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arXiv:1303.4026 · Physical Review A

Relative performance of ancilla verification and decoding in the [[7,1,3]] Steane code

Ali Abu-Nada, Ben Fortescue, Mark Byrd

Published Jun 3, 2014• 10 citations
arXivHTMLPDF

Abstract

Ancilla postselection is a common means of achieving fault tolerance in quantum error correction. However, it can lead to additional data errors due to movement or wait operations. Alternatives to postselection may achieve lower overall failure rates due to avoiding such errors. We present numerical simulation results comparing the logical error rates for the fault-tolerant [[7,1,3]] Steane code using techniques of ancilla verification vs the newer method of ancilla decoding, as described in Phys. Rev. Lett. 98, 020501 (2007). We simulate QEC procedures in which the possibility of ancilla verification failures requires the creation and storage of additional ancillae and/or additional waiting of the data until a new ancilla can be created. We find that the decoding method, which avoids verification failures, is advantageous in terms of overall error rate in various cases, even when measurement operations are no slower than others. We analyze the effect of different classes of physical error on the relative performance of these two methods.

DOI
10.1103/physreva.89.062304

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A common assumption is that one applies a fault-tolerant quantum error correction (FTQEC) after every gate during quantum computing. However, it is known that this is not always optimal, since the FTQEC procedure itself can introduce errors. Here we vary the number of logical gates between FTQEC operations given that a failure of a postselection condition may cause the FTQEC to be skipped. We derive an expression for the logical error rate as a function of the error-correction frequency and find the optimal frequency for the application of the FTQEC. Furthermore, we show that this is relatively insensitive to the postselection failure probability for a large range of such probabilities. We provide an example of the application of the analytic expression to the ⟦7,1,3⟧ Steane code and data derived from a Monte Carlo simulation.

arXiv:1504.04057•Physical Review Letters•Nov 9, 2017•2 citations
Ancilla postselection is a common means of achieving fault tolerance in quantum error correction. However, it can lead to additional data errors due to movement or wait operations. Alternatives to postselection may achieve lower overall failure rates due to avoiding such errors. We present numerical simulation results comparing the logical error rates for the fault-tolerant [[7,1,3]] Steane code using techniques of ancilla verification vs the newer method of ancilla decoding, as described in Phys. Rev. Lett. 98, 020501 (2007). We simulate QEC procedures in which the possibility of ancilla verification failures requires the creation and storage of additional ancillae and/or additional waiting of the data until a new ancilla can be created. We find that the decoding method, which avoids verification failures, is advantageous in terms of overall error rate in various cases, even when measurement operations are no slower than others. We analyze the effect of different classes of physical error on the relative performance of these two methods.
Communication is the prototypical application of error-correction methods. To communicate, a sender needs to convey information to a receiver over a noisy "communication channel." Such a channel can be thought of as a means of transmitting an information-carrying physical system from one place to another. During transmission, the physical system is subject to disturbances (noise) that can adversely affect the information carried. To use a communication channel, the sender needs to encode the information to be transmitted in the physical system. After transmission, the receiver decodes the information. Quantum error correction is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal with both noise on stored quantum information, and also with faulty quantum gates, faulty quantum preparation,and faulty measurements. In this dissertation, we look at how additional information about the structure of the quantum circuit and noise can improve or alter the performance of techniques in quantum error correction. Chapter 1 and 2, are an introduction to the quantum computation, quantum error correction codes and fault-tolerant quantum computing. These chapters are written to be a useful for students at the graduate and advanced undergraduate level. Also. The first two chapters of this dissertation will be useful to researchers in other fields who would like to understand how quantum error correction and fault-tolerant quantum computing are possible. In chapter 3, we present numerical simulation results comparing the logical error rates for the fault-tolerant [[7, 1, 3]] 's 7 code using the technique of ancilla verification vs. the newer method of ancilla decoding as described in [1]. In chapter 4, we determine how often one should apply error correction. Therefore, we provide a relationship between the logical error rate and the physical error rate for a sequence of logical gates, sometimes followed by noisy quantum error correction
Fault tolerant quantum error correction is a procedure which satisfies the feature that if one of the gates in the procedure has failed then the failure causes at most one error in the output qubits of the encoded block. Quantum computer is based on the idea of two quantum state systems (Qubits). However, the majority of systems are constructed from higher than two- level subspace. Bad control and environmental interactions in these systems lead to leakage fault. Leakage errors are errors that couple the states inside a code subspace to the states outside a code subspace. One example for leakage fault is loss errors. Since the fault tolerant procedure may be unable to recognize the leakage fault because it was designed to deal with Pauli errors. In that case a single leakage fault might disrupt the fault tolerant technique. In this thesis we investigate the effect of ancilla losses on fault-tolerant quantum error correction in the [[7,1,3]] Steane code. We proved that both Shor and Steane methods are still fault tolerant if loss errors occur.
Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [ ⁣[7,1,3] ⁣][\![7,1,3]\!][[7,1,3]] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors - a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we're trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor's cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane's encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM's Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4×\times× compared to standard approaches. Our implementations achieve logical error rates as low as 5.1×10−55.1 \times 10^{-5}5.1×10−5 under realistic noise conditions with physical error rates of 10−310^{-3}10−3, while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.
Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [[7,1,3]] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors—a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we’re trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor’s cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane’s encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM’s Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4×compared to standard approaches. Our implementations achieve logical error rates as low as 5.1 ×10^−5 under realistic noise conditions with physical error rates of 10^−3, while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.
Abstract Building reliable quantum computers requires protecting fragile quantum states from inevitable environmental noise and operational errors. While quantum error correction codes like the Steane [7,1,3] code provide elegant theoretical solutions, their practical success hinges critically on how we measure errors—a process called syndrome extraction. The challenge lies in the ancilla qubits used for measurement: when they fail, errors can cascade across the entire quantum system, destroying the very information we're trying to protect. We address this fundamental problem by implementing and comparing three sophisticated syndrome measurement strategies: Shor's cat-state approach, which distributes measurements across multiple entangled ancillas achieving 85-92% preparation success; Steane's encoded-ancilla method using complete error-corrected logical qubits reaching 97.8% syndrome fidelity; and a flexible unified framework that adapts strategies based on hardware capabilities. Through extensive simulations using IBM's Qiskit platform spanning randomized benchmarking and T-heavy circuits, we demonstrate that intelligent ancilla management improves error suppression by up to 2.4x compared to standard approaches. Our implementations achieve logical error rates as low as 5.1 x 10 -5 under realistic noise conditions with physical error rates of 10 -3 , while maintaining near-unity logical fidelity (0.99997) even for deep circuits. The threshold analysis reveals robust performance across distance-3 to distance-13 codes with characteristic threshold curves showing exponential error suppression below the critical physical error rate. These results provide immediately deployable tools for near-term quantum devices and establish practical design principles for scaling toward fault-tolerant quantum computers.
A common assumption is that one applies a fault-tolerant quantum error correction (FTQEC) after every gate during quantum computing. However, it is known that this is not always optimal, since the FTQEC procedure itself can introduce errors. Here we vary the number of logical gates between FTQEC operations given that a failure of a postselection condition may cause the FTQEC to be skipped. We derive an expression for the logical error rate as a function of the error-correction frequency and find the optimal frequency for the application of the FTQEC. Furthermore, we show that this is relatively insensitive to the postselection failure probability for a large range of such probabilities. We provide an example of the application of the analytic expression to the ⟦7,1,3⟧ Steane code and data derived from a Monte Carlo simulation.