Explicit Separations for One-Query Unitary Synthesis

2026-07-29

AI Summary

This work investigates whether every $n$-qubit unitary operator can be implemented by an efficient quantum circuit that queries a classical oracle only once. For permutation unitaries and alternating-basis phase unitaries, the authors establish lower bounds demonstrating their impossibility of exact synthesis with a single query; in contrast, they provide an upper bound showing that complex-phase unitaries can be approximated to constant precision by such single-query algorithms. Introducing two novel cryptographic game frameworks—“oracle state search” and “Choi state”—the paper more precisely characterizes the synthesis hardness of non-uniformly random unitaries and reveals a stronger separation between single-query synthesis and quantum program implementability. Leveraging quantum complexity theory, oracle models, and diamond norm analysis, the authors prove the unsynthesizability of explicit families of unitaries under single-query constraints and derive new hardness results for approximating quantum programs.

Abstract

The unitary synthesis problem (Aaronson-Kuperberg, CCC 2007) asks whether every $n$-qubit unitary $U$ is computable by efficient quantum circuits relative to some classical oracle $f = f_U$ depending on $U$. Recently, Lombardi-Ma-Wright (STOC 2024) proved that Haar-random unitaries cannot be efficiently synthesized by algorithms that make 1 query (or poly$(n)$ parallel queries) to an arbitrary classical oracle. In this work, we prove several results about the hardness (and easiness!) of variants of unitary synthesis. Our results include: (1) 1-query vs. 2-query unitary synthesis: we prove 1-query lower bounds for synthesizing random permutation unitaries $P\lvert x\rangle = \lvert π(x)\rangle$, as well as random alternating-basis phase unitaries $F_2 \cdot H^{\otimes n} \cdot F_1$. This gives 1-query lower bounds for "explicit" families of unitaries that have efficient (even 2-query) synthesis algorithms. (2) Upper bound for complex phase unitaries: we also consider complex phase unitaries $\lvert x\rangle\mapsto α_x \lvert x\rangle$, which have a clean 2-query synthesis algorithm with no obvious 1-query algorithm. In this case, we prove an upper bound: there are 1-query algorithms (relative to binary phase oracles) that constant-approximate these unitaries in diamond distance. In order to prove our lower bounds, we introduce and analyze two new cryptographic games: the oracle state search game and the oracle Choi state game. Compared to prior work, our framework is mathematically simple, more flexible in what it can prove, and more accurately captures the hardness of synthesizing unitaries that are not "fully random". Finally, we also use the search game to prove a new hardness-of-approximation result for quantum programs (synthesizing unitaries relative to quantum advice) for phase unitaries, giving a sharper separation between 1-query unitary synthesis and quantum programs.

Problem

• unitary synthesis
• one-query
• quantum circuits
• oracle
• hardness

Innovation

• unitary synthesis
• quantum oracle
• query complexity
• cryptographic games
• quantum advice