← 研踪 PaperTrailarXiv:2611.04381 · v1
On the Hardness of Approximate Quantum Circuit Membership and Optimization
首次提交 2026-11-03 14:12
版本更新 2026-11-03 14:12
雷达观测 2026-11-04 23:40
摘要
Quantum circuit optimization is an indispensable component of the quantum software stack, used to reduce noise, maximize performance, alleviate costs of quantum error correction, or target specific hardware. A related problem is approximate membership: for a fixed set of unitaries $S$, decide whether a given circuit $C$ is within distance $\varepsilon$ of a unitary in $S$ under some metric $m$. Membership is a special case of optimization, where $S$ is a set of unitaries implementable at no cost (e.g., with noise-minimal circuits, or without magic states). Janzing, Wocjan, and Beth (Int. J. Quantum Inf., 2005) showed that, for the operator metric, the case $S=\{I\}$ (approximate identity) is QMA-complete when $C$ is promised to be either close to the identity or far from it. We generalize this setting in three directions: (i) promise-free inputs, (ii) besides $S=\{I\}$, sets $S$ containing only Clifford unitaries, or only generalized permutation matrices (i.e., no superposition), and (iii) the trace-based, spectral, and operator metrics. We show that, in all cases, approximate membership is PP-hard for wide ranges of $\varepsilon$, provided $C$'s gate-set can implement Clifford+T, as the gate-sets of standard hardware do. If $S=\{I\}$, or if $S$ contains only generalized permutation matrices, $I\in S$, and $m$ is a trace-based distance, PP-hardness holds for all non-trivial $\varepsilon$. If $S=\{I\}$ and $m$ is a trace-based distance, PP-hardness holds even for circuits over classical-reversible gate-sets. Our results indicate that approximate membership is harder over promise-free inputs than the QMA-complete promise version, and that this hardness persists across quantum resources (any gates, non-Clifford, superposition) and distance metrics. They directly imply PP-hardness of circuit optimization when minimizing total, non-Clifford, or superposition gate depth or count.
作者备注：26 pages, 4 figures
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v1 · 2026-11-03 14:12 · 最新观测版本
On the Hardness of Approximate Quantum Circuit Membership and Optimization
Quantum circuit optimization is an indispensable component of the quantum software stack, used to reduce noise, maximize performance, alleviate costs of quantum error correction, or target specific hardware. A related problem is approximate membership: for a fixed set of unitaries $S$, decide whether a given circuit $C$ is within distance $\varepsilon$ of a unitary in $S$ under some metric $m$. Membership is a special case of optimization, where $S$ is a set of unitaries implementable at no cost (e.g., with noise-minimal circuits, or without magic states). Janzing, Wocjan, and Beth (Int. J. Quantum Inf., 2005) showed that, for the operator metric, the case $S=\{I\}$ (approximate identity) is QMA-complete when $C$ is promised to be either close to the identity or far from it. We generalize this setting in three directions: (i) promise-free inputs, (ii) besides $S=\{I\}$, sets $S$ containing only Clifford unitaries, or only generalized permutation matrices (i.e., no superposition), and (iii) the trace-based, spectral, and operator metrics. We show that, in all cases, approximate membership is PP-hard for wide ranges of $\varepsilon$, provided $C$'s gate-set can implement Clifford+T, as the gate-sets of standard hardware do. If $S=\{I\}$, or if $S$ contains only generalized permutation matrices, $I\in S$, and $m$ is a trace-based distance, PP-hardness holds for all non-trivial $\varepsilon$. If $S=\{I\}$ and $m$ is a trace-based distance, PP-hardness holds even for circuits over classical-reversible gate-sets. Our results indicate that approximate membership is harder over promise-free inputs than the QMA-complete promise version, and that this hardness persists across quantum resources (any gates, non-Clifford, superposition) and distance metrics. They directly imply PP-hardness of circuit optimization when minimizing total, non-Clifford, or superposition gate depth or count.
查看此版本解读 →查看此版本原文 ↗