Fault-tolerant quantum computation with quantum error-correcting codes has been considerably developed over the past decade. However, there are still difficult issues, particularly on the resource requirement. For further improvement of fault-tolerant quantum computation, here we propose a soft-decision decoder for quantum error correction and detection by teleportation. This decoder can achieve almost optimal performance for the depolarizing channel. Applying this decoder to Knill's C4/C6 scheme for fault-tolerant quantum computation, which is one of the best schemes so far and relies heavily on error correction and detection by teleportation, we dramatically improve its performance. This leads to substantial reduction of resources.
Fault-tolerant quantum computation with quantum error-correcting codes has been considerably developed over the past decade. However, there are still difficult issues, particularly on the resource requirement. For further improvement of fault-tolerant quantum computation, here we propose a soft-decision decoder for quantum error correction and detection by teleportation. This decoder can achieve almost optimal performance for the depolarizing channel. Applying this decoder to Knill's C4/C6 scheme for fault-tolerant quantum computation, which is one of the best schemes so far and relies heavily on error correction and detection by teleportation, we dramatically improve its performance. This leads to substantial reduction of resources.
A soft-decision decoder for quantum erasure errors with standard depolarizing errors is proposed for concatenated Calderbank-Shor-Steane codes. This decoding is computationally efficient as long as the block size of each concatenation level is sufficiently small. The numerical simulation with a specific quantum code called the C4/C6 code shows that this decoder can achieve almost optimal performance and the so-called hashing bound for the erasure and depolarizing error model. Since probabilistic-gate errors can be regarded as erasure errors, this decoder is useful for fault-tolerant quantum computation with probabilistic gates. To demonstrate this, we perform numerical simulation with the C4/C6 code. Consequently, it turns out that the error probability of a logical controlled-not gate with probabilistic physical two-qubit gates is improved by three orders of magnitude by the present decoder compared to a previous hard-decision decoder.
Quantum error mitigation (QEM) is typically viewed as a suite of practical techniques for today's noisy intermediate-scale quantum devices, with limited relevance once fault-tolerant quantum computers become available. In this work, we challenge this conventional wisdom by showing that QEM can continue to provide substantial benefits in the era of quantum error correction (QEC), and in an even more efficient manner than it does on current devices. We introduce a framework for logical-level QEM that leverages soft information naturally produced by QEC decoders, requiring no additional data, hardware modifications, or runtime overhead beyond what QEC protocols already provide. Within this framework, we develop and analyze three logical-level QEM techniques: post-selection and runtime abort policies, probabilistic error cancellation, and zero-noise extrapolation. Our techniques reduce logical error rates by more than 100x while discarding fewer than 0.1% of shots; they also provide in situ characterization of logical channels for QEM protocols. As a proof of principle, we benchmark our approach using a surface-code architecture and two state-of-the-art decoders based on tensor-network contraction and minimum-weight perfect matching. We evaluate logical-level QEM on random Clifford circuits and molecular simulation algorithms and find that, compared to previous approaches relying on QEC only or QEC combined with QEM, we can achieve up to 87.4% spacetime overhead savings. Our results demonstrate that logical-level QEM with QEC decoder soft information can reliably improve logical performance, underscoring the efficiency and usefulness of QEM techniques for fault-tolerant quantum computers.
Quantum computation promises to revolutionize information processing by exploiting superposition and entanglement, enabling applications such as quantum simulation, combinatorial optimization, and secure quantum communication. Beyond individual processors, the quantum internet aims to connect quantum nodes through entanglement distribution, enabling distributed computation. However, quantum states are fragile: environmental interactions, imperfect gate operations, and measurement errors can quickly degrade encoded information. Reliable quantum computation therefore requires carefully designed quantum error-correcting codes and fault-tolerant protocols. Unlike classical redundancy schemes, quantum error correction must preserve delicate phase relationships and relies on indirect error detection through syndrome extraction with ancillary qubits, enabling error identification without collapsing encoded states. Topological codes, in particular, emerged early as the preferred choice for many architectures due to their planar structure, local interactions, and natural support for fault-tolerant logical operations. Color codes, for instance, allow transversal implementation of all Clifford gates, whereas non-Clifford operations require magic-state injection. More recent constructions, including quantum low-density parity-check codes, offer higher encoding rates and reduced resource overhead, but require long-range qubit connectivity and more complex protocols. Despite experimental progress, several challenges remain. Many platforms exhibit biased noise, motivating codes that exploit channel asymmetry. Real-time decoding remains a critical bottleneck, as fault-tolerant computation requires low-latency processing of rapidly generated syndrome data. Fault-tolerant logical state initialization also remains resource-intensive and scales poorly with code distance. Addressing these challenges, this thesis develops a unified framework spanning theory and experiment. It provides precise performance analysis of quantum error-correcting codes in the low physical error-rate regime, introduces codes tailored to biased noise, develops fast low-complexity decoding algorithms for real-time operation, and presents modular and efficient protocols for fault-tolerant state preparation of CSS codes. The latter is demonstrated experimentally through the preparation of a logical zero state of the Golay code on the Quantinuum H2 trapped-ion quantum computer.
The decoder is a critical component of a fault-tolerant quantum computer, computing corrections based on parity-check measurements performed throughout the computation. A soft decoder supplements its output with a confidence score which, when used alongside post-selection, can substantially improve logical performance. We introduce a soft beam search decoder for quantum low-density parity-check codes that uses internal decoder data as a confidence metric, removing the need for extra computation. We perform circuit-level simulations of five quantum LDPC codes relevant for superconducting and trapped ion architectures equipped with our global soft decoder and we obtain up to 580× logical-error suppression at a physical error rate of 10−3 while rejecting only 0.1% of shots. Then, we simulate an error detected measurement, which is a core logical operation of the walking cat architecture, using a streaming version of soft beam decoder and we achieve up to 210× error suppression while increasing the rejection probability by only 0.5 percentage points. Finally, we revisit recent quantum LDPC code memory experiments on trapped ions, demonstrating that our soft decoder doubles the logical qubit lifetimes at the price of a mean rejection rate of 2.6%--5.6% per syndrome round, bringing all five codes into the beyond-breakeven regime.
Quantum computers have the potential to solve several interesting problems in polynomial \ntime for which no polynomial time classical algorithms have been found. However, \none of the major challenges in building quantum devices is that quantum systems are very \nsensitive to noise arising from undesired interactions with the environment. Noise can lead \nto errors which can corrupt the results of the computation. Quantum error correction is \none way to mitigate the effects of noise arising in quantum devices. \nWith a plethora of quantum error correcting codes that can be used in various settings, \none of the main challenges of quantum error correction is understanding how well various \ncodes perform under more realistic noise models that can be observed in experiments. \nThis thesis proposes a new decoding algorithm which can optimize threshold values of \nerror correcting codes under different noise models. The algorithm can be applied to \nany Markovian noise model. Further, it is shown that for certain noise models, logical \nClifford corrections can further improve a code's threshold value if the code obeys certain \nsymmetries. \n \nSince gates and measurements cannot in general be performed with perfect precision, \nthe operations required to perform quantum error correction can introduce more errors \ninto the system thus negating the benefits of error correction. Fault-tolerant quantum \ncomputing is a way to perform quantum error correction with imperfect operations while \nretaining the ability to suppress errors as long as the noise is below a code's threshold. \nOne of the main challenges in performing fault-tolerant error correction is the high resource \nrequirements that are needed to obtain very low logical noise rates. With the use of \nflag qubits, this thesis develops new fault-tolerant error correction protocols that are applicable \nto arbitrary distance codes. Various code families are shown to satisfy the requirements \nof flag fault-tolerant error correction. We also provide circuits using a constant number of \nqubits for these codes. It is shown that the proposed flag fault-tolerant method uses fewer \nqubits than previous fault-tolerant error correction protocols. \n \nIt is often the case that the noise afflicting a quantum device cannot be fully characterized. \nFurther, even with some knowledge of the noise, it can be very challenging to use \nanalytic decoding methods to improve the performance of a fault-tolerant scheme. This \nthesis presents decoding schemes using several state of the art machine learning techniques \nwith a focus on fault-tolerant quantum error correction in regimes that are relevant to near \nterm experiments. It is shown that even in low noise rate regimes and with no knowledge \nof the noise, noise can be further suppressed for small distance codes. Limitations of machine \nlearning decoders as well as the classical resources required to perform active error \ncorrection are discussed. \n \nIn many cases, gate times can be much shorter than typical measurement times of \nquantum states. Further, classical decoding of the syndrome information used in quantum \nerror correction to compute recovery operators can also be much slower than gate times. \nFor these reasons, schemes where error correction can be implemented in a frame (known \nas the Pauli frame) have been developed to avoid active error correction. In this thesis, we \ngeneralize previous Pauli frame schemes and show how Clifford frame error correction can \nbe implemented with minimal overhead. Clifford frame error correction is necessary if the \nlogical component of recovery operators were chosen from the Clifford group, but could \nalso be used in randomized benchmarking schemes.
Abstract Imperfect measurements are a prevalent source of error across quantum computing platforms, significantly degrading the logical error rates achievable on current hardware. To mitigate this issue, rich measurement data referred to as soft information has been proposed to efficiently identify and correct measurement errors as they occur. In this work, we model soft information decoding across a variety of physical qubit platforms and decoders and showcase how soft information can make error correction viable at lower code distances and higher physical error rates than is otherwise possible. We simulate the effects of soft information decoding on quantum memories for surface codes and bivariate bicycle codes, and evaluate the error suppression performance of soft decoders against traditional decoders. Our simulations show that soft information decoding on near-term devices can provide up to 11% higher error suppression on superconducting qubits and up to 20% stronger error suppression on neutral atom qubits. These accuracy gains correspond to 13% and 33% reductions in the physical qubit footprint of superconducting and neutral atom devices respectively when operating at a logical error rate of 10 −6 , showcasing that soft information is a powerful tool for reducing the cost and complexity of large-scale fault-tolerant quantum computers.