An efficient algorithm to compute entanglement in states with low magic

TL;DR

The paper addresses the challenge of computing entanglement in magic-rich quantum states by exploiting stabilizer-code structure to separate entanglement into a stabilizer contribution and a logical contribution for states of the form $|\Psi\rangle=\sum_{j=1}^K c_j |\psi_j\rangle$ with common stabilizers. For low stabilizer nullity $\nu$, the method delivers a polynomial-in-$n$ and exponential-in-$\nu$ classical algorithm to obtain the von Neumann entropy, the entanglement spectrum, and all Rényi entropies, using an RT-like decomposition $S(\rho_A)=S(\rho_a)+\mathscr{A}(\rho_A)$ where the area term is stabilizer-only. The bulk entropy $S(\rho_a)$ is computed from the logical state on a $2^\nu$-dimensional subspace, while $\mathscr{A}(\rho_A)$ is obtained from a reference stabilizer state, enabling efficient reconstruction of the full entanglement spectrum. The framework applies to sparse-magic circuits, MIPT studies, holographic stabilizer codes, and experimental learning of entropies, providing a practical tool for exploring entanglement in magic-rich quantum many-body systems.

Abstract

A bottleneck for analyzing the interplay between magic and entanglement is the computation of these quantities in highly entangled quantum many-body magic states. Efficient extraction of entanglement can also inform our understanding of dynamical quantum processes such as measurement-induced phase transition and approximate unitary designs. We develop an efficient classical algorithm to compute the von Neumann entropy and entanglement spectrum for such states under the condition that they have low stabilizer nullity. The algorithm exploits the property of stabilizer codes to separate entanglement into two pieces: one generated by the common stabilizer group and the other from the logical state. The low-nullity constraint ensures both pieces can be computed efficiently. Our algorithm can be applied to study the entanglement in sparsely $T$-doped circuits with possible Pauli measurements as well as certain classes of states that have both high entanglement and magic. Combining with stabilizer learning subroutines, it also enables the efficient learning of von Neumann entropies for low-nullity states prepared on quantum devices.

Figures (3)

• Figure 1: A holographic stabilizer code where red edges represent bulk logical qubits while the white circles represent physical qubits. All the stabilizer entanglement contributing to $S(\rho_A)$ comes from the minimal surface (dashed curve) area $\mathscr{A}$ captured by the in-plane edges (green). The bulk state can also be entangled across the $k$ logical qubits and only this entanglement can contain non-stabilizerness. The bulk logical qubits in the yellow wedge are in the state $\rho_a$ and are recoverable from subsystem $A$. The remaining bulk qubits are recoverable from $A^c$. Such a code can be built from a circuit on the right where $C_\chi$ creates the entanglement resource $|\chi\rangle$ (on the green wires) that facilitates the entanglement in green on the left.
• Figure 2: Clifford circuit with few number of non-Clifford single qubits gates, acting on initial product state.
• Figure 3: Clifford acting on initial tensor product of qubits, with $t$-number of magic states.

Theorems & Definitions (6)

• Definition 1: Complementary Recovery
• Theorem 1
• Theorem 2
• proof
• Theorem 3
• Theorem 4