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Qubit stabilizer states are complex projective 3-designs

Richard Kueng, David Gross

TL;DR

This work addresses whether stabilizer states form complex projective $t$-designs, providing explicit structure for Haar-moment replication via the frame potential. By developing a phase-space (discrete symplectic) framework and counting intersections of Lagrangian subspaces, the authors derive a recursion for the frame potential of stabilizer states in prime-power dimensions $D=d^n$. They prove that stabilizer states form a complex projective $2$-design for all prime-power dimensions, form a $3$-design exactly when $d=2$, and are not a $4$-design; they also connect these results to the Clifford group's action on symmetric powers. The work leverages counting arguments in discrete symplectic geometry and provides explicit formulas for overlaps between stabilizer states, with potential applications to derandomized measurements in quantum information, phase retrieval, and low-rank matrix recovery. Overall, it establishes stabilizer states as a highly structured, explicitly characterizable source of complex projective designs with practical implications for quantum tomography and signal processing.

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.

Qubit stabilizer states are complex projective 3-designs

TL;DR

This work addresses whether stabilizer states form complex projective -designs, providing explicit structure for Haar-moment replication via the frame potential. By developing a phase-space (discrete symplectic) framework and counting intersections of Lagrangian subspaces, the authors derive a recursion for the frame potential of stabilizer states in prime-power dimensions . They prove that stabilizer states form a complex projective -design for all prime-power dimensions, form a -design exactly when , and are not a -design; they also connect these results to the Clifford group's action on symmetric powers. The work leverages counting arguments in discrete symplectic geometry and provides explicit formulas for overlaps between stabilizer states, with potential applications to derandomized measurements in quantum information, phase retrieval, and low-rank matrix recovery. Overall, it establishes stabilizer states as a highly structured, explicitly characterizable source of complex projective designs with practical implications for quantum tomography and signal processing.

Abstract

A complex projective -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 . We show that the set of all -qubit stabilizer states forms a complex projective -design in dimension . Stabilizer states had previously only been known to constitute -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.

Paper Structure

This paper contains 12 sections, 11 theorems, 63 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Designs and frame potential
  4. Main results
  5. Applications and Outlook
  6. Relation to previous work and history
  7. Proof of the main statement
  8. Outline
  9. Phase Space Formalism
  10. Stabilizer States
  11. Grassmannian subspaces and discrete symplectic geometry
  12. Proof of the main Theorem

Key Result

Theorem 1

Fix a dimension $D$ and let $\mu$ be a probability measure on the unit sphere in $\mathbbm{C}^D$. The measure $\mu$ is a complex projective $t$-design if and only if its frame potential meets the Welch bound

Theorems & Definitions (20)

  • Definition 1
  • Theorem 1: benedetto_tight_2003scott_tight_2006Levenshtein_designs_1998koenig_cubature_1999
  • Theorem 2: Main Theorem
  • Corollary 1
  • Theorem 3: Stabilizer States
  • proof : Proof of \ref{['th:stabilizer_states']}
  • Lemma 1: Compatible bases
  • proof
  • Lemma 2: Overlap of stabilizer states
  • proof
  • ...and 10 more