Heuristics in quantum error correction

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 $[[5\leq n\leq12,1\leq k\leq3]]$ 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.

Quantum error correction is pivotal in advancing toward large-scale quantum computation, and efficient verification is crucial for ensuring the high fidelity of code states. Traditional methodologies, such as state tomography, direct fidelity estimation, and state verification, either fall short in measurement efficiency, especially for large-scale systems, or are restricted to some specific states. In this work, we introduce a general framework for quantum subspace verification, enabling efficient and measurement-noise-robust fidelity estimation between a given state and the target subspace with a specified confidence level. By integrating the proposed subspace verification with direct fidelity estimation, we develop a composite protocol that significantly improves the efficiency of verifying the fidelity of general magic logical states, as demonstrated by intuitive numerical results. This improvement stems from the use of subspace verification, which leverages the knowledge of code subspaces to significantly reduce measurement costs. Additionally, we detail the construction of verification operators for typical error correction codes, including general stabilizer codes and quantum low-density parity-check codes, enabling their efficient implementation using practical local measurements. Notably, for certain codes, such as the Calderbank-Shor-Steane codes and quantum low-density parity-check stabilizer codes, we reduce the number of required measurement settings and sample complexity to a constant level using graphical methods. Our approach facilitates efficient and feasible verification of error correction codes and generic magic logical states, advancing their practical implementations on quantum platforms.

Chapter 10 - Quantum LDPC Codes

Chapter 7 - Quantum error correction fundamentals

Erasures are the primary type of errors in physical systems dominated by leakage errors. While quantum error correction (QEC) using stabilizer codes can combat erasure errors, it remains unknown which constructions achieve capacity performance. If such codes exist, decoders with linear runtime in the code length are also desired. In this paper, we present erasure capacity-achieving quantum codes under maximum-likelihood decoding (MLD), though MLD requires cubic runtime in the code length. For QEC, using an accurate decoder with the shortest possible runtime will minimize the degradation of quantum information while awaiting the decoder’s decision. To address this, we propose belief propagation (BP) decoders that run in linear time and exploit error degeneracy in stabilizer codes, achieving capacity or near-capacity performance for a broad class of codes, including bicycle codes, product codes, and topological codes. We furthermore explore the potential of our BP decoders to handle mixed erasure and depolarizing errors, and also local deletion errors via concatenation with permutation invariant codes.

In this paper, we estimate the fidelity of stabilizer and CSS codes. First, we derive a lower bound on the fidelity of a stabilizer code via its quantum enumerator. Next, we find the average quantum enumerators of the ensembles of finite length stabilizer and CSS codes. We use the average quantum enumerators for obtaining lower bounds on the average fidelity of these ensembles. We further improve the fidelity bounds by estimating the quantum enumerators of expurgated ensembles of stabilizer and CSS codes. Finally, we derive fidelity bounds in the asymptotic regime when the code length tends to infinity. These results tell us which code rate we can afford for achieving a target fidelity with codes of a given length. The results also show that in symmetric depolarizing channel a typical stabilizer code has better performance, in terms of fidelity and code rate, compared with a typical CSS codes, and that balanced CSS codes significantly outperform other CSS codes. Asymptotic results demonstrate that CSS codes have a fundamental performance loss compared with stabilizer codes.

The subject of this chapter is the use of classical/quantum information theory and coding in genetics and evolution. The chapter starts with the description of using the concepts from both classical and quantum information theories to describe the evolution of biological channel capacity through generations. In order to do so, several classical and quantum biological channel models are employed including the Markovian classical and Markovian-like quantum model, hybrid quantum-classical model, multilevel symmetric channel model, and Kimura model-based Markovian process. In order to describe the reliable long-time storage of genetic information in DNA, the use of unequal error protection (UEP) coding is studied. Several classes of error-correction codes suitable for UEP on a cellular level are described including nested coding, multilevel coding (MLC), rate-adaptive coding, and generalized LDPC coding. The use of concepts of constrained coding to describe the genetic information flow from DNA to proteins is also described as well as joint-constrained and error-correction coding. After that, the use of quantum error-correction concepts to deal with environmental errors including canonical quantum error-correction and stabilizer codes is briefly described. One particular class of stabilizer codes, known as topological codes, is then described that might be relevant to biological processes as they only involve the local qubits in encoding process. Another relevant class of codes, the subsystem codes, is then described. The key idea behind subsystem codes is to decompose the quantum code as the tensor product of two subsystems, exon subsystem A and intron subsystem B, and we are concerned with correcting errors only on the exon subsystem. Finally, we describe the use of nonbinary quantum stabilizer codes to deal with nucleobase substitution errors, both random and burst errors. We also briefly discuss the possible use of both classical and quantum error-correction concepts to improve tolerance to tumor and cancer introducing errors.

Constructing quantum low-density parity-check (LDPC) codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large $X$-distance but a much smaller $Z$-distance. However, together with a classical expander LDPC code and a tensoring method that generalizes a construction of Hastings and also the Tillich--Zémor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance $n^{1/2}\log^{1/2}n$. We then exploit the expansion properties of the complex to devise the first polynomial-time algorithm that decodes above the square root barrier for quantum LDPC codes. Using a 3-dimensional Ramanujan complex, we also obtain an overall quantum code of minimum distance $n^{1/2}\log n$, which sets a new record for quantum LDPC codes.

Quantum error correction is a set of methods to protect quantum information—that is, quantum states—from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state. Most work on quantum error correction has focused on systems of quantum bits, or qubits, which are two-level quantum systems. These can be physically realized by the states of a spin-1/2 particle, the polarization of a single photon, two distinguished levels of a trapped atom or ion, the current states of a microscopic superconducting loop, or many other physical systems. The most widely used codes are the stabilizer codes, which are closely related to classical linear codes. The code space is the joint +1 eigenspace of a set of commuting Pauli operators on n qubits, called stabilizer generators; the error syndrome is determined by measuring these operators, which allows errors to be diagnosed and corrected. A stabilizer code is characterized by three parameters [[n,k,d]], where n is the number of physical qubits, k is the number of encoded logical qubits, and d is the minimum distance of the code (the smallest number of simultaneous qubit errors that can transform one valid codeword into another). Every useful code has n>k; this physical redundancy is necessary to detect and correct errors without disturbing the logical state. Quantum error correction is used to protect information in quantum communication (where quantum states pass through noisy channels) and quantum computation (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of fault-tolerant design. Other approaches to error mitigation in quantum systems include decoherence-free subspaces, noiseless subsystems, and dynamical decoupling.

Chapter 7 - Quantum Error Correction

Quantum error correction (QEC) is a way to protect quantum information against noise. It consists of encoding input information into entangled quantum states known as the code space. Furthermore, to classify if the encoded information is corrupted or intact, a step known as syndrome detection is performed. For stabilizer codes, this step consists of measuring a set of stabilizer operators. In this paper, inspired by the QEC approach, and specifically stabilizer codes, we propose a prototype quantum circuit model for classification of classical data. The core quantum circuit can be considered as a quantum perceptron where the classification is based on syndrome detection. In this proposal, a quantum perceptron is realized by one stabilizer as part of a stabilizer code, while a quantum neural network (QNN) layer is realized by a stabilizer code which consists of many stabilizers. The concatenation of stabilizer codes results in complex QNNs. The QNN is trained by performing measurements and optimization of a set of parametrized stabilizers. We demonstrate the concept numerically. In this paper we also consider the first challenge to most applications of quantum computers, including data classification, which is to load data into the memory of the quantum computer. This loading amounts to representing the data as a quantum state, i.e., quantum state preparation. An exact amplitude encoding algorithm requires a circuit of exponential depth. We introduce an alternative recursive algorithm which approximates amplitude encoding with only a polynomial number of elementary gates. We name it the recursive approximate-scheme algorithm.

Polar coding is the error correction coding proposed by Arikan. In polar coding, the error rate of the decoding result differs for each bit position owing to the channel polarization. In the case of polar coding with an infinite codeword length, the decoding error rate at each position of the decoded bit sequence is polarized to 0.5 or 0. Therefore, when a codeword of a sufficient length can be adopted, information bits will be decoded without errors by placing the frozen bits to bit positions where the decoding error rates are 0.5 at the transmitter. On the other hand, in the case of polar coding with a finite codeword length, the decoding error rate at each bit position takes a value between 0.5 and 0. Therefore, the decoding errors also occur at the bit positions where the information bits are placed. In this paper, we introduce half-frozen bits into polar coding. Half-frozen bits are the information bits which play roles like frozen bits. Half-frozen bits are placed at positions where errors hardly occur within the frozen bit positions as well as where errors prone to occur within the information bit positions. As half-frozen bits, a part of information bits is used after being applied another error correction coding. This improves the decoding error rates at information bit positions which result in high decoding error rates. Therefore, half-frozen bits improve the overall bit error rate (BER) performance of polar coding. Through computer simulation, we show that the polar coding with introducing half-frozen bits provides a higher error correction performance than the original polar coding.

Quantum error correction and the use of quantum error correction codes are likely to be essential for the realization of practical quantum computing. Because the error models of quantum devices vary widely, quantum codes that are tailored for a particular error model may have much better performance. In this work, we present a novel evolutionary algorithm that searches for an optimal stabilizer code for a given error model, number of physical qubits, and number of encoded qubits. We demonstrate an efficient representation of stabilizer codes as binary strings, which allows for random generation of valid stabilizer codes as well as mutation and crossing of codes. Our algorithm finds stabilizer codes whose distance closely matches the best-known-distance codes of Grassl (2007) for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n \leq 20$</tex-math></inline-formula> physical qubits. We perform a search for optimal distance Calderbank–Steane–Shor codes and compare their distance to the best known codes. Finally, we show that the algorithm can be used to optimize stabilizer codes for biased error models, demonstrating a significant improvement in the undetectable error rate for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[[12,1]]_{2}$</tex-math></inline-formula> codes versus the best-known-distance code with the same parameters. As part of this work, we also introduce an evolutionary algorithm QDistEvol for finding the distance of quantum error correction codes.

Chapter 8 - Quantum stabilizer codes and beyond

We analyze simple quantum error detection and quantum error correction protocols relevant to current experiments with superconducting qubits. We show that for qubits with energy relaxation the repetitive $N$-qubit codes cannot be used for quantum error correction, but can be used for quantum error detection. In the latter case it is sufficient to use only two qubits for the encoding. In the analysis we demonstrate a useful technique of unraveling the qubit energy relaxation into ``relaxation'' and ``no relaxation'' scenarios. Also, we propose and numerically analyze several two-qubit algorithms for quantum error detection and correction, which can be readily realized at the present-day level of the phase qubit technology.