Details
| Original language | English |
|---|---|
| Article number | 022304 |
| Journal | Physical Review A - Atomic, Molecular, and Optical Physics |
| Volume | 82 |
| Issue number | 2 |
| Publication status | Published - 5 Aug 2010 |
| Externally published | Yes |
Abstract
Recently, Ji disproved the local-unitary-local Clifford (LU-LC) conjecture and showed that the local unitary (LU) and local Clifford (LC) equivalence classes of the stabilizer states are not always the same. Despite the fact that this settles the LU-LC conjecture, a sufficient condition for stabilizer states that violate the LU-LC conjecture is not known. In this paper, we investigate further the properties of stabilizer states with respect to local equivalence. Our first result shows that there exist infinitely many stabilizer states that violate the LU-LC conjecture. In particular, we show that for all numbers of qubits n≥28, there exist distance-two stabilizer states which are counterexamples to the LU-LC conjecture. We prove that, for all odd n≥195, there exist stabilizer states with distance greater than two that are LU equivalent but not LC equivalent. Two important classes of stabilizer states that are of great interest in quantum computation are the cluster states and stabilizer states of the surface codes. We show that, under some minimal restrictions, both these classes of states preclude any counterexamples. In this context, we also show that the associated surface codes do not have any encoded non-Clifford transversal gates. We characterize the Calderbank-Shor-Steane (CSS) surface-code states in terms of a class of minor closed binary matroids. In addition to making a connection to an important open problem in binary matroid theory, this characterization does in some cases provide an efficient test for CSS states that are not counterexamples.
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Atomic and Molecular Physics, and Optics
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In: Physical Review A - Atomic, Molecular, and Optical Physics, Vol. 82, No. 2, 022304, 05.08.2010.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Local equivalence, surface-code states, and matroids
AU - Sarvepalli, Pradeep
AU - Raussendorf, Robert
PY - 2010/8/5
Y1 - 2010/8/5
N2 - Recently, Ji disproved the local-unitary-local Clifford (LU-LC) conjecture and showed that the local unitary (LU) and local Clifford (LC) equivalence classes of the stabilizer states are not always the same. Despite the fact that this settles the LU-LC conjecture, a sufficient condition for stabilizer states that violate the LU-LC conjecture is not known. In this paper, we investigate further the properties of stabilizer states with respect to local equivalence. Our first result shows that there exist infinitely many stabilizer states that violate the LU-LC conjecture. In particular, we show that for all numbers of qubits n≥28, there exist distance-two stabilizer states which are counterexamples to the LU-LC conjecture. We prove that, for all odd n≥195, there exist stabilizer states with distance greater than two that are LU equivalent but not LC equivalent. Two important classes of stabilizer states that are of great interest in quantum computation are the cluster states and stabilizer states of the surface codes. We show that, under some minimal restrictions, both these classes of states preclude any counterexamples. In this context, we also show that the associated surface codes do not have any encoded non-Clifford transversal gates. We characterize the Calderbank-Shor-Steane (CSS) surface-code states in terms of a class of minor closed binary matroids. In addition to making a connection to an important open problem in binary matroid theory, this characterization does in some cases provide an efficient test for CSS states that are not counterexamples.
AB - Recently, Ji disproved the local-unitary-local Clifford (LU-LC) conjecture and showed that the local unitary (LU) and local Clifford (LC) equivalence classes of the stabilizer states are not always the same. Despite the fact that this settles the LU-LC conjecture, a sufficient condition for stabilizer states that violate the LU-LC conjecture is not known. In this paper, we investigate further the properties of stabilizer states with respect to local equivalence. Our first result shows that there exist infinitely many stabilizer states that violate the LU-LC conjecture. In particular, we show that for all numbers of qubits n≥28, there exist distance-two stabilizer states which are counterexamples to the LU-LC conjecture. We prove that, for all odd n≥195, there exist stabilizer states with distance greater than two that are LU equivalent but not LC equivalent. Two important classes of stabilizer states that are of great interest in quantum computation are the cluster states and stabilizer states of the surface codes. We show that, under some minimal restrictions, both these classes of states preclude any counterexamples. In this context, we also show that the associated surface codes do not have any encoded non-Clifford transversal gates. We characterize the Calderbank-Shor-Steane (CSS) surface-code states in terms of a class of minor closed binary matroids. In addition to making a connection to an important open problem in binary matroid theory, this characterization does in some cases provide an efficient test for CSS states that are not counterexamples.
UR - http://www.scopus.com/inward/record.url?scp=77955822383&partnerID=8YFLogxK
U2 - 10.48550/arXiv.0911.1571
DO - 10.48550/arXiv.0911.1571
M3 - Article
AN - SCOPUS:77955822383
VL - 82
JO - Physical Review A - Atomic, Molecular, and Optical Physics
JF - Physical Review A - Atomic, Molecular, and Optical Physics
SN - 1050-2947
IS - 2
M1 - 022304
ER -