Multiqubit Clifford groups are unitary 3-designs

Abstract

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 ≥ 3 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.

Permutation Symmetry Determines the Discrete Wigner Function

“…Our study also reveals the group theoretical root why such a highly symmetric representation can only exist in odd prime power dimensions besides dimensions 2 and 8, thereby resolving the enigma on the discrete Wigner function that persists for the past three decades. The exception for dimension 8 is tied with a special symmetric informationally complete measurement (SIC) [13][14][15][16], known as Hoggar lines [17][18][19], which is of independent interest.In the course of our study, we show that an operator basis has doubly transitive permutation symmetry if and only if its symmetry group is a unitary 2-design [20][21][22][23][24]. Therefore, the basis of phase point operators is almost uniquely characterized by its symmetry group being a unitary 2-design.…”

“…In the course of our study, we show that an operator basis has doubly transitive permutation symmetry if and only if its symmetry group is a unitary 2-design [20][21][22][23][24]. Therefore, the basis of phase point operators is almost uniquely characterized by its symmetry group being a unitary 2-design.…”

“…Many of these applications are tied with the fact that the Clifford group is a unitary 2-design [20][21][22]. When p = 2, it is also a unitary 3-design according to a recent result of the author [23] and that of Webb [24]. Recall that a set of K unitary operators {U j } is a unitary t-design [20][21][22][23][24] if it satisfies…”

“…When p = 2, it is also a unitary 3-design according to a recent result of the author [23] and that of Webb [24]. Recall that a set of K unitary operators {U j } is a unitary t-design [20][21][22][23][24] if it satisfies…”

“…The basis {F j } is supersymmetric if and only if the group G (2) := {U ⊗ U |U ∈ G} has two orbits on {F j ⊗ F k }. According to Lemma 1, the latter is equivalent to the condition that G (2) has two inequivalent irreducible components, which holds if and only if G is a unitary 2-design [22,23].…”