Unitary designs from statistical mechanics in random quantum circuits

TL;DR

The paper addresses how fast one-dimensional local random quantum circuits converge to unitary $k$-designs by mapping the frame potential $\mathcal{F}^{(k)}_{\rm RQC}$ to a classical spin-lattice partition function on a triangular lattice and employing Weingarten calculus. It yields exact results for the second moment, showing that random circuits form an $\epsilon$-approximate 2-design in depth $t_2 = C(2n\log q + \log n + \log(1/\epsilon))$, with explicit constants and asymptotics (e.g., $t_2 \approx 6.2 n$ for qubits and $t_2 \to 2n$ as $q\to\infty$). For general moments $k$, the framework indicates that the leading contributions come from single domain-wall configurations, giving $t_k = O(nk)$ in the large-$q$ limit and, under a plausible dominance conjecture, $t_k \sim nk\log q + k\log k + \log(1/\epsilon)$, suggesting near-optimal depth for random-circuit $k$-designs. The results provide a controllable, analytic route to understanding randomness generation in 1D quantum circuits and motivate extensions to higher dimensions and symmetry-resolved circuit ensembles.

Abstract

Random quantum circuits are proficient information scramblers and efficient generators of randomness, rapidly approximating moments of the unitary group. We study the convergence of local random quantum circuits to unitary $k$-designs. Employing a statistical mechanical mapping, we give an exact expression of the distance to forming an approximate design as a lattice partition function. In the statistical mechanics model, the approach to randomness has a simple interpretation in terms of domain walls extending through the circuit. We analytically compute the second moment, showing that random circuits acting on $n$ qudits form approximate 2-designs in $O(n)$ depth, as is known. Furthermore, we argue that random circuits form approximate unitary $k$-designs in $O(nk)$ depth and are thus essentially optimal in both $n$ and $k$. We can show this in the limit of large local dimension, but more generally rely on a conjecture about the dominance of certain domain wall configurations.

Figures (4)

• Figure 1: The random circuits we consider are built from staggered layers of 2-site unitaries on $n$ qudits of local dimension $q$, where each gate is drawn randomly from $U(q^2)$.
• Figure 2: The $k$-th frame potential can be written as the partition function of a spin system on a hexagonal lattice with local $S_k$ spins (left). By summing over the blue nodes we can define effective plaquette terms and write the frame potential as the partition function on a triangular lattice (right). In both figures time runs from left to right and periodic boundary conditions in time means the red nodes on the ends of the circuit are identified.
• Figure 3: Examples of nonzero contributions to the $k=2$ partition function: a single and double domain wall configuration, where the domain walls separate regions of local identity and swap permutations.
• Figure 4: An example of a domain wall annihilating at a vertex, creating a closed loop in the circuit. The domain walls represent transpositions, generators of $S_k$, which we denoted with different colors.

Theorems & Definitions (3)

• Definition : approximate $k$-design
• Definition : frame potential
• Conjecture