Random unitaries in extremely low depth

TL;DR

The paper shows that random unitaries can be formed in extremely shallow circuits by gluing together small, locally random unitaries into a global design or pseudorandom unitary. It provides explicit depth bounds showing near-Haar behavior in log-depth regimes for 1D and polylogarithmic regimes for other geometries, and proves optimality in n-dependence. The construction yields both unitary designs and PRUs with applications to classical shadows, learning, and topological-order hardness, while highlighting fundamental quantum-classical distinctions in information scrambling. Under cryptographic assumptions, the work also establishes low-depth PRUs with strong indistinguishability properties, implying powerful quantum advantages at shallow depths. Overall, the approach delivers a simple, versatile framework for fast generation of pseudo-random quantum dynamics across geometries, with broad theoretical and practical impact.

Abstract

We prove that random quantum circuits on any geometry, including a 1D line, can form approximate unitary designs over $n$ qubits in $\log n$ depth. In a similar manner, we construct pseudorandom unitaries (PRUs) in 1D circuits in $\text{poly}(\log n)$ depth, and in all-to-all-connected circuits in $\text{poly}(\log \log n)$ depth. In all three cases, the $n$ dependence is optimal and improves exponentially over known results. These shallow quantum circuits have low complexity and create only short-range entanglement, yet are indistinguishable from unitaries with exponential complexity. Our construction glues local random unitaries on $\log n$-sized or $\text{poly}(\log n)$-sized patches of qubits to form a global random unitary on all $n$ qubits. In the case of designs, the local unitaries are drawn from existing constructions of approximate unitary $k$-designs, and hence also inherit an optimal scaling in $k$. In the case of PRUs, the local unitaries are drawn from existing PRU constructions. Applications of our results include proving that classical shadows with 1D log-depth Clifford circuits are as powerful as those with deep circuits, demonstrating superpolynomial quantum advantage in learning low-complexity physical systems, and establishing quantum hardness for recognizing phases of matter with topological order.

Figures (6)

• Figure 1: (a) The central question we seek to answer is: How shallow can a random quantum circuit be while replicating the behavior of a Haar-random unitary? A Haar-random unitary over $n$ qubits requires a circuit depth that grows exponentially in $n$. Approximate unitary $k$-designs replicate the behavior of Haar-random unitaries within any quantum experiment that queries the unitary $k$ times. Pseudorandom unitaries replicate the behavior of Haar-random unitaries within any efficient quantum experiment. (b) Any quantum experiment can be represented as follows: an observer prepares an initial state $\ket{\Psi_{\mathrm{init}}}$, applies the unitary $U$ many times, interleaved with many quantum circuits for quantum information processing, and concludes by performing a measurement (not shown).
• Figure 2: (a) Our random unitary ensemble corresponds to a two-layer brickwork circuit, where each small unitary acts on $2\xi$ qubits in each layer. (b) To generate $\varepsilon$-approximate unitary $k$-designs in $\log n$ depth, we draw each small unitary from an approximate unitary $k$-design on $2\xi = 2\log(nk^2/\varepsilon)$ qubits. (c) To generate pseudorandom unitaries in $\text{poly} \log n$ depth, we draw each small unitary from a pseudorandom unitary ensemble, such as the $PFC$ ensemble metger2024simplePRU2024SPRU2024, on $2\xi = \omega(\log n)$ qubits.
• Figure 3: (a) Shallow random classical circuits cannot look uniformly random. In contrast, our results show that shallow random quantum circuits can already look Haar-random. (b) To create random unitaries on any circuit geometry, we implement a 1D random circuit along a Hamiltonian path of the geometry. While Hamiltonian paths do not exist in any geometry, when jumping to constant-distance neighbor is allowed, they always exist and are efficient to construct.
• Figure 4: The low-depth unitary designs and pseudorandom unitaries that we construct have broad applications, several of which are depicted here. (a)Log-depth classical shadows: Our shallow unitary 3-designs enable provably-efficient classical shadow tomography using $\log(n)$-depth random Clifford circuits instead of linear depth. (b)Quantum hardness of recognizing topological order: By applying our shallow pseudorandom unitaries to product and toric code states, we generate pseudorandom states with trivial and topological order, respectively. This demonstrates that recognizing topological order in an unknown state is quantumly hard. (c)Power of time-reversal in learning: We establish that quantum experiments capable of reversing time-evolution can exhibit super-polynomial advantages over conventional experiments. We prove this in a simple example where one wishes to detect whether long-range couplings are present in a strongly-interacting dynamical quantum system.
• Figure 5: (a) A key step in our proof is to show that one can "glue" two approximate unitary $k$-designs into a larger $\varepsilon$-approximate unitary $k$-design. This holds whenever the two unitaries are applied in sequence and overlap on at least $\sim \! \text{log}(k/\varepsilon)$ qubits. (b) To prove Theorems \ref{['thm:main-design']}, \ref{['thm:main-PRUs']}, we apply this gluing lemma $n/\xi$ times as shown (arrows). At each application, we glue one additional small unitary (green) into a larger unitary design on all qubits to its left (pink). (c) The gluing lemma also enables us to immediately extend our theorems to 2D circuits, by applying the lemma in the order shown (arrows), as well as other geometries.

Theorems & Definitions (54)

• Theorem 1: Gluing small random unitary designs
• Corollary 1: Low-depth random unitary designs
• Proposition 1
• Theorem 2: Gluing small pseudorandom unitaries
• Corollary 2: Low-depth pseudorandom unitaries
• Corollary 3: Hardness of recognizing topological order
• Lemma 1: Approximate Haar twirl
• Lemma 2: Unitary designs from EPR states
• Lemma 3: Gluing two random unitaries; informal
• Definition 1: Approximate unitary design