General Entanglement-Assisted Quantum Error-Correcting CodesTodd A. Brun, Igor Devetak and Min-Hsiu Hsieh Communication Sciences InstituteQEC07n,k quantum error correcting codemeasure + correctPauli unitariesZXPauli groupDiscretization of errorsShor 95; Steane 96; Gottesman 96; Calderbank, Rains, Shor, Sloane 96 An n,k quantum error correcting code is described by a (n-k) 2n parity check matrix H. Its rowspace B(H) is an isotropic subspace of commuting stabilizer generatorsdual containing code and commuten=5, k=1 The symplectic product is defined by and commute (anti-commute) iff Classical symplectic codes The correctable error set E is defined by:degenerate code The code space is defined as the simultaneous +1 eigenspace of the stabilizer operators Correction involves measuring the “error syndrome” (i.e. the simultaneous eigenvector of the stabilizer generators) , distinct error syndromesY error on 4th q-bitQuantum stabilizer codesIf E1 and E2 are in E, then at least one of the two conditions hold:Properties of Stabilizer CodesWe can see that stabilizer codes have the following properties:The code corresponds to an isotropic (that is, dual- containing) classical code over a symplectic space.The error correcting conditions are almost the same as classical (except for the existence of degenerate quantum codes, in which distinct errors share the same error syndrome).Correction consists of measuring an error syndrome and performing an appropriate correcting action (a unitary).Entanglement-assisted error correctionn,k;c EA quantum error correcting codeAliceBobe-bitc e-bitsBowen 03; Brun, Devetak, Hsieh, Science 2006; quant-ph/0608027 Entanglement-assisted stabilizer formalismIt turns out that we can establish a simple extension of the usual stabilizer formalism to describe entanglement-assisted codes. We again establish a “stabilizer” which is a subgroup of the Pauli group on n q-bits; but we no longer require this subgroup to be Abelian. For such a subgroup, we can find a set of generators which fall into two groups:Isotropic generators, which commute with all other generators; andSymplectic generators, which come in anticommuting pairs; each pair commutes with all other generators. An n,k;c EA quantum error correcting code is described by a (n-k) 2n parity check matrix H. B = rowspace(H). Again, Take a general symplectic matrix H. Its rowspace B can be written as Canonical examplesymplectic pairsEntanglement-assisted stabilizer formalismThe isotropic generators generate SI and the symplectic generators generate SE.stabilized byMeasure in the simultaneous eigenbasis of n = 3, k = 1, c = 23, 1; 2 code The correctable error set E is defined by:degenerate code The code space is defined as the simultaneous +1 eigenspace of the stabilizer generators Decoding involves measuring the “error syndrome” (i.e. the simultaneous eigenvector of the stabilizer generators) , nnncccIf E1 and E2 are in E, then at least one of the two conditions hold:Properties of the entanglement-assisted stabilizer formalismWe can now compare the properties of an EAQECC to those of an ordinary QECC:The code corresponds to a classical code over a symplectic space. (No longer needs to be dual-containing!)The error correcting conditions are almost the same as classical (except for the existence of degenerate quantum codes, in which distinct errors share the same error syndrome).Correction consists of measuring an error syndrome and performing an appropriate correcting action (a unitary).The GF(4) Construction Natural isometry between GF(4) and (Z2)2 Any dual containing classical n,k,d4 code can be made into a n,2kn,d QECC Now: Any classical n,k,d4 code can be made into a n,2k-n+c,d;c catalytic QECC for some c When the classical code attains the Singleton bound n-k d-1 the quantum code attains the quantum Singleton bound n-k 2(d-1) When the classical code attains the Shannon limit 2 H4(1 3p, p,p,p) on a quaternary symmetric channel, the quantum code attains the Hashing limit 1-H2(1-3p, p,p,p). Modern classical codes (LDPC, turbo) can now be made quantum without having to be dual-containing. Operator QECCsPoulin; Bacon; Aly, Klappenecker and SarvepalliThe basic idea of operator QECCs is that part of the system (the noisy or gauge part) contains no information about either the quantum information to be transmitted or the errors which occur. We allow arbitrary noise to affect this gauge subsystem.Because the gauge subsystem can be in any arbitrary state, OQECCs are not subspaces. (That is, the superposition of two valid states in the OQECC is not a valid state unless they have the same state for the gauge subsystem.)An n,k;r OQECC: r = number of gauge q-bits Once again, write rowspace B of symplectic matrix H as OQECC Stabilizer Formalism Now, symplectic pair