发表机构
QuSoft, University of Amsterdam（阿姆斯特丹大学 QuSoft）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明，即使对于两个量子比特，Haar测度下几乎所有的酉算子都不存在有限维资源的精确非局域量子计算协议，并指出超越相位等自然目标无法实现，结论也适用于可局域化量子测量。
AI 中文摘要
非局域量子计算（NLQC）要求两方使用纠缠资源态和一轮同时量子通信，对其量子输入应用联合操作。NLQC在量子信息领域有广泛应用，包括对量子位置验证的攻击、通信复杂性和量子引力。每个二分酉算子都有一个具有有限纠缠的近似NLQC协议，并且已知若干结构化族类的精确协议。然而，是否每个固定酉算子都允许具有有限维资源的精确协议，这一问题仍然悬而未决。我们证明，即使对于两个量子比特，Haar测度下几乎所有的酉算子都不允许这样的协议，即使资源态和局域操作是针对目标定制的。证明首先表明，精确协议的平滑形变只能通过局域酉算子改变其目标。对于固定架构，半代数几何将精确协议限制为有限多个连通族，因此每个架构只能达到有限多个局域酉轨道。对所有架构取并集，精确可实现的酉算子形成可数个局域酉轨道。我们还表明，一些自然目标不允许协议：受控相位算子$\mathrm{diag}(1,1,1,e^{i\theta})$在$e^{i\theta}$为超越数时（例如$\theta = 1$）没有具有有限维资源的精确协议。我们的结果也适用于可局域化的量子测量，其结果是无需通信即可从输入和共享纠缠资源态的局域测量中获得。Haar随机基下的秩一投影测量几乎必然不能通过有限维资源态和有限多个局域结果进行局域化。
英文摘要
Non-local quantum computation (NLQC) asks two parties to apply a joint operation to their quantum inputs using an entangled resource state and one round of simultaneous quantum communication. NLQC has applications across quantum information, including attacks on quantum position verification, communication complexity, and quantum gravity. Every bipartite unitary has an approximate NLQC protocol with finite entanglement, and exact protocols are known for several structured families. However, it has remained open whether every fixed unitary admits an exact protocol with finite-dimensional resources. We show that, even for two qubits, Haar-almost every unitary does not admit such a protocol, even when the resource state and local operations are tailored to the target. The proof first shows that smooth deformations of an exact protocol can only change its target by local unitaries. For a fixed architecture, semialgebraic geometry limits the exact protocols to finitely many connected families, so each architecture reaches only finitely many local-unitary orbits. Taking the union over all architectures, the exactly implementable unitaries form countably many local-unitary orbits. We also show that some natural targets do not admit a protocol: the controlled phase $\mathrm{diag}(1,1,1,e^{iθ})$ has no exact protocol with finite-dimensional resources whenever $e^{iθ}$ is transcendental, for instance for $θ= 1$. Our results also apply to localizable quantum measurements, whose outcome is obtained from local measurements on inputs and a shared entangled resource state without communication. A rank-one projective measurement in a Haar-random basis is almost surely not localizable with a finite-dimensional resource state and finitely many local outcomes.
Comments29 pages, 2 figures
