58 research outputs found
Hardness of decoding quantum stabilizer codes
In this article we address the computational hardness of optimally decoding a
quantum stabilizer code. Much like classical linear codes, errors are detected
by measuring certain check operators which yield an error syndrome, and the
decoding problem consists of determining the most likely recovery given the
syndrome. The corresponding classical problem is known to be NP-complete, and a
similar decoding problem for quantum codes is also known to be NP-complete.
However, this decoding strategy is not optimal in the quantum setting as it
does not take into account error degeneracy, which causes distinct errors to
have the same effect on the code. Here, we show that optimal decoding of
stabilizer codes is computationally much harder than optimal decoding of
classical linear codes, it is #P
The Encoding and Decoding Complexities of Entanglement-Assisted Quantum Stabilizer Codes
Quantum error-correcting codes are used to protect quantum information from
decoherence. A raw state is mapped, by an encoding circuit, to a codeword so
that the most likely quantum errors from a noisy quantum channel can be removed
after a decoding process.
A good encoding circuit should have some desired features, such as low depth,
few gates, and so on. In this paper, we show how to practically implement an
encoding circuit of gate complexity for an
quantum stabilizer code with the help of pairs of maximally-entangled
states. For the special case of an stabilizer code with , the
encoding complexity is , which is previously known to be
. For this suggests that the benefits from shared
entanglement come at an additional cost of encoding complexity.
Finally we discuss decoding of entanglement-assisted quantum stabilizer codes
and extend previously known computational hardness results on decoding quantum
stabilizer codes.Comment: accepted by the 2019 IEEE International Symposium on Information
Theory (ISIT2019
Stabilizer codes from modified symplectic form
Stabilizer codes form an important class of quantum error correcting codes
which have an elegant theory, efficient error detection, and many known
examples. Constructing stabilizer codes of length is equivalent to
constructing subspaces of which are
"isotropic" under the symplectic bilinear form defined by . As a
result, many, but not all, ideas from the theory of classical error correction
can be translated to quantum error correction. One of the main theoretical
contribution of this article is to study stabilizer codes starting with a
different symplectic form.
In this paper, we concentrate on cyclic codes. Modifying the symplectic form
allows us to generalize the previous known construction for linear cyclic
stabilizer codes, and in the process, circumvent some of the Galois theoretic
no-go results proved there. More importantly, this tweak in the symplectic form
allows us to make use of well known error correcting algorithms for cyclic
codes to give efficient quantum error correcting algorithms. Cyclicity of error
correcting codes is a "basis dependent" property. Our codes are no more
"cyclic" when they are derived using the standard symplectic forms (if we
ignore the error correcting properties like distance, all such symplectic forms
can be converted to each other via a basis transformation). Hence this change
of perspective is crucial from the point of view of designing efficient
decoding algorithm for these family of codes. In this context, recall that for
general codes, efficient decoding algorithms do not exist if some widely
believed complexity theoretic assumptions are true
Improved belief propagation decoding algorithm based on decoupling representation of Pauli operators for quantum LDPC codes
We propose a new method called decoupling representation to represent Pauli
operators as vectors over GF(2), based on which we propose partially decoupled
belief propagation and fully decoupled belief propagation decoding algorithm
for quantum low density parity-check codes. Under the assumption that there is
no measurement error, compared with traditional belief propagation algorithm in
symplectic representation over GF(2), within the same number of iterations, the
decoding accuracy of partially decoupled belief propagation and fully decoupled
belief propagation algorithm is significantly improved in pure Y noise channel
and depolarizing noise channel, which supports that decoding algorithms of
quantum error correcting codes might have better performance in decoupling
representation than in symplectic representation. The impressive performance of
fully decoupled belief propagation algorithm might promote the realization of
quantum error correcting codes in engineering
Hardness results for decoding the surface code with Pauli noise
Real quantum computers will be subject to complicated, qubit-dependent noise,
instead of simple noise such as depolarizing noise with the same strength for
all qubits. We can do quantum error correction more effectively if our decoding
algorithms take into account this prior information about the specific noise
present. This motivates us to consider the complexity of surface code decoding
where the input to the decoding problem is not only the syndrome-measurement
results, but also a noise model in the form of probabilities of single-qubit
Pauli errors for every qubit.
In this setting, we show that Maximum Probability Error (MPE) decoding and
Maximum Likelihood (ML) decoding for the surface code are NP-hard and #P-hard,
respectively. We reduce directly from SAT for MPE decoding, and from #SAT for
ML decoding, by showing how to transform a boolean formula into a
qubit-dependent Pauli noise model and set of syndromes that encode the
satisfiability properties of the formula. We also give hardness of
approximation results for MPE and ML decoding. These are worst-case hardness
results that do not contradict the empirical fact that many efficient surface
code decoders are correct in the average case (i.e., for most sets of syndromes
and for most reasonable noise models). These hardness results are nicely
analogous with the known hardness results for MPE and ML decoding of arbitrary
stabilizer codes with independent and noise.Comment: 37 pages, 18 figures. 26 pages, 12 figures in main tex
Hardness of decoding stabilizer codes
RĂ©sumĂ© : Ce mĂ©moire porte sur lâĂ©tude de la complexitĂ© du problĂšme du dĂ©codage des codes stabilisateurs quantiques. Les trois premiers chapitres introduisent les notions nĂ©cessaires pour comprendre notre rĂ©sultat principal. Dâabord, nous rappelons les bases de la thĂ©orie de la complexitĂ© et illustrons les concepts qui sây rattachent Ă lâaide dâexemples tirĂ©s de la physique. Ensuite, nous expliquons le problĂšme du dĂ©codage des codes correcteurs classiques. Nous considĂ©rons les codes linĂ©aires sur le canal binaire symĂ©trique et nous discutons du cĂ©lĂšbre rĂ©sultat de McEliece et al. [1].
Dans le troisiĂšme chapitre, nous Ă©tudions le problĂšme de la communication quantique
sur des canaux de Pauli. Dans ce chapitre, nous introduisons le formalisme des codes stabilisateurs pour Ă©tudier la correction dâerreur quantique et mettons en Ă©vidence le concept de dĂ©gĂ©nĂ©rescence. Le problĂšme de dĂ©codage des codes stabilisateurs quantiques nĂ©gligeant la dĂ©gĂ©nĂ©rescence est appelĂ© «quantum maximum likelihood decoding»(QMLD). Il a Ă©tĂ© dĂ©montrĂ© que ce problĂšme est NP-complet par Min Hseiu Heish et al., dans [2]. Nous nous concentrons sur la stratĂ©gie optimale de dĂ©codage, appelĂ©e «degenerate quantum maximum likelihood decoding »(DQMLD), qui prend en compte la prĂ©sence de la dĂ©gĂ©nĂ©rescence et nous mettons en Ă©vidence quelques instances pour lesquelles les performances de ces deux mĂ©thodes
diffĂšrent drastiquement. La contribution principale de ce mĂ©moire est de prouver que DQMLD est considĂ©rablement plus difficile que ce que les rĂ©sultats prĂ©cĂ©dents indiquaient. Dans le dernier chapitre, nous prĂ©sentons notre rĂ©sultat principal (Thm. 5.1.1), Ă©tablissant que DQMLD est #P-complet. Pour le prouver, nous dĂ©montrons que le problĂšme de lâĂ©valuation de lâĂ©numĂ©rateur de poids dâun code linĂ©aire, qui est #P-complet, se rĂ©duit au problĂšme DQMLD. Le rĂ©sultat principal de ce mĂ©moire est prĂ©sentĂ© sous forme dâarticle dans [3] et est prĂ©sentement considĂ©rĂ© pour publication dans IEEE Transactions on Information Theory. Nous montrons Ă©galement que, sous certaines conditions, les rĂ©sultats de QMLD et DQMLD coĂŻncident. Il sâagit dâune amĂ©lioration par rapport aux rĂ©sultats obtenus dans [4, 5]. // Abstract : This thesis deals with the study of computational complexity of decoding stabilizer codes. The first three chapters contain all the necessary background to understand the main result of this thesis. First, we explain the necessary notions in computational complexity, introducing P, NP, #P classes of problems, along with some examples intended for physicists. Then, we explain the decoding problem in classical error correction, for linear codes on the binary symmetric channel and discuss the celebrated result of Mcleicee et al., in [1]. In the third chapter, we study the problem of quantum communication, over Pauli channels. Here, using the stabilizer formalism, we discuss the concept of degenerate errors. The decoding problem for stabilizer codes, which simply neglects the presence of degenerate errors, is called quantum maximum likelihood decoding (QMLD) and it was shown to be NP-complete, by Min Hseiu Heish et al., in [2]. We focus on the problem of optimal decoding, called degenerate quantum maximum likelihood decoding (DQMLD), which accounts for the presence of degenerate errors. We will highlight some instances of stabilizer codes, where the presence of degenerate errors causes drastic variations between the performances of DQMLD and QMLD. The main contribution of this thesis is to demonstrate that the optimal decoding problem for stabilizer codes is much harder than what the previous results had anticipated. In the last chapter, we present our own result (in Thm. 5.1.1), establishing that the optimal decoding problem for stabilizer codes, is #P-complete. To prove this, we demonstrate that the problem of evaluating the weight enumerator of a binary linear code, which is #P-complete, can be reduced (in polynomial time) to the DQMLD problem, see (Sec. 5.1). Our principal result is also presented as an article in [3], which is currently under review for publication in IEEE Transactions on Information Theory. In addition to the main result, we also show that under certain conditions, the outputs of DQMLD and QMLD always agree. We consider the conditions developed by us to be an improvement over the ones in [4, 5]
Exploiting Degeneracy in Belief Propagation Decoding of Quantum Codes
Quantum information needs to be protected by quantum error-correcting codes
due to imperfect physical devices and operations. One would like to have an
efficient and high-performance decoding procedure for the class of quantum
stabilizer codes. A potential candidate is Pearl's belief propagation (BP), but
its performance suffers from the many short cycles inherent in a quantum
stabilizer code, especially highly-degenerate codes. A general impression
exists that BP is not effective for topological codes. In this paper, we
propose a decoding algorithm for quantum codes based on quaternary BP with
additional memory effects (called MBP). This MBP is like a recursive neural
network with inhibitions between neurons (edges with negative weights), which
enhance the perception capability of a network. Moreover, MBP exploits the
degeneracy of a quantum code so that the most probable error or its degenerate
errors can be found with high probability. The decoding performance is
significantly improved over the conventional BP for various quantum codes,
including quantum bicycle, hypergraph-product, surface and toric codes. For MBP
on the surface and toric codes over depolarizing errors, we observe error
thresholds of 16% and 17.5%, respectively.Comment: 22 pages, 25 figures, 3 tables, and 3 algorithm
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