无界规模常数深度量子电路的能力

中文摘要 AI 辅助

具有无界扇入的经典电路在其规模不受限制时，可以在常数深度内计算任何布尔函数。我们提出疑问：移除电路规模和辅助量子比特的限制，是否也能使由任意单量子比特门和广义Toffoli门构建的量子电路在常数深度内实现任意酉变换。我们给出了计算基态任意置换、对角酉变换以及任意纯态制备的精确常数深度构造。这些构造将量子态制备与可逆经典计算及概率分布制备联系起来。当门集中包含扇出时，这些构造使用指数多的门和辅助量子比特，并将所有辅助量子比特归零。用原始门集上的精确电路替换扇出可保持常数深度，尽管规模界限可能变为双重指数。在常数深度内实现任意酉变换仍然开放。我们给出了等价表述，涉及复制指定正交基的向量、提取其标签以及实现受限酉变换族。我们还利用一个额外的干净量子比特，将任意酉变换的实现简化为无迹酉对合的实现。借助自适应测量，门隐形传态给出的深度与Clifford层级中门的层级成正比。为迈向常数深度内的任意酉变换实现，我们使用基于端口的隐形传态：对于输入维度$d$和$M\geq d^2-1$个端口，我们构造一个深度为$O(\sqrt d)$的酉电路，该深度与$M$无关，且包含资源制备和端口选择，纠缠保真度至少为$(1-(d^2-1)/(2M))^2$。因此，在固定$d$下，近似可以在不增加深度的情况下达到任意精度。对$d$的依赖是否也能被移除仍然开放。

英文摘要

Classical circuits with unbounded fan-in can compute any Boolean function in constant depth when their size is unrestricted. We ask whether removing the restrictions on circuit size and ancillary qubits also allows quantum circuits built from arbitrary single-qubit gates and generalised Toffoli gates to implement every unitary in constant depth. We give exact constant-depth constructions for arbitrary permutations of computational basis states, diagonal unitaries and the preparation of arbitrary pure states. These connect quantum state preparation to reversible classical computation and the preparation of probability distributions. With fanout in the gate set, they use exponentially many gates and ancillary qubits and return all ancillary qubits to zero. Replacing fanout by an exact circuit over the original gate set preserves constant depth, although the size bounds can become doubly exponential. The implementation of arbitrary unitaries in constant depth remains open. We give equivalent formulations in terms of copying the vectors of a specified orthonormal basis, extracting their labels and implementing restricted families of unitaries. We also reduce arbitrary unitary implementation to that of traceless unitary involutions using one additional clean qubit. With adaptive measurements, gate teleportation gives depth proportional to the level of a gate in the Clifford hierarchy. Towards arbitrary unitary implementation in constant depth, we use port-based teleportation: for input dimension $d$ and $M\geq d^2-1$ ports, we construct a unitary circuit of depth $O(\sqrt d)$, independent of $M$ and including resource preparation and port selection, with entanglement fidelity at least $(1-(d^2-1)/(2M))^2$. Thus, at fixed $d$, the approximation can be made arbitrarily accurate without increasing depth. Whether the dependence on $d$ can also be removed remains open.

发表机构

• Department of Computer Science, University of Oxford（牛津大学计算机系）

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