Fault-tolerant quantum computing (FTQC) uses quantum error correction (QEC) codes to execute large-scale quantum programs on noisy quantum computers. Quantum low-density parity-check (qLDPC) codes are promising as they use an order of magnitude fewer qubits than widely used surface codes. However, as qLDPC codes support only a limited set of operations, they require programs to be decomposed into many qLDPC-supported operations. This prohibitively increases the execution time of FTQC applications to tens of days, hindering the practical viability of qLDPC codes.
Fault-tolerant quantum computing (FTQC) uses quantum error correction (QEC) codes to execute large-scale quantum programs on noisy quantum computers. Quantum low-density parity-check (qLDPC) codes are promising as they use an order of magnitude fewer qubits than widely used surface codes. However, as qLDPC codes support only a limited set of operations, they require programs to be decomposed into many qLDPC-supported operations. This prohibitively increases the execution time of FTQC applications to tens of days, hindering the practical viability of qLDPC codes.
Quantum error correction (QEC) is a cornerstone of quantum computing, enabling reliable information processing in the presence of noise. Sparse stabilizer codes -- referred to generally as quantum low-density parity-check (QLDPC) codes -- have risen to the forefront of QEC research in recent years. This can be attributed to several key factors. First, classical LDPC codes admit low-complexity belief propagation iterative decoding and near-capacity performance, which contributed to the early interest in QLDPC codes. Then, the result promising constant overhead fault tolerance using QLDPC codes led to the search for code families that go beyond the long-holding n scaling barrier of minimum distance for codelength n. This resulted in recent breakthroughs in the construction of QLDPC codes, which, combined with efficient decoding algorithms and the development of fault-tolerant protocols operating on QLDPC-encoded quantum information, provide a promising pathway to low-overhead, fault-tolerant quantum computation. However, despite their potential, challenges remain, particularly in constructing and decoding finite-length codes that account for, or efficiently leverage, specific characteristics of quantum hardware, such as connectivity, topology, native gate sets, and noise models. This article provides an in-depth examination of QLDPC codes and their iterative decoders, catering to an information theory audience with no or limited background in quantum mechanics. We discuss the theoretical underpinnings, explore unique characteristics of quantum channels, and delineate key code constructions and decoding algorithms, ultimately highlighting the impact and future prospects of QLDPC codes in quantum information science.
Fault-tolerant quantum computing (FTQC) is required to execute quantum algorithms capable of tackling computationally intractable problems in chemistry, material science, and cryptography. Noisy Intermediate-Scale Quantum (NISQ) devices remain constrained by physical error rates that degrade quantum coherence over short circuit depths. Quantum error correction (QEC) protects fragile logical states by encoding them into non-local degrees of freedom across topological physical qubit arrays. This research presents a comprehensive architectural and theoretical evaluation of three primary QEC paradigms: planar surface codes, topological color codes, and asymptotic quantum low-density parity-check (qLDPC) codes. We examine error thresholds, transversal gate constraints governed by the Eastin-Knill theorem, magic state distillation (MSD) overheads, lattice surgery, and classical decoding latencies across superconducting transmons, trapped ions, and neutral atom platforms.
We propose a fault-tolerant quantum computation scheme that is broadly applicable to quantum low-density parity-check (qLDPC) codes. The scheme achieves constant qubit overhead and a time overhead of O ( d a + o ( 1 ) ) for any [ [ n , k , d ] ] qLDPC code with constant encoding rate and distance d = Ω ( n 1 / a ) . For good qLDPC codes, the time overhead is minimized and reaches O ( d 1 + o ( 1 ) ) . In contrast, code surgery based on gauging measurement and brute-force branching requires a time overhead of O ( d w 1 + o ( 1 ) ) , where d ≤ w ≤ n . Thus, our scheme is asymptotically faster for all codes with a < 2 . This speedup is achieved by developing techniques that enable parallelized code surgery under constant qubit overhead and leverage classical locally testable codes for efficient resource state preparation. These results establish a new paradigm for accelerating fault-tolerant quantum computation on qLDPC codes, while maintaining low overhead and broad applicability.
High-rate quantum low-density parity-check (qLDPC) codes are a leading candidate for fault-tolerant quantum computing. They feature higher encoding rates than planar alternatives such as the surface code, but their implementation often entails significant hardware hurdles like the need for long-range couplers. We leverage the flexibility of a trapped-ion quantum computer to demonstrate nine quantum error-correcting codes with starkly different qubit connectivity requirements on a single device without any hardware reconfiguration. These experiments span three families of quantum error-correcting codes: qLDPC codes, topological codes, and concatenated codes. With a qLDPC code encoding 4 logical qubits into 18 physical qubits, we achieve a logical error rate up to 9× better than a previous demonstration of a similar code on superconducting solid-state qubits. Moreover, our implementation exhibits breakeven performance, with some instances achieving qubit lifetimes comparable to or slightly exceeding that of our trapped-ion qubits. We use a novel implementation of the optical-metastable-ground (OMG) architecture for addressable mid-circuit measurement and reset, which enables us to perform these experiments without any ion transport or dedicated coolant ions, requirements that typically consume a large fraction of the runtime or ion count of trapped-ion quantum computers.
Quantum Error Correction (QEC) is an essential field of research towards the realization of large-scale quantum computers. On the theoretical side, a lot of effort is put into designing error-correcting codes that protect quantum data from errors, which inevitably happen due to the noisy nature of quantum hardware and quantum bits (qubits). Protecting data with an error-correcting code necessitates means to recover the original data, given a potentially corrupted data set---a task referred to as decoding. It is vital that decoding algorithms can recover error-free states in an efficient manner. While theoretical properties of certain QEC methods have been extensively studied, good techniques to analyze their performance in practically more relevant settings is still a widely unexplored area. In this work, we propose a set of software tools that facilitate numerical experiments with so-called Quantum Low-Density Parity-Check codes (QLDPC codes)---a broad class of codes, some of which have recently been shown to be asymptotically good. Based on that, we provide an implementation of a general decoder for QLDPC codes. On top of that, we propose a highly efficient heuristic decoder that eliminates the runtime bottlenecks of the general QLDPC decoder while still maintaining comparable decoding performance. These tools eventually make it possible to confirm theoretical results around QLDPC codes in a more practical setting and showcase the value of software tools (in addition to theoretical considerations) for investigating codes for practical applications. The resulting tool, which is publicly available at https://github.com/cda-tum/qecc as part of the Munich Quantum Toolkit (MQT), is meant to provide a playground for the search for "practically good" quantum codes.