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This research demonstrates that shallow, less-structured random quantum circuits can achieve approximate unitary k-designs with nearly logarithmic depth by leveraging relative entropy decay, removing the need for strict temporal or spatial architecture. The study confirms that specific circuit structures are not physically necessary to generate these ensembles and introduces new recombination methods of independent interest in information theory.
Shallow Approximate Unitary Designs from Relative Entropy Decay of Unstructured Random Circuits
arXiv:2510.08537v3 Announce Type: replace 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.
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.
27 pages, 3 figures. This major update corrects some earlier flaws, enhances results, and introduces new methods. It should be considered a replacement for the previous version
Quantum Physics (quant-ph); Information Theory (cs.IT)
arXiv:2510.08537 [quant-ph]
(or arXiv:2510.08537v3 [quant-ph] for this version)
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