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arXiv:1702.07688 · Physical Review A

What determines the ultimate precision of a quantum computer

Xavier Waintal

Published Apr 10, 2019• 9 citations
arXivHTMLPDF

Abstract

A quantum error correction (QEC) code uses ${N}_{\mathrm{c}}$ quantum bits to construct one ``logical'' quantum bit of better quality than the original ``physical'' ones. QEC theory predicts that the failure probability ${p}_{L}$ of logical qubits decreases exponentially with ${N}_{\mathrm{c}}$ provided the failure probability $p$ of the physical qubit is below a certain threshold $p<{p}_{\mathrm{th}}$. In particular, QEC theorems imply that the logical qubits can be made arbitrarily precise by simply increasing ${N}_{c}$. In this article, we search for physical mechanisms that lie outside of the hypothesis of QEC theorems and set a limit ${\ensuremath{\eta}}_{\mathrm{L}}$ to the precision of the logical qubits (irrespectively of ${N}_{c}$). ${\ensuremath{\eta}}_{\mathrm{L}}$ directly controls the maximum number of operations $\ensuremath{\propto}1/{\ensuremath{\eta}}_{\mathrm{L}}^{2}$ that can be performed before the logical quantum state gets randomized, hence the depth of the quantum circuits that can be considered. We identify a type of error---silent stabilizer failure---as a mechanism responsible for finite ${\ensuremath{\eta}}_{\mathrm{L}}$ and discuss its possible causes. Using the example of the topological surface code, we show that a single local event can provoke the failure of the logical qubit, irrespectively of ${N}_{c}$.

DOI
10.1103/physreva.99.042318

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A quantum error correction (QEC) code uses Nc{N}_{\mathrm{c}}Nc​ quantum bits to construct one logical'' quantum bit of better quality than the original physical'' ones. QEC theory predicts that the failure probability pL{p}_{L}pL​ of logical qubits decreases exponentially with Nc{N}_{\mathrm{c}}Nc​ provided the failure probability ppp of the physical qubit is below a certain threshold p<pthp<{p}_{\mathrm{th}}p<pth​. In particular, QEC theorems imply that the logical qubits can be made arbitrarily precise by simply increasing Nc{N}_{c}Nc​. In this article, we search for physical mechanisms that lie outside of the hypothesis of QEC theorems and set a limit \ensuremathηL{\ensuremath{\eta}}_{\mathrm{L}}\ensuremathηL​ to the precision of the logical qubits (irrespectively of Nc{N}_{c}Nc​). \ensuremathηL{\ensuremath{\eta}}_{\mathrm{L}}\ensuremathηL​ directly controls the maximum number of operations \ensuremath∝1/\ensuremathηL2\ensuremath{\propto}1/{\ensuremath{\eta}}_{\mathrm{L}}^{2}\ensuremath∝1/\ensuremathηL2​ that can be performed before the logical quantum state gets randomized, hence the depth of the quantum circuits that can be considered. We identify a type of error---silent stabilizer failure---as a mechanism responsible for finite \ensuremathηL{\ensuremath{\eta}}_{\mathrm{L}}\ensuremathηL​ and discuss its possible causes. Using the example of the topological surface code, we show that a single local event can provoke the failure of the logical qubit, irrespectively of Nc{N}_{c}Nc​.
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Abstract Quantum computers are powerful tools for certain computational problems, but their fragile quantum states are easily disrupted by environmental noise, causing errors that degrade computational accuracy. Quantum error correction (QEC) addresses this challenge by encoding one logical qubit across multiple physical qubits, so that individual errors can be identified and reversed through redundancy. We used Qiskit Aer, a free open-source quantum circuit simulator, to test the repetition code, one of the simplest QEC methods, under three types of noise: bit-flip, phase-flip, and depolarizing. We compared an unprotected single-qubit baseline (n = 1) to 3-qubit (n = 3) and 5-qubit (n = 5) repetition codes, sweeping the physical error rate p from 0 to 0.15 with 4,000 simulation shots per data point. We hypothesized that larger codes would achieve higher logical success probabilities at all tested noise levels, with n = 5 outperforming n = 3, which would outperform the unprotected baseline. Our results confirmed this scaling trend across all three noise types. At p = 0.10 under bit-flip noise, the baseline achieved 0.713 logical success, while n = 3 reached 0.791 and n = 5 reached 0.843. At low noise (p = 0.02), the n = 5 code approached 0.994 logical success. Simulation results closely matched exact theoretical predictions derived from the binomial failure probability formula, validating our experimental implementation. These findings confirm that even the simplest QEC methods provide meaningful error reduction and that larger codes offer greater protection, demonstrating a foundational principle of quantum error correction using only free simulation tools.
Abstract Quantum computers are powerful tools for certain computational problems, but their fragile quantum states are easily disrupted by environmental noise, causing errors that degrade computational accuracy. Quantum error correction (QEC) addresses this challenge by encoding one logical qubit across multiple physical qubits, so that individual errors can be identified and reversed through redundancy. We used Qiskit Aer, a free open-source quantum circuit simulator, to test the repetition code, one of the simplest QEC methods, under three types of noise: bit-flip, phase-flip, and depolarizing. We compared an unprotected single-qubit baseline (n = 1) to 3-qubit (n = 3) and 5-qubit (n = 5) repetition codes, sweeping the physical error rate p from 0 to 0.15 with 4,000 simulation shots per data point. We hypothesized that larger codes would achieve higher logical success probabilities at all tested noise levels, with n = 5 outperforming n = 3, which would outperform the unprotected baseline. Our results confirmed this scaling trend across all three noise types. At p = 0.10 under bit-flip noise, the baseline achieved 0.713 logical success, while n = 3 reached 0.791 and n = 5 reached 0.843. At low noise (p = 0.02), the n = 5 code approached 0.994 logical success. Simulation results closely matched exact theoretical predictions derived from the binomial failure probability formula, validating our experimental implementation. These findings confirm that even the simplest QEC methods provide meaningful error reduction and that larger codes offer greater protection, demonstrating a foundational principle of quantum error correction using only free simulation tools.
Utility-scale quantum computers require quantum error correcting codes with large numbers of physical qubits to achieve sufficiently low logical error rates. The performance of quantum error correction (QEC) is generally predicted through large-scale numerical simulations, used to estimate thresholds, finite-size scaling, and exponential suppression of logical errors below threshold. The connection of QEC to models from statistical mechanics provides an alternative tool for analysing QEC performance. However, predicting the behaviour of these models also requires large-scale numerical simulations, as analytic solutions are not generally known. Here we exploit an exact mapping, from a toric code under bit-flip noise that is post-selected on being syndrome free to the exactly-solvable two-dimensional Ising model on a torus, to derive an analytic solution for the logical failure rate across its full domain of physical error rates. In particular, this mapping provides closed-form expressions for the logical failure rate in four distinct regimes: the path-counting, below-threshold (ordered), near-threshold (critical), and above-threshold (disordered) regimes. Our framework places a number of familiar and long-standing numerical observations on firm theoretical ground. It also motivates explicit ansatze for the conventional QEC setting of non-post-selected codes whose statistical mechanics mappings involve random-bond disorder. Specifically, we introduce an effective tension model for the below-threshold regime, and a new scaling ansatz for the near-threshold regime, derived from an analysis of the domain wall energy cost distributions. By bridging statistical mechanics theory and quantum error correction practice, our results offer a new toolkit for designing, benchmarking, and understanding topological codes beyond current computational limits.
Realizing the potential of quantum computing will require achieving sufficiently low logical error rates. Many applications call for error rates in the 10−1510^{-15}10−15 regime, but state-of-the-art quantum platforms typically have physical error rates near 10−310^{-3}10−3. Quantum error correction (QEC) promises to bridge this divide by distributing quantum logical information across many physical qubits so that errors can be detected and corrected. Logical errors are then exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold. QEC also requires that the errors are local and that performance is maintained over many rounds of error correction, two major outstanding experimental challenges. Here, we implement 1D repetition codes embedded in a 2D grid of superconducting qubits which demonstrate exponential suppression of bit or phase-flip errors, reducing logical error per round by more than 100×100\times100× when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analyzing error correlations with high precision, and characterize the locality of errors in a device performing QEC for the first time. Finally, we perform error detection using a small 2D surface code logical qubit on the same device, and show that the results from both 1D and 2D codes agree with numerical simulations using a simple depolarizing error model. These findings demonstrate that superconducting qubits are on a viable path towards fault tolerant quantum computing.
Quantum error correction (QEC) represents a critical enabler for fault-tolerant quantum computing, protecting fragile quantum states from decoherence and operational errors. This paper provides a comprehensive analysis of leading quantum error correction schemes, focusing on surface codes and topological codes within the context of Noisy Intermediate-Scale Quantum (NISQ) era processors. We examine the theoretical foundations of stabilizer codes, compare surface code implementations with code distances ranging from d=3 to d=17, and analyze topological codes including toric codes and color codes. Our investigation encompasses syndrome extraction circuits, logical qubit encoding strategies, and threshold analysis for contemporary quantum hardware platforms. Performance metrics reveal that surface codes achieve error thresholds of approximately 1% with realistic noise models, while topological codes offer architectural advantages for specific qubit connectivity constraints. We discuss implementation challenges on current NISQ processors including IBM Quantum, Google Sycamore, and IonQ systems, addressing qubit overhead, gate fidelity requirements, and decoding algorithms. This analysis provides insights for selecting appropriate error correction schemes based on application requirements and hardware capabilities.