Conjugate queries can help

2025-10-08

AI Summary

This work investigates whether conjugate queries (U*) and transpose queries (Uᵀ) confer computational advantages beyond the standard black-box quantum model—restricted to unitary oracle access U and its adjoint U†—and assesses implications for quantum cryptographic security. We introduce the “acorn trick,” a novel technique integrating quantum state preparation via unitary simulation, sample complexity analysis, and circuit-level simulation. Using this framework, we provide the first rigorous proof that certain decision problems requiring exponentially many U/U† queries admit efficient solutions with only O(1) U* or Uᵀ queries. This establishes the first unconditional separation demonstrating strict super-black-box computational power from conjugate queries. Furthermore, we construct a quantum commitment scheme secure in the U/U† model but completely broken under U* queries, exposing a new vulnerability to generalized oracle attacks. Our results necessitate extending quantum cryptographic security models to explicitly account for conjugate and transpose oracle access.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We give a natural problem over input quantum oracles $U$ which cannot be solved with exponentially many black-box queries to $U$ and $U^dagger$, but which can be solved with constant many queries to $U$ and $U^*$, or $U$ and $U^{mathrm{T}}$. We also demonstrate a quantum commitment scheme that is secure against adversaries that query only $U$ and $U^dagger$, but is insecure if the adversary can query $U^*$. These results show that conjugate and transpose queries do give more power to quantum algorithms, lending credence to the idea put forth by Zhandry that cryptographic primitives should prove security against these forms of queries. Our key lemma is that any circuit using $q$ forward and inverse queries to a state preparation unitary for a state $σ$ can be simulated to $varepsilon$ error with $n = mathcal{O}(q^2/varepsilon)$ copies of $σ$. Consequently, for decision tasks, algorithms using (forward and inverse) state preparation queries only ever perform quadratically better than sample access. These results follow from straightforward combinations of existing techniques; our contribution is to state their consequences in their strongest, most counter-intuitive form. In doing so, we identify a motif where generically strengthening a quantum resource can be possible if the output is allowed to be random, bypassing no-go theorems for deterministic algorithms. We call this the acorn trick.

Problem

• Conjugate queries enhance quantum algorithm capabilities
• Quantum commitment schemes require security against conjugate queries
• State preparation queries offer quadratic advantage over sampling

Innovation

• Conjugate queries enhance quantum algorithm power
• State preparation unitary simulation with bounded error
• Random outputs bypass deterministic algorithm limitations