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We introduce a purely graph-theoretical object, namely the coding clique, to construct quantum errorcorrecting codes. Almost all quantum codes constructed so far are stabilizer (additive) codes and the construction of nonadditive codes, which are potentially more efficient, is not as well understood as that of stabilizer codes. Our graphical approach provides a unified and classical way to construct both stabilizer and nonadditive codes. In particular we have explicitly constructed the optimal ((10,24,3)) code and a family of 1-error detecting nonadditive codes with the highest encoding rate so far. In the case of stabilizer codes a thorough search becomes tangible and we have classified all the extremal stabilizer codes up to 8 qubits.
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In this paper, based on the nonbinary graph state, we present a systematic way of constructing good non-binary quantum codes, both additive and nonadditive, for systems with integer dimensions. With the help of computer search, which results in many interesting codes including some nonadditive codes meeting the Singleton bounds, we are able to construct explicitly four families of optimal codes, namely, $[[6,2,3]]_p$, $[[7,3,3]]_p$, $[[8,2,4]]_p$ and $[[8,4,3]]_p$ for any odd dimension $p$ and a family of nonadditive code $((5,p,3))_p$ for arbitrary $p>3$. In the case of composite numbers as dimensions, we also construct a family of stabilizer codes $((6,2cdot p^2,3))_{2p}$ for odd $p$, whose coding subspace is {em not} of a dimension that is a power of the dimension of the physical subsystem.
We present a general formalism for quantum error-correcting codes that encode both classical and quantum information (the EACQ formalism). This formalism unifies the entanglement-assisted formalism and classical error correction, and includes encoding, error correction, and decoding steps such that the encoded quantum and classical information can be correctly recovered by the receiver. We formally define this kind of quantum code using both stabilizer and symplectic language, and derive the appropriate error-correcting conditions. We give several examples to demonstrate the construction of such codes.
Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of pre-existing entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard QECCs which can only be constructed from dual-containing codes. Operator quantum error-correcting codes (OQECCs) allow certain errors to be corrected (or prevented) passively, reducing the complexity of the correction procedure. We combine these two extensions of standard quantum error correction into a unified entanglement-assisted quantum error correction formalism. This new scheme, which we call entanglement-assisted operator quantum error correction (EAOQEC), is the most general and powerful quantum error-correcting technique known, retaining the advantages of both entanglement-assistance and passive correction. We present the formalism, show the considerable freedom in constructing EAOQECCs from classical codes, and demonstrate the construction with examples.
We study stabilizer quantum error correcting codes (QECC) generated under hybrid dynamics of local Clifford unitaries and local Pauli measurements in one dimension. Building upon 1) a general formula relating the error-susceptibility of a subregion to its entanglement properties, and 2) a previously established mapping between entanglement entropies and domain wall free energies of an underlying spin model, we propose a statistical mechanical description of the QECC in terms of entanglement domain walls. Free energies of such domain walls generically feature a leading volume law term coming from its surface energy, and a sub-volume law correction coming from thermodynamic entropies of its transverse fluctuations. These are most easily accounted for by capillary-wave theory of liquid-gas interfaces, which we use as an illustrative tool. We show that the information-theoretic decoupling criterion corresponds to a geometric decoupling of domain walls, which further leads to the identification of the contiguous code distance of the QECC as the crossover length scale at which the energy and entropy of the domain wall are comparable. The contiguous code distance thus diverges with the system size as the subleading entropic term of the free energy, protecting a finite code rate against local undetectable errors. We support these correspondences with numerical evidence, where we find capillary-wave theory describes many qualitative features of the QECC; we also discuss when and why it fails to do so.
The recently introduced detected-jump correcting quantum codes are capable of stabilizing qubit-systems against spontaneous decay processes arising from couplings to statistically independent reservoirs. These embedded quantum codes exploit classical information about which qubit has emitted spontaneously and correspond to an active error-correcting code embedded in a passive error-correcting code. The construction of a family of one detected jump-error correcting quantum codes is shown and the optimal redundancy, encoding and recovery as well as general properties of detected jump-error correcting quantum codes are discussed. By the use of design theory multiple jump-error correcting quantum codes can be constructed. The performance of one jump-error correcting quantum codes under non-ideal conditions is studied numerically by simulating a quantum memory and Grovers algorithm.
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