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Stabilizer Entanglement Enhances Magic Injection

Zong-Yue Hou, ChunJun Cao, Zhi-Cheng Yang

Abstract

Non-stabilizerness is a key resource for fault-tolerant quantum computation, yet its interplay with entanglement in dynamical settings remains underexplored. We address this by analyzing a well-controlled, analytically tractable setup, where we show that entanglement acts as a conduit that teleports magic across the system, thereby enhancing magic injection. Using exact calculations, we prove that when a Haar-random unitary $U_A$ is applied to a subsystem $A$ of an entangled stabilizer state, the total injected magic increases with the entanglement between $A$ and its complement. More generally, for any unitary $U_A$, we show that this enhancement is maximized when $A$ is maximally entangled with its complement, in which case the total injected magic is exactly given by the unitary stabilizer Rényi entropy we introduce. This quantity provides both a directly computable measure of unitary magic and a lower bound on the minimum number of $T$ gates required to synthesize $U_A$. We further extend our analysis to tripartite stabilizer entanglement, non-stabilizer entanglement, and magic injection via shallow-depth brickwork circuits, finding that the qualitative picture remains unchanged.

Stabilizer Entanglement Enhances Magic Injection

Abstract

Non-stabilizerness is a key resource for fault-tolerant quantum computation, yet its interplay with entanglement in dynamical settings remains underexplored. We address this by analyzing a well-controlled, analytically tractable setup, where we show that entanglement acts as a conduit that teleports magic across the system, thereby enhancing magic injection. Using exact calculations, we prove that when a Haar-random unitary is applied to a subsystem of an entangled stabilizer state, the total injected magic increases with the entanglement between and its complement. More generally, for any unitary , we show that this enhancement is maximized when is maximally entangled with its complement, in which case the total injected magic is exactly given by the unitary stabilizer Rényi entropy we introduce. This quantity provides both a directly computable measure of unitary magic and a lower bound on the minimum number of gates required to synthesize . We further extend our analysis to tripartite stabilizer entanglement, non-stabilizer entanglement, and magic injection via shallow-depth brickwork circuits, finding that the qualitative picture remains unchanged.

Paper Structure

This paper contains 20 sections, 5 theorems, 97 equations, 10 figures, 1 table.

Table of Contents

  1. Additional numerical results on the second stabilizer Rényi entropy $\overline{M_2}$
  2. Tripartite entanglement
  3. Analytical method
  4. A diagrammatic approach
  5. A useful quantity and its approximations
  6. Derivation of the key results presented in the main text
  7. Eq. (\ref{['eq:Y_bi']})
  8. Eq. (\ref{['eq:UaUb']})
  9. Eq. (\ref{['eq:tripartite_UaUb']})
  10. An example of the decomposition of the stabilizer group
  11. Factorized unitary acting on three subregions
  12. Magic injection with shallow-depth brickwork circuits
  13. Unitary stabilizer Rényi entropy
  14. The relationship between unitary SRE and state SRE
  15. Lower bound on the unitary nullity and $T$ count
  16. ...and 5 more sections

Key Result

Theorem 1

For any $\alpha\geq 0$, where the unitary stabilizer nullity is defined as $v(U):= 2N-{\rm log} |s(U)|$, where $s(U):=\{P_i\in P_N| U P_i U^\dagger = \pm P_j \}$Jiang_2023.

Figures (10)

  • Figure 1: Schematics of the setup considered in this work. The initial states are bipartite stabilizer pure states. (a) A Haar random unitary injects magic by acting on a subregion $A$. (b) A factorized unitary $U=U_A \otimes U_B$, where $U_A$ and $U_B$ are independently Haar-random, acts on the initial stabilizer state. (c) Schematic of gate teleportation. Because of the Bell pairs between $A$ and $B$, $U_A$ acting on $A$ is equivalent to ${U'_A}$ and ${U'_B}$ acting on $A$ and $B$.
  • Figure 2: Numerical simulations of $\overline{Y^{\rm lin}}$ for the setup depicted in (a) Fig. \ref{['fig:Fig1']}(a), and (b) Fig. \ref{['fig:Fig1']}(b), respectively. We consider stabilizer initial states with varying system sizes and bipartite entanglement $E$. The analytical results Eqs. (\ref{['eq:Y_bi']}) and (\ref{['eq:UaUb']}) are shown as dashed lines.
  • Figure 3: Numerical simulations of magic injection to initial states with non-stabilizer entanglement. (a)&(b): imperfect Bell states; (c)&(d): states with engineered entanglement spectra (see text for a precise description). In (a)&(c), we plot the amount of preexisting magic in the initial states, while in (b)&(d) we plot the amount of injected magic: $\Delta \overline{M_2} = \overline{M_2^{\rm final}}-\overline{M_2^{\rm init}}$.
  • Figure 4: Numerical simulations of $\overline{Y^{\rm lin}}$ and $\overline{M_2}$ for the setup depicted in Fig.1(a) of the main text. $1-\overline{Y^{\rm lin}}$ is shown as upper triangles and $2^{-\overline{M_2}}$ is shown as lower triangles.
  • Figure 5: (a) Schematic of the initial stabilizer state used for tripartite systems. Notice that any tripartite stabilizer state can be brought into this form via local Clifford unitaries acting on subsystems $A$, $B$ and $C$. (b) Schematic of the setup for $U_A\otimes U_B$ acting on tripartite systems. (c) Numerical results for $U_A\otimes U_B$ acting on tripartite systems. The dashed lines represent the quantity shown in Eq. (\ref{['eq:tripartite_UaUb']}) and the squares represent the exact result Eq. (\ref{['eq:SMp_23']}).
  • ...and 5 more figures

Theorems & Definitions (10)

  • Definition
  • Theorem 1
  • Theorem 2
  • Definition G.1
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Corollary G.1
  • proof