2,066 research outputs found

    Towards efficient decoding of classical-quantum polar codes

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    Known strategies for sending bits at the capacity rate over a general channel with classical input and quantum output (a cq channel) require the decoder to implement impractically complicated collective measurements. Here, we show that a fully collective strategy is not necessary in order to recover all of the information bits. In fact, when coding for a large number N uses of a cq channel W, N I(W_acc) of the bits can be recovered by a non-collective strategy which amounts to coherent quantum processing of the results of product measurements, where I(W_acc) is the accessible information of the channel W. In order to decode the other N (I(W) - I(W_acc)) bits, where I(W) is the Holevo rate, our conclusion is that the receiver should employ collective measurements. We also present two other results: 1) collective Fuchs-Caves measurements (quantum likelihood ratio measurements) can be used at the receiver to achieve the Holevo rate and 2) we give an explicit form of the Helstrom measurements used in small-size polar codes. The main approach used to demonstrate these results is a quantum extension of Arikan's polar codes.Comment: 21 pages, 2 figures, submission to the 8th Conference on the Theory of Quantum Computation, Communication, and Cryptograph

    Belief propagation decoding of quantum channels by passing quantum messages

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    Belief propagation is a powerful tool in statistical physics, machine learning, and modern coding theory. As a decoding method, it is ubiquitous in classical error correction and has also been applied to stabilizer-based quantum error correction. The algorithm works by passing messages between nodes of the factor graph associated with the code and enables efficient decoding, in some cases even up to the Shannon capacity of the channel. Here we construct a belief propagation algorithm which passes quantum messages on the factor graph and is capable of decoding the classical-quantum channel with pure state outputs. This gives explicit decoding circuits whose number of gates is quadratic in the blocklength of the code. We also show that this decoder can be modified to work with polar codes for the pure state channel and as part of a polar decoder for transmitting quantum information over the amplitude damping channel. These represent the first explicit capacity-achieving decoders for non-Pauli channels.Comment: v3: final version for publication; v2: improved discussion of the algorithm; 7 pages & 2 figures. v1: 6 pages, 1 figur

    Sequential decoding of a general classical-quantum channel

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    Since a quantum measurement generally disturbs the state of a quantum system, one might think that it should not be possible for a sender and receiver to communicate reliably when the receiver performs a large number of sequential measurements to determine the message of the sender. We show here that this intuition is not true, by demonstrating that a sequential decoding strategy works well even in the most general "one-shot" regime, where we are given a single instance of a channel and wish to determine the maximal number of bits that can be communicated up to a small failure probability. This result follows by generalizing a non-commutative union bound to apply for a sequence of general measurements. We also demonstrate two ways in which a receiver can recover a state close to the original state after it has been decoded by a sequence of measurements that each succeed with high probability. The second of these methods will be useful in realizing an efficient decoder for fully quantum polar codes, should a method ever be found to realize an efficient decoder for classical-quantum polar codes.Comment: 12 pages; accepted for publication in the Proceedings of the Royal Society

    Polar codes in network quantum information theory

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    Polar coding is a method for communication over noisy classical channels which is provably capacity-achieving and has an efficient encoding and decoding. Recently, this method has been generalized to the realm of quantum information processing, for tasks such as classical communication, private classical communication, and quantum communication. In the present work, we apply the polar coding method to network quantum information theory, by making use of recent advances for related classical tasks. In particular, we consider problems such as the compound multiple access channel and the quantum interference channel. The main result of our work is that it is possible to achieve the best known inner bounds on the achievable rate regions for these tasks, without requiring a so-called quantum simultaneous decoder. Thus, our work paves the way for developing network quantum information theory further without requiring a quantum simultaneous decoder.Comment: 18 pages, 2 figures, v2: 10 pages, double column, version accepted for publicatio

    Magic state distillation with punctured polar codes

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    We present a scheme for magic state distillation using punctured polar codes. Our results build on some recent work by Bardet et al. (ISIT, 2016) who discovered that polar codes can be described algebraically as decreasing monomial codes. Using this powerful framework, we construct tri-orthogonal quantum codes (Bravyi et al., PRA, 2012) that can be used to distill magic states for the TT gate. An advantage of these codes is that they permit the use of the successive cancellation decoder whose time complexity scales as O(Nlog(N))O(N\log(N)). We supplement this with numerical simulations for the erasure channel and dephasing channel. We obtain estimates for the dimensions and error rates for the resulting codes for block sizes up to 2202^{20} for the erasure channel and 2162^{16} for the dephasing channel. The dimension of the triply-even codes we obtain is shown to scale like O(N0.8)O(N^{0.8}) for the binary erasure channel at noise rate 0.010.01 and O(N0.84)O(N^{0.84}) for the dephasing channel at noise rate 0.0010.001. The corresponding bit error rates drop to roughly 8×10288\times10^{-28} for the erasure channel and 7×10157 \times 10^{-15} for the dephasing channel respectively.Comment: 18 pages, 4 figure

    Bounds on Information Combining With Quantum Side Information

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    "Bounds on information combining" are entropic inequalities that determine how the information (entropy) of a set of random variables can change when these are combined in certain prescribed ways. Such bounds play an important role in classical information theory, particularly in coding and Shannon theory; entropy power inequalities are special instances of them. The arguably most elementary kind of information combining is the addition of two binary random variables (a CNOT gate), and the resulting quantities play an important role in Belief propagation and Polar coding. We investigate this problem in the setting where quantum side information is available, which has been recognized as a hard setting for entropy power inequalities. Our main technical result is a non-trivial, and close to optimal, lower bound on the combined entropy, which can be seen as an almost optimal "quantum Mrs. Gerber's Lemma". Our proof uses three main ingredients: (1) a new bound on the concavity of von Neumann entropy, which is tight in the regime of low pairwise state fidelities; (2) the quantitative improvement of strong subadditivity due to Fawzi-Renner, in which we manage to handle the minimization over recovery maps; (3) recent duality results on classical-quantum-channels due to Renes et al. We furthermore present conjectures on the optimal lower and upper bounds under quantum side information, supported by interesting analytical observations and strong numerical evidence. We finally apply our bounds to Polar coding for binary-input classical-quantum channels, and show the following three results: (A) Even non-stationary channels polarize under the polar transform. (B) The blocklength required to approach the symmetric capacity scales at most sub-exponentially in the gap to capacity. (C) Under the aforementioned lower bound conjecture, a blocklength polynomial in the gap suffices.Comment: 23 pages, 6 figures; v2: small correction

    The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

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    Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing and protecting fragile qubits against the undesirable effects of quantum decoherence. Similar to classical codes, hashing bound approaching QECCs may be designed by exploiting a concatenated code structure, which invokes iterative decoding. Therefore, in this paper we provide an extensive step-by-step tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided concatenated quantum codes based on the underlying quantum-to-classical isomorphism. These design lessons are then exemplified in the context of our proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the outer component of a concatenated quantum code. The proposed QIRCC can be dynamically adapted to match any given inner code using EXIT charts, hence achieving a performance close to the hashing bound. It is demonstrated that our QIRCC-based optimized design is capable of operating within 0.4 dB of the noise limit

    Superadditivity of Quantum Channel Coding Rate with Finite Blocklength Joint Measurements

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    The maximum rate at which classical information can be reliably transmitted per use of a quantum channel strictly increases in general with NN, the number of channel outputs that are detected jointly by the quantum joint-detection receiver (JDR). This phenomenon is known as superadditivity of the maximum achievable information rate over a quantum channel. We study this phenomenon for a pure-state classical-quantum (cq) channel and provide a lower bound on CN/NC_N/N, the maximum information rate when the JDR is restricted to making joint measurements over no more than NN quantum channel outputs, while allowing arbitrary classical error correction. We also show the appearance of a superadditivity phenomenon---of mathematical resemblance to the aforesaid problem---in the channel capacity of a classical discrete memoryless channel (DMC) when a concatenated coding scheme is employed, and the inner decoder is forced to make hard decisions on NN-length inner codewords. Using this correspondence, we develop a unifying framework for the above two notions of superadditivity, and show that for our lower bound to CN/NC_N/N to be equal to a given fraction of the asymptotic capacity CC of the respective channel, NN must be proportional to V/C2V/C^2, where VV is the respective channel dispersion quantity.Comment: To appear in IEEE Transactions on Information Theor

    Polar codes for degradable quantum channels

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    Channel polarization is a phenomenon in which a particular recursive encoding induces a set of synthesized channels from many instances of a memoryless channel, such that a fraction of the synthesized channels becomes near perfect for data transmission and the other fraction becomes near useless for this task. Mahdavifar and Vardy have recently exploited this phenomenon to construct codes that achieve the symmetric private capacity for private data transmission over a degraded wiretap channel. In the current paper, we build on their work and demonstrate how to construct quantum wiretap polar codes that achieve the symmetric private capacity of a degraded quantum wiretap channel with a classical eavesdropper. Due to the Schumacher-Westmoreland correspondence between quantum privacy and quantum coherence, we can construct quantum polar codes by operating these quantum wiretap polar codes in superposition, much like Devetak's technique for demonstrating the achievability of the coherent information rate for quantum data transmission. Our scheme achieves the symmetric coherent information rate for quantum channels that are degradable with a classical environment. This condition on the environment may seem restrictive, but we show that many quantum channels satisfy this criterion, including amplitude damping channels, photon-detected jump channels, dephasing channels, erasure channels, and cloning channels. Our quantum polar coding scheme has the desirable properties of being channel-adapted and symmetric capacity-achieving along with having an efficient encoder, but we have not demonstrated that the decoding is efficient. Also, the scheme may require entanglement assistance, but we show that the rate of entanglement consumption vanishes in the limit of large blocklength if the channel is degradable with classical environment.Comment: 12 pages, 1 figure; v2: IEEE format, minor changes including new figure; v3: minor changes, accepted for publication in IEEE Transactions on Information Theor
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