TY - JOUR

T1 - Entropic lens on stabilizer states

AU - Keeler, Cynthia

AU - Munizzi, William

AU - Pollack, Jason

N1 - Funding Information:
The authors thank S. Aaronson, N. Bao, C. Cao, and S. Hernandez-Cuenca for helpful discussions. J.P. is supported by the Simons Foundation through It from Qubit: Simons Collaboration on Quantum Fields, Gravity, and Information. C.K. and W.M. are supported by the U.S. Department of Energy under Grant No. DE-SC0019470 and by the Heising-Simons Foundation “Observational Signatures of Quantum Gravity” collaboration Grant No. 2021-2818.
Publisher Copyright:
© 2022 American Physical Society.

PY - 2022/12

Y1 - 2022/12

N2 - The n-qubit stabilizer states are those left invariant by a 2n-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 controlled-not (cnot) gates which comprise the Clifford gates, impose a graph structure on the set of stabilizers. We explicitly construct these structures, the "reachability graphs,"at n≤5. 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 nonholographic states within the stabilizer reachability graphs.

AB - The n-qubit stabilizer states are those left invariant by a 2n-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 controlled-not (cnot) gates which comprise the Clifford gates, impose a graph structure on the set of stabilizers. We explicitly construct these structures, the "reachability graphs,"at n≤5. 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 nonholographic states within the stabilizer reachability graphs.

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U2 - 10.1103/PhysRevA.106.062418

DO - 10.1103/PhysRevA.106.062418

M3 - Article

AN - SCOPUS:85145259270

SN - 1050-2947

VL - 106

JO - Physical Review A - Atomic, Molecular, and Optical Physics

JF - Physical Review A - Atomic, Molecular, and Optical Physics

IS - 6

M1 - 062418

ER -