Quantum volume

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.{{Cite web|title=Honeywell claims to have built the highest-performing quantum computer available|url=https://phys.org/news/2020-06-honeywell-built-highest-performing-quantum.html|access-date=2020-06-22|website=phys.org|language=en}}{{Cite web|last=Smith-Goodson|first=Paul|title=Quantum Volume: A Yardstick To Measure The Performance Of Quantum Computers|url=https://www.forbes.com/sites/moorinsights/2019/11/23/quantum-volume-a-yardstick-to-measure-the-power-of-quantum-computers/|access-date=2020-06-22|website=Forbes|language=en}}{{Cite web|last=|first=|date=|title=Measuring Quantum Volume|url=https://community.qiskit.org/textbook/ch-quantum-hardware/measuring-quantum-volume.html|access-date=2020-08-21|website=Qiskit.org|language=en}}

IBM defined its Quantum Volume metric{{cite journal |last1=Cross |first1=Andrew W. |last2=Bishop |first2=Lev S. |last3=Sheldon |first3=Sarah |last4=Nation |first4=Paul D. |last5=Gambetta |first5=Jay M. |year=2019 |title=Validating quantum computers using randomized model circuits |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.032328 |journal=Phys. Rev. A |volume=100 |issue=3 |pages=032328 |doi=10.1103/PhysRevA.100.032328 |access-date=2020-10-02|arxiv=1811.12926 |bibcode=2019PhRvA.100c2328C |s2cid=119408990 }} 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.{{Cite web|last=Mandelbaum|first=Ryan F.|date=2020-08-20|title=What Is Quantum Volume, Anyway?|url=https://medium.com/qiskit/what-is-quantum-volume-anyway-a4dff801c36f|access-date=2020-08-21|website=Medium Qiskit|language=en}}{{Cite web|last=Sanders|first=James|first4=2:00 Pm|date=August 12, 2019|title=Why quantum volume is vital for plotting the path to quantum advantage|url=https://www.techrepublic.com/article/why-quantum-volume-is-vital-for-plotting-the-path-to-quantum-advantage/|access-date=2020-08-22|website=TechRepublic|language=en}}

Generally, the larger the quantum volume, the more complex the problems a quantum computer can solve.{{Cite web|last=Patty|first=Lee|date=2020|others=Chief Scientist for Honeywell Quantum Solutions|title=Quantum Volume: The Power of Quantum Computers|url=https://www.honeywell.com/content/honeywell/us/en/newsroom/news/2020/03/quantum-volume-the-power-of-quantum-computers.html|access-date=2020-08-21|website=www.honeywell.com|language=en}}

Alternative benchmarks, such as Cross-entropy benchmarking, reliable Quantum Operations per Second (rQOPS) proposed by Microsoft, Circuit Layer Operations Per Second (CLOPS) proposed by IBM and IonQ's Algorithmic Qubits, have also been proposed.{{Cite web|last=Yirka |first=Bob|date=2023-06-24|title=Microsoft claims to have achieved first milestone in creating a reliable and practical quantum computer |url=https://phys.org/news/2023-06-microsoft-milestone-reliable-quantum.html|access-date=2024-07-01|website=phys.org|language=en}}{{Cite web|last=Leprince-Ringuet |first=Daphne|date=2021-11-02|title=Quantum computing: IBM just created this new way to measure the speed of quantum processors|url=https://www.zdnet.com/article/quantum-computing-this-is-how-ibm-is-now-measuring-the-speed-of-its-quantum-processors/|access-date=2024-07-01|website=ZDNet|language=en}}

The quantum volume of a quantum computer was originally defined in 2018 by Nikolaj Moll et al.{{Cite journal

| first1 = Nikolaj | last1 = Moll

| first2 = Panagiotis | last2 = Barkoutsos

| first3 = Lev S | last3 = Bishop

| first4 = Jerry M | last4 = Chow

| first5 = Andrew | last5 = Cross

| first6 = Daniel J | last6 = Egger

| first7 = Stefan | last7 = Filipp

| first8 = Andreas | last8 = Fuhrer

| first9 = Jay M | last9 = Gambetta

| first10 = Marc | last10 = Ganzhorn

| first11 = Abhinav | last11 = Kandala

| first12 = Antonio | last12 = Mezzacapo

| first13 = Peter | last13 = Müller

| first14 = Walter | last14 = Riesswe introd

| first15 = Gian | last15 = Salis

| first16 = John | last16 = Smolin

| first17 = Ivano | last17 = Tavernelli

| first18 = Kristan | last18 = Temme

| title = Quantum optimization using variational algorithms on near-term quantum devices

| journal = Quantum Science and Technology

| volume = 3

| year = 2018

| issue = 3

| page = 030503

| doi = 10.1088/2058-9565/aab822

| arxiv = 1710.01022

| bibcode = 2018QS&T....3c0503M

| doi-access= free

}}

However, since around 2021 that definition has been supplanted by IBM's 2019 redefinition.{{cite journal |last1=Baldwin |first1=Charles |last2=Mayer |first2=Karl |date=2022 |title=Re-examining the quantum volume test: Ideal distributions, compiler optimizations, confidence intervals, and scalable resource estimations |journal=Quantum |volume=6 |pages=707 |doi=10.22331/q-2022-05-09-707 |arxiv=2110.14808 |bibcode=2022Quant...6..707B |s2cid=240070758 }}{{cite arXiv |last=Miller |first=Keith |date=2022-07-14 |title=An Improved Volumetric Metric for Quantum Computers via more Representative Quantum Circuit Shapes |class=quant-ph |eprint=2207.02315}}

The original definition depends on the number of qubits {{mvar|N}} as well as the number of steps that can be executed, the circuit depth {{mvar|d}}

:$$

\tilde{V}_Q = \min[N, d(N)]^2.

The circuit depth depends on the effective error rate {{math|{{var|ε}}{{sub|eff}}}} as

:$$

d \simeq \frac{1}{N\varepsilon_\mathrm{eff}}.

The effective error rate {{math|{{var|ε}}{{sub|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 {{math|{{var|ε}}{{sub|eff}} > {{var|ε}}}}, where {{mvar|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 {{math|{{var|ε}}{{sub|eff}} < {{var|ε}}}}.

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

:$$

V_Q = \max_{n

where the maximum is taken over an arbitrary choice of {{mvar|n}} qubits.

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:https://pennylane.ai/qml/demos/quantum_volume.html ([https://web.archive.org/web/20201216123226/https://pennylane.ai/qml/demos/quantum_volume.html archived])

:$\backslash log\_2\; V\_Q\; =\; \backslash underset\{n\; \backslash le\; N\}\{\backslash operatorname\{arg\backslash ,max\}\}\backslash left\backslash \{\backslash min\backslash left[n,\; d(n)\backslash right]\backslash right\backslash \}$

The world record, {{As of|2025|05|lc=y}}, for the highest quantum volume is 2{{sup|23}}.{{Cite web |date=August 11, 2024 |url=https://github.com/CQCL/quantinuum-hardware-quantum-volume |title=quantinuum-hardware-quantum-volume |website=GitHub }} Here is an overview of historically achieved quantum volumes:

The quantum volume benchmark defines a family of square circuits, whose number of qubits {{mvar|N}} and depth {{mvar|d}} are the same. Therefore, the output of this benchmark is a single number. However, a proposed generalization is the volumetric benchmark{{cite journal | last1=Blume-Kohout | first1=Robin | last2=Young | first2=Kevin C. | title=A volumetric framework for quantum computer benchmarks | journal=Quantum | volume=4 | date=2020-11-15 | issn=2521-327X | doi=10.22331/q-2020-11-15-362 | page=362| arxiv=1904.05546 | bibcode=2020Quant...4..362B }} framework, which defines a family of rectangular quantum circuits, for which {{mvar|N}} and {{mvar|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 {{mvar|N}} and {{mvar|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,{{cite journal | last1=Proctor | first1=Timothy | last2=Rudinger | first2=Kenneth | last3=Young | first3=Kevin | last4=Nielsen | first4=Erik | last5=Blume-Kohout | first5=Robin | title=Measuring the capabilities of quantum computers | journal=Nature Physics | publisher=Springer Science and Business Media LLC | volume=18 | issue=1 | date=2021-12-20 | issn=1745-2473 | doi=10.1038/s41567-021-01409-7 | pages=75–79| arxiv=2008.11294 }} 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 {{mvar|N}} versus {{mvar|d}} trade-off for the processor being benchmarked. This Pareto front provides information on the largest depth {{mvar|d}} a patch of a given number of qubits {{mvar|N}} can withstand, or, alternatively, the biggest patch of {{mvar|N}} qubits that can withstand executing a circuit of given depth {{mvar|d}}.

• Noisy intermediate-scale quantum era
• Quantum fidelity

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