Long-range nonstabilizerness and phases of matter

Abstract

Long-range nonstabilizerness can be defined as the amount of nonstabilizerness which cannot be removed by shallow local quantum circuits. In this work, we study long-range nonstabilizerness in the context of many-body quantum physics, a task with possible implications for quantum-state preparation protocols and implementation of quantum-error correcting codes. After presenting a simple argument showing that long-range nonstabilizerness is a generic property of many-body states, we restrict to the class of ground states of gapped local Hamiltonians. We focus on one-dimensional systems and present rigorous results in the context of translation-invariant matrix product states (MPSs). By analyzing the fixed points of the MPS renormalization-group flow, we provide a sufficient condition for long-range nonstabilizerness, which depends entirely on the local MPS tensors. Physically, our condition captures the fact that the mutual information between distant regions of stabilizer fixed points is quantized, and this fact is not changed after applying shallow quantum circuits. We also discuss possible ramifications in the classification of phases of matter and quantum error correction.

Figures (3)

• Figure 1: Partition considered in the proof of Theorem \ref{['th:main_result']}. The $1D$ periodic chain is divided into three disjoint regions $A$, $B$, and $C$, where $A$ and $B$ are sufficiently separated intervals, while $C$ is the complement of $A\cup B$.
• Figure 1: Graphical proof of Lemma \ref{['lemma:main_lemma_tripartion']}. The figure shows a QC of depth $D=4$. For each output region, we identify a backward causal cone. For $C_1$ and $C_2$, the gates are removed by taking the trace, while for $A$ and $B$ the gates can be removed by the unitaries $U_A$ and $U_B$. The size of the causal cones depends on whether the boundaries fall in between two unitaries or not. The two cases are displayed for $A/B$. Region $A$ covers $2D+2$, whereas region $B$ covers $2D$ sites. Note the periodic boundary conditions.
• Figure 2: Graphical representation of the quantum channel $\mathcal{E}_A$ appearing in \ref{['eq:sigma_ab']}. Here we use a tensor-network notation where lower and upper legs correspond to input and output degrees of freedom, respectively. The double lines in the upper and lower left legs denote that they are contracted, i.e. the trace is taken over the corresponding qubits.

Theorems & Definitions (17)

• Definition 1
• Theorem 1
• Theorem 2
• Lemma 1
• Definition 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4