Quantum Benchmarking Methods: Evaluating the Performance of Quantum Devices

Table of Contents

1. Introduction
2. What Is Quantum Benchmarking?
3. Need for Benchmarking in Quantum Computing
4. Key Metrics in Quantum Benchmarking
5. Gate Fidelity and Process Fidelity
6. Coherence Times: T₁ and T₂
7. Quantum Volume
8. Randomized Benchmarking
9. Interleaved Randomized Benchmarking
10. Cross-Entropy Benchmarking
11. Cycle Benchmarking
12. Quantum Tomography
13. Gate Set Tomography (GST)
14. State Fidelity and Process Fidelity
15. Noise Characterization Techniques
16. SPAM Errors and Their Impact
17. Benchmarking in Noisy Intermediate-Scale Quantum (NISQ) Devices
18. Device-Level vs System-Level Benchmarking
19. Challenges and Best Practices
20. Conclusion

1. Introduction

Quantum benchmarking encompasses techniques used to assess the accuracy, stability, and performance of quantum hardware. It is vital for comparing devices, validating error correction, and guiding system improvements.

2. What Is Quantum Benchmarking?

Benchmarking quantifies the real-world performance of quantum gates, circuits, or systems by comparing expected outcomes with experimental results under realistic noise.

3. Need for Benchmarking in Quantum Computing

• Ensures hardware meets fidelity thresholds
• Enables cross-platform comparison
• Informs calibration and system tuning
• Assesses quantum supremacy or advantage claims

4. Key Metrics in Quantum Benchmarking

• Gate fidelity
• State fidelity
• Quantum volume
• T₁ and T₂ coherence times
• Error per gate or per layer

5. Gate Fidelity and Process Fidelity

• Gate fidelity (F₉): overlap between ideal and implemented gate
• Process fidelity: overlaps for complete quantum channels
  These are measured via tomography or indirect methods.

6. Coherence Times: T₁ and T₂

• T₁: energy relaxation time
• T₂: dephasing time
  Measured via pulse sequences like inversion recovery and Ramsey fringes. Important for estimating noise strength and gate lifetimes.

7. Quantum Volume

A single-number benchmark proposed by IBM:

• Accounts for gate fidelity, connectivity, and circuit depth
• Measures capability of running random circuits of increasing size
• Exponentially increases with system improvement

8. Randomized Benchmarking

Uses random Clifford gate sequences:

• Reduces sensitivity to SPAM errors
• Estimates average gate fidelity
• Robust and scalable to many qubits

9. Interleaved Randomized Benchmarking

Tests a specific gate’s fidelity by interleaving it within randomized benchmarking sequences. Reveals gate-specific errors.

10. Cross-Entropy Benchmarking

Used in Google’s Sycamore experiment:

• Compares measured outcomes with simulated ideal probabilities
• Useful for circuits near quantum advantage threshold
• Defines cross-entropy fidelity as:
  \[
  F_{ ext{XEB}} = 2^n \langle P_{ ext{ideal}}
  angle – 1
  \]

11. Cycle Benchmarking

Evaluates entire layers of gates (cycles), especially in NISQ-era devices. Accounts for parallelism, crosstalk, and multi-qubit error effects.

12. Quantum Tomography

Full reconstruction of quantum states or processes:

• State tomography: reconstructs density matrix
• Process tomography: reconstructs superoperator
  Limited scalability due to exponential resource requirements.

13. Gate Set Tomography (GST)

A self-consistent method to characterize all gates, SPAM operations, and measurement errors simultaneously. Offers detailed error modeling but requires long sequences and processing.

14. State Fidelity and Process Fidelity

Fidelity measures are:
\[
F(
ho, \sigma) = \left( ext{Tr}\sqrt{\sqrt{
ho} \sigma \sqrt{
ho}}
ight)^2
\]
These quantify overlap between experimental and ideal quantum states or processes.

15. Noise Characterization Techniques

Noise modeling includes:

• Pauli twirling
• Kraus operators
• Markovian vs non-Markovian noise
  Important for developing error mitigation and correction strategies.

16. SPAM Errors and Their Impact

SPAM = State Preparation and Measurement errors:

• Often dominate in tomography
• Randomized benchmarking helps suppress them
• GST includes SPAM in its model

17. Benchmarking in Noisy Intermediate-Scale Quantum (NISQ) Devices

NISQ benchmarking must:

• Handle circuit noise and gate imperfections
• Assess fidelity under realistic workloads
• Quantify mitigation efficacy

18. Device-Level vs System-Level Benchmarking

• Device-level: focuses on qubits, gates, and measurement fidelity
• System-level: evaluates end-to-end performance (e.g., running algorithms)

19. Challenges and Best Practices

• Avoid overfitting to benchmark-specific metrics
• Use complementary techniques (RB, tomography, GST)
• Track performance over time and calibrations
• Calibrate against known benchmarks for consistency

20. Conclusion

Quantum benchmarking methods provide critical insights into quantum hardware performance and scalability. As quantum processors evolve, robust, scalable, and noise-resilient benchmarking techniques will remain essential for progress in quantum computing.