Random-Oracle Unitary Synthesis is Impossible

2026-10-05

AI Summary

This study addresses the challenge of efficient unitary synthesis based on random oracles in quantum complexity by formulating the average-case unitary synthesis problem. By integrating Haar-random unitaries, Boolean function query techniques, and unitary design theory, this work establishes the first impossibility theorem within this model, overcoming prior limitations confined to semi-classical designs. Specifically, it proves a super-polynomial query lower bound, demonstrating that the ROM-PRU model cannot realize scalable pseudorandom unitaries. Furthermore, it reveals that $\Omega(N^{1+\gamma})$-designs are not efficiently implementable, while simultaneously constructing $O(N)$-designs achievable with polynomial queries. These results precisely delineate the complexity boundaries of unitary synthesis.

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 $\Omega(N^{1+\gamma})$ unitary design on $n$ qubits, where $N=2^n$, can be efficiently achieved for any $\gamma>0$. On the other hand, we construct an $O(N)$-design in $\text{poly}(n)$ queries in the same model, surpassing the previously best known results which constructed $O(\sqrt{N})$-designs.

Problem

• Unitary Synthesis
• Quantum Complexity
• Random Oracle
• Pseudorandom Unitaries
• Query Lower Bound

Innovation

• Unitary Synthesis
• Random Oracle Model
• Pseudorandom Unitaries
• Query Lower Bound
• Unitary Design