High-Rate Quantum Low-Density Parity-Check Codes Assisted by Reliable Qubits

Abstract

Abstract-Quantum error correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error-correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum information science, where in principle any binary linear code can be turned into a quantum error-correcting code by assuming a small number of reliable quantum bits. This paper studies how best to take advantage of these latest ideas to construct desirable quantum error-correcting codes of very high information rate. Our methods exploit structured high-rate low-density parity-check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths. Our approach to designing high-rate quantum error-correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes, making it possible to deal with a situation where promising quantum analogues of low-density parity-check codes are difficult to find.Index Terms-Quantum error correction, low-density parity-check code, combinatorial design, entanglement-assisted quantum error-correcting code.

A Survey of Quantum Error Correction

“…Codes in [34] and [35] are derived from [36]. In the latter case, it is shown that the performance is close to conventional LDPC codes [37]. It is worth mentioning that the first authors of four references [34]- [37] are Japanese researchers.…”

“…To overcome this problem, a method of giving up the commutativity and using EA instead has been proposed [39]. This method has been reported to achieve high performance close to that of conventional LDPC codes [37].…”

Practical Limits of Error Correction for Quantum Metrology

“…We overcome these restrictions by finding an exact solution of a specific error correction strategy: a parity check error correction code [53][54][55] involving a single The input state ρ in , is composed of n sensing qubits and one ancillary qubit. The sensing qubits evolve according to the signal ω and noise γ.…”

2-Designs and Redundant Syndrome Extraction for Quantum Error Correction

“…In the last few years, more sophisticated schemes have been proposed. Fujiwara et al noted that measuring more than a minimal set of stabilizers could be a more efficient way to extract reliable information and noted a possible connection to designs [5,6]. Ashikmin et al generalized existence theorems and the quantum Singleton bound of standard quantum error correction (QEC) theory so that they take into account redundancy in measurements [7].…”