The Clifford group fails gracefully to be a unitary 4-design

TL;DR

This paper characterizes how the Clifford group nearly forms a unitary 4-design, revealing its limitations and potential for constructing high-precision designs relevant to quantum information and signal recovery.

Contribution

It provides an explicit characterization of the Clifford group's failure to be a 4-design and demonstrates how Clifford orbits can be used to construct approximate higher-order designs.

Findings

Clifford group is a 3-design but not a 4-design.

The 4th tensor power of the Clifford group has one more invariant subspace than the unitary group.

Clifford orbits can be used to construct approximate 4-designs and possibly 5-designs.

Abstract

A unitary t-design is a set of unitaries that is "evenly distributed" in the sense that the average of any t-th order polynomial over the design equals the average over the entire unitary group. In various fields -- e.g. quantum information theory -- one frequently encounters constructions that rely on matrices drawn uniformly at random from the unitary group. Often, it suffices to sample these matrices from a unitary t-design, for sufficiently high t. This results in more explicit, derandomized constructions. The most prominent unitary t-design considered in quantum information is the multi-qubit Clifford group. It is known to be a unitary 3-design, but, unfortunately, not a 4-design. Here, we give a simple, explicit characterization of the way in which the Clifford group fails to constitute a 4-design. Our results show that for various applications in quantum information theory and in the theory of convex signal recovery, Clifford orbits perform almost as well as those of true 4-designs. Technically, it turns out that in a precise sense, the 4th tensor power of the Clifford group affords only one more invariant subspace than the 4th tensor power of the unitary group. That additional subspace is a stabilizer code -- a structure extensively studied in the field of quantum error correction codes. The action of the Clifford group on this stabilizer code can be decomposed explicitly into previously known irreps of the discrete symplectic group. We give various constructions of exact complex projective 4-designs or approximate 4-designs of arbitrarily high precision from Clifford orbits. Building on results from coding theory, we give strong evidence suggesting that these orbits actually constitute complex projective 5-designs.