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AIP conference proceedings

Continuous Quantum Error Correction

Mohan Sarovar

Published Jan 1, 2004• 3 citations
Publisher

Abstract

Abstract. We describe new implementations of quantum error correction that are continuous in time, and thus described by continuous dynamical maps. We evaluate the performance of such schemes using numerical simulations, and comment on the effectiveness and applicability of continuous error correction for quantum computing.
DOI
10.1063/1.1834397

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Quantum error correction is an essential ingredient for quantum computation. The standard descriptions of how to implement active error correction assume ideal resources such as projective measurements and instantaneous gate operations. Unfortunately in practice such resources are not realizable in most quantum computing architectures and it is not clear how such error correction implementations will perform under more realistic conditions. Motivated by this we examine schemes for implementing active error correction that use a more modest set of resources. This leads to new implementations of error correction that are continuous in time, and thus described by continuous dynamical maps. We evaluate the performance of such schemes using numerical simulations and comment on the applicability and effectiveness of continuous error correction for quantum computing.
Abstract: Design a quantum error correction scheme optimized for the unique characteristics of dynamical systems, enabling more reliable computation of complex models. Core Claim: Design a quantum error correction scheme optimized for the unique characteristics of dynamical systems, enabling more reliable computation of complex models. Implement a quantum error correction protocol specifically tailored to the dynamic nature of the system, leveraging quantum entanglement for error detection. Core Mechanism: Implement a quantum error correction protocol specifically tailored to the dynamic nature of the system, leveraging quantum entanglement for error detection. Why It Is New: Current error correction is largely ineffective against dynamic systems. This approach offers a fundamentally new approach.
Quantum mechanical applications range from quantum computers to quantum key distribution to teleportation. In these applications, quantum error correction is extremely important for protecting quantum states against decoherence. Here I present two main results regarding quantum error correction protocols. The first main topic I address is the development of continuous-time quantum error correction protocols via combination with techniques from quantum control. These protocols rely on weak measurement and Hamiltonian feedback instead of the projective measurements and unitary gates usually assumed by canonical quantum error correction. I show that a subclass of these protocols can be understood as a quantum feedback protocol, and analytically analyze the general case using the stabilizer formalism; I show that in this case perfect feedback can perfectly protect a stabilizer subspace. I also show through numerical simulations that another subclass of these protocols does better than canonical quantum error correction when the time between corrections is limited. The second main topic is development of improved overhead results for fault-tolerant computation. In particular, through analysis of topological quantum error correcting codes, it will be shown that the required blowup in depth of a noisy circuit performing a fault-tolerant computation can be reduced to a factor of O(log log L), an improvement over previous results. Showing this requires investigation into a local method of performing fault-tolerant correction on a topological code of arbitrary dimension.
Quantum computing promises revolutionary advancements across diverse fields, but the inherent fragility of quantum information necessitates robust error correction techniques. Current methods often rely on meticulous calibration procedures, limiting their applicability and hindering scalability. This paper presents a novel dynamical quantum error correction scheme leveraging adaptive thresholding, designed to mitigate the impact of system noise through a continuous feedback loop. We detail the proposed mechanism, its underlying mathematical principles, and a comprehensive evaluation of its performance against established techniques. The core claim highlights the significant improvement in resilience to noise, offering a more adaptable and robust strategy for quantum error correction.
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.