured random circuits satisfying a solvable condition are more random than Haar random circuits with the same architecture (c).
• Figure 2: Numerical results on $|\lambda_3|$ of $M_{L,u}$ (LRC) and $M_{B,u}$ (BWC) for general $e_u$ and $g_u$ in qubit system ($d=2$) and qutrit system ($d=3$) in the case of $n=14$. The x-axis and the y-axis refer to the entangling power $e_u$ and the gate typicality $g_u$, respectively. We derive analytical solutions of them on the solvable lines (the broken red lines) satisfying the condition Eq. \ref{['eq-def-solvable-condition-maintext']} in Theorem \ref{['thm-spectral_maintext']}. At the Haar points (the orange dots), the second-moment operators of structured random circuits are equal to those of the Haar random circuits with the same architecture, and the points are on the solvable lines. Only the values between the thick white lines are feasible for $e_u$ and $g_u$, as discussed in Sec. S3 A in the SM and also in Ref. PhysRevResearch.2.043126.
• Figure S1: Vectorization of the identity and SWAP gates. The identity and SWAP gates are graphically described (a), where each arc is the unnormalized Bell state vector $\sum_{j=1}^d\ket{j,j}$. The equality $u_{i,i+1}^{\otimes 2,2} \ket{I}\otimes \ket{I}=\ket{I}\otimes \ket{I}$ is graphically described (b), where the green and blue boxes are $u_{i,i+1}$ and $\bar{u}_{i,i+1}$, respectively, and $i$ and $i+1$ are the site indices of the qudits.
• Figure S2: The frame potential (left-hand side) and the classical spin model (right-hand side).
• Figure S3: Non-conservation of the number of domain walls.

Theorems & Definitions (11)

• Definition 1: More random with respect to spectral gaps
• Definition 2: More random with respect to positivity
• Theorem 1: More randomness in local random circuits
• Theorem 2: More randomness in brick-wall random circuits
• Theorem 3: Spectral values
• Definition S1: More randomness
• Theorem S1: Restatement of Theorem 1 and 2 of the main text
• Theorem S2: Diagonalization for structured local random circuits
• Corollary S3: Spectral gap of structured local random circuits
• Theorem S4: Diagonalization for structured brick-wall circuits