A graph-based approach to entanglement entropy of quantum error correcting codes

TL;DR

A graph-based method is developed to study the entanglement entropy of Calderbank-Shor-Steane quantum codes through graph-theoretical concepts, shedding light on the origins of both the local and long-range entanglement.

Abstract

We develop a graph-based method to study the entanglement entropy of Calderbank-Shor-Steane quantum codes. This method offers a straightforward interpretation for the entanglement entropy of quantum error correcting codes through graph-theoretical concepts, shedding light on the origins of both the local and long-range entanglement. Furthermore, it inspires an efficient computational scheme for evaluating the entanglement entropy. We illustrate the method by calculating the von Neumann entropy of subsystems in toric codes and two types of quantum low-density-parity check codes, and by comparing the scaling behavior of the entanglement entropy with respect to the subsystem size. Our method provides a new perspective for understanding the entanglement structure in quantum many-body systems.

Figures (7)

• Figure 1: Illustration of subsystems and their spanning trees in a toric code embedded in a square lattice with periodic boundary conditions, initialized in the logical $Z$ eigenstate. (a) A $Z$-type stabilizer generator $S_Z$ acts on four qubits surrounding a face, while an $X$-type stabilizer generator $S_X$ acts on four qubits incident to a vertex. (b) Subsystem $A$ consists of qubits on blue edges and contains one cycle, highlighted by the orange face. Subsystem $B$ includes all remaining qubits on black edges. The exterior boundary of subsystem $A$ is indicated by the deep black edges. Stabilizers shared by subsystems $A$ and $B$ are marked by gray faces. (c) The spanning tree of subsystem $A$ is formed by removing one qubit (blue cross) from the orange face. The spanning tree of subsystem $B$ is constructed by removing qubits indicated by black crosses. (d) Blue edges $E_{T_A}$ indicate the spanning tree of subsystem $A$ and purple edges $E_{T_B}$ mark the spanning tree of subsystem $B$. The gray faces represent the joint cycles and the red vertices labeled by $v_i$ denote the common vertices of the two spanning trees. The entanglement entropy equals $5$, which is the number of independent joint cycles, as determined by $|\mathcal{C}_{T_A\cup T_B}|$ or computed via Eq. \ref{['eq:graphic-form']}.
• Figure 2: Scaling of entanglement entropy for toric codes, bivariate bicycle codes and quasi-cyclic codes. The size of the subsystem is chosen to be less than half of the entire system.
• Figure 3: Entanglement entropy for qLDPC codes and toric codes. (a) Average entanglement entropy for randomly selecting subsystems. (b) Derivative of the discrepancy of entropy $I_A$ for BB codes. The inset shows the same quantity for toric codes. (c) Derivative of $I_A$ for QC codes with stabilizer weight 6. (d) Derivative of $I_A$ for QC codes with stabilizer weight 8.
• Figure 4: Illustration of the edge space and cycle space in a graph. Consider a given graph, where a basis of the edge space is comprised of the edges $e_1,e_2,e_3,e_4,e_5$. If we consider the edge subsets in $\text{pow}(E)$, $\omega_1=\left\{e_1,e_4,e_5\right\}$ and $\omega_2 = \left\{e_2,e_3,e_4\right\}$, the symmetric difference of the two edge subsets is given by $\omega_1 \Delta \omega_2 =\left\{e_1,e_2,e_3,e_5\right\}$. This operation demonstrates the combination of edges resulting from the symmetric difference, which is a key concept in understanding the edge space and its algebraic structure. The corresponding vectors for $\omega_1$ and $\omega_2$ in $\Gamma$ are $(1,0,0,1,1)$ and $(0,1,1,1,0)$, respectively. $\omega_1 \Delta \omega_2$ corresponds to the addition (mod 2) of $(1,0,0,1,1)$ and $(0,1,1,1,0)$. $(1,0,0,1,1)+(0,1,1,1,0)=(1,1,1,0,1)$ means that $\omega_1 \Delta \omega_2 = \left\{e_1,e_2,e_3,e_5\right\}$. The cycle space contains three cycles $\left\{e_1,e_4,e_5\right\},\left\{e_2,e_3,e_4\right\},\left\{e_1,e_2,e_3,e_5\right\}$. The dimension of the cycle space is 2.
• Figure 5: Illustration of subsystems in the toric code, and their spanning trees and joint cycles. These subsystems are the qubits on the red edges. (a) The spanning tree of subsystem $A$ is obtained by removing one of the $d$ qubits (red cross) along the non-contractible qubit chain. The spanning tree of subsystem $B$ is obtained by removing appropriate qubits (black cross) in subsystem $B$. The joint cycles are indicated by the orange faces. (b) The spanning forest of subsystem $A$ consists of the qubits on the red edges. The spanning tree of subsystem $B$ is obtained by removing appropriate qubits (black cross) in subsystem $B$. The qubits of $B$ on the blue edges form non-contractible joint cycles with the edges in the spanning forest of $A$. (c) The red edges, with the exception of the one that has been removed (red cross), constitute the spanning forest of subsystem $A$. By removing the edges indicated by the black cross, one can obtain the spanning tree of subsystem $B$. The spanning forest of $A$ and the spanning tree of $B$ share the stabilizer operators that are indicated by one pink face and $d-2$ orange faces. The red edges and the blue edges also form non-contractible joint cycles. (d) The spanning forest of subsystem $A$ and that of subsystem $B$ form a joint cycle space. Two spanning forests share the stabilizer operators which are indicated by the orange faces.