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Leibniz international proceedings in informatics

Random Unitaries in Constant (Quantum) Time

Ben Foxman, Natalie Parham, Francisca Vasconcelos, Henry Yuen

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Random unitaries are a central object of study in quantum information, with applications to quantum computation, quantum many-body physics, and quantum cryptography. Recent work has constructed unitary designs and pseudorandom unitaries (PRUs) using Θ(log log n)-depth unitary circuits with two-qubit gates. In this work, we show that unitary designs and PRUs can be efficiently constructed in several well-studied models of constant-time quantum computation (i.e., the time complexity on the quantum computer is independent of the system size). These models are constant-depth circuits augmented with certain nonlocal operations, such as (a) many-qubit TOFFOLI gates, (b) many-qubit FANOUT gates, or (c) mid-circuit measurements with classical feedforward control. Recent advances in quantum computing hardware suggest experimental feasibility of these models in the near future. Our results demonstrate that unitary designs and PRUs can be constructed in much weaker circuit models than previously thought. Furthermore, our construction of PRUs in constant-depth with many-qubit TOFFOLI gates shows that, under cryptographic assumptions, there is no polynomial-time learning algorithm for the circuit class QAC⁰. Finally, our results suggest a new approach towards proving that PARITY is not computable in QAC⁰, a long-standing question in quantum complexity theory.

DOI
10.4230/lipics.itcs.2026.61

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Random unitaries are a central object of study in quantum information, with applications to quantum computation, quantum many-body physics, and quantum cryptography. Recent work has constructed unitary designs and pseudorandom unitaries (PRUs) using Θ(log log n)-depth unitary circuits with two-qubit gates. In this work, we show that unitary designs and PRUs can be efficiently constructed in several well-studied models of constant-time quantum computation (i.e., the time complexity on the quantum computer is independent of the system size). These models are constant-depth circuits augmented with certain nonlocal operations, such as (a) many-qubit TOFFOLI gates, (b) many-qubit FANOUT gates, or (c) mid-circuit measurements with classical feedforward control. Recent advances in quantum computing hardware suggest experimental feasibility of these models in the near future. Our results demonstrate that unitary designs and PRUs can be constructed in much weaker circuit models than previously thought. Furthermore, our construction of PRUs in constant-depth with many-qubit TOFFOLI gates shows that, under cryptographic assumptions, there is no polynomial-time learning algorithm for the circuit class QAC⁰. Finally, our results suggest a new approach towards proving that PARITY is not computable in QAC⁰, a long-standing question in quantum complexity theory.
The nature of randomness and complexity growth in systems governed by unitary dynamics is a fundamental question in quantum many-body physics. This problem has motivated the study of models such as local random circuits and their convergence to Haar-random unitaries in the long-time limit. However, these models do not correspond to any family of physical time-independent Hamiltonians. In this work, we address this gap by studying the indistinguishability of time-independent Hamiltonian dynamics from truly random unitaries. On one hand, we establish a no-go result showing that for any ensemble of constant-local Hamiltonians and any evolution times, the resulting time-evolution unitary can be efficiently distinguished from Haar-random and fails to form a 222-design or a pseudorandom unitary (PRU). On the other hand, we prove that this limitation can be overcome by increasing the locality slightly: there exist ensembles of random polylog-local Hamiltonians in one-dimension such that under constant evolution time, the resulting time-evolution unitary is indistinguishable from Haar-random, i.e. it forms both a unitary kkk-design and a PRU. Moreover, these Hamiltonians can be efficiently simulated under standard cryptographic assumptions.
Random unitaries sampled from the Haar measure serve as fundamental models for generic quantum many-body dynamics. Under standard cryptographic assumptions, recent works have constructed polynomial-size quantum circuits that are computationally indistinguishable from Haar-random unitaries, establishing the concept of pseudorandom unitaries (PRUs). While PRUs have found broad implications in many-body physics, they fail to capture the energy conservation that governs physical systems. In this work, we investigate the computational complexity of generating PRUs that conserve energy under a fixed and known Hamiltonian HHH. We provide an efficient construction of energy-conserving PRUs when HHH is local and commuting with random coefficients. Conversely, we prove that for certain translationally invariant one-dimensional HHH, there exists an efficient quantum algorithm that can distinguish truly random energy-conserving unitaries from any polynomial-size quantum circuit. This establishes that energy-conserving PRUs cannot exist for these Hamiltonians. Furthermore, we prove that determining whether energy-conserving PRUs exist for a given family of one-dimensional local Hamiltonians is an undecidable problem. Our results reveal an unexpected computational barrier that fundamentally separates the generation of generic random unitaries from those obeying the basic physical constraint of energy conservation.
Quantum computational pseudorandomness has emerged as a fundamental notion that spans connections to complexity theory, cryptography and fundamental physics. However, all known constructions of efficient quantum-secure pseudorandom objects rely on complexity theoretic assumptions. In this work, we establish the first unconditionally secure efficient pseudorandom constructions against shallow-depth quantum circuit classes. We prove that: - Any quantum state 2-design yields unconditional pseudorandomness against both QNC⁰ circuits with arbitrarily many ancillae and AC⁰∘QNC⁰ circuits with nearly linear ancillae. - Random phased subspace states, where the phases are picked using a 4-wise independent function, are unconditionally pseudoentangled against the above circuit classes. - Any unitary 2-design yields unconditionally secure parallel-query pseudorandom unitaries against geometrically local QNC⁰ adversaries, even with limited AC⁰ postprocessing. Our results stand in stark contrast to the standard guarantee of the 2-design property, which only ensures that they cannot be distinguished from Haar random ensembles using two copies or queries. Our work demonstrates that quantum computational pseudorandomness can be achieved unconditionally for natural classes of restricted adversaries, opening new directions in quantum complexity theory.
The existence of pseudorandom unitaries (PRUs) -- efficient quantum circuits that are computationally indistinguishable from Haar-random unitaries -- has been a central open question, with significant implications for cryptography, complexity theory, and fundamental physics. In this work, we close this question by proving that PRUs exist, assuming that any quantum-secure one-way function exists. We establish this result for both (1) the standard notion of PRUs, which are secure against any efficient adversary that makes queries to the unitary UUU, and (2) a stronger notion of PRUs, which are secure even against adversaries that can query both the unitary UUU and its inverse U†U^\daggerU†. In the process, we prove that any algorithm that makes queries to a Haar-random unitary can be efficiently simulated on a quantum computer, up to inverse-exponential trace distance.
We consider the task of constructing pseudorandom unitaries (PRUs) with scalable security, i.e. families in which the security parameter may vary independently of the dimension (or input bit-length). It is not known whether scalable PRUs can be constructed. In this work we show that if scalable PRUs can be constructed via the prevailing paradigm for analyzing PRUs, then there would be a positive solution to the Aaronson-Kuperberg unitary synthesis problem, a longstanding question in quantum complexity theory about whether implementing arbitrary unitaries can be efficiently reduced to computing a Boolean function. Specifically, we formalize the notion of ROM-PRUs, which are statistically secure PRUs in the random oracle model (ROM). All prior known constructions of cryptographically secure PRUs are based on a ROM-PRU construction. We prove novel connections between ROM-PRUs, approximate unitary designs, epsilon-nets over the unitary group, and the unitary synthesis problem. In particular, we prove that any unitary synthesis algorithm (and thus any ROM-PRU) must use a classical oracle with input length (2 - o(1)) log d bits, where d is the dimension of the unitary to be implemented. This bound rules out all existing candidates for scalable PRUs in the literature. Together, these connections indicate that ROM-PRUs provide a fruitful idealized model for studying pseudorandom unitaries.
Pseudorandomness is increasingly recognized as a key property of ensembles in quantum information theory, statistical mechanics, and quantum many-body physics. Yet it appears in two conceptually different forms: statistical pseudorandomness, embodied by unitary designs, and computational pseudorandomness captured by pseudorandom unitaries (PRUs). The relationship between these two forms of pseudorandomness remains surprisingly poorly understood. Existing PRU constructions reveal this interplay where a statistically randomizing ingredient, a unitary design, is combined with classical cryptographic primitives to produce computational pseudorandomness. We show that statistical pseudorandomness is not necessary for computationally pseudorandom unitaries. We do this by replacing the unitary 222-design layer in the existing constructions with ensembles that are not even state 111-designs, yet are sufficiently {\em distinct}, a property we identify to be necessary for any PRU. This yields new non-adaptively secure PRU ensembles whose computational pseudorandomness is obtained without an underlying statistically pseudorandom quantum ensemble, such as a 222-design. We characterize distinctness via an entangled analogue of anticoncentration and use it to show that distinctness already captures constraints on coherence and imaginarity of PRUs, while identifying broad classes of inputs for which the latter obstruction disappears, enabling real-valued PRUs even for certain (maximally) entangled states. As an application, we use lack of distinctness to constrain the conjectured pseudorandomness of the random phase-Hadamard ensemble to form a PRU.