Local unitary versus local Clifford equivalence of stabilizer and graph states

Bei Zeng, Hyeyoun Chung, Andrew W. Cross, and Isaac L. Chuang
Phys. Rev. A 75, 032325 – Published 19 March 2007

Abstract

The equivalence of stabilizer states under local transformations is of fundamental interest in understanding properties and uses of entanglement. Two stabilizer states are equivalent under the usual stochastic local operations and classical communication criterion if and only if they are equivalent under local unitary (LU) operations. More surprisingly, under certain conditions, two LU-equivalent stabilizer states are also equivalent under local Clifford (LC) operations, as was shown by Van den Nest et al. [Phys. Rev. A 71, 062323 (2005)]. Here, we broaden the class of stabilizer states for which LU equivalence implies LC equivalence (LULC) to include all stabilizer states represented by graphs with cycles of length neither 3 nor 4. To compare our result with Van den Nest et al.’s, we show that any stabilizer state of distance δ=2 is beyond their criterion. We then further prove that LULC holds for a more general class of stabilizer states of δ=2. We also explicitly construct graphs representing δ>2 stabilizer states which are beyond their criterion: we identify all 58 graphs with up to 11 vertices and construct graphs with 2m1 (m4) vertices using quantum error-correcting codes which have non-Clifford transversal gates.

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  • Received 11 December 2006

DOI:https://doi.org/10.1103/PhysRevA.75.032325

©2007 American Physical Society

Authors & Affiliations

Bei Zeng1, Hyeyoun Chung2, Andrew W. Cross2,3, and Isaac L. Chuang1,2

  • 1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
  • 2Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
  • 3IBM Research Division, T. J. Watson Research Center, P. O. Box 218, Yorktown Heights, New York 10598, USA

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Vol. 75, Iss. 3 — March 2007

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