Cross-Entropy Benchmarking (XEB): Verifying Quantum Supremacy and Circuit Fidelity

Table of Contents

1. Introduction
2. Motivation for Cross-Entropy Benchmarking (XEB)
3. Principles of Cross-Entropy Benchmarking
4. Classical vs Quantum Sampling
5. XEB Fidelity Metric
6. Mathematical Definition
7. Practical Implementation Steps
8. Random Quantum Circuit Generation
9. Circuit Depth and Qubit Count
10. Measuring Output Probabilities
11. Estimating Ideal Probabilities Classically
12. Cross-Entropy Difference and Fidelity
13. Advantages Over Other Benchmarks
14. Sources of Error and Noise
15. Application to Quantum Supremacy Experiments
16. Google Sycamore and XEB
17. Limitations of Cross-Entropy Benchmarking
18. Relation to Quantum Volume and RB
19. Systematic Bias and Error Mitigation
20. Conclusion

1. Introduction

Cross-entropy benchmarking (XEB) is a powerful technique used to assess the fidelity of quantum circuits, particularly in the context of random circuit sampling. It gained prominence through quantum supremacy experiments and offers a way to compare quantum and classical outputs.

2. Motivation for Cross-Entropy Benchmarking (XEB)

As quantum systems increase in size, traditional benchmarks like tomography become impractical. XEB provides a scalable way to estimate how closely a quantum circuit’s output matches theoretical expectations.

3. Principles of Cross-Entropy Benchmarking

The idea is to:

• Run a random quantum circuit on real hardware
• Measure output bitstrings
• Compare their probabilities with the ideal theoretical distribution

4. Classical vs Quantum Sampling

For a random quantum circuit of sufficient depth, classical simulation becomes intractable. XEB exploits this to detect if a quantum computer is outperforming classical algorithms.

5. XEB Fidelity Metric

Fidelity \( F_{ ext{XEB}} \) is computed by comparing measured bitstring probabilities with ideal values:

• Higher \( F_{ ext{XEB}} \) indicates better quantum circuit performance
• Fidelity drops in the presence of noise or decoherence

6. Mathematical Definition

Let \( {x_i} \) be the set of sampled bitstrings and \( P(x_i) \) their ideal probabilities:
\[
F_{ ext{XEB}} = 2^n \langle P(x_i)
angle – 1
\]
where \( n \) is the number of qubits and \( \langle P(x_i)
angle \) is the average over sampled bitstrings.

7. Practical Implementation Steps

1. Generate random circuit of given depth and size
2. Run it on quantum hardware
3. Measure a large number of output bitstrings
4. Calculate ideal probabilities for those bitstrings (on a classical simulator)
5. Compute \( F_{ ext{XEB}} \)

8. Random Quantum Circuit Generation

Circuits use:

• Randomized single- and two-qubit gates
• Repeated layers (cycles)
• Architecture-specific connectivity constraints

9. Circuit Depth and Qubit Count

Deeper circuits and more qubits increase sampling complexity. XEB becomes more powerful as classical simulation becomes infeasible for verifying results.

10. Measuring Output Probabilities

Each bitstring occurs with probability \( P(x) \). Frequencies are tallied over many trials (~10⁵ or more) to estimate the distribution.

11. Estimating Ideal Probabilities Classically

Ideal probabilities are computed via exact simulation:

• Feasible for \( \leq 40 \) qubits
• Approximation or truncation required beyond that

12. Cross-Entropy Difference and Fidelity

• Ideal uniform distribution yields \( F_{ ext{XEB}} pprox 0 \)
• Perfect quantum output gives \( F_{ ext{XEB}} = 1 \)
• Noisy hardware results in intermediate values

13. Advantages Over Other Benchmarks

• Scalable to large systems
• Tolerant to SPAM errors
• Captures global system fidelity, not just per-gate errors

14. Sources of Error and Noise

• Gate infidelity
• Crosstalk
• Measurement errors
• SPAM noise (partially suppressed)

15. Application to Quantum Supremacy Experiments

Google’s 2019 Sycamore experiment used XEB to:

• Demonstrate 53-qubit sampling fidelity
• Show 200s quantum runtime vs 10⁵ years classically
• Declare quantum supremacy for random circuit sampling

16. Google Sycamore and XEB

The team achieved \( F_{ ext{XEB}} pprox 0.0024 \), sufficient for claiming quantum advantage. Cross-entropy validated the quantum-classical gap.

17. Limitations of Cross-Entropy Benchmarking

• Sensitive to simulator accuracy
• Assumes availability of ideal probabilities
• Not applicable to structured circuits (only random circuits)

18. Relation to Quantum Volume and RB

• RB (Randomized Benchmarking): gate-level fidelity
• QV (Quantum Volume): system-level capability
• XEB: circuit-level fidelity on unstructured workloads

19. Systematic Bias and Error Mitigation

• Averaging over multiple circuits
• Circuit compilation with noise mitigation
• Crosstalk-aware layout and scheduling

20. Conclusion

Cross-entropy benchmarking provides a robust and scalable method for evaluating quantum circuits, especially in the context of quantum supremacy. As devices grow, XEB will remain central to certifying and comparing quantum processors.