Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound

Abstract

Quantum error correction has recently been shown to benefit greatly from specific physical encodings of the code qubits. In particular, several researchers have considered the individual code qubits being encoded with the continuous variable Gottesman-Kitaev-Preskill (GKP) code, and then imposed an outer discrete-variable code such as the surface code on these GKP qubits. Under such a concatenation scheme, the analog information from the inner GKP error correction improves the noise threshold of the outer code. However, the surface code has vanishing rate and demands a lot of resources with growing distance. In this work, we concatenate the GKP code with generic quantum low-density parity-check (QLDPC) codes and demonstrate a natural way to exploit the GKP analog information in iterative decoding algorithms. We first show the noise thresholds for two lifted product QLDPC code families, and then show the improvements of noise thresholds when the iterative decoder -a hardwarefriendly min-sum algorithm (MSA) -utilizes the GKP analog information. We also show that, when the GKP analog information is combined with a sequential update schedule for MSA, the scheme surpasses the well-known CSS Hamming bound for these code families. Furthermore, we observe that the GKP analog information helps the iterative decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves. Finally, we discuss new fundamental and practical questions that arise from this work on channel capacity under GKP analog information, and on improving decoder design and analysis.N. Raveendran, N. Rengaswamy and B. Vasić are with the

Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem

“…In refs. [32,33,9,34,35], known decoding algorithms for quantum error correcting codes such as minimum-weight-perfectmatching (MWPM, which is an MED decoder) have been adapted to decode diverse concatenations of the single mode square GKP code with the popular surface, toric, color, and quantum low-densityparity-check (QLDPC) codes, where the corresponding GKP-lattices can all be understood as Construction A lattices. As noted previously, we denote the underlying multi-mode square-GKP code as L N and the full lattice corresponding to the concatenated code as L = Λ (Q), such that we have L N ⊆ L and reversely L ⊥ ⊆ L ⊥ N .…”

Soft Syndrome Decoding of Quantum LDPC Codes for Joint Correction of Data and Syndrome Errors

“…We choose QC-LDPC codes as constituent classical LDPC codes to construct LP code families for demonstrating our simulation results. For example, from the [155, 64, 20] Tanner code [28] with quasicyclic base matrix of size 3 × 5 and circulant size L = 31, we obtain the [[1054, 140, 20]] LP Tanner code [29]. To show the decoding threshold plots, we also chose the LP code family described in [29,Table II].…”

“…For example, from the [155, 64, 20] Tanner code [28] with quasicyclic base matrix of size 3 × 5 and circulant size L = 31, we obtain the [[1054, 140, 20]] LP Tanner code [29]. To show the decoding threshold plots, we also chose the LP code family described in [29,Table II]. LP-QLDPC codes with increasing code length and minimum distance (d = 10, 16, 20, 24) are chosen to demonstrate the thresholds.…”