Measuring magic on a quantum processor

The ability to characterize Noisy Intermediate-Scale Quantum (NISQ) computers is paramount to assessing their quantum computational advantage. This paper focuses on measuring "magic," a resource beyond the Clifford group necessary for quantum advantage, using the stabilizer Rényi entropy. While Clifford gates and measurements in the computational basis are fault-tolerant and classically simulable, quantum advantage demands resources like the Phase #0 magic state. The challenge lies in the fragility and difficulty of implementing these resources, necessitating accurate measurement and calibration. Decoherence impacts magic, potentially increasing or decreasing it, underscoring the need for precise quantification. Inaccurate Clifford gates can also produce unwanted magic, indicating noise. Measuring magic thus provides a way to quantify and characterize this noise. Previous magic measures relied on extremization, lacking an experimental scheme. This work introduces a protocol based on randomized measurements, superior to state tomography in resource efficiency, to measure quantum and classical magic in qubits and characterize quantum hardware. The magic measure, denoted as Me, is defined using Rényi entropy and involves randomized one-qubit measurements instead of expensive multi-qubit measurements. The protocol estimates correlations between multiple copies of the state by applying random Clifford operations, avoiding full state tomography. Randomized measurement protocols are preferable to state tomography due to their efficiency and reduced error.

The paper reviews existing literature on fault-tolerant quantum computation, emphasizing the classical simulability of Clifford circuits and the necessity of non-Clifford resources for quantum advantage. It highlights the existing challenges in accurately implementing and measuring magic states, referencing prior work on magic state distillation and the theoretical framework for quantifying magic. The authors note the limitations of previous magic measures, which lacked experimental protocols and relied on computationally expensive extremization procedures. This sets the stage for the introduction of their novel randomized measurement approach.

The proposed protocol employs randomized measurements to efficiently estimate the stabilizer 2-Rényi entropy, a measure of magic. The protocol involves applying random sequences of one-qubit Clifford operators to four copies of the quantum state of interest and then measuring the results in the computational basis. The stabilizer 2-Rényi entropy is calculated using the measured probabilities. The protocol leverages the property that Clifford rotations form a 2-design, allowing simultaneous estimation of both W(ϕ) (related to purity) and the purity P(ϕ) from occupation probabilities obtained from randomized measurements. This differs from previous methods that required global multi-qubit measurements. The experiments are conducted on IBM Quantum processors (ibmq_lima and ibmq_quito), involving preparation of single-qubit and entangled states. For entangled states, a specific circuit is employed, which involves Hadamard gates, T-gates to inject magic, and CX gates to create entanglement. Different numbers of T-gates (magic seeds) are used to prepare states with varying levels of magic. A noise model is incorporated to account for imperfections in gate implementation and decoherence, with parameters of the noise model tuned to match experimental data. The noise model incorporates both decoherence and imperfections in Clifford gates, the latter being particularly important in the creation of low-magic states. The analytical and numerical scaling of resources required is assessed, with a detailed statistical analysis to determine the number of measurements needed for a given level of accuracy.

Experiments on IBM quantum processors demonstrate the feasibility and accuracy of the proposed randomized measurement protocol for quantifying magic. The experimental data for single-qubit magic states show good agreement with theoretical predictions, indicating low decoherence. Experiments on multi-qubit entangled states reveal that the measured magic often exceeds theoretical predictions, particularly for low-magic states. This discrepancy is attributed to imperfections in Clifford gate implementation, introducing unwanted magic. A noise model is developed to account for these imperfections and decoherence effects, demonstrating that the proposed method allows characterization of the quantum processor's noise properties. The noise model provides a better approximation to experimental data as the number of gates in the circuit increases. The analysis shows that high-magic states are more robust to noise compared to low-magic states. Finally, the authors provide a statistical analysis, showing the scaling of resources needed for accurate estimation of magic, demonstrating that the proposed method is computationally efficient compared to full state tomography. Figures 4, 5, and 6 illustrate the experimental results for 3, 4, and 5 qubits respectively, comparing experimental values to theoretical predictions and the values given by the noise model.

The results demonstrate the efficacy of the proposed protocol in measuring magic and characterizing noise in quantum hardware. The higher-than-predicted magic in experimental results for multi-qubit states highlights the importance of considering gate imperfections and decoherence when aiming to produce specific magic levels. The developed noise model allows for a more realistic assessment of quantum processor performance. The method's efficiency compared to full state tomography opens the way for large-scale characterization of NISQ devices. The study provides a practical tool for assessing and improving the quality of quantum computations by quantifying the amount of magic and characterizing the noise sources affecting its creation and manipulation. The improved understanding of noise facilitates the development of mitigation strategies.

This paper presents a novel protocol for measuring magic in quantum states using randomized measurements, significantly improving upon previous approaches. Experimental validation on IBM Quantum processors confirms the protocol's accuracy and efficiency. The ability to characterize noise sources using this method offers a valuable tool for advancing NISQ technology. Future work could explore extensions to larger qubit systems and investigate the application of this protocol in optimizing quantum algorithms and developing advanced noise mitigation techniques.

The study focuses on specific IBM quantum processors, and the results might not be directly generalizable to other hardware platforms. The noise model used is a simplified representation of the complex noise processes in quantum processors. While the protocol offers significant improvements in efficiency over full state tomography, the number of required measurements still scales with the number of qubits, limiting scalability to very large systems. Further research could explore more sophisticated noise models and investigate the protocol's performance on different quantum computing architectures.