Approximate unitary $t$-designs by short random quantum circuits using nearest-neighbor and long-range gates

TL;DR

The paper demonstrates that short random quantum circuits acting on $D$-dimensional lattices achieve approximate unitary $t$-designs in several operational senses, improving upon the prior linear-depth bounds. It introduces two circuit models—nearest-neighbor lattice circuits and long-range all-to-all gates—and proves that lattice designs require depth $\text{poly}(t)\cdot n^{1/D}$, while all-to-all circuits yield strong anti-concentration at size $O(n\ln^2 n)$ with explicit Markov-chain analyses. By connecting moments to Haar projectors and employing overlap and quasi-orthogonality arguments, the work provides both dimension-specific ($D=2$) and general-$D$ bounds, and even an alternative route to anti-concentration in $O(\ln n\ln\ln n)$ depth under a different model. The results have implications for classical hardness of sampling from quantum circuits under PH and $"#P$-hardness assumptions, and they inform near-term quantum supremacy experiments, notably Google/USTC-type architectures, by showing sub-linear depth suffices for anti-concentration in certain geometries. Overall, the paper advances the understanding of how random circuits approximate Haar properties at modest depths, with concrete depth/size scaling and multiple design notions.

Abstract

We prove that $poly(t) \cdot n^{1/D}$-depth local random quantum circuits with two qudit nearest-neighbor gates on a $D$-dimensional lattice with n qudits are approximate $t$-designs in various measures. These include the "monomial" measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was $poly(t)\cdot n$ due to Brandao-Harrow-Horodecki (BHH) for $D=1$. We also improve the "scrambling" and "decoupling" bounds for spatially local random circuits due to Brown and Fawzi. One consequence of our result is that assuming the polynomial hierarchy (PH) is infinite and that certain counting problems are $\#P$-hard on average, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under the assumption that PH is infinite. However, to show the hardness of approximate sampling using this strategy requires that the quantum circuits have a property called "anti-concentration", meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Thus our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent proposal by the Google Quantum AI group to perform such a sampling task with 49 qubits on a two-dimensional lattice and confirms their conjecture that $O(\sqrt n)$ depth suffices for anti-concentration. We also prove that anti-concentration is possible in depth O(log(n) loglog(n)) using a different model.

Figures (3)

• Figure 1: The architecture proposed by the quantum AI group at Google to demonstrate quantum supremacy consists of a 2D lattice of superconducting qubits. This figure depicts two illustrative timesteps in this proposal. At each timestep, 2-qubit gates (blue) are applied across some pairs of neighboring qubits.
• Figure 2: The random circuit model in definition \ref{['def:modelsD']}. Each black circle is a qudit and each blue link is a random $\text{SU}(d^2)$ gate. The model does $O(\sqrt n\mathop{\mathrm{poly}}\nolimits(t))$ rounds alternating between applying (1) and (2). Then for $O(\sqrt n\mathop{\mathrm{poly}}\nolimits(t))$ rounds it alternates between (3) and then (4). This entire loop is then repeated $O(\mathop{\mathrm{poly}}\nolimits(t))$ times.
• Figure :

Theorems & Definitions (164)

• Definition 1: Monomial definition of $t$-designs
• Definition 2: Strong definition of $t$-designs
• Theorem 3
• Definition 4: Random circuits on one-dimensional lattices
• Definition 5: Random circuits on two-dimensional lattices
• Definition 6: Random circuits on $D$-dimensional lattices
• Definition 7: Random circuit models on complete graphs
• Theorem 8
• proof : Proof sketch for part \ref{['p1']}
• Theorem 9