Unitary designs from statistical mechanics in random quantum circuits

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.

25 pages, many figures

References in corpus (8)

• Black holes as mirrors: quantum information in random subsystems
• Evenly distributed unitaries: on the structure of unitary designs
• Optimizing quantum process tomography with unitary 2-designs
• Exact convergence times for generation of random bipartite entanglement
• Scrambling and Complexity in Phase Space
• Comment on the paper "Random Quantum Circuits are Approximate 2-designs"
• Operator growth in random quantum circuits with symmetry
• Pseudo-randomness and Learning in Quantum Computation