Why Is IBM’s Notion of Quantum Volume Only Valid up to About 50 Qubits?

  • Measurement of quantum volume is limited to the capacity of available quantum simulators running on classical computers.
  • At present, 50 qubits is viewed as the current practical limit for classical simulation of a quantum computer.
  • The 50-qubit limit appears to be the number of qubits which are simulated and executed by the generated circuits, not the total number of qubits on the system.
  • For example, IBM recently announced a quantum volume of 32 for a 65-qubit system. Only five (or so) qubits out of 65 were used in that test. And six (or so) qubits were used to calculate the quantum volume of 64 for their 29-qubit Falcon processor.
  1. The IBM protocol in a nutshell
  2. Simulation of randomly-generated circuits
  3. Square circuits
  4. Measuring connectivity
  5. Any-to-any connectivity
  6. Swap networks
  7. Calculation of quantum volume
  8. QV vs. VQ
  9. Limits of quantum simulators
  10. 50 is the base 2 logarithm of one quadrillion
  11. One quadrillion quantum states pushes the limits of classical simulation
  12. The IBM paper
  13. Sorry, but this paper won’t dive into all details of the IBM protocol or all aspects of quantum volume
  14. Quantum volume in the Qiskit Textbook
  15. Unclear whether the 50-qubit limit is for the entire system or only for the circuit being tested
  16. What if your quantum computer has more than 50 qubits?
  17. How might logical qubits impact the 50-qubit limit?
  18. Possible need for an artificial limit on quantum volume
  19. What does the IBM paper say about lower error rates?
  20. So what is the real limit?
  21. What is the practical limit?
  22. How much storage is needed for quantum state?
  23. What are the prospects for a quantum performance metric that doesn’t require classical simulation?
  24. Potential for application-specific benchmarks
  25. Benchmarks based on verifiable algorithms
  26. How many qubits are likely to be used for production-scale applications?
  27. Quantum volume is a deadend
  28. Why doesn’t IBM refer to quantum volume in their quantum roadmap?
  29. Why doesn’t IBM publish the quantum volume of their 53-qubit machine?
  30. But IBM does say that their 65-qubit system has a quantum volume of 32
  31. How does one configure a simulator for a system with more than 50 qubits?
  32. Did IonQ achieve a quantum volume of 4 million for 32 qubits?
  33. Other related topics
  34. The future

The IBM protocol in a nutshell

Simulation of randomly-generated circuits

Square circuits

Measuring connectivity

Any-to-any connectivity

Swap networks

Calculation of quantum volume

  • log2(VQ) = argmax min(m, d(m))
  1. m = 3, VQ = 2³ = 8.
  2. m = 4, VQ = 2⁴ = 16.
  3. m = 5, VQ = 2⁵ = 32.
  4. m = 6, VQ = 2⁶ = 64.
  5. m = 7, VQ = 2⁷ = 128.
  6. m = 20, VQ = 2²⁰ = 1,048,576.

QV vs. VQ

Limits of quantum simulators

50 is the base 2 logarithm of one quadrillion

  • 50 qubits = 2⁵⁰ = 1,125,899,906,842,624 (one quadrillion) quantum states.
  • Or, log2(1,125,899,906,842,624 quantum states) = 50 qubits.

One quadrillion quantum states pushes the limits of classical simulation

The IBM paper

  • We introduce a single-number metric, quantum volume, that can be measured using a concrete protocol on near-term quantum computers of modest size (n≲50), and measure it on several state-of-the-art transmon devices, finding values as high as 16. The quantum volume is linked to system error rates, and is empirically reduced by uncontrolled interactions within the system. It quantifies the largest random circuit of equal width and depth that the computer successfully implements. Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting toolchains are expected to have higher quantum volumes. The quantum volume is a pragmatic way to measure and compare progress toward improved system-wide gate error rates for near-term quantum computation and error-correction experiments.
  • near-term quantum computers of modest size (n≲50)

Sorry, but this paper won’t dive into all details of the IBM protocol or all aspects of quantum volume

Quantum volume in the Qiskit Textbook

What if your quantum computer has more than 50 qubits?

Unclear whether the 50-qubit limit is for the entire system or only for the circuit being tested

How might logical qubits impact the 50-qubit limit?

Possible need for an artificial limit on quantum volume

  1. 2⁵⁰. A reasonable default — if sufficient resources are available.
  2. 2⁵⁵. Probably a general practical limit for current classical technology.
  3. 2⁶⁰. Probably an absolute limit for current classical technology.
  4. 2⁴⁰. May be a more reasonable default since 2⁵⁰ or higher would consume extreme resources.
  5. 2³⁸. If that’s what Google’s cloud-based simulator supports and if it does so efficiently.
  6. 2³⁵. May be a reasonable default unless the quantum hardware is exceptionally good.
  7. 2³². May be a good default for average, commodity hardware.

What does the IBM paper say about lower error rates?

  • For error rates as low at 10E−4, we anticipate that model circuits U that can be successfully implemented will involve few enough qubits and/or low enough depth to compute HU classically. For lower error rates than this, the quantum volume can be superseded by new volume metrics or modified so classical simulations are not necessary.
  • quantum volume can be superseded by new volume metrics
  • or modified so classical simulations are not necessary

So what is the real limit?

What is the practical limit?

  1. 16 qubits = 2¹⁶ = 65,536 quantum states.
  2. 20 qubits = 2²⁰ = 1,048,576 (one million) quantum states.
  3. 24 qubits = 2²⁴ = 16,777,216 (16 million) quantum states.
  4. 28 qubits = 2²⁸ = 268,435,456 (268 million) quantum states.
  5. 32 qubits = 2³² = 4,294,967,296 (4 billion) quantum states.
  6. 36 qubits = 2³⁶ = 68,719,476,736 (68 billion) quantum states.
  7. 38 qubits = 2³⁸ = 274,877,906,943 (274 billion) quantum states.
  8. 40 qubits = 2⁴⁰ = 1,099,511,627,776 (one trillion) quantum states.
  9. 41 qubits = 2⁴¹ = 2,199,023,255,552 (two trillion) quantum states.
  10. 42 qubits = 2⁴² = 4,398,046,511,104 (four trillion) quantum states.
  11. 45 qubits = 2⁴⁵ = 35,184,372,088,832 (35 trillion) quantum states.
  12. 50 qubits = 2⁵⁰ = 1,125,899,906,842,624 (one quadrillion) quantum states.
  13. 53 qubits = 2⁵³ = 9,007,199,254,740,992 (nine quadrillion) quantum states.
  14. 54 qubits = 2⁵⁴ = 18,014,398,509,481,984 (18 quadrillion) quantum states.
  15. 55 qubits = 2⁵⁵ = 36,028,797,018,963,968 (36 quadrillion) quantum states.
  16. 60 qubits = 2⁶⁰ = 1,152,921,504,606,846,976 (one quintillion) quantum states.
  17. 64 qubits = 2⁶⁴ = 18,446,744,073,709,551,616 (18 quintillion) quantum states.
  18. 65 qubits = 2⁶⁵ = 36,893,488,147,419,103,232 (37 quintillion) quantum states.
  19. 72 qubits = 2⁷² = 4,722,366,500,000,000,000,000 (4 sextillion) quantum states.

How much storage is needed for quantum state?

What are the prospects for a quantum performance metric that doesn’t require classical simulation?

Potential for application-specific benchmarks

Benchmarks based on verifiable algorithms

How many qubits are likely to be used for production-scale applications?

Quantum volume is a deadend

Why doesn’t IBM refer to quantum volume in their quantum roadmap?

Why doesn’t IBM publish the quantum volume of their 53-qubit machine?

But IBM does say that their 65-qubit system has a quantum volume of 32

How does one configure a simulator for a system with more than 50 qubits?

Did IonQ achieve a quantum volume of 4 million for 32 qubits?

Other related topics

  1. General plain-language explanation of quantum volume.
  2. Quantum volume metric doesn’t appear to have any direct utility to designers of quantum algorithms or developers of quantum applications. No guidance is offered for how to interpret or parse a quantum volume metric in a way that yields useful information for algorithm designers or application developers.
  3. The utility of quantum volume seems to be limited to marketing — to allow a vendor to show that their machine is more “powerful” than a competitor’s machine.
  4. Why is quantum volume a power of 2? Why not just use the exponent alone — 6 instead of 64 (2⁶)? The original IBM paper offers no rationale.
  5. A logarithmic metric would be much more appropriate for quantum computers where the advantage is expected to be exponential, so that relatively small numbers can be used rather than very large numbers. An ideal quantum computer with 65 perfect qubits would have a quantum volume of 36,893,488,147,419,103,232 (37 quintillion), which is not a number which a typical quantum algorithm designer or application developer could use in a practical manner, as opposed to 65 qubits and a sense of maximum circuit depth, coherence, and any limits to connectivity.
  6. What can you infer about the number of qubits from the quantum volume metric? log2(QV) is the minimum of the number of qubits — the machine may have more qubits, and currently is likely to have significantly more qubits.
  7. Is quantum volume too strict or too lenient? Does an average algorithm need that much connectivity, accuracy, or coherence? Some algorithms might need more qubits but shallow depth circuits, while others may need fewer qubits but much deeper circuits. Should there be levels of quantum volume, so algorithm designers and application developers can match their particular needs to the most appropriate metric?
  8. Impact of SWAP networks when the machine does not have direct any-to-any connectivity. How much swapping of quantum states was needed to achieve a particular value of quantum volume? What qubit and circuit sizes failed due to excessive SWAP errors?
  9. Using quantum Fourier transform (QFT) for benchmarking.
  10. Using quantum phase estimation (QPE) for benchmarking.
  11. Using electronic complexity of an atom for benchmarking.
  12. Using complexity of a molecule for benchmarking.
  13. Using various specific optimization scenarios for benchmarking.
  14. Using portfolio size for benchmarking for financial applications.
  15. Families or groups of application-specific benchmarking.
  16. Benchmarking metrics that can be expressed both in terms of raw machine resources and application domain-specific terms. Such as the “n” used in Big-O notation for algorithmic complexity.
  17. Multi-factor benchmarking, rather than a single, all-encompassing benchmark metric.
  18. How to benchmark machines with hundreds, thousands, or millions of qubits.
  19. Whether quantum volume might apply to any of the machines on IBM’s quantum roadmap — 65, 127, 433, 1,121, or one million qubits.
  20. Benchmarks for algorithms which don’t require square circuits — more qubits but shallow circuits, or deeper circuits with fewer qubits.
  21. Circuits which use a lot of nearest-neighbor connectivity, but don’t require full any-to-any connectivity.
  22. Future quantum architectures which are a dramatic departure from current architectures.
  23. Modular and distributed quantum machine architectures. Impact of quantum interconnects or qubit shuttling.
  24. Benchmarking in terms of higher-level programming models using algorithmic building blocks other than raw qubits and quantum logic gates.
  25. Contemplate future quantum simulator capacity and performance for much larger simulations.
  26. It hardly seems worthwhile to characterize the performance of a quantum computer by anything short of its relative advantage compared to a classical computer. It would be more compelling to have a metric whose value at 1.0 would mean the same performance as a classical computer.

The future

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Freelance Consultant

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Jack Krupansky

Jack Krupansky

Freelance Consultant

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