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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Logical zeros for the seven-qubit quantum error correction code

Gerald N. Gilbert, Yaakov S. Weinstein

Published May 13, 2011• 0 citations
Publisher

Abstract

In this work we compare the accuracy of two methods used to construct a logical zero state appropriate for the [7, 1, 3] CSS quantum error correction code in a non-equiprobable Pauli operator error environment. The first method is to apply error correction, via syndrome measurement, on seven physical qubits all in the state zero. To do the syndrome measurements in a fault-tolerant fashion requires the construction of four qubit Shor states. These Shor states are also assumed to be constructed in a non-equiprobable Pauli operator error environment and it is these that are used to implement the syndrome measurement. The second construction method is to implement the [7, 1, 3] encoding gate sequence, also in the nonequiprobable Pauli operator error environment. The fidelity of the output states is calculated for each of these methods. With respect to the Shor state construction we find that the implementation of (noisy) parity based verifications does not necessarily raise the fidelity of the resulting Shor state. We also find that the second logical zero construction method outputs a seven qubit state with a respectfully higher fidelity than the first (fault tolerant) method. However, the fidelity of the single qubit of stored information has almost equivalent fidelity from the two construction methods.

DOI
10.1117/12.884024

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In this work we compare the accuracy of two methods used to construct a logical zero state appropriate for the [7, 1, 3] CSS quantum error correction code in a non-equiprobable Pauli operator error environment. The first method is to apply error correction, via syndrome measurement, on seven physical qubits all in the state zero. To do the syndrome measurements in a fault-tolerant fashion requires the construction of four qubit Shor states. These Shor states are also assumed to be constructed in a non-equiprobable Pauli operator error environment and it is these that are used to implement the syndrome measurement. The second construction method is to implement the [7, 1, 3] encoding gate sequence, also in the nonequiprobable Pauli operator error environment. The fidelity of the output states is calculated for each of these methods. With respect to the Shor state construction we find that the implementation of (noisy) parity based verifications does not necessarily raise the fidelity of the resulting Shor state. We also find that the second logical zero construction method outputs a seven qubit state with a respectfully higher fidelity than the first (fault tolerant) method. However, the fidelity of the single qubit of stored information has almost equivalent fidelity from the two construction methods.
I calculate the fidelity of a [7,1,3][7,1,3][7,1,3] Calderbank-Shor-Steane quantum error correction code logical zero state constructed in a nonequiprobable Pauli operator error environment for two methods of encoding. The first method is to apply fault-tolerant error correction to an arbitrary state of seven qubits utilizing Shor states for syndrome measurement. The Shor states are themselves constructed in the nonequiprobable Pauli operator error environment, and their fidelity depends on the number of verifications done to ensure multiple errors will not propagate into the encoded quantum information. Surprisingly, performing these verifications may lower the fidelity of the constructed Shor states. The second encoding method is to simply implement the [7,1,3][7,1,3][7,1,3] encoding gate sequence also in the nonequiprobable Pauli operator error environment. Perfect error correction is applied after both methods to determine the correctability of the implemented errors. I find that which method attains higher fidelity depends on which of the Pauli operators errors is dominant. Nevertheless, perfect error correction applied after the encoding suppresses errors to at least first order for both methods.
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 calculate the fidelity with which an arbitrary state can be encoded into a [7,1,3] CSS quantum error correction code in a non-equiprobable Pauli operator error environment with the goal of determining whether this encoding can be used for practical implementations of quantum computation. This determination is accomplished by applying ideal error correction to the encoded state which demonstrates the correctability of errors that occurred during the encoding process. We then apply single-qubit Clifford gates to the encoded state and determine the accuracy with which these gates can be applied. Finally, fault tolerant noisy error correction is applied to the encoded states in the non-equiprobable Pauli operator error environment allowing us to compare noisy (realistic) and perfect error correction implementations. We note that this maintains the fidelity of the encoded state for certain error-probability values. These results have implications for when non-fault tolerant procedures may be used in practical quantum computation and whether quantum error correction should be applied at every step in a quantum protocol.
We explore basic necessary protocols to achieve fault tolerance for quantum computation in the cluster state and circuit models. For the cluster state model we simulate the implementation of an arbitrary rotation via only measurement on a decohered cluster state. Fidelity is used to quantify the accuracy of the initial cluster state and a gate fidelity for the arbitrary rotation is determined. In the circuit model we compare the accuracy of two methods that can be used to construct a logical zero state appropriate for the [7, 1, 3] Steane quantum error correction code in a non-equiprobable Pauli operator error environment: a fault tolerant method done by applying error correction on seven qubits all in the state zero, and a non-fault tolerant method done by implementing the encoding gate sequence. We find that the latter construction method, in spite of its lack of "fault tolerance," outputs a seven qubit state with a higher fidelity than the first (fault tolerant) method. However, the fidelity of the single qubit of stored information exhibits almost equivalent values between the two construction methods.
In this work we explore the accuracy of quantum error correction depending of the order of the implemented syndrome measurements. CSS codes require bit-flip and phase flip-syndromes be measured separately. To comply with fault tolerant demands and to maximize accuracy this set of syndrome measurements should be repeated allowing for flexibility in the order of their implementation. We examine different possible orders of Shor state and Steane state syndrome measurements for the [[7,1,3]] quantum error correction code. We find that the best choice of syndrome order, determined by the fidelity of the state after noisy error correction, will depend on the error environment. We also compare the fidelity when syndrome measurements are done with Shor states versus Steane states and find that Steane states generally, but not always, lead to final states with higher fidelity. Together, these results allow a quantum computer programmer to choose the optimal syndrome measurement scheme based on the system's error environment.
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.