[Paper Review] Simulating the Effects of Quantum Error-correction Schemes

J. Niwa, Keiji Matsumoto+1 more|ArXiv.org|Nov 13, 2002

Quantum Computing Algorithms and Architecture17 references5 citations

TL;DR

This paper simulates the performance of three quantum error-correcting codes—five, seven, and nine qubit codes—under realistic conditions where both decoherence and operational errors occur, including errors introduced during the error-correction process itself. The study finds that the seven-qubit code is most effective when decoherence rates are below 10⁻⁴ and operational error standard deviations are below 10⁻³, especially when error correction is applied every 50–2000 gates.

ABSTRACT

It is important to protect quantum information against decoherence and operational errors, and quantum error-correcting (QEC) codes are the keys to solving this problem. Of course, just the existence of codes is not efficient. It is necessary to perform operations fault-tolerantly on encoded states because error-correction process (i.e., encoding, decoding, syndrome measurement and recovery) itself induces an error. By using simulation, this paper investigates the effects of some important QEC codes (the five qubit code, the seven qubit code and the nine qubit code) and their fault-tolerant operations when the error-correction process itself induces an error. The corresponding results, statistics and analyses are presented in this paper.

Motivation & Objective

• To evaluate the real-world effectiveness of simple quantum error-correcting codes (QECC) under combined decoherence and operational errors.
• To investigate how error-correction operations themselves introduce errors, potentially degrading performance.
• To compare the fidelity of different QECC schemes—five, seven, and nine qubit codes—under realistic simulation conditions.
• To determine optimal error-correction frequency and operational thresholds for maintaining high fidelity in quantum computation.
• To assess the impact of transversal operations and ancilla design on error correction performance.

Proposed method

• Implemented a scalable quantum computer simulation system (QCSS) capable of simulating up to 30 qubits using parallel computing on multi-computers.
• Modeled decoherence using a depolarizing channel where each qubit experiences X, Y, or Z errors with equal probability p/3, and no error with probability 1−p.
• Modeled operational errors by introducing Gaussian-distributed deviations in rotation and phase shift angles, with standard deviation σ.
• Simulated fault-tolerant circuits for the five, seven, and nine qubit codes, including encoding, syndrome measurement, and recovery operations.
• Measured fidelity by averaging repeated simulations to estimate performance under error accumulation.
• Evaluated performance across varying decoherence rates (10⁻⁵ to 10⁻³) and operational error standard deviations (10⁻⁴ to 10⁻²), with error correction applied at intervals of 50 to 2000 main gates.

Experimental results

Research questions

• RQ1 How effective are the five, seven, and nine qubit quantum error-correcting codes when both decoherence and operational errors are present?
• RQ2 What is the impact of error-correction operations themselves on the overall fidelity of quantum computation?
• RQ3 At what decoherence rate and operational error standard deviation does the QEC scheme begin to degrade performance?
• RQ4 How does the frequency of error-correction operations affect the fidelity of quantum computation?
• RQ5 Which code—five, seven, or nine qubit—provides the highest fidelity under realistic error conditions?

Key findings

• The seven-qubit code outperforms both the five- and nine-qubit codes in terms of fidelity across all tested decoherence rates and operational error levels.
• When decoherence rate is below 10⁻⁴ and operational error standard deviation is below 10⁻³, the seven-qubit code maintains high fidelity even with error correction applied every 50–2000 main gates.
• The combined effect of decoherence and operational errors is multiplicative, meaning performance degrades more severely than with either error source alone.
• For decoherence rates of 10⁻³, the nine-qubit code achieves the highest fidelity during the first 200 computation steps, though this is not sustainable over longer computations.
• The five-qubit code requires at least 21 steps to complete one main computation step due to encoding and decoding overhead, significantly reducing its effective fidelity compared to the seven-qubit code.
• The nine-qubit code’s simpler QEC circuit results in fewer induced errors during correction, contributing to better performance than the five-qubit code despite higher qubit count.