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Hardness of decoding stabilizer codes

Pavithran Sridharan Iyer

Published Jan 1, 2014• 0 citations
OpenAlex

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].

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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].
In this paper, 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 are appropriate 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 consider error degeneracy, which causes distinct errors to have the same effect on the code. Here, we show that optimal decoding of stabilizer codes (previously known to be NP-hard) is in fact computationally much harder than optimal decoding of classical linear codes, it is #P-complete.
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.
We classify the complexity classes of several important decoding problems for quantum stabilizer codes. First, regardless of the channel model, quantum bounded distance decoding is shown to be NP-hard, like what Berlekamp, McEliece and Tilborg did for classical binary linear codes in 1978. Then under the depolarizing channel, the decoding problems for finding a most likely error and for minimizing the decoding error probability are also shown to be NP-hard. Our results indicate that finding a polynomial-time decoding algorithm for general stabilizer codes may not be possible, but this, on the other hand, strengthens the foundation of quantum code-based cryptography.
We classify the time complexities of three important decoding problems for quantum stabilizer codes. First, regardless of the channel model, quantum bounded distance decoding is shown to be NP-hard, like what Berlekamp, McEliece and Tilborg did for classical binary linear codes in 1978. Then over the depolarizing channel, the decoding problems for finding a most likely error and for minimizing the decoding error probability are also shown to be NP-hard. Our results indicate that finding a polynomial-time decoding algorithm for general stabilizer codes may be impossible, but this, on the other hand, strengthens the foundation of quantum code-based cryptography.
Noise is a major obstacle in the development of practical schemes for quantum computation and communication. Similar to the case of classical communication, this noise can be protected against by employing a code, which provides a means for encoding quantum states prior to transmission and allows for errors to be inferred, and hopefully corrected, by a decoder at the receiver. Unfortunately, designing good codes and decoders is typically a difficult problem. This thesis focuses on developing low-complexity heuristic approaches to three such problems: the design of modified belief propagation decoders for quantum low-density parity-check codes, the design of stabilizer codes for asymmetric channels, and the design of codeword stabilized codes. Quantum low-density parity-check codes are stabilizer codes with low-weight generators. Such codes permit low-complexity decoding via the use of belief propagation, which is an iterative message passing algorithm that takes place on a factor graph defined by the code. However, the performance of such a decoder is limited both by code structure and the degenerate nature of quantum errors. To overcome these limitations, at least in part, a number of modifications to belief propagation are developed. Central among these is the augmented decoder, which in the case of a decoding error, iteratively reattempts decoding using modified factor graph. This heuristic modification simply involves the duplication of a randomly selected subset of the graph’s check nodes, which are in one-to-one correspondence with the code’s stabilizer generators. Across a range of codes, it is shown that the decoders developed perform as well as or better than other modified decoders presented in literature. For a number of channels of physical interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these so-called asymmetric channels, the decoding error rate can be minimized by tailoring the code used to the channel. However, assessing the performance of codes on a given channel is made difficult by the #P-completeness of optimal decoding. To address this complexity, it is shown that the decoding error rate can be accurately approximated using only a small fraction of the possible errors caused by the channel. This approximation is then used to identify a number of cyclic stabilizer codes that perform well on two different asymmetric channels. To further build on this, a heuristic is demonstrated for assessing code performance based on the decoding error rate of an associated classical code. The complexity of calculating this classical error rate is relatively low, and it is shown that it can be used as the basis for a hill-climbing search algorithm. Such searches have yielded a large number of highly performant codes satisfying various structure constraints. The family of codeword stabilized codes encompasses both the stabilizer codes as well as many of the best known nonadditive codes. Constructing a standard form codeword stabilized code is a matter of selecting a simple undirected graph and a binary classical code. This makes designing optimal codes difficult as the number of possible graphs grows exponentially with code length, and the clique search required to construct the classical code is NP-hard. To address the exponential growth of the search space, a heuristic is developed for assessing graphs. This heuristic is then employed by a genetic algorithm that also makes use of a novel crossover operation based on spectral bisection, which is show to be superior to more standard crossover operations. With a graph selected, it is demonstrated that the complexity of the clique search required to construct the associated classical code can be mitigated through the use of a heuristic clique finding algorithm. A number of best known codes are presented that have been found using this approach.
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 quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, 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 QMLD and DQMLD. 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 QMLD and DQMLD for arbitrary stabilizer codes with independent X and Z noise.