Abstract:Approximate unitary k-designs are ensembles of unitary matrices whose first k statistical moments approximate the Haar (uniform) measure. Shallow, random quantum circuits with temporally and spatially structured architectures are known to yield approximate k-designs in depth that is logarithmic in the number of qubits. It was open whether this structure is physically necessary. Shown here is shallow relative entropy convergence for standard notions of less structured, time-periodic circuits, implying additive-error approximate k-designs in nearly logarithmic depth (up to sub-logarithmic factors). These architectures include 1-dimensional brickwork and gates placed randomly according to a connected, bounded-degree graph. Relative entropy recombination methods derived herein are potentially of independent interest.
From: Nicholas LaRacuente [view email]
[v1]
Thu, 9 Oct 2025 17:53:27 UTC (95 KB)
[v2]
Mon, 13 Oct 2025 15:23:25 UTC (105 KB)
[v3]
Tue, 18 Aug 2026 22:51:50 UTC (102 KB)
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