Asymmetric quantum codes: constructions, bounds and performance

Abstract

Recently, quantum error-correcting codes have been proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit-and phase-flip errors. An example for a channel that exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit and phase flips can be related to relaxation and dephasing time, respectively. We study asymmetric quantum codes that are obtained from the Calderbank-Shor-Steane (CSS) construction. For such codes, we derive upper bounds on the code parameters using linear programming. A central result of this paper is the explicit construction of some new families of asymmetric quantum stabilizer codes from pairs of nested classical codes. For instance, we derive asymmetric codes using a combination of Bose-ChaudhuriHocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes. We show that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allow a higher performance when compared with symmetric codes of comparable rates. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.

Rewriting flash memories by message passing

“…Theorem 5 implies that C 2 needs to be an LDPC code with a good performance over a binary erasure channel (under message passing decoding). Constructions of conjugate code pairs in which C 2 is an LDPC code are studied in [6][9] [19]. Sarvepalli et al [19] construct a pair of codes such that C 1 is a BCH code and C 2 is a Euclidean geometry LDPC code, which is particularly useful for our purpose.…”

On the construction of optimal asymmetric quantum codes

“…For a quantum error e = ω c X(a)Z(b) ∈ G n , the quantum weight w Q (e), the X-weight w X (e) and the Z-weight w Z (e) of e, are defined respectively as The following well-known CSS construction shown in [18][19][20] …”

“…Loffe et al 17 utilize BCH codes to correct qubit-flip errors and LDPC codes to correct more frequently phase-shift errors. AQECC derived from LDPC codes and BCH codes were also constructed in [18][19][20][21]. Stephens et al 22 consider the investigation of AQECC via code conversion.…”

Soft-pulse dynamical decoupling with Markovian decoherence

“…(28), or partial] would prohibit the use of special QECCs designed for strongly asymmetric error rates between the channels [43,44,45,46,47,48].…”