For a number of quantum channels of interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these asymmetric channels, the decoding error rate can be reduced by tailoring the code used to the channel. However, analyzing the performance of stabilizer codes on these channels is made difficult by the #P-completeness of optimal decoding. To address this, at least for short codes, we demonstrate that the decoding error rate can be approximated by considering only a fraction of the possible errors caused by the channel. Using this approximate error rate calculation, we extend a recent result to show that there are a number of $[[5\ensuremath{\le}n\ensuremath{\le}12,1\ensuremath{\le}k\ensuremath{\le}3]]$ cyclic stabilizer codes that perform well on two different asymmetric channels. We also demonstrate that an indication of a stabilizer code's error rate is given by considering the error rate of a classical binary code related to the stabilizer. This classical error rate is far less complex to calculate, and we use it as the basis for a hill-climbing algorithm, which we show to be effective at optimizing codes for asymmetric channels. Furthermore, we demonstrate that simple modifications can be made to our hill-climbing algorithm to search for codes with desired structure requirements.
For a number of quantum channels of interest, phase-flip errors occur far more frequently than bit-flip errors. When transmitting across these asymmetric channels, the decoding error rate can be reduced by tailoring the code used to the channel. However, analyzing the performance of stabilizer codes on these channels is made difficult by the #P-completeness of optimal decoding. To address this, at least for short codes, we demonstrate that the decoding error rate can be approximated by considering only a fraction of the possible errors caused by the channel. Using this approximate error rate calculation, we extend a recent result to show that there are a number of [[5\ensuremath≤n\ensuremath≤12,1\ensuremath≤k\ensuremath≤3]] cyclic stabilizer codes that perform well on two different asymmetric channels. We also demonstrate that an indication of a stabilizer code's error rate is given by considering the error rate of a classical binary code related to the stabilizer. This classical error rate is far less complex to calculate, and we use it as the basis for a hill-climbing algorithm, which we show to be effective at optimizing codes for asymmetric channels. Furthermore, we demonstrate that simple modifications can be made to our hill-climbing algorithm to search for codes with desired structure requirements.
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.
The performance of quantum error correction can be significantly improved if detailed information about the noise is available, allowing to optimize both codes and decoders. It has been proposed to estimate error rates from the syndrome measurements done anyway during quantum error correction. While these measurements preserve the encoded quantum state, it is currently not clear how much information about the noise can be extracted in this way. So far, apart from the limit of vanishing error rates, rigorous results have only been established for some specific codes. In this work, we rigorously resolve the question for arbitrary stabilizer codes. The main result is that a stabilizer code can be used to estimate Pauli channels with correlations across a number of qubits given by the pure distance. This result does not rely on the limit of vanishing error rates, and applies even if high weight errors occur frequently. Moreover, it also allows for measurement errors within the framework of quantum data-syndrome codes. Our proof combines Boolean Fourier analysis, combinatorics and elementary algebraic geometry. It is our hope that this work opens up interesting applications, such as the online adaptation of a decoder to time-varying noise.
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].
Efficient high-performance decoding of topological stabilizer codes has the potential to crucially improve the balance between logical failure rates and the number and individual error rates of the constituent qubits. High-threshold maximum-likelihood decoders require an explicit error model for Pauli errors to decode a specific syndrome, whereas lower-threshold heuristic approaches such as minimum-weight matching are error agnostic. Here we consider an intermediate approach, formulating a decoder that depends on the bias, i.e., the relative probability of phase-flip to bit-flip errors, but is agnostic to error rate. Our decoder is based on counting the number and effective weight of the most likely error chains in each equivalence class of a given syndrome. We use Metropolis-based Monte Carlo sampling to explore the space of error chains and find unique chains that are efficiently identified using a hash table. Using the error-rate invariance, the decoder can sample chains effectively at an error rate which is higher than the physical error rate and without the need for thermalization between chains in different equivalence classes. Applied to the surface code and the XZZX code, the decoder matches maximum-likelihood decoders for moderate code sizes or low error rates. We anticipate that, because of the compressed information content per syndrome, it can be taken full advantage of in combination with machine-learning methods to extrapolate Monte Carlo--generated data.
The family of codeword stabilized codes encompasses the stabilizer codes as well as many of the best known nonadditive codes. However, constructing optimal n-qubit codeword stabilized codes is made difficult by two main factors. The first of these is the exponential growth with n of the number of graphs on which a code can be based. The second is the NP-hardness of the maximum clique search required to construct a code from a given graph. We address the second of these issues through the use of a heuristic clique finding algorithm. This approach has allowed us to find $((9,97\ensuremath{\le}K\ensuremath{\le}100,2))$ and $((11,387\ensuremath{\le}K\ensuremath{\le}416,2))$ codes, which are larger than any previously known codes. To address the exponential growth of the search space, we demonstrate that graphs that give large codes typically yield clique graphs with a large number of nodes. The number of such nodes can be determined relatively efficiently, and we demonstrate that n-node graphs yielding large clique graphs can be found using a genetic algorithm. This algorithm uses a crossover operation based on spectral bisection that we demonstrate to be superior to more standard crossover operations. Using this genetic algorithm approach, we have found ((13, 18, 4)) and ((13, 20, 4)) codes that are larger than any previously known code. We also consider codes for the amplitude damping channel. We demonstrate that for $n\ensuremath{\le}9$, optimal codeword stabilized codes correcting a single amplitude damping error can be found by considering standard form codes that detect one of only three of the 3n possible equivalent error sets. By combining this error-set selection with the genetic algorithm approach, we have found ((11, 68)) and ((11, 80)) codes capable of correcting a single amplitude damping error and ((11, 4)), ((12, 4)), ((13, 8)), and ((14, 16)) codes capable of correcting two amplitude damping errors.
The quantum hashing bound guarantees that rates up to 1−H(pI,pX,pY,pZ) are achievable for memoryless Pauli channels, but it is not generally tight. A known way to improve achievable rates for certain asymmetric Pauli channels is to apply a small inner stabilizer code to a few channel uses, decode, and treat the resulting logical noise as an induced Pauli channel; reapplying the hashing argument to this induced channel can beat the baseline hashing bound. We generalize this induced-channel viewpoint to arbitrary stabilizer codes used purely as channel transforms. Given any $ [\![ n, k ]\!] $ stabilizer generator set, we construct a full symplectic tableau, compute the induced joint distribution of logical Pauli errors and syndromes under the physical Pauli channel, and obtain an achievable rate via a hashing bound with decoder side information. We perform a structured search over small transforms and report instances that improve the baseline hashing bound for a family of Pauli channels with skewed and independent errors studied in prior work.