An Entropic Lens on Stabilizer States

TL;DR

This paper explores the structure and entropic properties of stabilizer states in quantum computing, constructing reachability graphs for small qubit systems and analyzing the growth of complexity and entropy characteristics as the system size increases.

Contribution

It explicitly constructs and analyzes reachability graphs of stabilizer states for up to five qubits, revealing how entropic structures evolve and identifying the emergence of non-holographic entropy vectors.

Findings

Reachability graphs are constructed for up to five qubits.

Entropic structures become more complex with increasing qubits.

Some four-qubit stabilizer states violate holographic entropy inequalities.

Abstract

The $n$-qubit stabilizer states are those left invariant by a $2^n$-element subset of the Pauli group. The Clifford group is the group of unitaries which take stabilizer states to stabilizer states; a physically--motivated generating set, the Hadamard, phase, and CNOT gates which comprise the Clifford gates, imposes a graph structure on the set of stabilizers. We explicitly construct these structures, the "reachability graphs," at $n\le5$. When we consider only a subset of the Clifford gates, the reachability graphs separate into multiple, often complicated, connected components. Seeking an understanding of the entropic structure of the stabilizer states, which is ultimately built up by CNOT gate applications on two qubits, we are motivated to consider the restricted subgraphs built from the Hadamard and CNOT gates acting on only two of the $n$ qubits. We show how the two subgraphs already present at two qubits are embedded into more complicated subgraphs at three and four qubits. We argue that no additional types of subgraph appear beyond four qubits, but that the entropic structures within the subgraphs can grow progressively more complicated as the qubit number increases. Starting at four qubits, some of the stabilizer states have entropy vectors which are not allowed by holographic entropy inequalities. We comment on the nature of the transition between holographic and non-holographic states within the stabilizer reachability graphs.