We consider a global phase-invariant metric in the projective unitary group PUn, relevant for universal quantum computing. We obtain the volume and measure of small metric ball in PUn and derive the Gilbert-Varshamov and Hamming bounds in PUn. In addition, we provide upper and lower bounds for the kissing radius of the codebooks in PUn as a function of the minimum distance. Using the lower bound of the kissing radius, we find a tight Hamming bound. Also, we establish bounds on the distortion-rate function for quantizing a source uniformly distributed over PUn. As example codebooks in PUn, we consider the projective Pauli and Clifford groups, as well as the projective group of diagonal gates in the Clifford hierarchy, and find their minimum distances. For any code in PUn with given cardinality we provide a lower bound of covering radius. Also, we provide expected value of the covering radius of randomly distributed points on PUn, when cardinality of code is sufficiently large. We discuss codebooks at various stages of the projective Clifford + T and projective Clifford + S constructions in PU2, and obtain their minimum distance, distortion, and covering radius. Finally, we verify the analytical results by simulation.

翻译：本文研究了与通用量子计算相关的射影酉群PUn中的全局相位不变度量。我们获得了PUn中小度量球的体积和测度，并推导了PUn中的Gilbert-Varshamov界和Hamming界。此外，我们给出了PUn中码本亲吻半径关于最小距离函数的上下界。利用亲吻半径的下界，我们得到了一个紧致的Hamming界。同时，我们建立了均匀分布于PUn上的信源量化失真率函数的界。作为PUn中的示例码本，我们考虑了射影Pauli群和Clifford群，以及Clifford层级中对角门构成的射影群，并计算了它们的最小距离。对于PUn中任意给定基数的码，我们给出了覆盖半径的下界。当码基数足够大时，我们还提供了PUn上随机分布点覆盖半径的期望值。我们讨论了PU2中射影Clifford+T和射影Clifford+S构造各阶段的码本，并计算了它们的最小距离、失真和覆盖半径。最后，通过仿真验证了理论分析结果。