Random-Oracle Unitary Synthesis is Impossible
Abstract
A major open problem in quantum complexity is the question of unitary synthesis---namely, is it possible to efficiently implement any unitary, given the ability to evaluate any classical function in superposition? Inspired by a recent work by Brakerski and Yuen (CRYPTO 2026), we propose an ``average-case'' unitary synthesis problem, which asks whether it is possible to efficiently implement a Haar random unitary given access to a random Boolean function. We then demonstrate a superpolynomial query lower bound, establishing that unitary synthesis in this model is impossible. Along with our main result, we settle in the negative a question Brakerski and Yuen pose, of whether scalable pseudorandom unitaries (PRUs) can be implemented in the ROM-PRU model. In particular, we show that in the ROM-PRU model, no Ω(N1+γ) unitary design on n qubits, where N=2n, can be efficiently achieved for any γ> 0. On the other hand, we construct an O(N)-design in poly(n) queries in the same model, surpassing the previously best known results which constructed O(√N)-designs.
Related
- Impossible to conjugate or reflect an unknown quantum state via a unitary evolution
- Quantum Time-Lock Puzzles in the Quantum Random Oracle Model
- Black Hole Radiation Decoding in the Haar Random Oracle Model
- State Synthesis with an NP oracle
- Beyond NISQ Assumptions: One-time Memory in the Classically Accessible Random-Oracle Model
BibTeX
@misc{huang2026randomoracle,
title = {{Random-Oracle Unitary Synthesis is Impossible}},
author = {Andrew Huang and Akshar Ramkumar and John Wright},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06700},
eprint = {2610.06700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06700},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Random-Oracle Unitary Synthesis is Impossible
AU - Andrew Huang
AU - Akshar Ramkumar
AU - John Wright
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06700
UR - https://arxiv.org/abs/2610.06700
AB - A major open problem in quantum complexity is the question of unitary synthesis---namely, is it possible to efficiently implement any unitary, given the ability to evaluate any classical function in superposition? Inspired by a recent work by Brakerski and Yuen (CRYPTO 2026), we propose an ``average-case'' unitary synthesis problem, which asks whether it is possible to efficiently implement a Haar random unitary given access to a random Boolean function. We then demonstrate a superpolynomial query lower bound, establishing that unitary synthesis in this model is impossible. Along with our main result, we settle in the negative a question Brakerski and Yuen pose, of whether scalable pseudorandom unitaries (PRUs) can be implemented in the ROM-PRU model. In particular, we show that in the ROM-PRU model, no Ω(N^1+γ) unitary design on n qubits, where N=2^n, can be efficiently achieved for any γ> 0. On the other hand, we construct an O(N)-design in poly(n) queries in the same model, surpassing the previously best known results which constructed O(√N)-designs.
ER -