Long-range nonstabilizerness and quantum codes, phases, and complexity

Abstract

As a necessary resource for quantum computational advantage, quantum magic (nonstabilizerness) is of fundamental importance in the study of quantum computation and physics. We develop a systematic theory of \emph{long-range magic (LRM)} -- nonstabilizerness that cannot be erased by shallow unitary circuits -- and demonstrate its broad relevance. By bridging LRM with fault-tolerant logical gate theory, we show the emergence of LRM families from quantum error-correcting codes and devise a simple yet powerful method for testing transversal logical gates. Further, we introduce and characterize \emph{LRM phases} in which all ground states exhibit LRM, and identify certain non-Abelian topological orders as representative examples. Then, adopting a complexity theory perspective, we demonstrate the classicality of non-LRM systems in e.g.~preparation and learning settings, and present a ``no low-energy trivial magic'' (NLTM) conjecture with key motivation in the quantum PCP context, for which our LRM results suggest a promising route. Additionally, we demonstrate how to diagnose LRM with correlation functions. The concepts and results admit substantive extensions to approximate (robust) and nongeometric scenarios. Our LRM theory illuminates profound new connections among quantum resources, computational advantage, error correction and fault tolerance, many-body physics, and complexity theory.

Figures (5)

• Figure 1: Correlations in SRM systems. For two distant subsystems $A$ and $B$ with nonintersecting backward lightcones, the bipartite reduced state can only exhibit correlations mediated by a stabilizer state $|S\rangle$. This constraint can be used to diagnose LRM. (Here we depict the 1D case for simplicity but the same reasoning applies to arbitrary connectivity.)
• Figure 2: Diagnosing LRM with two-point correlation functions. Here we plot the $K=2,3$ cases for illustration; the general cases are analogous. $P\in\{X,Y,Z\}$ is a single-qubit Pauli operator. The blue areas enclosed by dark blue boundaries are "feasible regions" of $(b,c)$ given by Proposition \ref{['prop:correlations_EPR']} in Appendix \ref{['app:LRM_from_correlation']} (the boundaries consist of two linear functions and $2^K$ quadratic functions): if a two-qubit state $\rho$ has $(b,c)$ lying outside the feasible region for $K$, then $\rho$ cannot be obtained by local operations on $K$ EPR pairs. For the $|\mathrm{C}^{n-1}Z\rangle$ example, choosing $P=X$, the corresponding $(b,c)=(1-2^{2-n},1-2^{2-n})$ lie on the orange line $b=c$ and exit the feasible region for any fixed $K$ upon increasing $n$, signifying LRM.
• Figure 3: LRM from logical nonstabilizerness. A set of $n$ stabilizer generators of the logical state $|\overline{T0}\rangle$ in toric code is shown, one of them is non-Pauli. We define $|T\rangle:=\operatorname{cos}(\pi/8)|0\rangle+\operatorname{sin}(\pi/8)|1\rangle$, the $+1$ eigenstate of $\frac{1}{\sqrt{2}}(X+Z)$. The Pauli generators (blue) consist of $n-2$ local ones and a global $\overline{Z}_2$, which is a tensor product of Pauli $Z$'s around the blue handle. The orange one is a global non-Pauli generator $\frac{1}{\sqrt{2}}(\overline{X}_1+\overline{Z}_1)$, which introduces global $\frac{1}{\sqrt{2}}$'s into the Pauli spectrum of $|\overline{T0}\rangle$, producing "topological nonstabilizerness". Tracing out the entire region outside an arbitrary gray "cross" will not decrease the amount of nonstabilizerness contained in $|\overline{T0}\rangle$.
• Figure 4: The qubit layout and the stabilizer generators of the $[\![144,12,12]\!]$ gross code Bravyi2024Highyoder2025tourgrossmodularquantum. For illustration, the $X$-check $x^5y^4X(A, B)$ and the $Z$-check $x^7Z(B^\mathsf{T},A^\mathsf{T})$ are highlighted; they commute since their supports overlap on exactly two qubits, namely $(L,x^8y^3)$ and $(R,x^4y)$.
• Figure 5: The logical Pauli $\overline{X}_1$ and the $X$-check $xy^3X(A,B)$ overlap at only one qubit. The purple squares label the qubits that $\overline{X}_1$ acts on. The only Pauli string that commutes with the stabilizer group of the gross code and has the same support as $\overline{X}_1$ is $\overline{X}_1$ itself (up to a global phase).

Theorems & Definitions (69)

• Definition 1
• Theorem 1
• Theorem 2: Robust Bravyi--König
• Theorem 3: Transversal gate/symmetry testing (informal)
• Example 1
• Example 2
• Definition 2
• Theorem 4
• Corollary 5
• Example 3