Approximate Unitary $k$-Designs from Shallow, Low-Communication Circuits

TL;DR

The paper demonstrates that relative-error approximate unitary $k$-designs can be constructed with sublinear circuit depth and limited quantum communication by exploiting overlapping Haar twirls and alternating projections. It introduces two main protocols, Twirl-Swap-Twirl and Twirl-Crosstwirl, which, via 2-norm contraction and a conversion leveraging von Neumann subalgebras, achieve relative-error designs with depth scaling polylogarithmically in system size and $k$. The authors also extend these constructions to lattice architectures, achieving log-depth designs with entanglement obeying area laws up to logarithmic corrections, and provide a rigorous framework that unifies TPE bounds, representation theory, and norm-conversion techniques. These results address open questions on sublinear-depth design generation and have implications for near-term quantum information tasks where entanglement and communication are resource-limited. The work broadens the toolkit for designing pseudo-random unitaries with provable moment-matching properties under realistic circuit constraints.

Abstract

Random unitaries are useful in quantum information and related fields, but hard to generate with limited resources. An approximate unitary $k$-design is an ensemble of unitaries with an underlying measure over which the average is close to a Haar random ensemble up to the first $k$ moments. A particularly strong notion of approximation bounds the distance from Haar randomness in relative error. Such relative-error approximate designs are secure against queries by an adaptive adversary trying to distinguish it from a Haar ensemble. We construct relative-error approximate unitary $k$-design ensembles for which communication between subsystems is $O(1)$ in the system size. These constructions use the alternating projection method to analyze overlapping Haar averages, giving a bound on the convergence speed to the full averaging with respect to the $2$-norm. Using von Neumann subalgebra indices to replace system dimension, the 2-norm distance converts to relative error without introducing any additional dimension dependence. We use these constructions as the building blocks of a two-step protocol that achieves a relative-error design in $O \big ( (\log m + \log(1/ε) + k \log k ) k\, \text{polylog}(k) \big )$ depth, where $m$ is the number of qudits in the complete system and $ε$ the approximation error. This sublinear depth construction answers a variant of [Harrow and Mehraban 2023, Section 1.5, Open Questions 1 and 7]. Moreover, entanglement generated by the sublinear depth scheme follows area laws on spatial lattices up to corrections logarithmic in the full system size.

Figures (2)

• Figure 1: Concrete example of the Twirl-Crosstwirl protocol for $P=2$ parties each consisting of $K=3$ copies of 9-qudit blocks $A_{p,k}$. Each circle represents a qudit, and unitaries acting on these qudits are indicated by boxes around them. The first $\ell_1=4$ qudits in the first-row blocks $A_{1,*}$ and the first $\ell_2=2$ qudits in the second-row blocks $A_{2,*}$ participate in the Crosstwirl $\mathcal{Q}_{\mathrm{mct}}$ in panel ( B).
• Figure 2: Illustrations of the 2-layer procedure in Protocol \ref{['alg:2-layer']} designs on lattices in one spatial dimension (panels ( A) and ( B)) and in two spatial dimensions (panel ( C)). In each case the blocks participating in the first step of Protocol \ref{['alg:2-layer']} are shown as white squares with black outlines, whereas the blocks participating in the second step are shown as colored squares. Panel ( B) depicts the first four iterations of crosstwirls in the second step of Protocol \ref{['alg:2-layer']} on a $1$-dimensional lattice. In each step, a crosstwirl joins a new first-layer block (shown in white) to the previously merged block (shown at the left edge of the line in color).

Theorems & Definitions (35)

• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 4.1
• proof
• Proposition 4.2