Efficient pseudorandom unitaries with general (adaptive) query security

Establish the existence of efficient pseudorandom unitary ensembles that are computationally indistinguishable from Haar measure against any polynomial-time quantum adversary making general (adaptive) oracle queries, under standard cryptographic assumptions such as the existence of quantum-secure one-way functions.

Background

Pseudorandom unitaries (PRUs) are the unitary analog of pseudorandom functions, but constructing them under standard assumptions has been challenging. While several pseudorandom state constructions exist and intermediate objects (e.g., pseudorandom state scramblers and pseudorandom isometries) have been realized, full PRUs with general query security have remained elusive.

This paper achieves a construction of PRUs with nonadaptive (parallel) security from quantum-secure one-way functions, narrowing the gap. However, full adaptive-query security—where the adversary can choose queries based on prior outcomes—remains unresolved and is explicitly identified as open.

References

However, the existence of efficient pseudorandom unitaries with general query security remains open.

— Efficient unitary designs and pseudorandom unitaries from permutations  (2404.16751 - Chen et al., 2024) in Introduction (Section 1), Computational pseudorandomness discussion

Finally, the optimal depth of strong pseudorandom unitaries remains open. The present theorem gives a logarithmic-depth information-theoretic strong-design construction, but it does not provide a keyed family or computational security against efficient distinguishers. A logarithmic-depth, system-only construction of strong pseudorandom unitaries would require a suitable keyed local primitive together with a security-composition theorem that remains valid under inverse, transpose, and complex-conjugate queries.

— Strong unitary designs in optimal depth and space  (2608.13491 - Parella-Dilmé et al., 13 Aug 2026) in Section Discussion

A natural next step is to extend the theory of distinctness to stronger query models, including inverse, transpose, conjugate, controlled, and adaptive access. Is there a hierarchy of distinctness conditions corresponding to these increasingly powerful notions of security? Can such conditions again be separated from unitary-design properties and used both to construct and to rule out PRUs and $t$-designs secure against the corresponding query models?

— Distinctness threshold for pseudorandom unitaries  (2609.03065 - Raza et al., 2 Sep 2026) in Discussion and open questions, paragraph “Distinctness in stronger query models and beyond”

Another similar question is whether the real- and complex-valued PRUs and $t$-designs proposed here remain secure against stronger forms of query accesses.

— Distinctness threshold for pseudorandom unitaries  (2609.03065 - Raza et al., 2 Sep 2026) in Discussion and open questions, paragraph “Real-valued pseudorandomness and applications”

These questions remain largely open. We do not know how deep these circuits need to be to generate pseudorandom quantum states, how ancillary workspace affects the depth, or whether local Haar randomness suffices or structured cryptographic ensembles are required.

— Learning Random Quantum Circuits and the Emergence of Pseudorandomness  (2609.39821 - Arunachalam et al., 30 Sep 2026) in Section 1, Introduction