Multiqubit Clifford groups are unitary 3-designs

Unitary $t$-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers, little is known about unitary $t$-designs with $t\geq3$ in the literature. We show that the multiqubit Clifford group in any even prime-power dimension is not only a unitary 2-design, but also a 3-design. Moreover, it is a minimal 3-design except for dimension~4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3-design; in particular, the set of stabilizer states forms a 3-design. In addition, our study is helpful to studying higher moments of the Clifford group, which are useful in many research areas ranging from quantum information science to signal processing. Furthermore, we reveal a surprising connection between unitary 3-designs and the physics of discrete phase spaces and thereby offer a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group, which is of intrinsic interest to studying quantum computation.Comment: 7 pages, published in Phys. Rev.

Large Sets of t-Designs

We investigate the existence of large sets of t-designs. We introduce t-wise equivalence and (n, t)-partitionable sets. We propose a general approach to construct large sets of t-designs. Then, we consider large sets of a prescribed size n. We partition the set of all k-subsets of a v-set into several parts, each can be written as product of two trivial designs. Utilizing these partitions we develop some recursive methods to construct large sets of t-designs. Then, we direct our attention to the large sets of prime size. We prove two extension theorems for these large sets. These theorems are the only known recursive constructions for large sets which do not put any additional restriction on the parameters, and work for all t and k. One of them, has even a further advantage; it increase the strength of the large set by one, and it can be used recursively which makes it one of a kind. Then applying this theorem recursively, we construct large sets of t-designs for all t and some blocksizes k. Hartman conjectured that the necessary conditions for the existence of a large set of size two are also sufficient. We suggest a recursive approach to the Hartman conjecture, which reduces this conjecture to the case that the blocksize is a power of two, and the order is very small. Utilizing this approach, we prove the Hartman conjecture for t = 2. For t = 3, we prove that this conjecture is true for infinitely many k, and for the rest of them there are at most k/2 exceptions. In Chapter 4 we consider the case k = t + 1. We modify the recursive methods developed by Teirlinck, and then we construct some new infinite families of large sets of t-designs (for all t), some of them are the smallest known large sets. We also prove that if k = t + 1, then the Hartman conjecture is asymptotically correct.</p

Some 33-designs and shortened codes from binary cyclic codes with three zeros

Linear codes and $t$-designs are interactive with each other. It is well known that some $t$-designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended codes of linear codes holding an infinite family of $t$-designs with $t\geq 3$ are reported in the literature. In this paper, we study the extended codes of the augmented codes of a class of binary cyclic codes with three zeros and their dual codes, and show that those codes hold $3$-designs. Furthermore, we obtain some shortened codes from the studied cyclic codes and explicitly determine their parameters. Some of those shortened codes are optimal or almost optimal.Comment: 20 pages. arXiv admin note: text overlap with arXiv:2110.03881, arXiv:2007.0592