Quantum Volume Benchmark

List of acronyms

CE: Constructor Evaluation (checked if the evaluation is done by the chip manufacturer)
SP: Scientific paper (checked if a scientific paper explain the results)

For clarity the Quantum Volum is expressed in a logarithmic basis.

Quantum Volume protocol

The Quantum Volume (QV) [16], [3] is a benchmarking protocol used to measure the ability of circuit-based quantum computers to simulate quantum circuits. This protocol gathers in a single metric number, the maximum width and depth that a quantum computer can successfully implement. A quantum computer has to sucessfully solve the Heavy Output Generation (HOG) problem [17] of size \(n\) to validate a quantum volume of size \(2^n\).

Receipe for a Quantum Volume Experiment:

• Choose a number \(n\) as the width of the circuit (i.e., number of qubits in the circuit).
• Set \(n=d\) with \(d\) the number of layers of the quantum circuit.
• Generate a model circuit composed of \(d\) layers. Each layer is composed of a random permutation of qubits \(\pi\) and a random unitary sampled from \(SU(4)\).

This circuit is then used as input for the sampling task associated to the Heavy Output Generation (HOG) problem. If the quantum computer is able to sample the right distribution, it validates the associated quantum volume score of \(2^n\).

Assumptions

• The circuit compiler may use all the possible tricks to improve the mapping of the quantum circuit, which can lead to possibly high extra-processing time.
• The Quantum computer should make an honest attempt to solve the HOG problem and not choose an implementation that is far from the initial model of circuit (i.e., the approximation error should be limited).

Elements of the quantum stack being benchmarked

• Gate fidelity
• Coherence time
• Chip topology
• Efficiency of the transpilation method

Protocol complexity

The complexity of the HOG problem is exponential either in time or space for a classical computer. For further details on the complexity the reader may refer to [17].

References

1. [1]AQT, “State of quantum computing in Europe: AQT pushing performance with a quantum volume of 128.” 2023 [Online]. Available at: https://www.aqt.eu/aqt-pushing-performance-with-a-quantum-volume-of-128/
2. [2]N. Sundaresan, I. Lauer, E. Pritchett, E. Magesan, P. Jurcevic, and J. M. Gambetta, “Reducing unitary and spectator errors in cross resonance with optimized rotary echoes,” PRX Quantum, vol. 1, no. 2, p. 020318, 2020.
3. [3]A. W. Cross, L. S. Bishop, S. Sheldon, P. D. Nation, and J. M. Gambetta, “Validating quantum computers using randomized model circuits,” Physical Review A, vol. 100, no. 3, p. 032328, 2019.
4. [4]P. Jurcevic et al., “Demonstration of quantum volume 64 on a superconducting quantum computing system,” Quantum Science and Technology, vol. 6, no. 2, p. 025020, 2021.
5. [5]IBM, “Processor types.” 2024 [Online]. Available at: https://docs.quantum.ibm.com/run/processor-types
6. [6]E. Pelofske, A. Bärtschi, and S. Eidenbenz, “Quantum volume in practice: What users can expect from nisq devices,” IEEE Transactions on Quantum Engineering, vol. 3, pp. 1–19, 2022.
7. [7]J. M. Pino et al., “Demonstration of the trapped-ion quantum CCD computer architecture,” Nature, vol. 592, no. 7853, pp. 209–213, 2021.
8. [8]Quantinuum, “Quantuum Hardware Quantum Volume Data.” 2024 [Online]. Available at: https://github.com/CQCL/quantinuum-hardware-quantum-volume
9. [9]Honeywell, “Achieving quantum volume 128 on the Honeywell Quantum Computer.” 2020 [Online]. Available at: https://www.honeywell.com/us/en/news/2020/09/achieving-quantum-volume-128-on-the-honeywell-quantum-computer
10. [10]Quantinuum, “Demonstrating Benefits of Quantum Upgradable Design Strategy: System Model H1-2 Frist to Prove 2048 Quantum Volume.” 2021 [Online]. Available at: https://www.quantinuum.com/news/demonstrating-benefits-of-quantum-upgradable-design-strategy-system-model-h1-2-first-to-prove-2-048-quantum-volume
11. [11]Quantinuum, “Quantinuum Announces Quantum Volume 4096 Achievement.” 2022 [Online]. Available at: https://www.quantinuum.com/news/quantinuum-announces-quantum-volume-4096-achievement
12. [12]Quantinuum, “Quantinuum Sets New Record with Highest Ever Quantum Volume.” 2022 [Online]. Available at: https://www.quantinuum.com/news/quantinuum-sets-new-record-with-highest-ever-quantum-volume
13. [13]C. H. Baldwin, K. Mayer, N. C. Brown, C. Ryan-Anderson, and D. Hayes, “Re-examining the quantum volume test: Ideal distributions, compiler optimizations, confidence intervals, and scalable resource estimations,” Quantum, vol. 6, p. 707, 2022.
14. [14]Quantinuum, “Quantum Volume reaches 5 digits for the first time: 5 perspectives on what it means for quantum computing.” 2023 [Online]. Available at: https://www.quantinuum.com/news/quantum-volume-reaches-5-digits-for-the-first-time-5-perspectives-on-what-it-means-for-quantum-computing
15. [15]S. A. Moses et al., “A race-track trapped-ion quantum processor,” Physical Review X, vol. 13, no. 4, p. 041052, 2023.
16. [16]L. S. Bishop, S. Bravyi, A. Cross, J. M. Gambetta, and J. Smolin, “Quantum volume,” Quantum Volume. Technical Report, 2017.
17. [17]S. Aaronson and L. Chen, “Complexity-theoretic foundations of quantum supremacy experiments,” arXiv preprint arXiv:1612.05903, 2016.