Quantum volume is a metric that measures the capabilities and error rates of a quantum computer. It expresses the maximum size of square quantum circuits that can be implemented successfully by the computer. The form of the circuits is independent from the quantum computer architecture, but compiler can transform and optimize it to take advantage of the computer's features. Thus, quantum volumes for different architectures can be compared.

The current world record for highest quantum volume as of April 2024 is 220, accomplished by Quantinuum's H1-1 20-qubit ion trap quantum computer.[1]

## Introduction

Quantum computers are difficult to compare. Quantum volume is a single number designed to show all around performance. It is a measurement and not a calculation, and takes into account several features of a quantum computer, starting with its number of qubits—other measures used are gate and measurement errors, crosstalk and connectivity.[2][3][4]

IBM defined its Quantum Volume metric[5] because a classical computer's transistor count and a quantum computer's quantum bit count aren't the same. Qubits decohere with a resulting loss of performance so a few fault tolerant bits are more valuable as a performance measure than a larger number of noisy, error-prone qubits.[6][7]

Generally, the larger the quantum volume, the more complex the problems a quantum computer can solve.[8]

Alternative benchmarks, such as Cross-entropy benchmarking and IonQ's Algorithmic Qubits, have also been proposed.

## Definition

### Original Definition

The quantum volume of a quantum computer was originally defined in 2018 by Nikolaj Moll et al.[9] However, since around 2021 that definition has been supplanted by IBM's 2019 redefinition.[10][11] The original definition depends on the number of qubits N as well as the number of steps that can be executed, the circuit depth d

${\displaystyle {\tilde {V))_{Q}=\min[N,d(N)]^{2}.}$

The circuit depth depends on the effective error rate εeff as

${\displaystyle d\simeq {\frac {1}{N\varepsilon _{\mathrm {eff} ))}.}$

The effective error rate εeff is defined as the average error rate of a two-qubit gate. If the physical two-qubit gates do not have all-to-all connectivity, additional SWAP gates may be needed to implement an arbitrary two-qubit gate and εeff > ε, where ε is the error rate of the physical two-qubit gates. If more complex hardware gates are available, such as the three-qubit Toffoli gate, it is possible that εeff < ε.

The allowable circuit depth decreases when more qubits with the same effective error rate are added. So with these definitions, as soon as d(N) < N, the quantum volume goes down if more qubits are added. To run an algorithm that only requires n < N qubits on an N-qubit machine, it could be beneficial to select a subset of qubits with good connectivity. For this case, Moll et al. [9] give a refined definition of quantum volume.

${\displaystyle V_{Q}=\max _{n

where the maximum is taken over an arbitrary choice of n qubits.

### IBM's redefinition

In 2019, IBM's researchers modified the quantum volume definition to be an exponential of the circuit size, stating that it corresponds to the complexity of simulating the circuit on a classical computer:[5][12]

${\displaystyle \log _{2}V_{Q}={\underset {n\leq N}{\operatorname {arg\,max} ))\left\{\min \left[n,d(n)\right]\right\))$

## Achievement history

Date Quantum
volume[a]
Qubit
count
Manufacturer System name and reference
2020, January 25 28 IBM "Raleigh"[13]
2020, June 26 6 Honeywell [14]
2020, August 26 27 IBM Falcon r4 "Montreal"[15]
2020, November 27 10 Honeywell "System Model H1"[16]
2020, December 27 27 IBM Falcon r4 "Montreal"[17]
2021, March 29 10 Honeywell "System Model H1"[18]
2021, July 210 10 Honeywell "Honeywell System H1"[19]
2021, December 211 12 Quantinuum
(previously Honeywell)
"Quantinuum System Model H1-2"[20]
2022, April 28 27 IBM Falcon r10 "Prague"[21]
2022, April 212 12 Quantinuum "Quantinuum System Model H1-2"[22]
2022, May 29 27 IBM Falcon r10 "Prague"[23]
2022, September 213 20 Quantinuum "Quantinuum System Model H1-1"[24]
2023, February 27 24 Alpine Quantum Technologies "Compact Ion-Trap Quantum Computing Demonstrator"[25]
2023, February 215 20 Quantinuum "Quantinuum System Model H1-1"[26]
2023, May 216 32 Quantinuum "Quantinuum System Model H2"[27]
2023, June 219 20 Quantinuum "Quantinuum System Model H1-1"[28]
2024, February 25 20 IQM "IQM 20-qubit system"[29]
2024, April 220 20 Quantinuum "Quantinuum System Model H1-1"[1]

## Volumetric benchmarks

The quantum volume benchmark defines a family of square circuits, whose number of qubits N and depth d are the same. Therefore, the output of this benchmark is a single number. However, a proposed generalization is the volumetric benchmark[30] framework, which defines a family of rectangular quantum circuits, for which N and d are uncoupled to allow the study of time/space performance trade-offs, thereby sacrificing the simplicity of a single-figure benchmark.

Volumetric benchmarks can be generalized not only to account for uncoupled N and d dimensions, but also to test different types of quantum circuits. While quantum volume benchmarks the quantum computer's ability to implement a specific type of randomized circuits, these can, in principle, be substituted by other families of random circuits, periodic circuits,[31] or algorithm-inspired circuits. Each benchmark must have a success criterion that defines whether a processor has "passed" a given test circuit.

While these data can be analyzed in many ways, a simple method of visualization is illustrating the Pareto front of the N versus d trade-off for the processor being benchmarked. This Pareto front provides information on the largest depth d a patch of a given number of qubits N can withstand, or, alternatively, the biggest patch of N qubits that can withstand executing a circuit of given depth d.

## References

1. ^ a b "Quantinuum extends its significant lead in quantum computing, achieving historic milestones for hardware fidelity and Quantum Volume". www.quantinuum.com. Retrieved 2024-04-17.
2. ^ "Honeywell claims to have built the highest-performing quantum computer available". phys.org. Retrieved 2020-06-22.
3. ^ Smith-Goodson, Paul. "Quantum Volume: A Yardstick To Measure The Performance Of Quantum Computers". Forbes. Retrieved 2020-06-22.
4. ^ "Measuring Quantum Volume". Qiskit.org. Retrieved 2020-08-21.
5. ^ a b Cross, Andrew W.; Bishop, Lev S.; Sheldon, Sarah; Nation, Paul D.; Gambetta, Jay M. (2019). "Validating quantum computers using randomized model circuits". Phys. Rev. A. 100 (3): 032328. arXiv:1811.12926. Bibcode:2019PhRvA.100c2328C. doi:10.1103/PhysRevA.100.032328. S2CID 119408990. Retrieved 2020-10-02.
6. ^ Mandelbaum, Ryan F. (2020-08-20). "What Is Quantum Volume, Anyway?". Medium Qiskit. Retrieved 2020-08-21.
7. ^ Sanders, James (August 12, 2019). "Why quantum volume is vital for plotting the path to quantum advantage". TechRepublic. Retrieved 2020-08-22.
8. ^ Patty, Lee (2020). "Quantum Volume: The Power of Quantum Computers". www.honeywell.com. Chief Scientist for Honeywell Quantum Solutions. Retrieved 2020-08-21.
9. ^ a b Moll, Nikolaj; Barkoutsos, Panagiotis; Bishop, Lev S; Chow, Jerry M; Cross, Andrew; Egger, Daniel J; Filipp, Stefan; Fuhrer, Andreas; Gambetta, Jay M; Ganzhorn, Marc; Kandala, Abhinav; Mezzacapo, Antonio; Müller, Peter; Riesswe introd, Walter; Salis, Gian; Smolin, John; Tavernelli, Ivano; Temme, Kristan (2018). "Quantum optimization using variational algorithms on near-term quantum devices". Quantum Science and Technology. 3 (3): 030503. arXiv:1710.01022. Bibcode:2018QS&T....3c0503M. doi:10.1088/2058-9565/aab822.
10. ^ Baldwin, Charles; Mayer, Karl (2022). "Re-examining the quantum volume test: Ideal distributions, compiler optimizations, confidence intervals, and scalable resource estimations". Quantum. 6: 707. arXiv:2110.14808. Bibcode:2022Quant...6..707B. doi:10.22331/q-2022-05-09-707. S2CID 240070758.
11. ^ Miller, Keith (2022-07-14). "An Improved Volumetric Metric for Quantum Computers via more Representative Quantum Circuit Shapes". arXiv:2207.02315 [quant-ph].
12. ^
13. ^ "IBM Doubles Its Quantum Computing Power Again". Forbes. 2020-01-08.
14. ^ Samuel K. Moore (2020-06-24). "Honeywell Claims It Has Most Powerful Quantum Computer". IEEE Spectrum.
15. ^ Condon, Stephanie (August 20, 2020). "IBM hits new quantum computing milestone". ZDNet. Retrieved 2020-08-21.
16. ^ Samuel K. Moore (2020-11-10). "Rapid Scale-Up of Commercial Ion-Trap Quantum Computers". IEEE Spectrum.
17. ^ Gambetta, Jay [@jaygambetta] (December 3, 2020). "On the same system (IBM Q System One - Montreal) that we hit a quantum volume of 64 the team recently achieved a quantum volume of 128. This year's progress in the quality of quantum circuits has been amazing. https://t.co/pBYmLkmSoS" (Tweet). Archived from the original on 2022-10-21. Retrieved 2022-12-04 – via Twitter.
18. ^ Leprince-Ringuet, Daphne. "Quantum computing: Honeywell just quadrupled the power of its computer". ZDNet. Retrieved 2021-03-11.
19. ^ "Honeywell and Cambridge Quantum Reach New Milestones". www.honeywell.com. Retrieved 2021-07-23.
20. ^ "Demonstrating Benefits of Quantum Upgradable Design Strategy: System Model H1-2 First to Prove 2,048 Quantum Volume". www.quantinuum.com. Retrieved 2022-01-04.
21. ^ "Pushing quantum performance forward with our highest quantum volume yet". IBM Research Blog. 6 April 2022.
22. ^ "Quantinuum Announces Quantum Volume 4096 Achievement". www.quantinuum.com. Retrieved 2022-04-14.
23. ^ Gambetta, Jay [@jaygambetta] (May 25, 2022). "Just a little update from the IBM Quantum team. QV of 512 achieved😀. Our new gate architecture (Falcon R10) continues to allow higher fidelity and low crosstalk and as a result better quality circuits. Two jumps in QV in the last 2 months. https://t.co/szAKCAD4gA" (Tweet). Archived from the original on 2022-05-28. Retrieved 2022-12-04 – via Twitter.
24. ^ Smith-Goodson, Paul (2022-10-06). "Quantinuum Is On A Roll – 17 Significant Quantum Computing Achievements In 12 Months". Forbes. Archived from the original on 2022-10-06. Retrieved 2023-02-24.((cite web)): CS1 maint: bot: original URL status unknown (link)
25. ^ Monz, Thomas (2023-02-10). "State of Quantum Computing in Europe: AQT pushing performance with a Quantum Volume of 128". techmonitor.ai. Retrieved 2023-05-09.
26. ^ Morrison, Ryan (2023-02-23). "Quantinuum hits quantum performance milestone". techmonitor.ai. Retrieved 2023-02-24.
27. ^ Moses, S.A. (2023-05-09). "A Race-Track Trapped-Ion Quantum Processor". Physical Review X. 13 (4): 041052. arXiv:2305.03828. Bibcode:2023PhRvX..13d1052M. doi:10.1103/PhysRevX.13.041052.
28. ^ Morrison, Ryan (2023-06-30). "Quantinuum H-Series quantum computer accelerates through 3 more performance records for quantum volume". quantinuum. Retrieved 2023-06-30.
29. ^ "IQM Quantum Reports Benchmarks on 20-Qubit System". www.meetiqm.com. 20 February 2024. Retrieved 2024-02-20.
30. ^ Blume-Kohout, Robin; Young, Kevin C. (2020-11-15). "A volumetric framework for quantum computer benchmarks". Quantum. 4: 362. arXiv:1904.05546. Bibcode:2020Quant...4..362B. doi:10.22331/q-2020-11-15-362. ISSN 2521-327X.
31. ^ Proctor, Timothy; Rudinger, Kenneth; Young, Kevin; Nielsen, Erik; Blume-Kohout, Robin (2021-12-20). "Measuring the capabilities of quantum computers". Nature Physics. 18 (1). Springer Science and Business Media LLC: 75–79. arXiv:2008.11294. doi:10.1038/s41567-021-01409-7. ISSN 1745-2473.