The Resource Theory of Stabilizer Computation

TL;DR

The paper builds a resource theory of magic for stabilizer quantum computation, introducing two central monotones—the relative entropy of magic and the computable mana derived from the discrete Wigner function—to bound distillation efficiency and analyze asymptotic interconversion. It proves faithfulness of the regularized relative entropy and establishes a practical, additive bound via mana, enabling explicit lower bounds on the number of resource states needed to distill target magic states. The work clarifies the uniqueness of sum negativity as a phase-space measure and provides detailed qutrit results, including maximal sum-negativity states and distillation analyses. By connecting negativity in the Wigner representation to contextuality and classical simulability, it offers a concrete, computable toolkit for evaluating and optimizing magic-state distillation in prime-dimensional systems, with broad implications for fault-tolerant quantum computation.

Abstract

Recent results on the non-universality of fault-tolerant gate sets underline the critical role of resource states, such as magic states, to power scalable, universal quantum computation. Here we develop a resource theory, analogous to the theory of entanglement, for resources for stabilizer codes. We introduce two quantitative measures - monotones - for the amount of non-stabilizer resource. As an application we give absolute bounds on the efficiency of magic state distillation. One of these monotones is the sum of the negative entries of the discrete Wigner representation of a quantum state, thereby resolving a long-standing open question of whether the degree of negativity in a quasi-probability representation is an operationally meaningful indicator of quantum behaviour.

Figures (5)

• Figure 1: Efficiency of the $[[5,1,3]]_3$ qutrit code of AnwarQutritMSD. We generate 50000 inputs of the form $\rho_{\text{in}}=\left(1-p_{1}-p_{2}\right)|H_{+}\rangle\!\langle H_{+}|+p_{1}|H_{-}\rangle\!\langle H_{-}|+p_{2}|H_{i}\rangle\!\langle H_{i}|,$ which is the form $\rho_{\text{in}}$ takes after the twirling step of the protocol. The mana of the $5$ input states is computed and plotted against the effective mana output following one round of the protocol, $\mathbb{E}\left[\mathscr{M}\left(\rho_{\text{out}}\right)\right]=\text{Pr} \left(\text{protocol succeeds}\right)\cdot{\rm \mathscr{M}\left(\rho_{\text{out}}\right)}.$ We used $p_{1}\in_{R}\left[0,0.4\right]$ and $p_{2}\in_{R}\left[0,0.3\right]$, and the twirling basis states are the eigenstates of the qutrit Hadamard operatorAnwarQutritMSD, with eigenvalues $\left\{ 1,-1,\imath\right\}$.
• Figure 2: Efficiency of the $[[8,1,3]]_3$ qutrit code of CampbellMSDAllPrimeDim. We generate 50000 inputs of the form $\rho_{\text{in}}=\left(1-p_{1}-p_{2}\right)|M_{0}\rangle\!\langle M_{0}|+p_{1}|M_{1}\rangle\!\langle M_{1}|+p_{2}|M_{2}\rangle\!\langle M_{2}|,$ which is the form $\rho_{\text{in}}$ takes after the twirling step of the protocol. The mana of the $8$ input states is computed and plotted against the effective mana output following one round of the protocol, $\mathbb{E}\left[\mathscr{M}\left(\rho_{\text{out}}\right)\right]=\text{Pr} \left(\text{protocol succeeds}\right)\cdot{\rm \mathscr{M}\left(\rho_{\text{out}}\right)}.$ We used $p_{1}\in_{R}\left[0,0.3\right]$, $p_{2}\in_{R}\left[0,0.3\right]$, and the twirling basis states are $\left| M_{0} \right\rangle =\frac{1}{\sqrt{3}}\left(\mathrm{e}^{\frac{4}{9}\pi\mathrm{i}}\left| 0 \right\rangle +\mathrm{e}^{\frac{2}{9}\pi\mathrm{i}}\left| 1 \right\rangle +\left| 2 \right\rangle \right),\ \left| M_{1} \right\rangle =\frac{1}{\sqrt{3}}\left(\mathrm{e}^{\frac{16}{9}\pi\mathrm{i}}\left| 0 \right\rangle +\mathrm{e}^{\frac{8}{9}\pi\mathrm{i}}\left| 1 \right\rangle +\left| 2 \right\rangle \right),\ \left| M_{2} \right\rangle =\frac{1}{\sqrt{3}}\left(\mathrm{e}^{\frac{10}{9}\pi\mathrm{i}}\left| 0 \right\rangle +\mathrm{e}^{\frac{14}{9}\pi\mathrm{i}}\left| 1 \right\rangle +\left| 2 \right\rangle \right)$.
• Figure 3: Efficiency of the $[[4,1,2]]_5$ ququint code of CampbellMSDAllPrimeDim. We generate 50000 inputs of the form $\rho_{\text{in}}=\left(1-p_{1}-p_{2}-p_{3}-p_{4}\right)|M_{0}\rangle\!\langle M_{0}|+\sum_{i=1}^{4}p_{i}|M_{i}\rangle\!\langle M_{i}|,$ which this is the form $\rho_{\text{in}}$ takes after the twirling step of the protocol. The mana of the $4$ input states is computed and plotted against the effective mana output following one round of the protocol, $\mathbb{E}\left[\mathscr{M}\left(\rho_{\text{out}}\right)\right]=\text{Pr} \left(\text{protocol succeeds}\right)\cdot{\rm \mathscr{M}\left(\rho_{\text{out}}\right)}.$ We used $p_{i}\in_{R}\left[0,0.2\right]$, and the twirling basis states are the eigenstates of the $CM$ ququint operator defined in CampbellMSDAllPrimeDim.
• Figure 4: The Wigner representations of two qutrit states, $\left| \mathbb{S} \right\rangle =\frac{1}{\sqrt{2}}\left(\left| 1 \right\rangle -\left| 2 \right\rangle \right)$ (left) and $\left| \mathbb{N} \right\rangle =\frac{1}{\sqrt{6}}\left(-\left| 0 \right\rangle +2\left| 1 \right\rangle -\left| 2 \right\rangle \right)$ (right). $\left| \mathbb{S} \right\rangle$ has sum negativity $\left|-\frac{1}{3}\right|$ and the $\left| \mathbb{N} \right\rangle$ has sum negativity $\left|-\frac{1}{6}-\frac{1}{6}\right|=\frac{1}{3}$.
• Figure 5: The plane $\left(1-x-y\right)\frac{\mathbb{I}}{3}+x|\mathbb{S}\rangle\!\langle\mathbb{S}|+y|\mathbb{N}\rangle\!\langle\mathbb{N}|$. The heat map shows the value of the mana. The light grey ($0$ mana) region is the set of states in the Wigner simplex, ie. states with positive Wigner representation. The stabilizer polytope is delineated by a dashed line.

Theorems & Definitions (34)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 5
• proof
• Definition 6
• Lemma 7
• proof
• Theorem 8