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 about 0.28. This finding illustrates the effectiveness of using QLDPC codes in entanglement distillation.
Many entanglement distillation schemes use either universal random hashing or breeding as their final step to obtain almost perfect shared EPR pairs. In spite of a high yield, the hardness of decoding a random linear code makes the use of random hashing and breeding infeasible in practice. In this pilot study, we analyze the performance of the recurrence method, a well-known entanglement distillation scheme, with its final random hashing or breeding procedure being replaced by various efficiently decodable quantum codes. Among all the replacements investigated, the one using a certain adaptive…
Single photons with tailored temporal profiles are a vital resource for future quantum networks. Here we distill them out of custom-shaped laser pulses that reflect from a single atom strongly coupled to an optical resonator. A subsequent measurement on the atom is employed to herald a successful distillation. Out of vacuum-dominated light pulses, we create single photons with fidelity $66(1)\%$, two-and-more-photon suppression $95.5(6)\%$, and a Wigner function with negative value $-0.125(6)$. Our scheme applied to state-of-the-art fiber resonators could boost the single-photon fidelity to up…
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 …
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 …
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…
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 …
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…
In the shallow sub-threshold regime, fault-tolerant quantum computation requires a tremendous amount of qubits. In this paper, we study the error correction in the deep sub-threshold regime. We estimate the physical error rate for achieving the logical error rates of $10^{-6} - 10^{-15}$ using few-qubit codes, i.e. short repetition codes, small surface codes and the Steane code. Error correction circuits that are efficient for biased error channels are identified. Using the Steane code, when error channels are biased with a ratio of $10^{-3}$, the logical error rate of $10^{-15}$ can be achiev…
Quantum low-density parity-check codes can be decoded using a syndrome based $\mathrm{GF}(4)$ belief propagation decoder. However, the performance of this decoder is limited both by unavoidable $4$-cycles in the code's factor graph and the degenerate nature of quantum errors. For the subclass of CSS codes, the number of $4$-cycles can be reduced by breaking an error into an $X$ and $Z$ component and decoding each with an individual $\mathrm{GF}(2)$ based decoder. However, this comes at the expense of ignoring potential correlations between these two error components. We present a number of mod…
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 unders…
Nearest-neighbour gates are all you need: High-rate quantum low-density parity-check codes on a planar grid
Dahlem Center for Complex Quantum Systems, Freie Universität Berlin（达勒姆复杂量子系统中心，柏林自由大学）；Quantum Software Lab, The University of Edinburgh（爱丁堡大学量子软件实验室）；IQM Quantum Computers（IQM量子计算机公司）；Institute of Mathematics, Johannes Gutenberg-Universität Mainz（美因茨约翰内斯·古腾堡大学数学研究所）；Helmholtz-Zentrum Berlin für Materialien und Energie（柏林亥姆霍兹材料与能源中心）
We demonstrate a direct mapping of max k-SAT problems (and weighted max k-SAT) to a Chimera graph, which is the non-planar hardware graph of the devices built by D-Wave Systems Inc. We further show that this mapping can be used to map a similar class of maximum satisfiability problems where the clauses are replaced by parity checks over potentially large numbers of bits. The latter is of specific interest for applications in decoding for communication. We discuss an example in which the decoding of a turbo code, which has been demonstrated to perform near the Shannon limit, can be mapped to a …
Coherent parity check (CPC) codes are a new framework for the construction of quantum error correction codes that encode multiple qubits per logical block. CPC codes have a canonical structure involving successive rounds of bit and phase parity checks, supplemented by cross-checks to fix the code distance. In this paper, we provide a detailed introduction to CPC codes using conventional quantum circuit notation. We demonstrate the implementation of a CPC code on real hardware, by designing a [[4,2,2]] detection code for the IBM 5Q superconducting qubit device. Whilst the individual gate-error …
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…
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…
Extensive quantum error correction is necessary in order to perform a useful computation on a noisy quantum computer. Moreover, quantum error correction must be implemented based on imperfect parity check measurements that may return incorrect outcomes or inject additional faults into the qubits. To achieve fault-tolerant error correction, Shor proposed to repeat the sequence of parity check measurements until the same outcome is observed sufficiently many times. Then, one can use this information to perform error correction. A basic implementation of this fault tolerance strategy requires $Ω(…
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 informat…
We face the following dilemma for designing low-density parity-check codes (LDPC) for quantum error correction. 1) The row weights of parity-check should be large: The minimum distances are bounded above by the minimum row weights of parity-check matrices of constituent classical codes. Small minimum distance tends to result in poor decoding performance at the error-floor region. 2) The row weights of parity-check matrices should not be large: The sum-product decoding performance at the water-fall region is degraded as the row weight increases. Recently, Kudekar et al. showed spatially-coupled…
We consider a quantum key expansion (QKE) protocol based on entanglement-assisted quantum error-correcting codes (EAQECCs). In these protocols, a seed of a previously shared secret key is used in the post-processing stage of a standard quantum key distribution protocol like the Bennett-Brassard 1984 protocol, in order to produce a larger secret key. This protocol was proposed by Luo and Devetak, but codes leading to good performance have not been investigated. We look into a family of EAQECCs generated by classical finite geometry (FG) low-density parity-check (LDPC) codes, for which very effi…
