Efficient Unitary T-designs from Random Sums

TL;DR

This work presents an efficient construction of unitary $T$-designs for $n$-qudit systems by harnessing random-matrix theory. The core idea is to approximate GUE via an i.i.d. sum of random Hermitian matrices and then show that the product of two exponentiated Gaussians is already near Haar, enabling a $T$-design through Hamiltonian simulation. A novel connection between the polynomial method and large-$N$ expansions in random matrix theory yields exponentially improved high-moment bounds without heavy Weingarten calculus, crystallized through a finite-unitary-moment problem. The authors implement the design efficiently via QSVT, achieving a runtime of $\tilde{O}(T^2n^2)$ gates with $\tilde{O}(Tn^2)$ bits of randomness and a boosting procedure to reach small diamond-norm error, significantly improving over prior $O(T^{5+o(1)})$-scaling approaches. This approach broadens the toolkit for quantum benchmarking, tomography, and randomized algorithms by delivering scalable, high-order unitary designs with practical resource costs.

Abstract

Unitary $T$-designs play an important role in quantum information, with diverse applications in quantum algorithms, benchmarking, tomography, and communication. Until now, the most efficient construction of unitary $T$-designs for $n$-qudit systems has been via random local quantum circuits, which have been shown to converge to approximate $T$-designs in the diamond norm using $O(T^{5+o(1)} n^2)$ quantum gates. In this work, we provide a new construction of $T$-designs via random matrix theory using $\tilde{O}(T^2 n^2)$ quantum gates. Our construction leverages two key ideas. First, in the spirit of central limit theorems, we approximate the Gaussian Unitary Ensemble (GUE) by an i.i.d. sum of random Hermitian matrices. Second, we show that the product of just two exponentiated GUE matrices is already approximately Haar random. Thus, multiplying two exponentiated sums over rather simple random matrices yields a unitary $T$-design, via Hamiltonian simulation. A central feature of our proof is a new connection between the polynomial method in quantum query complexity and the large-dimension ($N$) expansion in random matrix theory. In particular, we show that the polynomial method provides exponentially improved bounds on the high moments of certain random matrix ensembles, without requiring intricate Weingarten calculations. In doing so, we define and solve a new type of moment problem on the unit circle, asking whether a finite number of equally weighted points, corresponding to eigenvalues of unitary matrices, can reproduce a given set of moments.

Figures (4)

• Figure 1: Proof dependency graph. An arrow $A \rightarrow B$ means that the proof of $B$ depends on $A$.
• Figure 2: (a) An illustration of a full wiring diagram for $\mathop{\mathbb{E}}_{\bm{U}\leftarrow \mu} \left[\operatorname{\overline{Tr}}[\bm{W}^p]\right]$. The top wires all contract with $\bm{D}_2$ while the bottom wires all contract with $\bm{D}_1$. (b) A sample wiring diagram for $p=4$.
• Figure 3: An illustration of the $p=4$ contraction for the term corresponding to $\tau=(1)(3)(24)$, which are shown via the red lines, and $\sigma=(12)(34)$, which are shown via the blue lines.
• Figure 4: This illustrates one of the terms in the $r=4$ Weingarten expansion for $F_r(\bm{D})$. This term corresponds to to $\tau=(1)(3)(24)$, which are shown via the red lines, and $\sigma=(12)(34)$, which are shown via the blue lines.

Theorems & Definitions (137)

• Theorem 1.1: Informal
• Corollary 2.1
• Definition 3.1: Queries Models
• Definition 3.2: Unitary $T$-design
• Definition 3.3: $\delta$-approximate parallel unitary $T$-design
• Definition 3.4: $\delta$-approximate $s(n)$-space adaptive unitary $T$-designs
• Definition 3.5: Gaussian Unitary Ensemble (GUE)
• Definition 3.6: GUE $T$-design
• Theorem 4.1: Efficient T-Designs
• Lemma 4.1: From small moments to Haar