What role does the Porter-Thomas distribution play in the analysis of quantum circuits using cross-entropy benchmarking (XEB)?

by EITCA Academy / Tuesday, 11 June 2024 / Published in Artificial Intelligence, EITC/AI/TFQML TensorFlow Quantum Machine Learning, Quantum supremacy, Extracting coherence information from random circuits, Examination review

The Porter-Thomas distribution plays a important role in the analysis of quantum circuits using cross-entropy benchmarking (XEB), particularly in the context of quantum supremacy and the extraction of coherence information from random circuits. To understand this role comprehensively, it is essential to consider several foundational concepts, including the nature of random quantum circuits, the principles of XEB, and the statistical properties encapsulated by the Porter-Thomas distribution.

Random Quantum Circuits and Quantum Supremacy

Random quantum circuits are a class of quantum circuits where the sequence of quantum gates is chosen according to some random distribution. These circuits are significant in the study of quantum supremacy, which refers to the point where quantum computers can solve problems that classical computers practically cannot. The randomness in these circuits is designed to create complex entanglement patterns and chaotic behavior, making them difficult to simulate classically.

Cross-Entropy Benchmarking (XEB)

Cross-entropy benchmarking is a technique used to evaluate the performance of quantum circuits, particularly in the context of quantum supremacy experiments. XEB involves comparing the output distribution of a quantum circuit to the expected distribution, which is typically derived from classical simulations of the quantum circuit for small systems or inferred from theoretical models for larger systems.

The cross-entropy benchmarking process can be summarized as follows:
1. Circuit Execution: Execute the random quantum circuit on a quantum processor multiple times to collect a set of output bitstrings.
2. Probability Estimation: Estimate the probability of each observed bitstring based on the ideal output distribution of the circuit.
3. Cross-Entropy Calculation: Compute the cross-entropy between the observed distribution and the ideal distribution. The cross-entropy is a measure of the similarity between two probability distributions, with lower values indicating better agreement.

The Porter-Thomas Distribution

The Porter-Thomas distribution is a probability distribution that describes the statistical properties of the output probabilities of random quantum circuits. Specifically, it characterizes the distribution of the squared amplitudes of the output states of a random quantum circuit. For a quantum system with $N$ possible states, the squared amplitude $P_i$ of each state $i$ follows the Porter-Thomas distribution, which is given by:

$P(P_i) = (N-1) (1 - P_i)^{N-2}$

where $P_i$ is the probability of measuring the system in state $i$ .

In the context of random quantum circuits, the Porter-Thomas distribution predicts that most output probabilities will be close to zero, with a few probabilities being significantly larger. This distribution arises due to the high-dimensional Hilbert space and the chaotic nature of random quantum circuits, which lead to an exponential suppression of most output probabilities.

Role of the Porter-Thomas Distribution in XEB

The Porter-Thomas distribution is integral to XEB for several reasons:

1. Benchmarking Ideal Output Distributions: The ideal output distribution of a random quantum circuit is expected to follow the Porter-Thomas distribution. By comparing the experimentally obtained distribution to the Porter-Thomas distribution, researchers can assess how closely the quantum processor's behavior matches the theoretical predictions. A close match indicates that the quantum processor is functioning correctly and generating the expected complex entanglement patterns.

2. Error Detection and Characterization: Deviations from the Porter-Thomas distribution can indicate the presence of errors in the quantum circuit, such as decoherence, gate errors, or readout errors. By analyzing these deviations, researchers can identify and characterize the sources of errors, enabling them to improve the fidelity of the quantum processor.

3. Validation of Quantum Supremacy: Demonstrating quantum supremacy requires showing that a quantum processor can solve a problem that is infeasible for classical computers. One way to validate this is by showing that the output distribution of the quantum processor follows the Porter-Thomas distribution, which is difficult to simulate classically. This provides strong evidence that the quantum processor is operating in a regime beyond the reach of classical computation.

Practical Example

Consider a random quantum circuit with 50 qubits and a depth of 20 layers, where each layer consists of randomly chosen single-qubit gates and two-qubit entangling gates. The circuit is executed on a quantum processor, and the output bitstrings are collected. The ideal output distribution, derived from classical simulations for a smaller version of the circuit, is expected to follow the Porter-Thomas distribution.

To perform XEB, the following steps are taken:

1. Collect Data: Execute the quantum circuit multiple times (e.g., 1 million shots) to collect a set of output bitstrings.
2. Estimate Probabilities: Estimate the probability of each observed bitstring based on the ideal output distribution.
3. Compute Cross-Entropy: Calculate the cross-entropy between the observed distribution and the ideal distribution. This involves computing the logarithm of the estimated probabilities and averaging over the observed bitstrings.
4. Compare to Porter-Thomas: Compare the observed distribution to the Porter-Thomas distribution to assess the agreement. If the observed distribution closely follows the Porter-Thomas distribution, it indicates that the quantum processor is functioning correctly and generating the expected complex entanglement patterns.

Extracting Coherence Information

The coherence of a quantum processor refers to its ability to maintain quantum superposition and entanglement over time. Coherence is important for the accurate execution of quantum circuits, as decoherence leads to loss of quantum information and errors.

The Porter-Thomas distribution can be used to extract coherence information from random circuits in the following ways:

1. Fidelity Estimation: By comparing the observed output distribution to the ideal Porter-Thomas distribution, researchers can estimate the fidelity of the quantum circuit. Fidelity is a measure of how closely the experimental output matches the ideal output. High fidelity indicates good coherence, while low fidelity suggests the presence of decoherence or other errors.

2. Error Rates: Deviations from the Porter-Thomas distribution can be analyzed to estimate error rates in the quantum circuit. For example, if the observed distribution shows a higher probability of certain bitstrings than expected, it may indicate specific types of errors, such as gate errors or readout errors. By quantifying these deviations, researchers can estimate the error rates and identify areas for improvement.

3. Temporal Coherence: By executing the same random quantum circuit at different times and comparing the output distributions, researchers can study the temporal coherence of the quantum processor. Consistent agreement with the Porter-Thomas distribution over time indicates good temporal coherence, while variations suggest time-dependent decoherence or other temporal errors.

Conclusion

The Porter-Thomas distribution is a fundamental tool in the analysis of quantum circuits using cross-entropy benchmarking. It provides a theoretical benchmark for the ideal output distribution of random quantum circuits, enabling researchers to assess the performance, fidelity, and coherence of quantum processors. By comparing the observed output distribution to the Porter-Thomas distribution, researchers can validate quantum supremacy, detect and characterize errors, and extract valuable coherence information from random circuits.

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