Qubit stabilizer states are complex projective 3-designs

Abstract

A complex projective t-design is a configuration of vectors which is "evenly distributed" on a sphere in the sense that sampling uniformly from it reproduces the moments of Haar measure up to order 2t. We show that the set of all n-qubit stabilizer states forms a complex projective 3-design in dimension 2 n . Stabilizer states had previously only been known to constitute 2-designs. The main technical ingredient is a general recursion formula for the so-called frame potential of stabilizer states. To establish it, we need to compute the number of stabilizer states with pre-described inner product with respect to a reference state. This, in turn, reduces to a counting problem in discrete symplectic vector spaces for which we find a simple formula. We sketch applications in quantum information and signal analysis.

Quantum Measurements in the Light of Quantum State Estimation

“…Here Haar random pure states involved in the estimation problem can be replaced by any ensemble of pure states that form a 3-design. When the dimension d is a power of 2 for example, the set of stabilizer states is qualified [48][49][50]. This observation is quite helpful to devising experiments to demonstrate the above results.…”

“…III B). When the dimension d is a power of 2, any orbit of the Clifford group is a 3-design; in particular, the set of stabilizer states forms a 3-design [48][49][50]. In addition, special orbits of the Clifford group can form 4-designs [51,52].…”

Symmetry-resolved entanglement detection using partial transpose moments

“…More generic single-qubit ensembles (like Haar-random unitaries) are also an option -what matters is that the single qubit ensemble forms a 3-design [74,75]. The (single-and multi-qubit) Clifford group is one ensemble with this feature [76][77][78]. As demonstrated in Ref.…”

Quantum scrambling with classical shadows

“…That is, one must apply a more random (chaotic) unitary ensemble to access higher-point scrambling features. Note that the Clifford group forms a 3-design [60][61][62], but not a 4-design [61]. One can generate an approximate t-design through a random local circuit [63], in particular, by inserting few T gates into Clifford circuits [64].…”

“…Even with the optimistic estimation by taking the trace distance comparable to the infidelity, K = Ω(d 2 / ) = Ω(d 3 / ), which is still worse than Eq. (62). Furthermore, if one is restricted to independent measurements on a single copy of ρ V (like in the classical shadow protocols), the scaling worsens: K = Ω(d 3 / 2 ) = Ω(d 5 / 2 ).…”