Photonic circuits in which stateful components are coupled via guided electromagnetic fields are natural candidates for native implementation of iterative stochastic algorithms based on propagation of information around a graph. Conversely, such message passing algorithms suggest novel circuit architectures for signal processing and computation that are well matched to nanophotonic device physics. Here we construct and analyze a quantum optical model of a photonic circuit for iterative decoding of a class of low-density parity-check (LDPC) codes called expander codes. Our circuit can be understood as an open quantum system whose autonomous dynamics map straightforwardly onto the subroutines of an LDPC decoding scheme, with several attractive features: it can operate in the ultra-low power regime of photonics in which quantum fluctuations become significant, is robust to noise and component imperfections, achieves comparable performance to known iterative algorithms for this class of codes, and provides an instructive example of how nanophotonic cavity quantum electrodynamic components can enable useful new information technology even if the solid-state qubits on which they are based are heavily dephased and cannot support large-scale entanglement.
Magic states are eigenstates of non-Pauli operators. One way of suppressing errors present in magic states is to perform parity measurements in their non-Pauli eigenbasis and postselect on even parity. Here we develop new protocols based on non-Pauli parity checking, where the measurements are implemented with the aid of pre-distilled multiqubit resource states. This leads to a two step process: pre-distillation of multiqubit resource states, followed by implementation of the parity check. These protocols can prepare single-qubit magic states that enable direct injection of single-qubit axial …
Learning a hidden parity function from noisy data, known as learning parity with noise (LPN), is an example of intelligent behavior that aims to generalize a concept based on noisy examples. The solution to LPN immediately leads to decoding a random binary linear code in the presence of classification noise. This problem is thought to be intractable classically, but can be solved efficiently if a quantum oracle can be queried. However, in practice, a learner is more likely to receive data from classical oracles. In this work, we show that a naive application of the quantum LPN algorithm to cla…
We present an universal way to concentrate an arbitrary $N$-particle less-entangled $W$ state into a maximally entangled $W$ state with different parity check gates. It comprises two protocols. The first protocol is based on the linear optical elements say the partial parity check gate and the second one uses the quantum nondemolition (QND) to construct the complete parity check gate. Both of which can achieve the concentration task. These protocols have several advantages. First, it can obtain a maximally entangled W state only with the help of some single photons, which greatly reduces the n…
We present an optimal entanglement concentration ECP for an arbitrary less-entangled W state. By two of the parties say Alice and Charlie performing one parity check measurements, they can obtain the maximally entangled W state with a certain probability. Otherwise, they can obtain another lesser-entangled W state with another probability, which can be used to reconcentrated into a maximally entangled W state. By iterating this ECP several times, it has the maximal success probability. This ECP maybe an optimal one and is useful in current quantum information processing.
Quantum error correction is an indispensable ingredient for scalable quantum computing. In this Perspective we discuss a particular class of quantum codes called low-density parity-check (LDPC) quantum codes. The codes we discuss are alternatives to the surface code, which is the currently leading candidate to implement quantum fault-tolerance. We introduce the zoo of LDPC quantum codes and discuss their potential for making quantum computers robust against noise. In particular, we explain recent advances in the theory of LDPC quantum codes related to certain product constructions and discuss …
Quantum low density parity check (QLDPC) codes are useful primitives for quantum information processing because they can be encoded and decoded efficiently. Besides, the error correcting capability of a few QLDPC codes exceeds the quantum Gilbert-Varshamov bound. Here, we report a numerical performance analysis of an adaptive entanglement distillation scheme using QLDPC codes. In particular, we find that the expected yield of our adaptive distillation scheme to combat depolarization errors exceed that of Leung and Shor whenever the error probability is less than about 0.07 or greater than abou…
We introduce a "hyperbicycle" ansatz for quantum codes which gives the hypergraph-product (generalized toric) codes by Tillich and Zémor and generalized bicycle codes by MacKay et al. as limiting cases. The construction allows for both the lower and the upper bounds on the minimum distance; they scale as a square root of the block length. Many of thus defined codes have finite rate and a limited-weight stabilizer generators, an analog of classical low-density parity check (LDPC) codes. Compared to the hypergraph-product codes, hyperbicycle codes generally have wider range of parameters; in par…
We discuss error-correction properties for families of quantum low-density parity check (LDPC) codes with relative distance that tends to zero in the limit of large blocklength. In particular, we show that any family of LDPC codes, quantum or classical, where distance scales as a positive power of the block length, $d \propto n^α$, $α>0$, can correct all errors with certainty if the error rate per (qu)bit is sufficiently small. We specifically analyze the case of LDPC version of the quantum hypergraph-product codes recently suggested by Tillich and Zémor. These codes are a finite-rate generali…
We report the experimental demonstration of a transmission scheme of photonic qubits over unstabilized optical fibers, which has the plug-and-play feature as well as the ability to transmit any state of a qubit, regardless of whether it is known, unknown, or entangled to other systems. A high fidelity to the noiseless quantum channel was achieved by adding an ancilla photon after the signal photon within the correlation time of the fiber noise and by performing quantum parity checking. Simplicity, maintenance-free feature and robustness against path-length mismatches among the nodes make our s…
We investigate the construction of quantum low-density parity-check (LDPC) codes from classical quasi-cyclic (QC) LDPC codes with girth greater than or equal to 6. We have shown that the classical codes in the generalized Calderbank-Shor-Steane (CSS) construction do not need to satisfy the dual-containing property as long as pre-shared entanglement is available to both sender and receiver. We can use this to avoid the many 4-cycles which typically arise in dual-containing LDPC codes. The advantage of such quantum codes comes from the use of efficient decoding algorithms such as sum-product alg…
Benchmarking simulation of hybrid decoding scheme for parity-encoded spin systems
NEC-AIST Quantum Technology Cooperative Research Laboratory（NEC-AIST量子技术联合研究实验室）；National Institute of Advanced Industrial Science and Technology（国立产业技术综合研究所）
In quantum coding theory, stabilizer codes are probably the most important class of quantum codes. They are regarded as the quantum analogue of the classical linear codes and the properties of stabilizer codes have been carefully studied in the literature. In this paper, a new but simple construction of stabilizer codes is proposed based on syndrome assignment by classical parity-check matrices. This method reduces the construction of quantum stabilizer codes to the construction of classical parity-check matrices that satisfy a specific commutative condition. The quantum stabilizer codes from …
The recent success in constructing asymptotically good quantum low-density parity-check (QLDPC) codes makes this family of codes a promising candidate for error-correcting schemes in quantum computing. However, conventional belief propagation (BP) decoding of QLDPC codes does not yield satisfying performance due to the presence of unavoidable short cycles in their Tanner graph and the special degeneracy phenomenon. In this work, we propose to decode QLDPC codes based on a check matrix with redundant rows, generated from linear combinations of the rows in the original check matrix. This approac…
Practical hybrid decoding scheme for parity-encoded spin systems
NEC-AIST Quantum Technology Cooperative Research Laboratory（NEC-AIST量子技术联合研究实验室）；National Institute of Advanced Industrial Science and Technology（产业技术综合研究所）
Accelerating A*-Based Algorithms for Decoding Quantum Low-Density Parity-Check Codes
School of Electrical and Electronic Engineering, University College Dublin（都柏林大学学院电气与电子工程学院）；Department of Computer Architecture and Automatics, Complutense University of Madrid（马德里康普顿斯大学计算机体系结构与自动系）
Recently a framework for assisted quantum error correction was proposed in which a specific type of error is allowed to occur on auxiliary qubits, which is in contrast to standard entanglement assistance that requires noiseless auxiliary qubits. However, while the framework maintains the ability to import any binary or quaternary linear code without sacrificing active error correction power, it requires the code designer to turn a parity-check matrix of the underlying classical code into an equivalent one in standard form. This means that classical coding theoretic techniques that require pari…
Quantum Error Correction (QEC) is an essential field of research towards the realization of large-scale quantum computers. On the theoretical side, a lot of effort is put into designing error-correcting codes that protect quantum data from errors, which inevitably happen due to the noisy nature of quantum hardware and quantum bits (qubits). Protecting data with an error-correcting code necessitates means to recover the original data, given a potentially corrupted data set-a task referred to as decoding. It is vital that decoding algorithms can recover error-free states in an efficient manner. …
An information reconciliation method for continuous-variable quantum key distribution with Gaussian modulation that is based on non-binary low-density parity-check (LDPC) codes is presented. Sets of regular and irregular LDPC codes with different code rates over the Galois fields $GF(8)$, $GF(16)$, $GF(32)$, and $GF(64)$ have been constructed. We have performed simulations to analyze the efficiency and the frame error rate using the sum-product algorithm. The proposed method achieves an efficiency between $0.94$ and $0.98$ if the signal-to-noise ratio is between $4$ dB and $24$ dB.
The decoding throughput in the postprocessing is one of the bottlenecks for a continuous-variable quantum key distribution (CV-QKD) system. In this paper, we propose a layered decoder to decode quasi-cyclic multi-edge type LDPC (QC-METLDPC) codes based on graphic processing unit (GPU) in continuous-variable quantum key distribution (CV-QKD) systems. We optimize the storage method of the parity check matrix, merge the sub-matrices which are unrelated, and decode multiple codewords in parallel on GPU. Simulation results demonstrate that the average decoding speed of LDPC codes with three typical…
As in classical coding theory, quantum analogues of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane (CSS) construction. However, special requirements in the quantum setting severely limit the structures such quantum codes can have. While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation. This paper gives necessary and sufficient conditions for the …
This paper develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes (EAQECCs) with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error correction performance, high rates, and low decoding complexity. The proposed method produces infinitely many new codes with a wide variety of parameters and entanglement requirements. Our framework encompass…
