# Quantum volume

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 July 2023 is 219, 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 ${\displaystyle \epsilon _{\mathrm {eff} }}$ as

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

The effective error rate ${\displaystyle \epsilon _{\mathrm {eff} }}$ 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 ${\displaystyle \epsilon _{\mathrm {eff} }>\epsilon }$, where ${\displaystyle \epsilon }$ 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 ${\displaystyle \epsilon _{\mathrm {eff} }<\epsilon }$.

The allowable circuit depth decreases when more qubits with the same effective error rate are added. So with these definitions, as soon as ${\displaystyle d(N), the quantum volume goes down if more qubits are added. To run an algorithm that only requires ${\displaystyle 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] Manufacturer Notes
2020, January ${\displaystyle 2^{5}}$ = 32 IBM "Raleigh" (28 qubits)[13]
2020, June ${\displaystyle 2^{6}}$ = 64 Honeywell 6 qubits[14]
2020, August ${\displaystyle 2^{6}}$ = 64 IBM Falcon r4 "Montreal" (27 qubits)[15]
2020, November ${\displaystyle 2^{7}}$ = 128 Honeywell "System Model H1" (10 qubits)[16]
2020, December ${\displaystyle 2^{7}}$ = 128 IBM Falcon r4 "Montreal" (27 qubits)[17]
2021, March ${\displaystyle 2^{9}}$ = 512 Honeywell "System Model H1" (10 qubits)[18]
2021, July ${\displaystyle 2^{10}}$ = 1024 Honeywell "Honeywell System H1" (10 qubits) [19]
2021, December ${\displaystyle 2^{11}}$ = 2048 Quantinuum (previously Honeywell) "Quantinuum System Model H1-2" (12 qubits) [20]
2022, April ${\displaystyle 2^{8}}$ = 256 IBM Falcon r10 "Prague" (27 qubits) [21]
2022, April ${\displaystyle 2^{12}}$ = 4096 Quantinuum "Quantinuum System Model H1-2" (12 qubits) [22]
2022, May ${\displaystyle 2^{9}}$ = 512 IBM Falcon r10 "Prague" (27 qubits) [23]
2022, September ${\displaystyle 2^{13}}$ = 8192 Quantinuum "Quantinuum System Model H1-1" (20 qubits)[24]
2023, February ${\displaystyle 2^{7}}$ = 128 Alpine Quantum Technologies "Compact Ion-Trap Quantum Computing Demonstrator" (24 qubits) [25]
2023, February ${\displaystyle 2^{15}}$ = 32,768 Quantinuum "Quantinuum System Model H1-1" (20 qubits) [26]
2023, May ${\displaystyle 2^{16}}$ = 65,536 Quantinuum "Quantinuum System Model H2" (32 qubits) [27]
2023, June ${\displaystyle 2^{19}}$ = 524,288 Quantinuum "Quantinuum System Model H1-1" (20 qubits) [1]

## References

1. ^ a b Morrison, Ryan (2023-06-30). "Quantinuum H-Series quantum computer accelerates through 3 more performance records for quantum volume". quantinuum. Retrieved 2023-06-30.
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] (2020-12-03). "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. 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] (2022-05-25). "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. 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". arXiv:2305.03828 [quant-ph].