Fifteen years of quantum LDPC coding and improved decoding strategies

The near-capacity performance of classical low-density parity check (LDPC) codes and their efficient iterative decoding makes quantum LDPC (QLPDC) codes a promising candidate for quantum error correction. In this paper, we present a comprehensive survey of QLDPC codes from the perspective of code design as well as in terms of their decoding algorithms. We also conceive a modified non-binary decoding algorithm for homogeneous Calderbank-Shor-Steane-type QLDPC codes, which is capable of alleviating the problems imposed by the unavoidable length-four cycles. Our modified decoder outperforms the state-of-the-art decoders in terms of their word error rate performance, despite imposing a reduced decoding complexity. Finally, we intricately amalgamate our modified decoder with the classic uniformly reweighted belief propagation for the sake of achieving an improved performance

Decoding Across the Quantum LDPC Code Landscape

We show that belief propagation combined with ordered statistics post-processing is a general decoder for quantum low density parity check codes constructed from the hypergraph product. To this end, we run numerical simulations of the decoder applied to three families of hypergraph product code: topological codes, fixed-rate random codes and a new class of codes that we call semi-topological codes. Our new code families share properties of both topological and random hypergraph product codes, with a construction that allows for a finely-controlled trade-off between code threshold and stabilizer locality. Our results indicate thresholds across all three families of hypergraph product code, and provide evidence of exponential suppression in the low error regime. For the Toric code, we observe a threshold in the range $9.9\pm0.2\%$. This result improves upon previous quantum decoders based on belief propagation, and approaches the performance of the minimum weight perfect matching algorithm. We expect semi-topological codes to have the same threshold as Toric codes, as they are identical in the bulk, and we present numerical evidence supporting this observation.Comment: The code for the BP+OSD decoder used in this work can be found on Github: https://github.com/quantumgizmos/bp_os

Quantum error correction protects quantum search algorithms against decoherence

When quantum computing becomes a wide-spread commercial reality, Quantum Search Algorithms (QSA) and especially Grover’s QSA will inevitably be one of their main applications, constituting their cornerstone. Most of the literature assumes that the quantum circuits are free from decoherence. Practically, decoherence will remain unavoidable as is the Gaussian noise of classic circuits imposed by the Brownian motion of electrons, hence it may have to be mitigated. In this contribution, we investigate the effect of quantum noise on the performance of QSAs, in terms of their success probability as a function of the database size to be searched, when decoherence is modelled by depolarizing channels’ deleterious effects imposed on the quantum gates. Moreover, we employ quantum error correction codes for limiting the effects of quantum noise and for correcting quantum flips. More specifically, we demonstrate that, when we search for a single solution in a database having 4096 entries using Grover’s QSA at an aggressive depolarizing probability of 10-3, the success probability of the search is 0.22 when no quantum coding is used, which is improved to 0.96 when Steane’s quantum error correction code is employed. Finally, apart from Steane’s code, the employment of Quantum Bose-Chaudhuri-Hocquenghem (QBCH) codes is also considered

Tailoring three-dimensional topological codes for biased noise

Tailored topological stabilizer codes in two dimensions have been shown to exhibit high storage threshold error rates and improved subthreshold performance under biased Pauli noise. Three-dimensional (3D) topological codes can allow for several advantages including a transversal implementation of non-Clifford logical gates, single-shot decoding strategies, parallelized decoding in the case of fracton codes as well as construction of fractal lattice codes. Motivated by this, we tailor 3D topological codes for enhanced storage performance under biased Pauli noise. We present Clifford deformations of various 3D topological codes, such that they exhibit a threshold error rate of $50\%$ under infinitely biased Pauli noise. Our examples include the 3D surface code on the cubic lattice, the 3D surface code on a checkerboard lattice that lends itself to a subsystem code with a single-shot decoder, the 3D color code, as well as fracton models such as the X-cube model, the Sierpinski model and the Haah code. We use the belief propagation with ordered statistics decoder (BP-OSD) to study threshold error rates at finite bias. We also present a rotated layout for the 3D surface code, which uses roughly half the number of physical qubits for the same code distance under appropriate boundary conditions. Imposing coprime periodic dimensions on this rotated layout leads to logical operators of weight $O(n)$ at infinite bias and a corresponding $\exp[-O(n)]$ subthreshold scaling of the logical failure rate, where $n$ is the number of physical qubits in the code. Even though this scaling is unstable due to the existence of logical representations with $O(1)$ low-rate Pauli errors, the number of such representations scales only polynomially for the Clifford-deformed code, leading to an enhanced effective distance.Comment: 51 pages, 34 figure

Joint quantum-assisted channel estimation and data detection

Joint Channel Estimation (CE) and Multi-User Detection (MUD) has become a crucial part of iterative receivers. In this paper we propose a Quantum-assisted Repeated Weighted Boosting Search (QRWBS) algorithm for CE and we employ it in the uplink of MIMO-OFDM systems, in conjunction with the Maximum A posteriori Probability (MAP) MUD and a near-optimal Quantum-assisted MUD (QMUD). The performance of the QRWBS-aided CE is evaluated in rank-deficient systems, where the number of receive Antenna Elements (AE) at the Base Station (BS) is lower than the number of supported users. The effect of the Channel Impulse Response (CIR) prediction filters, of the Power Delay Profile (PDP) of the channels and of the Doppler frequency have on the attainable system performance is also quantified. The proposed QRWBS-aided CE is shown to outperform the RWBS-aided CE, despite requiring a lower complexity, in systems where iterations are invoked between the MUD, the CE and the channel decoders at the receiver. In a system, where U=7 users are supported with the aid of P=4 receive AEs, the joint QRWBS-aided CE and QMUD achieves a 2 dB gain, when compared to the joint RWBS-aided CE and MAP MUD, despite imposing 43% lower complexity