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1. Learning Shallow Quantum Circuits

   https://doi.org/10.1145/3618260.3649722  

   Huang, Hsin-Yuan; Liu, Yunchao; Broughton, Michael; Kim, Isaac; Anshu, Anurag; Landau, Zeph; McClean, Jarrod R (June 2024, ACM)

   Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown 𝑛-qubit shallow quantum circuit 𝑈 (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of 𝑈. We also provide a polynomial-time classical algorithm for learning the description of any unknown 𝑛-qubit state |𝜓⟩ = 𝑈|0^𝑛⟩ prepared by a shallow quantum circuit 𝑈 (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of |𝜓⟩. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate. 

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2. Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

   https://doi.org/10.1145/3313276.3316404  

   Watts, Adam Bene; Kothari, Robin; Schaeffer, Luke; Tal, Avishay (June 2019, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing)

   Recently, Bravyi, Gosset, and Konig (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constant-depth quantum circuit using bounded fan-in gates (or QNC^0 circuits), but cannot be solved by any constant-depth classical circuit using bounded fan-in AND, OR, and NOT gates (or NC^0 circuits). In other words, they exhibited a search problem in QNC^0 that is not in NC^0. We strengthen their result by proving that the 2D HLF problem is not contained in AC^0, the class of classical, polynomial-size, constant-depth circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates. We also supplement this worst-case lower bound with an average-case result: There exists a simple distribution under which any AC^0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem. Our results are shown by constructing a new problem in QNC^0, which we call the Parity Halving Problem, which is easier to work with. We prove our AC^0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem. 

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3. Signatures of interacting Floquet phases in shallow quantum circuits

   https://doi.org/10.1103/PhysRevB.106.214310  

   Su, Kaixiang; Lawler, Michael J. (December 2022, Physical Review B)
4. Efficient Classical Simulation of Random Shallow 2D Quantum Circuits

   https://doi.org/10.1103/PhysRevX.12.021021  

   Napp, John C.; La Placa, Rolando L.; Dalzell, Alexander M.; Brandão, Fernando G. S. L.; Harrow, Aram W. (April 2022, Physical Review X)
5. Approximate Unitary k-Designs from Shallow, Low-Communication Circuits (Extended Abstract)

   https://doi.org/10.4230/LIPIcs.ITCS.2025.69  

   LaRacuente, Nicholas; Leditzky, Felix (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   Random unitaries are useful in quantum information and related fields but hard to generate with limited resources. An approximate unitary k-design is a measure over an ensemble of unitaries such that the average is close to a Haar (uniformly) random ensemble up to the first k moments. A strong notion of approximation bounds the distance from Haar randomness in relative error: the weighted twirl induced by an approximate design can be written as a convex combination involving that of an exact design and vice versa. The main focus of our work is on efficient constructions of approximate designs, in particular whether relative-error designs in sublinear depth are possible. We give a positive answer to this question as part of our main results: 1. Twirl-Swap-Twirl: Let A and B be systems of the same size. Consider a protocol that locally applies k-design unitaries to A^k and B^k respectively, then exchanges l qudits between each copy of A and B respectively, then again applies local k-design unitaries. This protocol yields an ε-approximate relative k-design when l = O(k log k + log(1/ε)). In particular, this bound is independent of the size of A and B as long as it is sufficiently large compared to k and 1/ε. 2. Twirl-Crosstwirl: Let A_1, … , A_P be subsystems of a multipartite system A. Consider the following protocol for k copies of A: (1) locally apply a k-design unitary to each A_p for p = 1, … , P; (2) apply a "crosstwirl" k-design unitary across a joint system combining l qudits from each A_p. Assuming each A_p’s dimension is sufficiently large compared to other parameters, one can choose l to be of the form 2 (Pk + 1) log_q k + log_q P + log_q(1/ε) + O(1) to achieve an ε-approximate relative k-design. As an intermediate step, we show that this protocol achieves a k-tensor-product-expander, in which the approximation error is in 2 → 2 norm, using communication logarithmic in k. 3. Recursive Crosstwirl: Consider an m-qudit system with connectivity given by a lattice in spatial dimension D. For every D = 1, 2, …, we give a construction of an ε-approximate relative k-design using unitaries of spatially local circuit depth O ((log m + log(1/ε) + k log k ) k polylog(k)). Moreover, across the boundaries of spatially contiguous sub-regions, unitaries used in the design ensemble require only area law communication up to corrections logarithmic in m. Hence they generate only that much entanglement on any product state input. These constructions use the alternating projection method to analyze overlapping Haar twirls, giving a bound on the convergence speed to the full twirl 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 system size. The Recursive Crosstwirl construction answers one variant of [Harrow and Mehraban, 2023, Open Problem 1], showing that with a specific, layered architecture, random circuits produce relative error k-designs in sublinear depth. Moreover, it addresses [Harrow and Mehraban, 2023, Open Problem 7], showing that structured circuits in spatial dimension D of depth << m^{1/D} may achieve approximate k-designs. 

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