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Fourier Analysis of Noisy Quantum Circuit Sampling: Insights into Quantum Supremacy Experiments


Core Concepts
Fourier analysis is used to study the effect of noise, including readout and gate errors, on random quantum circuit sampling, with applications to the Google quantum supremacy experiment.
Abstract
The paper focuses on using Fourier analysis to study the effect of noise on random quantum circuit sampling, with applications to the Google quantum supremacy experiment. Key highlights: Fourier-Walsh expansion is used to analyze the probability distributions of random quantum circuits, which are modeled as Porter-Thomas distributions. The effect of different noise models, including symmetric and asymmetric readout errors as well as gate errors, is studied using both analytical methods and simulations. The Fourier-based analysis is used to refine the linear cross-entropy fidelity estimator (FXEB) used in the Google experiment, and to study the samples from the experiment. Simulations using the Google and IBM quantum circuit simulators are performed to further understand the impact of gate errors on the Fourier-Walsh coefficients. Fourier-based estimators for fidelity and readout errors are developed and applied to the experimental data and simulations. The analysis suggests that gate errors have a larger effect on higher-degree Fourier-Walsh coefficients, in addition to the impact of readout errors.
Stats
The paper uses samples of N = 500,000 bitstrings for every circuit studied.
Quotes
"Considerable effort in experimental quantum computing is devoted to noisy intermediate scale quantum computers (NISQ computers). Understanding the effect of noise is important for various aspects of this endeavor including notable claims for achieving quantum supremacy and attempts to demonstrate quantum error correcting codes." "The main novelty of this paper is the use of Fourier analysis (the Fourier–Walsh expansion) for statistical analysis of samples coming from NISQ computers and from simulations and, in particular, to refine the FXEB fidelity estimator according to Fourier degrees." "Gate errors, similar to readout errors (but to a smaller extent), have a larger effect on high-degree Fourier coefficients."

Key Insights Distilled From

by Gil Kalai,Yo... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.00935.pdf
Random Circuit Sampling

Deeper Inquiries

How can the Fourier-based analysis be extended to study the effect of noise on other types of quantum circuits beyond random circuit sampling?

The Fourier-based analysis developed in the paper can be extended to study the effect of noise on various types of quantum circuits beyond random circuit sampling by applying similar principles to different circuit architectures and noise models. Here are some ways this extension can be achieved: Different Circuit Architectures: The Fourier analysis can be applied to circuits with different architectures, such as trapped ions, superconducting qubits, or topological qubits. By analyzing the Fourier coefficients of these circuits, researchers can understand how noise affects the performance of each architecture. Varied Noise Models: The analysis can be adapted to different noise models, including depolarizing noise, amplitude damping, and phase flip errors. By studying the Fourier coefficients under these noise models, researchers can gain insights into the specific impact of each type of noise on the quantum circuits. Quantum Error Correction: The Fourier analysis can be used to evaluate the effectiveness of quantum error correction codes in mitigating the impact of noise on quantum circuits. By analyzing how the Fourier coefficients change with the application of error correction, researchers can assess the robustness of different error correction schemes. Logical Circuits: The analysis can be extended to logical circuits based on neutral atoms or other quantum computing platforms. By studying the Fourier behavior of logical circuits, researchers can evaluate the performance of error correction techniques in the context of fault-tolerant quantum computation. Overall, by applying Fourier-based techniques to a variety of quantum circuits and noise models, researchers can gain a deeper understanding of how noise affects quantum systems and develop strategies to improve the reliability and performance of quantum computers in diverse settings.

How can the Fourier-based analysis be extended to study the effect of noise on other types of quantum circuits beyond random circuit sampling?

The finding that gate errors have a larger impact on higher-degree Fourier-Walsh coefficients has significant implications for the design of quantum error correction schemes. Here are some key implications: Error Correction Strategies: Understanding that gate errors affect higher-degree Fourier coefficients more significantly can guide the development of error correction strategies. Error correction codes can be designed to specifically target and correct errors that have a greater impact on the higher-order coefficients, thereby improving the overall error mitigation capabilities of the system. Resource Allocation: The knowledge that gate errors have a larger impact on certain Fourier coefficients can help in optimizing resource allocation for error correction. Resources such as qubits, gates, and computational power can be allocated more efficiently to address the errors that have the most significant impact on the system's performance. Fault-Tolerant Quantum Computing: The insight that gate errors affect higher-degree coefficients can inform the design of fault-tolerant quantum computing systems. By focusing on mitigating errors that have a disproportionate impact on the system's fidelity, researchers can develop more robust fault-tolerant architectures. Performance Evaluation: The impact of gate errors on higher-degree Fourier coefficients can serve as a metric for evaluating the performance of quantum systems. By monitoring the behavior of these coefficients, researchers can assess the effectiveness of error correction schemes and identify areas for improvement in system reliability. In conclusion, the finding that gate errors have a larger impact on higher-degree Fourier-Walsh coefficients underscores the importance of considering these effects in the design and implementation of quantum error correction schemes to enhance the overall performance and reliability of quantum computing systems.

How can the Fourier-based analysis be extended to study the effect of noise on other types of quantum circuits beyond random circuit sampling?

The Fourier-based techniques developed in this paper can be applied to analyze the performance of quantum computers in various application domains beyond quantum supremacy experiments by adapting the analysis to different types of quantum algorithms and tasks. Here are some ways in which these techniques can be extended: Quantum Algorithm Analysis: The Fourier-based analysis can be used to evaluate the impact of noise on the performance of specific quantum algorithms, such as quantum machine learning algorithms, quantum cryptography protocols, or quantum optimization algorithms. By studying the Fourier coefficients in the context of these algorithms, researchers can assess how noise affects their execution and accuracy. Quantum Communication: The techniques can be applied to analyze the effects of noise on quantum communication protocols, such as quantum key distribution or quantum teleportation. By examining the Fourier behavior of quantum communication channels under noisy conditions, researchers can optimize the protocols for enhanced reliability and security. Quantum Sensing and Metrology: The Fourier analysis can be extended to study the influence of noise on quantum sensing and metrology applications. By analyzing the Fourier coefficients in the context of quantum sensors and measurement devices, researchers can understand how noise impacts the precision and accuracy of quantum measurements. Quantum Simulation: The techniques can be utilized to investigate the effects of noise on quantum simulation tasks, such as simulating complex quantum systems or materials. By applying Fourier-based analysis to quantum simulation algorithms, researchers can evaluate the resilience of simulations to noise and optimize the performance of quantum simulators. Overall, by adapting the Fourier-based analysis to different application domains in quantum computing, researchers can gain insights into the impact of noise on various quantum tasks and develop strategies to enhance the robustness and efficiency of quantum systems in diverse settings.
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