What is cross-entropy benchmarking, and how is it used to evaluate the performance of quantum gates on the Sycamore processor?
Cross-entropy benchmarking (XEB) is a critical technique employed to evaluate the performance of quantum gates, particularly on quantum processors such as Google's Sycamore processor. This benchmarking method is instrumental in the field of quantum computing, where it serves as a robust tool to measure how well a quantum processor can perform complex quantum operations, which is essential for demonstrating quantum supremacy.
Quantum supremacy refers to the point where a quantum computer can perform a calculation that is infeasible for classical computers to execute within a reasonable timeframe. Achieving this milestone requires not only sophisticated quantum algorithms but also highly reliable quantum gates, which are the fundamental building blocks of quantum circuits. The Sycamore processor, a 54-qubit quantum processor developed by Google, has been at the forefront of this endeavor.
Cross-entropy benchmarking provides a quantitative measure of the fidelity of quantum gates. Fidelity, in this context, refers to the accuracy with which a quantum operation is performed compared to the ideal theoretical operation. The XEB method involves the following steps:
1. Circuit Generation: Random quantum circuits are generated. These circuits are composed of a series of quantum gates applied to the qubits. The randomness ensures that the circuits are complex enough to challenge both the quantum processor and classical simulation methods.
2. Execution on Quantum Processor: The generated circuits are executed on the quantum processor. The output of these circuits is a set of measurement results, which are the probabilities of the different possible states of the qubits after the circuit has been applied.
3. Classical Simulation: The same circuits are simulated on a classical computer to obtain the expected probability distribution of the measurement results. This step is important for benchmarking, as it provides a reference against which the quantum processor's performance can be compared.
4. Cross-Entropy Calculation: The core of the XEB method lies in comparing the probability distributions from the quantum processor and the classical simulation. Cross-entropy is a measure from information theory that quantifies the difference between two probability distributions. Specifically, it measures the average number of bits needed to identify an event from one distribution when using a code optimized for another distribution.
Mathematically, the cross-entropy 
between two probability distributions 
and 
is defined as:
![\[ H(p, q) = -\sum_{x} p(x) \log q(x) \]](https://dev-temp3.eitca.eu/wp-content/ql-cache/quicklatex.com-c4fca38d272faf66100932f4e03771bd_l3.png "Rendered by QuickLaTeX.com")
where 
is the probability of outcome 
in the ideal distribution (from classical simulation), and 
is the probability of the same outcome in the distribution obtained from the quantum processor.
5. Fidelity Estimation: The fidelity of the quantum processor is estimated by comparing the cross-entropy of the quantum processor's output with the ideal distribution. High fidelity indicates that the quantum processor's output closely matches the expected results, signifying that the quantum gates are performing well.
The XEB method is particularly advantageous for several reasons:
– Scalability: Cross-entropy benchmarking can be applied to large quantum systems, making it suitable for processors with many qubits, such as Sycamore.
– Comprehensive Assessment: By using random circuits, XEB tests the quantum processor's performance over a wide range of operations and configurations, providing a thorough evaluation of its capabilities.
– Benchmarking Quantum Supremacy: XEB is a key component in demonstrating quantum supremacy. For instance, in Google's landmark experiment, the Sycamore processor executed a specific random quantum circuit that was infeasible for classical supercomputers to simulate within a reasonable time frame. The fidelity of this execution, as measured by XEB, provided strong evidence of quantum supremacy.
An example of cross-entropy benchmarking in action can be drawn from Google's experiment with the Sycamore processor. In this experiment, a random quantum circuit with 53 active qubits (one qubit was disabled due to a defect) and 20 cycles of gate operations was executed. The resulting output distribution from the Sycamore processor was compared to the ideal distribution obtained from classical simulations. The cross-entropy benchmarking revealed a fidelity of approximately 0.002, which, despite appearing small, was significantly higher than what could be achieved by any known classical method.
This high fidelity indicated that the Sycamore processor was accurately implementing the quantum circuit, thus validating its performance and supporting the claim of quantum supremacy. The XEB method provided a rigorous and quantitative measure of the processor's capabilities, underscoring its role as a powerful tool in the evaluation and development of quantum technologies.
Cross-entropy benchmarking is an essential technique for assessing the performance of quantum gates on quantum processors like Sycamore. By comparing the output of random quantum circuits executed on the quantum processor with the ideal distribution from classical simulations, XEB provides a precise measure of fidelity. This measure is important for demonstrating quantum supremacy and advancing the field of quantum computing.
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