Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Third moments of qudit Clifford orbits and 3-designs based on magic orbits

Huangjun Zhu, Chengsi Mao, Changhao Yi

Abstract

When the local dimension $d$ is an odd prime, the qudit Clifford group is only a 2-design, but not a 3-design, unlike the qubit counterpart. This distinction and its extension to Clifford orbits have profound implications for many applications in quantum information processing. In this work we systematically delve into general qudit Clifford orbits with a focus on the third moments and potential applications in shadow estimation. First, we introduce the shadow norm to quantify the deviations of Clifford orbits from 3-designs and clarify its properties. Then, we show that the third normalized frame potential and shadow norm are both $\mathcal{O}(d)$ for any Clifford orbit, including the orbit of stabilizer states, although the operator norm of the third normalized moment operator may increase exponentially with the number $n$ of qudits when $d\neq 2\mod 3$. Moreover, we prove that the shadow norm of any magic orbit is upper bounded by the constant $15/2$, so a single magic gate can already eliminate the $\mathcal{O}(d)$ overhead in qudit shadow estimation and bridge the gap between qudit systems and qubit systems. Furthermore, we propose simple recipes for constructing approximate and exact 3-designs (with respect to three figures of merit simultaneously) from one or a few Clifford orbits. Notably, accurate approximate 3-designs can be constructed from only two Clifford orbits. For an infinite family of local dimensions, exact 3-designs can be constructed from two or four Clifford orbits. In the course of study, we clarify the key properties of the commutant of the third Clifford tensor power and the underlying mathematical structures.

Third moments of qudit Clifford orbits and 3-designs based on magic orbits

Abstract

When the local dimension is an odd prime, the qudit Clifford group is only a 2-design, but not a 3-design, unlike the qubit counterpart. This distinction and its extension to Clifford orbits have profound implications for many applications in quantum information processing. In this work we systematically delve into general qudit Clifford orbits with a focus on the third moments and potential applications in shadow estimation. First, we introduce the shadow norm to quantify the deviations of Clifford orbits from 3-designs and clarify its properties. Then, we show that the third normalized frame potential and shadow norm are both for any Clifford orbit, including the orbit of stabilizer states, although the operator norm of the third normalized moment operator may increase exponentially with the number of qudits when . Moreover, we prove that the shadow norm of any magic orbit is upper bounded by the constant , so a single magic gate can already eliminate the overhead in qudit shadow estimation and bridge the gap between qudit systems and qubit systems. Furthermore, we propose simple recipes for constructing approximate and exact 3-designs (with respect to three figures of merit simultaneously) from one or a few Clifford orbits. Notably, accurate approximate 3-designs can be constructed from only two Clifford orbits. For an infinite family of local dimensions, exact 3-designs can be constructed from two or four Clifford orbits. In the course of study, we clarify the key properties of the commutant of the third Clifford tensor power and the underlying mathematical structures.

Paper Structure

This paper contains 103 sections, 120 theorems, 574 equations, 12 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Outline of the main results and technical innovation
  3. Complex projective and unitary $t$-designs
  4. Complex projective $t$-designs
  5. The shadow norm and third frame potential
  6. Unitary $t$-designs
  7. Qudit stabilizer formalism
  8. Heisenberg-Weyl group and Clifford Group
  9. Clifford orbits, stabilizer codes, and stabilizer states
  10. Commutant of the $t$th Clifford tensor power
  11. Stochastic Lagrangian subspaces and a spanning set of the commutant
  12. Stochastic orthogonal group
  13. Defect subspaces
  14. Connection with Calderbank-Shor-Steane codes
  15. Moments of stabilizer states
  16. ...and 88 more sections

Key Result

Proposition 1

The following four statements about the state ensemble $\mathscr{E}$ are equivalent:

Figures (12)

  • Figure 1: The deviation of the third normalized frame potential $\bar{\Phi}_3(n,d,3)$ of the ensemble $\mathrm{Stab}(n,d)$ of stabilizer states as a function of the local dimension $d$ and qudit number $n$. Note that $d$ is a prime in this figure and all other figures in this paper. The dashed lines/curves are guides for the eye, which is similar for other figures. In the left plot, the results on $n=5,10,50$ almost coincide.
  • Figure 2: The operator norm of the third normalized moment operator $\bar{Q}(n,d,3)$ of $\mathrm{Stab}(n,d)$ as a function of $d$ and $n$. Here $d$ is a prime with $d \neq 2 \!\mod 3$ in the first row, while it is a prime with $d=2 \!\mod 3$ in the second row because the properties of $\|\bar{Q}(n,d,3)\|$ in the two cases are dramatically different.
  • Figure 3: The squared shadow norm of $\mathfrak{O}_0=\mathfrak{O}-\tr(\mathfrak{O})\mathbb{I}/D$ with respect to $\mathrm{Stab}(n,d)$. In the left plot, $\mathfrak{O}$ is the projector onto a stabilizer state; in the right plot, $n=50$ and $\mathfrak{O}$ is a stabilizer projector of rank $K=d^\iota$.
  • Figure 4: A circuit for sampling from a magic (Clifford) orbit. Here $F$ is the Fourier gate, $T_{f_1},\ldots, T_{f_k}$ are diagonal magic gates defined in Eq. (\ref{['eq:MagicGate']}), and $C$ is a random Clifford gate. The gates in the dashed box generate a fiducial state of the magic orbit; such a fiducial state belongs to the set $\mathscr{M}_{n,k}(d)$.
  • Figure 5: The deviations of $\bar{\Phi}_3(\mathrm{orb}(\Psi))$, $\|\bar{Q}(\mathrm{orb}(\Psi))\|$, and $\|\mathfrak{O}_0\|_\mathrm{sh}^2=\|\mathfrak{O}_0\|_{\mathrm{orb}(\Psi)}^2$ associated with the canonical magic state $|\Psi\rangle$ in $\mathscr{M}_{n,k}^\mathrm{can}(d)$. Here $n=10$ and $\mathfrak{O}$ is the projector onto a stabilizer state. A small deviation means the Clifford orbit is close to a 3-design with respect to the corresponding figure of merit.
  • ...and 7 more figures

Theorems & Definitions (216)

  • Proposition 1
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • ...and 206 more