Nines of Qubit Fidelity

  1. 90% error-free operation = 10% error rate (0.10) = one nine of qubit fidelity.
  2. 99% error-free operation = 1% error rate (0.01) = two nines of qubit fidelity.
  3. 99.9% error-free operation = 0.1% error rate (0.001) = three nines of qubit fidelity.
  4. 99.99% error-free operation = 0.01% error rate (0.0001) = four nines of qubit fidelity.
  5. 98% error-free operation = 2% error rate (0.02) = 1.8 nines of qubit fidelity.
  6. 95% error-free operation = 5% error rate (0.05) = 1.5 nines of qubit fidelity.
  7. 99.3% error-free operation = 0.7% error rate (0.07) = 2.3 nines of qubit fidelity.
  1. Qubit fidelity is all about getting correct results and minimizing errors.
  2. Qubit fidelity is freedom from worry about errors in the results of a quantum computation.
  3. Qubit fidelity is the degree of confidence in the results of a quantum computation.
  4. Perfect qubits would be best but they aren’t available.
  5. Quantum error correction (QEC) might achieve perfect qubits but it isn’t yet available.
  6. Types of errors and their sources.
  7. Failure versus error.
  8. Qubit fidelity includes gate fidelity.
  9. Qubit fidelity is gate fidelity.
  10. Fidelity and reliability are approximate synonyms.
  11. Qubit fidelity and qubit reliability are approximate synonyms.
  12. Error rate.
  13. Error rate as an integer.
  14. Error rate as a fraction.
  15. Error rate as a decimal number.
  16. Error rate as a percentage.
  17. Error-free operation.
  18. Error-free operation as a decimal number.
  19. Error-free operation as a percentage.
  20. Nines of error-free operation.
  21. Qubit fidelity (reliability).
  22. Nines of qubit fidelity — the degree of perfection.
  23. Fractional nines of qubit fidelity.
  24. Nines of qubit reliability.
  25. Can error rate ever be zero (or nines ever be infinite)?
  26. Roots of nines in classical hardware availability.
  27. Low, typical, high for error rates
  28. Per-qubit error rate.
  29. Per-qubit fidelity.
  30. Overall qubit fidelity.
  31. Single versus two qubit gate fidelity.
  32. Measurement fidelity.
  33. Why is measurement so error-prone?
  34. Composite fidelity.
  35. Benchmark test for composite fidelity.
  36. Effective qubit fidelity.
  37. Should bad qubits be ignored when calculating qubit fidelity?
  38. A major fly in the ointment: SWAP networks for connectivity.
  39. Optimizing qubit placement to reduce SWAP networks.
  40. Qubit fidelity for Google Sycamore Weber processor.
  41. Nuances of nines.
  42. Application-specific nines of qubit fidelity.
  43. Application requirements for qubit fidelity.
  44. What will qubit fidelity indicate about accuracy and gradations of probability amplitudes?
  45. What will qubit fidelity indicate about accuracy and gradations of phase?
  46. Might probability and phase have different qubit fidelities?
  47. Noisy qubits.
  48. Perfect qubits.
  49. Logical qubits.
  50. Near-perfect qubits.
  51. How close to perfect is a near-perfect qubit?
  52. Qubit fidelity vs. result accuracy.
  53. Relevance of qubit fidelity to The ENIAC Moment of quantum computing.
  54. Relevance of qubit fidelity to quantum error correction.
  55. Relevance of qubit fidelity to The FORTRAN moment of quantum computing.
  56. How many nines of qubit fidelity will be needed for quantum Fourier transform and quantum phase estimation?
  57. Relevance of qubit fidelity to achieving dramatic quantum advantage.
  58. Impact of qubit fidelity on shot count (circuit repetitions).
  59. Quantum error correction requires less qubit fidelity, but it’s a tradeoff with capacity due to limited numbers of physical qubits.
  60. Regular calibration needed for qubit fidelity.
  61. Impact of coherence relaxation curves on qubit fidelity for deeper circuits.
  62. Configure simulators to match nines of real machines.
  63. Where should vendors report error rates and qubit fidelity?
  64. Nines for quantum computer system availability.
  65. Nines of reliability.
  66. Need a roadmap for nines.
  67. IBM’s Quantum Volume versus qubit fidelity.
  68. Separating qubit fidelity from Quantum Volume enables greater flexibility in characterizing performance.
  69. Summary and conclusions.

Qubit fidelity is all about getting correct results and minimizing errors

Qubit fidelity is freedom from worry about errors in the results of a quantum computation

Qubit fidelity is the degree of confidence in the results of a quantum computation

Perfect qubits would be best but they aren’t available

Quantum error correction (QEC) might achieve perfect qubits but it isn’t yet available

Types of errors and their sources

  1. Errors which occur within individual qubits, even when completely idle.
  2. Errors which occur when operations are performed on qubits. Qubits in action.
  1. Decoherence. Gradual decay of values (quantum state) over time. Even when idle.
  2. Gate errors. Each operation on a qubit introduces another potential degree of error.
  3. Measurement errors. Simply measuring a qubit has some chance of failure.
  1. Environmental interference. Electromagnetic radiation, thermal, acoustical, mechanical (shocks, subtle vibrations, and earth tremors), electrical (power supplies and power sources.) Even despite the best available shielding.
  2. Crosstalk between supposedly isolated devices. Absolute isolation is not assured.
  3. Noise in control circuitry. Noise in the classical digital and analog circuitry which controls execution of gates on qubits.
  4. Imperfections in the manufacture of qubits.

Failure versus error

Qubit fidelity includes gate fidelity

Qubit fidelity is gate fidelity

Fidelity and reliability are approximate synonyms

Qubit fidelity and qubit reliability are approximate synonyms

Error rate

  1. The number of errors which occur per unit of time.
  2. The amount of time before an error can be expected to occur.
  3. The number of operations which can be performed before an error might be expected to occur.
  4. The fraction or percentage of operations which are error-free.
  5. The fraction or percentage of operations which fail or are in some way faulty — in error.
  1. 1.0 or 100% — all operations fail (or have errors). 0% qubit fidelity.
  2. 0.50 or 50% — half of operations fail (or have errors). 50% qubit fidelity.
  3. 0.10 or 10% — one in ten operations fail (or have errors). 90% qubit fidelity.
  4. 0.05 or 5% — one in twenty operations fail (or have errors). 95% qubit fidelity.
  5. 0.02 or 2% — one in fifty operations fail (or have errors). 98% qubit fidelity.
  6. 0.01 or 10E-2 or 1% — one in a hundred operations fail (or have errors). 99% qubit fidelity.
  7. 0.001 or 10E-3 or 0.1% — one in a thousand operations fail (or have errors). 99.9% (three nines) qubit fidelity.
  8. 0.0001 or 10E-4 or 0.01% — one in ten thousand operations fail (or have errors). 99.99% (four nines) qubit fidelity.
  9. 0.00001 or 10E-5 or 0.001% — one a hundred thousand operations fail (or have errors). 99.999% (five nines) qubit fidelity.
  10. 0.000001 or 10E-6 or 0.0001% — one in a million operations fail (or have errors). 99.9999% (six nines) qubit fidelity.
  11. 0.0000001 or 10E-7 or 0.00001% — one in ten million operations fail (or have errors). 99.99999% (seven nines) qubit fidelity.
  12. 0.00000001 or 10E-8 or 0.000001% — one in a hundred million operations fail (or have errors). 99.999999% eight nines) qubit fidelity.
  13. 0.000000001 or 10E-9 or 0.0000001% — one in a billion operations fail (or have errors). 99.9999999% (nine nines) qubit fidelity.

Forms for expressing fidelity

  1. Error rate as an integer.
  2. Error rate as a fraction.
  3. Error rate as a decimal number.
  4. Error rate as a percentage.
  5. Error-free operation as a decimal number.
  6. Error-free operation as a percentage.
  7. Nines of error-free operation.

Error rate as an integer

  1. 1 in 10. An error every 10 operations (on average.)
  2. 1 in 100. An error every 100 operations.
  3. 1 in 1,000. An error every 1,000 operations.
  4. 1 in 1,000,000. An error every one million operations.

Error rate as a fraction

  1. 1/10. An error every 10 operations.
  2. 1/100. An error every 100 operations.
  3. 1/1,000. An error every 1,000 operations.
  4. 1/1,000,000. An error every one million operations.

Error rate as a decimal number

  1. 0.1. An error every 10 operations.
  2. 0.01. An error every 100 operations.
  3. 0.001. An error every 1,000 operations.
  4. 0.000001. An error every one million operations.

Error rate as a percentage

  1. 10%. An error every 10 operations.
  2. 1%. An error every 100 operations.
  3. 0.1%. An error every 1,000 operations.
  4. 0.0001%. An error every one million operations.

Error-free operation

  1. Error-free operation as a decimal number = 1.0 minus the error rate as a decimal number.
  2. Error-free operation as a percentage = 100% minus the error rate as a percentage.

Error-free operation as a decimal number

  1. Error rate of 0.1. Error-free operation = 1.0 minus 0.1 = 0.9.
  2. Error rate of 0.01. Error-free operation = 1.0 minus 0.01 = 0.99.
  3. Error rate of 0.001. Error-free operation = 1.0 minus 0.001 = 0.999.

Error-free operation as a percentage

  1. Error rate of 10%. Error-free operation = 100% — 10% = 90%.
  2. Error rate of 1%. Error-free operation = 100% — 1% = 99%.
  3. Error rate of 0.1%. Error-free operation = 100% — 0.1% = 99.9%.

Nines of error-free operation

  1. 90% error free = one nine.
  2. 99% error free = two nines.
  3. 99.9% error free = three nines.
  4. 99.96% error free = 3.6 nines. See Fractional nines of qubit fidelity.

Qubit fidelity (reliability)

  1. Qubit fidelity for an error rate of 0.1 = 90%.
  2. Qubit fidelity for an error rate of 0.01 = 99%.
  3. Qubit fidelity for an error rate of 0.001 = 99.9%.
  4. Qubit fidelity for an error rate of 0.0001 = 99.99%.

Qubit fidelity as a decimal number

  1. Qubit fidelity for an error rate of 0.1 = 0.9.
  2. Qubit fidelity for an error rate of 0.01 = 0.99.
  3. Qubit fidelity for an error rate of 0.001 = 0.999.
  4. Qubit fidelity for an error rate of 0.0001 = 0.9999.

Nines of qubit fidelity — the degree of perfection

  1. One nine. Such as 90%, 98%, 97%, or maybe even 95%. One error in 10, 50, 33, or 20 operations.
  2. Two nines. Such as 99%, 99.5%, or even 99.8%. One error in 100 operations.
  3. Three nines. Such as 99.9%, 99.95%, or even 99.98%. One error in 1,000 operations.
  4. Four nines. Such as 99.99%, 99.995%, or even 99.998%. One error in 10,000 operations.
  5. Five nines. Such as 99.999%, 99.9995%, or even 99.9998%. One error in 100,000 operations.
  6. Six nines. Such as 99.9999%, 99.99995%, or even 99.99998%. One error in one million operations.
  7. Seven nines. Such as 99.99999%, 99.999995%, or even 99.999998%. One error in ten million operations.
  8. And so on. As many nines as you wish.

Fractional nines of qubit fidelity

  1. 98%, 97%, 95% — 1.8, 1.7, or 1.5 nines. One error in 50, 30, or 20 operations, in contrast to 90% (1 nine) which is one error in 10 operations or 99% (2 nines) which is one error in one hundred operations.
  2. 99.8%, 99.7%, 99.5% — 2.8, 2.7, or 2.5 nines. One error in 500, 300, or 200 operations, in contrast to 99% (2 nines) which is one error in 100 operations or 99.9% (3 nines) which is one error in 1,000 operations.
  3. 99.98%, 99.97%, 99.95% — 3.8, 3,7, 3.5 nines. One error in 5,000, 3,000, or 2,000 operations, in contrast to 99.9% (3 nines) which is one error in 1,000 operations or 99.99% (4 nines) which is one error in 10,000 operations.

Nines of qubit reliability

Can error rate ever be zero (or nines ever be infinite)?

Roots of nines in classical hardware availability

Low, typical, high for error rates

  1. Low. The lowest error rate.
  2. High. The highest error rate.
  3. Typical. The average or most typical error rate.
  1. Per-qubit. Each qubit can potentially have a different error rate.
  2. Multiple runs. Some number of runs of the measurement must be performed for each qubit. There may be variability between runs.

Per-qubit error rate

Per-qubit fidelity

  1. Per-qubit fidelity. One measure for each qubit, measuring only that qubit.
  2. Overall qubit fidelity. An overall measure which averages (in some unspecified fashion) the fidelities of all qubits to come up with a single overall measure of qubit fidelity. Technically, it should be a distribution with standard statistical characteristics — low, typical, high, etc., and maybe 50% and 90% measures as well.

Overall qubit fidelity

Single versus two qubit gate fidelity

  1. The best case.
  2. The worst case.
  3. The average of the best and worst cases.
  4. The range of best to worst case.

Measurement fidelity

  1. The best case.
  2. The worst case.
  3. The average of the best and worst cases.
  4. The range of best to worst case.

Why is measurement so error-prone?

Composite qubit fidelity

  1. The best case.
  2. The worst case.
  3. Some sort of blended or weighted average of the full range of cases.
  4. The range of best to worst case.
  1. Typical two-qubit error rate (fidelity). Any interesting circuit will have plenty of two-qubit gates. Presume sufficient depth that measurement error is the lesser issue.
  2. Measurement error rate. For shallow circuits.
  1. Select one of the many metrics as the preferred metric.
  2. Some magic formula to combine all of the many metrics into a single, composite metric.

Benchmark test for composite qubit fidelity

  1. A modest number of qubits. Possibly even only three, or maybe five.
  2. A circuit of modest depth. Possible five to ten gates deep. Enough to accumulate errors.
  3. A mix of both single and two-qubit gates. Maybe three quarters single-qubit gates.
  4. Measurement of a fair fraction of the qubits. Possibly as few as three to five of the qubits, possibly half of them, possibly three quarters of them, or maybe all of them. Enough to see measurement errors.
  5. A moderate number of circuit repetitions (shot count). Enough to achieve a reasonable statistical distribution of results.
  6. Calculate the overall composite error rate and qubit fidelity. Compare actual measured results to expected results. Takes into account single-qubit fidelity, two-qubit fidelity, and measurements.
  1. Small. 5–8 qubits.
  2. Medium. 12 to 28 qubits.
  3. Large. 32 to 40 qubits.
  4. Extra large. 50 to 80 qubits.

Effective qubit fidelity

  1. Select one of the many metrics as the preferred metric.
  2. Some magic formula to combine all of the many metrics into a single, composite metric.
  3. A benchmark test to empirically derive effective average qubit fidelity across all of the individual metrics.

Should bad qubits be ignored when calculating qubit fidelity?

  1. Some applications require higher qubit fidelity.
  2. Some have lower required fidelity.
  3. Some may not have a threshold at all.
  1. Maybe look at nines for the best 20% of the overall machine. Anything more than two nines from that 20% would be atypical.
  2. Or maybe 50% — or allow the system operator to configure the minimum number of qubits to set the atypical threshold.

A major fly in the ointment: SWAP networks for connectivity

  1. n steps are needed — quantum state must be moved a distance of n qubits.
  2. Each step, a SWAP operation, is actually three CNOT gates.
  3. Each CNOT gate is a 2-qubit gate which has a typical error rate.
  4. Finally, the desired 2-qubit gate can be executed.

Optimizing qubit placement to reduce SWAP networks

Qubit fidelity for Google Sycamore Weber processor

  1. Low, typical, and high error rates. Given as percentages. For each of the following categories.
  2. Single-qubit gate error rates. Isolated.
  3. Two-qubit gate error rates. Both isolated and parallel.
  4. Readout (measurement) error for the 0 state. Both isolated and parallel.
  5. Readout (measurement) error for the 1 state. Both isolated and parallel. Roughly three times greater than readout error for the 0 state.
  1. Typical isolated single-qubit error rate: 0.1% = 99.9% = three nines.
  2. Typical isolated two-qubit error rate: 0.9% = 99.1% = 2.1 nines.
  3. Typical parallel two-qubit error rate: 1.4% = 98.6% = 1.86 nines.
  4. Typical readout 0 isolated error rate: 1.1% = 98.9% = 1.89 nines.
  5. Typical readout 0 simultaneous error rate: 2.0% = 98.0% = 1.8 nines.
  6. Typical readout 1 isolated error rate: 5.0% = 95.0% = 1.5 nines.
  7. Typical readout 1 simultaneous error rate: 7.0% = 93.0% = 1.3 nines.

Nuances of nines

Application-specific nines of qubit fidelity

Application requirements for qubit fidelity

  • How many nines of qubit fidelity does your algorithm or application require?

What will qubit fidelity indicate about accuracy and gradations of probability amplitudes?

What will qubit fidelity indicate about accuracy and gradations of phase?

Might probability and phase have different qubit fidelities?

Noisy qubits

  1. 70% — not even a single nine.
  2. 85% — still not even a single nine.
  3. 90% — 1 nine.
  4. 95% — still only a single nine, or 1.5 nines.
  5. 99% — 2 nines. Still fairly noisy for many applications.
  6. 99.9% — 3 nines. Not terribly noisy, but still too noisy for some applications

Perfect qubits

Logical qubits

Near-perfect qubits

How close to perfect is a near-perfect qubit?

  1. To enable quantum error correction for logical qubits.
  2. To enable applications using raw physical qubits on NISQ devices.
  1. Shallow depth circuits will require fewer nines.
  2. Deeper circuits will require more nines.

Qubit fidelity vs. result accuracy

Relevance of qubit fidelity to The ENIAC Moment of quantum computing

Relevance of qubit fidelity to quantum error correction

Relevance of qubit fidelity to The FORTRAN moment of quantum computing

How many nines of qubit fidelity will be needed for quantum Fourier transform and quantum phase estimation?

Relevance of qubit fidelity to achieving dramatic quantum advantage

  1. The ENIAC Moment. The first significant production-scale application with a dramatic quantum advantage. But super-elite technical staff will be required to cope with the technical challenges.
  2. The FORTRAN Moment. Sufficient to enable quantum error correction (QEC) and logical qubits for production-scale applications. Advanced hardware and sophisticated algorithm libraries will enable non-elite technical staff to make dramatic progress and easily achieve dramatic quantum advantage for a wide range of applications.
  3. Quantum Fourier transform (QFT) and quantum phase estimation (QPE) for production-scale applications. Many applications will need QFT and QPE to achieve sufficient accuracy of results — and to achieve dramatic quantum advantage.

Impact of qubit fidelity on shot count (circuit repetitions)

  1. Low qubit fidelity. Errors which affect and corrupt the results.
  2. Probabilistic nature of most interesting quantum computations. Even if qubits and gates were ideal and perfect, superposition causes probabilistic results.
  1. Low qubit fidelity (few nines). Higher shot count needed.
  2. High qubit fidelity (more nines). Lower shot count needed.
  3. Very high qubit fidelity (many nines). Few circuit repetitions needed.
  4. Perfect qubits (logical qubits). Only a single execution of a circuit is needed.

Quantum error correction requires less qubit fidelity, but it’s a tradeoff with capacity due to limited numbers of physical qubits

Regular calibration needed for qubit fidelity

  1. Once a day.
  2. Twice a day.
  3. Every eight hours.
  4. Every six hours.
  5. Every four hours.
  6. Every two hours.
  7. Every hour. Probably too frequent, especially if calibration is expensive.
  8. Set a threshold for results of a test application and recalibrate whenever a regular and frequent run of the test application (hourly?) fails to deliver correct results some percentage of the time for some reasonable number of circuit repetitions.

Impact of coherence relaxation curve on qubit fidelity for deeper circuits

  1. T1 — energy relaxation time. How long before a fair percentage of qubits in the 1 state will have decayed to the 0 state.
  2. T2 — dephasing time measured using the Hahn experiment. How long before a fair percentage of qubits will have their phase randomized.
  3. T2* — dephasing time measured using the Ramsey experiment. How long before a fair percentage of qubits will have their phase randomized.

Configure simulators to match nines of real machines

Where should vendors report error rates and qubit fidelity?

  1. Principles of operation. Everything algorithm designers and application developers need to know about how a machine works so that they can develop functioning algorithms and applications. But not performance or implementation details.
  2. Implementation specification. All details about implementation of the machine, particularly limitations and performance.

Nines for quantum computer system availability

Nines of reliability

  1. Nines of qubit fidelity.
  2. Nines of quantum computer system availability.

Need a roadmap for nines

  1. 1 nine — 90% — we may actually be there.
  2. 1.5 nines — 95% — possibly within a year.
  3. 2 nines — 99% — probably reachable within a year or two.
  4. 2.5 nines — 99.5% — maybe 2–3 years.
  5. 3 nines — 99.9% — more of a 3 to 4-year goal.
  6. 3.5 nines — 99.95% — 4-year goal.
  7. 4 nines — 99.99% — more of a 4 to 5-year goal.
  8. 5 nines — 99.999% — a 5-year goal.
  9. 6 nines — 99.9999% — a 5 to 7-year goal.
  10. 8 nines — a pipe dream for now.
  11. 9 nines — ditto.
  12. 12 nines — will this be possible? Maybe, but some serious redesign would be needed.

IBM’s Quantum Volume versus qubit fidelity

Separating qubit fidelity from Quantum Volume enables greater flexibility in characterizing performance

Summary and conclusions

  1. Quantum computers are very error prone.
  2. This isn’t going to change much anytime soon.
  3. Near-perfect and logical qubits are coming, but not so soon.
  4. There are many sources of errors and different types of errors.
  5. Many metrics of fidelity (error rates) are needed.
  6. All of these metrics should be clearly documented
  7. Difficult to come up with a magic formula to reduce all of these metrics into a single, composite metric of qubit fidelity.
  8. But there is value in reducing all of the metrics into a single metric.
  9. Potential for a benchmark test to deduce effective average qubit fidelity.
  10. SWAP networks to overcome connectivity limitations can greatly complicate modeling of average gate fidelity.
  11. Coherence relaxation curves drive qubit fidelity for deeper circuits.
  12. Different applications may have a different focus on which single metric matters most.
  13. Nines are a convenient and easy to use summary of qubit fidelity.
  14. Vendors need to do a much better job of documenting the qubit fidelity of both their current hardware and each milestone in their roadmaps for future hardware.
  15. Algorithm designers and application developers should endeavor to characterize the qubit fidelity requirements of their algorithms and applications.
  16. IBM’s Quantum Volume metric is insufficient — it doesn’t expose qubit fidelity as a separate metric which is a critical need for algorithm design and application development.
  17. Much more research is required, both to improve the fidelity of future qubit hardware and to better characterize the qubit hardware we have today and in the near future.
  18. Much more research is needed for tools and techniques for algorithm designers and application developers as they grapple with qubit fidelity issues, including for advanced techniques such as Quantum Fourier transform (QFT) and quantum phase estimation (QPE).

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Jack Krupansky

Jack Krupansky

Freelance Consultant

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