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How to Construct Random Unitaries

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arxiv 2410.10116 v3 pith:J4Z2KR27 submitted 2024-10-14 quant-ph cs.CCcs.CLmath-phmath.MP

classification quant-phcs.CCcs.CLmath-phmath.MP
keywords prusunitariesunitaryefficienthaar-randommakesnotionquantum
verification ladder T0 review T1 audit T2 compute T3 formal
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abstract

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 $U$, and (2) a stronger notion of PRUs, which are secure even against adversaries that can query both the unitary $U$ and its inverse $U^\dagger$. 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.

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Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Information-Computation Gaps in Quantum Learning via Low-Degree Likelihood

    quant-ph 2025-05 conditional novelty 8.0 of 10

    A quantum extension of the low-degree method shows that state designs imply computational hardness for many single-copy quantum measurement strategies, yielding new information-computation gaps.

  2. Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms

    quant-ph 2025-09 accept novelty 7.0 of 10

    Clebsch-Gordan transforms give exact compressed oracles for Haar-random unitary group actions, with efficient circuits for U(d).

  3. MicroCrypt Assumptions with Quantum Input Sampling and Pseudodeterminism: Constructions and Separations

    quant-ph 2025-05 conditional novelty 7.0 of 10

    Quantum input sampling turns several MicroCrypt primitives into equivalent weak forms, and black-box separations show these forms are strictly weaker than uniform-sampling primitives.

  4. Pseudorandomness Properties of Random Reversible Circuits

    cs.CR 2025-02 accept novelty 7.0 of 10

    Random 3-bit gates in a fixed 2D nearest-neighbor brickwork produce approximate k-wise independent permutations of n bits in depth sqrt(n) e^{O(k^3)}.

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