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arXiv:1510.00121 · Physical Review A

Method for quantum-jump continuous-time quantum error correction

Kung-Chuan Hsu, Todd A. Brun

Published Feb 16, 2016• 12 citations
arXivHTMLPDF

Abstract

Continuous-time quantum error correction (CTQEC) is a technique for protecting quantum information against decoherence, where both the decoherence and error correction processes are considered continuous in time. Given any $[[n,k,d]]$ quantum stabilizer code, we formulate a class of protocols to implement CTQEC, involving weak coherent measurements and weak unitary corrections. Under this formalism, we show that the minimal required size of the ancillary system is $n\ensuremath{-}k+1$ qubits, and we propose one scheme that meets this minimal requirement. Furthermore, we compare our method with other known schemes, and show that a particular measure of performance described in this paper is better when using our method.

DOI
10.1103/physreva.93.022321

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Continuous-time quantum error correction (CTQEC) is an approach to protecting quantum information from noise in which both the noise and the error correcting operations are treated as processes that are continuous in time. This chapter investigates CTQEC based on continuous weak measurements and feedback from the point of view of the subsystem principle, which states that protected quantum information is contained in a subsystem of the Hilbert space. We study how to approach the problem of constructing CTQEC protocols by looking at the evolution of the state of the system in an encoded basis in which the subsystem containing the protected information is explicit. This point of view allows us to reduce the problem to that of protecting a known state, and to design CTQEC procedures from protocols for the protection of a single qubit. We show how previously studied CTQEC schemes with both direct and indirect feedback can be obtained from strategies for the protection of a single qubit via weak measurements and weak unitary operations. We also review results on the performance of CTQEC with direct feedback in cases of Markovian and non-Markovian decoherence, where we have shown that due to the existence of a Zeno regime in non-Markovian dynamics, the performance of CTQEC can exhibit a quadratic improvement if the time resolution of the weak error-correcting operations is high enough to reveal the non-Markovian character of the noise process.
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.
Canonical discrete quantum error correction (DQEC) schemes use projective von Neumann measurements on stabilizers to discretize the error syndromes into a finite set, and fast unitary gates are applied to recover the corrupted information. Quantum error correction (QEC) based on continuous measurement, known as continuous quantum error correction (CQEC), in principle, can be executed faster than DQEC and can also be resource efficient. However, CQEC requires meticulous filtering of noisy continuous measurement data to reliably extract error syndromes on the basis of which errors could be detected. In this paper, we show that by constructing a measurement-based estimator (MBE) of the logical qubit to be protected, which is driven by the noisy continuous measurement currents of the stabilizers, it is possible to accurately track the errors occurring on the physical qubits in real time. We use this MBE to develop a continuous quantum error correction (MBE-CQEC) scheme that can protect the logical qubit to a high degree, surpassing the performance of DQEC, and also allows QEC to be conducted either immediately or in delayed time with instantaneous feedbacks.
Quantum Error Correction (QEC) is an important technique for protecting quantum information against decoherence and errors. This involves the design and implementation of algorithms and techniques to minimize error rates and increase the stability of quantum circuits. One of the key parameters in QEC is the distance of the error- correcting code, which determines the number of errors that can be corrected. Another important parameter is the error probability, which quantifies the likelihood of errors occurring in the quantum system. In this context, the goal of a simulation sweeps like the one performed in the code is to study the performance of the QEC code for different values of the distance and error probability, and to optimize the code for maximum accuracy. By varying these parameters and observing the performance of the code, researchers can gain insights into how to design better codes and improve the reliability of quantum computing systems. We also discuss the challenges that need to be addressed for quantum computing to realize its potential in solving practical Error-correction problems.
Quantum Error Correction (QEC) is an important technique for protecting quantum information against decoherence and errors. This involves the design and implementation of algorithms and techniques to minimize error rates and increase the stability of quantum circuits. One of the key parameters in QEC is the distance of the error- correcting code, which determines the number of errors that can be corrected. Another important parameter is the error probability, which quantifies the likelihood of errors occurring in the quantum system. In this context, the goal of a simulation sweep like the one performed in the code is to study the performance of the QEC code for different values of the distance and error probability, and to optimize the code for maximum accuracy. By varying these parameters and observing the performance of the code, researchers can gain insights into how to design better codes and improve the reliability of quantum computing systems. We also discuss the challenges that need to be addressed for quantum computing to realize its potential in solving practical Error-correction problems.