TY - JOUR T1 - Symmetries of Codeword Stabilized Quantum Codes JF - 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013) Y1 - 2013 A1 - Salman Beigi A1 - Jianxin Chen A1 - Markus Grassl A1 - Zhengfeng Ji A1 - Qiang Wang A1 - Bei Zeng AB - Symmetry is at the heart of coding theory. Codes with symmetry, especially cyclic codes, play an essential role in both theory and practical applications of classical error-correcting codes. Here we examine symmetry properties for codeword stabilized (CWS) quantum codes, which is the most general framework for constructing quantum error-correcting codes known to date. A CWS code Q can be represented by a self-dual additive code S and a classical code C, i.,e., Q=(S,C), however this representation is in general not unique. We show that for any CWS code Q with certain permutation symmetry, one can always find a self-dual additive code S with the same permutation symmetry as Q such that Q=(S,C). As many good CWS codes have been found by starting from a chosen S, this ensures that when trying to find CWS codes with certain permutation symmetry, the choice of S with the same symmetry will suffice. A key step for this result is a new canonical representation for CWS codes, which is given in terms of a unique decomposition as union stabilizer codes. For CWS codes, so far mainly the standard form (G,C) has been considered, where G is a graph state. We analyze the symmetry of the corresponding graph of G, which in general cannot possess the same permutation symmetry as Q. We show that it is indeed the case for the toric code on a square lattice with translational symmetry, even if its encoding graph can be chosen to be translational invariant. VL - 22 U4 - 192-206 UR - http://arxiv.org/abs/1303.7020v2 U5 - 10.4230/LIPIcs.TQC.2013.192 ER - TY - JOUR T1 - The Power of Unentanglement Y1 - 2008 A1 - Scott Aaronson A1 - Salman Beigi A1 - Andrew Drucker A1 - Bill Fefferman A1 - Peter Shor AB - The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k)=QMA(2) for k>=2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given ~O(sqrt(n)) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on the existence of very short PCPs. Second, we show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all k>=2. Third, we prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one. UR - http://arxiv.org/abs/0804.0802v2 ER -