The construction of large, coherent quantum systems necessary for quantum computation remains an entreating but elusive goal, due to the ubiquitous nature of decoherence. Recent progress in quantum error correction schemes have given new hope to this field, but thus far, the codes presented in the literature assume a restricted number of errors and error free encoding, decoding, and measurement. We investigate a specific scenario without these assumptions; in particular, we evaluate a scheme to preserve a single quantum bit against phase damping using a three-qubit encoding based on Shor. By applying a new formalism which gives simple operators for decoherence and noisy logic gates, we find the fidelity of the stored qubit as a function of time, including decoherence which occurs not only during storage but also during processing. We generalize our results to include any source of error, and derive an upper limit on the allowable decoherence per timestep. Physically, our results suggest the feasibility of engineering artificial metastable states through repeated error correction.The construction of large, coherent quantum systems necessary for quantum computation remains an entreating but elusive goal, due to the ubiquitous nature of decoherence. Recent progress in quantum error correction schemes have given new hope to this field, but thus far, the codes presented in the literature assume a restricted number of errors and error free encoding, decoding, and measurement. We investigate a specific scenario without these assumptions; in particular, we evaluate a scheme to preserve a single quantum bit against phase damping using a three-qubit encoding based on Shor. By applying a new formalism which gives simple operators for decoherence and noisy logic gates, we find the fidelity of the stored qubit as a function of time, including decoherence which occurs not only during storage but also during processing. We generalize our results to include any source of error, and derive an upper limit on the allowable decoherence per timestep. Physically, our results suggest the feasibility of engineering artificial metastable states through repeated error correction.

arXiv:quant-ph/9604030Apr 25, 19962 citations

A naturally error suppressing quantum memory

Fumiko Yamaguchi, Y. Yamamoto

We propose a method to construct quantum storage wherein the phase error due to decoherence is naturally suppressed without constant error detection and correction. As an example, we describe a quantum memory made of two physical qubits encoded in the ground state of a two-qubit phase-error detecting code. Such a system can be simulated by introducing a coupling between the two physical qubits. This method is effective for physical systems in which the $T_1$ decay process is negligible but coherence is limited by the $T_2$ decay process. We take trapped ions as a possible example to apply the natural suppression method and show that the $T_2$ decay time due to slow ambient fluctuating fields at the physical qubits can be lengthened as much as $10^4$ .We propose a method to construct quantum storage wherein the phase error due to decoherence is naturally suppressed without constant error detection and correction. As an example, we describe a quantum memory made of two physical qubits encoded in the ground state of a two-qubit phase-error detecting code. Such a system can be simulated by introducing a coupling between the two physical qubits. This method is effective for physical systems in which the $T_1$ decay process is negligible but coherence is limited by the $T_2$ decay process. We take trapped ions as a possible example to apply the natural suppression method and show that the $T_2$ decay time due to slow ambient fluctuating fields at the physical qubits can be lengthened as much as $10^4$.

arXiv:quant-ph/0603159Mar 18, 20060 citations

Repeated quantum error correction on a continuously encoded qubit by real-time feedback

Julia Cramer, Norbert Kalb, Michiel Adriaan Rol +6 more

Reliable quantum information processing in the face of errors is a major fundamental and technological challenge. Quantum error correction protects quantum states by encoding a logical quantum bit (qubit) in multiple physical qubits. To be compatible with universal fault-tolerant computations, it is essential that states remain encoded at all times and that errors are actively corrected. Here we demonstrate such active error correction on a continuously protected logical qubit using a diamond quantum processor. We encode the logical qubit in three long-lived nuclear spins, repeatedly detect phase errors by non-destructive measurements, and apply corrections by real-time feedback. The actively error-corrected qubit is robust against errors and encoded quantum superposition states are preserved beyond the natural dephasing time of the best physical qubit in the encoding. These results establish a powerful platform to investigate error correction under different types of noise and mark an important step towards fault-tolerant quantum information processing.Reliable quantum information processing in the face of errors is a major fundamental and technological challenge. Quantum error correction protects quantum states by encoding a logical quantum bit (qubit) in multiple physical qubits. To be compatible with universal fault-tolerant computations, it is essential that states remain encoded at all times and that errors are actively corrected. Here we demonstrate such active error correction on a continuously protected logical qubit using a diamond quantum processor. We encode the logical qubit in three long-lived nuclear spins, repeatedly detect phase errors by non-destructive measurements, and apply corrections by real-time feedback. The actively error-corrected qubit is robust against errors and encoded quantum superposition states are preserved beyond the natural dephasing time of the best physical qubit in the encoding. These results establish a powerful platform to investigate error correction under different types of noise and mark an important step towards fault-tolerant quantum information processing.

arXiv:1508.01388Nature CommunicationsMay 5, 2016247 citations

Quantum Decoherence-Based Quantum Computing Error Correction Code

Jincheng Zhang

This paper proposes a novel quantum computing error correction code predicated on the principles of quantum decoherence. The core concept leverages the inherent fragility of quantum states to create a robust error detection and correction mechanism. Specifically, the code aims to control and manipulate the decoherence time of individual qubits, utilizing this temporal constraint within a structured error correction scheme. The theoretical framework details the encoding, decoding, and error correction processes, incorporating mathematical formulations to quantify the redundancy introduced and the resulting improvement in qubit fidelity. The proposed code's effectiveness is demonstrated through simulations, showcasing its potential to significantly enhance the reliability of quantum computations, a critical step towards achieving scalable quantum computing. The key contributions lie in the direct integration of decoherence dynamics into the error correction strategy, offering a fundamentally different approach compared to traditional methods. The paper concludes with a discussion of potential optimizations and future research directions.This paper proposes a novel quantum computing error correction code predicated on the principles of quantum decoherence. The core concept leverages the inherent fragility of quantum states to create a robust error detection and correction mechanism. Specifically, the code aims to control and manipulate the decoherence time of individual qubits, utilizing this temporal constraint within a structured error correction scheme. The theoretical framework details the encoding, decoding, and error correction processes, incorporating mathematical formulations to quantify the redundancy introduced and the resulting improvement in qubit fidelity. The proposed code's effectiveness is demonstrated through simulations, showcasing its potential to significantly enhance the reliability of quantum computations, a critical step towards achieving scalable quantum computing. The key contributions lie in the direct integration of decoherence dynamics into the error correction strategy, offering a fundamentally different approach compared to traditional methods. The paper concludes with a discussion of potential optimizations and future research directions.

Sep 3, 20260 citations

Effects of noise on quantum error correction algorithms

Adriano Barenco, Todd A. Brun, Rüdiger Schack +1 more

It has recently been shown that there are efficient algorithms for quantum computers to solve certain problems, such as prime factorization, which are intractable to date on classical computers. The chances for practical implementation, however, are limited by decoherence, in which the effect of an external environment causes random errors in the quantum calculation. To combat this problem, quantum error correction schemes have been proposed, in which a single quantum bit (qubit) is ``encoded'' as a state of some larger number of qubits, chosen to resist particular types of errors. Most such schemes are vulnerable, however, to errors in the encoding and decoding itself. We examine two such schemes, in which a single qubit is encoded in a state of $n$ qubits while subject to dephasing or to arbitrary isotropic noise. Using both analytical and numerical calculations, we argue that error correction remains beneficial in the presence of weak noise, and that there is an optimal time between error correction steps, determined by the strength of the interaction with the environment and the parameters set by the encoding.It has recently been shown that there are efficient algorithms for quantum computers to solve certain problems, such as prime factorization, which are intractable to date on classical computers. The chances for practical implementation, however, are limited by decoherence, in which the effect of an external environment causes random errors in the quantum calculation. To combat this problem, quantum error correction schemes have been proposed, in which a single quantum bit (qubit) is ``encoded'' as a state of some larger number of qubits, chosen to resist particular types of errors. Most such schemes are vulnerable, however, to errors in the encoding and decoding itself. We examine two such schemes, in which a single qubit is encoded in a state of $n$ qubits while subject to dephasing or to arbitrary isotropic noise. Using both analytical and numerical calculations, we argue that error correction remains beneficial in the presence of weak noise, and that there is an optimal time between error correction steps, determined by the strength of the interaction with the environment and the parameters set by the encoding.

arXiv:quant-ph/9612047Physical Review AAug 1, 199734 citations

Use of Shor states for the [7,1,3] quantum error-correcting code

Yaakov S. Weinstein, Sidney Buchbinder

We explore the effect of Shor-state construction methods on logical state encoding and quantum error-correction for the [7,1,3] Calderbank-Shor-Steane quantum error-correction code in a nonequiprobable-error environment. We determine the optimum number of verification steps to be used in Shor-state construction and whether Shor states without verification are usable for practical quantum computation. These results are compared to the same processes of encoding and error correction where Shor states are not used. We demonstrate that the construction of logical zero states with no first-order error terms may not require the complete edifice of quantum fault tolerance. With respect to error correction, we show for a particular initial state that error correction using a single qubit for the syndrome measurement yields a similar output-state accuracy to error correction using Shor states as syndrome qubits. In addition, we demonstrate that error correction with Shor states has an inherent sensitivity to bit-flip errors. Finally, we suggest that in this type of error-correction scenario one should always repeat a syndrome measurement until attaining an all-zero readout (twice in row).We explore the effect of Shor-state construction methods on logical state encoding and quantum error-correction for the [7,1,3] Calderbank-Shor-Steane quantum error-correction code in a nonequiprobable-error environment. We determine the optimum number of verification steps to be used in Shor-state construction and whether Shor states without verification are usable for practical quantum computation. These results are compared to the same processes of encoding and error correction where Shor states are not used. We demonstrate that the construction of logical zero states with no first-order error terms may not require the complete edifice of quantum fault tolerance. With respect to error correction, we show for a particular initial state that error correction using a single qubit for the syndrome measurement yields a similar output-state accuracy to error correction using Shor states as syndrome qubits. In addition, we demonstrate that error correction with Shor states has an inherent sensitivity to bit-flip errors. Finally, we suggest that in this type of error-correction scenario one should always repeat a syndrome measurement until attaining an all-zero readout (twice in row).

arXiv:1111.3930Physical Review ANov 30, 201214 citations

T_1

decay process is negligible but coherence is limited by the

T_2

decay process. We take trapped ions as a possible example to apply the natural suppression method and show that the

T_2

decay time due to slow ambient fluctuating fields at the physical qubits can be lengthened as much as

10^4

n

qubits while subject to dephasing or to arbitrary isotropic noise. Using both analytical and numerical calculations, we argue that error correction remains beneficial in the presence of weak noise, and that there is an optimal time between error correction steps, determined by the strength of the interaction with the environment and the parameters set by the encoding.