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

A quantum computer achieves results by executing quantum logic gates on the quantum state of qubits. As long as the correct results are produced and measured, we would say that the computer and its qubits have high fidelity.

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

More succinctly, 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

Another way of putting it is that qubit fidelity is the degree of confidence in the results of a quantum computation.

Perfect qubits would be best but they aren’t available

Classical computer hardware is very close to being perfect (most of the time), but quantum computer hardware is far from that reality. Perfect qubits would be great, but they simply cannot be realized with current quantum computing technology.

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

Quantum error correction (QEC) is a clever scheme to implement so-called logical qubits which are virtually perfect qubits using some type of coding scheme to use many imperfect physical qubits to implement each logical qubit.

Types of errors and their sources

There are two broad areas of errors in quantum computations:

  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

Generally, failure is an all or nothing proposition — an operation is a complete success or a complete failure, with no room for shades of gray or partial failure. But in quantum computing, values are represented as quantum states which are represented by complex numbers, which are pairs of real numbers, not even discrete integers, and certainly not strict binary 0 or 1, which admits the possibility of a wide range of errors beyond the simple binary possibilities of complete success and complete failure.

Qubit fidelity includes gate fidelity

Technically, qubit fidelity in the purest sense is the capacity of the qubit to maintain its quantum state over time — nominally referred to as coherence, and the separate concept of gate fidelity — the ability to assure that a quantum logic gate is correctly executed, are distinct, as is the distinct concept of measurement error, but this paper views them collectively as qubit fidelity — covering any factor that affects the correctness of the results of a quantum computation.

Qubit fidelity is gate fidelity

Just to be clear, when people refer to qubit fidelity they are for all intents and purposes referring to gate fidelity. Sure, the fidelity of the actual qubit alone does matter as well, but it is the fidelity of execution of quantum logic gates that drives overall qubit fidelity for algorithms and applications.

Fidelity and reliability are approximate synonyms

Fidelity and reliability are approximate synonyms and can generally be used interchangeably.

Qubit fidelity and qubit reliability are approximate synonyms

Qubit fidelity and qubit reliability are approximate synonyms as well. This paper generally refers to qubit fidelity.

Error rate

The error rate for a quantum computer or even a single qubit — or of any system in general, can be conceptualized in one of five ways:

  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

Qubit fidelity can be expressed in a number of ways:

  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

The error rate can be expressed as an integer to indicate how many operations could be expected to occur before an error is likely. Such as:

  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

The error rate can be expressed as a fraction with a numerator of 1 and a denominator of the error rate as an integer. Such as:

  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

The error rate as a fraction can be expressed as the decimal equivalent of the fraction:

  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

The decimal error rate can be expressed as a percentage by multiplying by 100. This is the percentage of operations which are expected to fail or otherwise have errors:

  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

The goal is for a quantum computer to be able to execute a large number of operations over an extended period of time with minimal, or even no, errors. So, ultimately, this is the true metric we are interested in.

  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

Error-free operation as a decimal number is 1.0 minus the error rate 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

Error-free operation as a percentage is 100% minus the error rate 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

A rough approximation of fidelity (reliability) is the number of nines (digit “9”) in the error-free operation as a percentage.

  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)

The reliability (fidelity) of a qubit is characterized as the percentage of error-free operations:

  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

In some contexts, qubit fidelity might be expressed 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

The degree of perfection of a qubit can be measured using so-called nines — 9’s, which is the qubit fidelity (reliability) expressed as a percentage of error-free operation, such as:

  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

Error rates are not always as clean and tidy as 1 in N operations where N is an integer power of ten. In such cases we can have fractional nines of qubit fidelity, where we have some number of nines followed by one or a few decimal digits which are less than 9 (1 to 8), such as:

  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

Nines of qubit reliability is simply a reference to nines of qubit fidelity.

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

In keeping with the principles of quantum mechanics, I’m wondering if the error rate for a qubit can ever even theoretically be absolutely 0.0, and similarly whether the nines of qubit fidelity can ever be infinite. Or, as I suspect, there is some minimum error rate, call it epsilon, which can never be exceeded since it is really simply the quantum uncertainty of performing any operation or any measurement whenever a quantum effect is involved.

Roots of nines in classical hardware availability

I first encountered this concept of nines in the context of availability of classical computing systems, particularly uptime — where service without interruption for 99.999% of the time is considered 5 nines reliability.

Low, typical, high for error rates

Alas, there is no single observable which can be measured to get a single overall error rate for all qubits of an entire quantum computer or even a single qubit.

  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

Technically, each qubit can have its own error rate, a per-qubit error rate.

Per-qubit fidelity

Since each qubit can have its own error rate, each qubit should have its own qubit fidelity. So we should two distinct measures of 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

As just mentioned in the preceding section, we will have both per-qubit measures of qubit fidelity and some overall average of qubit fidelity.

Single versus two qubit gate fidelity

Unfortunately not all quantum logic gates have the same qubit fidelity. In particular, single-qubit gates are usually significantly higher fidelity than two-qubit gates.

  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

For whatever reasons, measuring a qubit, sometimes called readout, tends to have a significantly lower fidelity than even two-qubit gates.

  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?

Just a placeholder to reemphasize how disturbing it is that measurement is so prone to error compared to the execution of quantum logic gates.

Composite qubit fidelity

Given the significant number of disparate metrics for qubit fidelity, it would still be useful to derive a single, rough metric that summarizes qubit fidelity in a single number.

  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

Rather than some simplistic formula for calculating composite qubit fidelity from all of the raw qubit fidelities, maybe a simple benchmark test could be designed which is used to calculate the effective qubit fidelity in terms of how close the final results from the test match expected results.

  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

Effective qubit fidelity is simply a synonym for the composite qubit fidelity described in the preceding sections, either:

  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?

Some qubits simply don’t work well at all. It would be a shame to drag down the nines of the entire machine just due to a few bad qubits. So it would seem to make sense to discount, ignore, and block out the bad qubits. In fact, preferably, flat out don’t use them at all.

  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

My analysis above concerning calculation or derivation of composite qubit fidelity ignored a major factor: the need for SWAP networks to compensate for very limited qubit 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

Another important design consideration for quantum circuits is placement of qubits, so that the need for SWAP networks can be reduced or even eliminated. Optimal placement can dramatically reduce the size of each SWAP network.

Qubit fidelity for Google Sycamore Weber processor

Google recently published the technical datasheet for its 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

As illustrated in the preceding section on qubit fidelity for the Google Sycamore Weber processor, there can be quite a few nuances of qubit fidelity.

Application-specific nines of qubit fidelity

Every application and application category will tend to have its own pattern of usage of qubits and quantum logic gates. The nines of qubit fidelity will be the same in all applications, but the particular qubit usage of quantum circuit patterns will give each application its own overall application error rate. Although it won’t be terribly useful to compare the aggregate error rate between disparate applications, it will be useful to compare the aggregate error rates for similar applications, multiple runs of the same application, or runs of the same application on new versions of the quantum hardware.

Application requirements for qubit fidelity

Part of designing any new quantum algorithm or quantum application should be consideration for estimating at least the general ballpark of qubit fidelity which will be required for the algorithm or application to deliver acceptable results. Such application requirements for qubit fidelity can then be compared against the specifications of candidate hardware to determine if it is even worth trying to run the algorithm or application on a particular quantum computer system.

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

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

The two basis states of each qubit have probabilities which are continuous values such that the sum of the two probabilities is exactly 1.0 — the probability for each basis state being the square of its probability amplitude. Presumably the fidelity of a qubit will determine how accurate these probabilities are maintained.

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

Similar to the preceding question about the accuracy and gradations of probability for the basis states, a similar question arises for the accuracy and gradations of the phase of a qubit based on the qubit fidelity.

Might probability and phase have different qubit fidelities?

It’s unclear whether the fidelities of qubit basis state probabilities and phase are similar, identical, or relatively different.

Noisy qubits

A noisy qubit, characteristic of noisy intermediate-scale quantum (NISQ) quantum computers, has a relatively low fidelity (reliability.) Certainly not a high number of nines. Maybe not even two or three nines. Four or more nines of qubit fidelity is likely to no longer be considered a noisy qubit.

  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

The ideal qubit, the perfect qubit, which currently does not exist and may never exist, at least not in the next 10 years, would have absolutely no errors for 100% fidelity (reliability.) The concept of nines will no longer be relevant, but you could say that a perfect qubit has infinite nines of fidelity.

Logical qubits

Although truly perfect qubits will almost certainly not be feasible, ever, the advent of quantum error correction (QEC) will enable logical qubits which for all intents and purposes will be considered perfect qubits. Or at least they will have a very large number of nines of qubit fidelity (very low error rate.)

Near-perfect qubits

We can’t really expect to achieve a perfect qubit, but we can come close, maybe even close enough that some, many, or even most applications can make do with such a near-perfect qubit.

How close to perfect is a near-perfect qubit?

There are two distinct purposes for near-perfect qubits:

  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

Qubit fidelity is not the same as application result accuracy. Every application category, every application, and every user of every application will have their own requirements for the accuracy of the results of a quantum computation. But that doesn’t tell you anything about what fidelity a qubit or gate will require to achieve the desired result accuracy. For example, very deep circuits will quickly add up errors so that an incredibly high fidelity will be required to achieve any kind of accuracy. A relatively shallow circuit may not require much qubit fidelity at all to achieve modest to moderate result accuracy.

Relevance of qubit fidelity to The ENIAC Moment of quantum computing

The ENIAC Moment of quantum computing would mark the milestone of the first quantum computer capable of running a production-scale application and achieving a dramatic quantum advantage over classical computers. It is expected that quantum error correction (QEC) will not yet be available, at least not with sufficient capacity of logical qubits needed for a production-scale application. This means that the algorithm designers and application developers will have to make do with less than perfect qubits. Outright noisy qubits are unlikely to be of sufficient fidelity to support production-scale applications, so near-perfect qubits will be needed.

Relevance of qubit fidelity to quantum error correction

Although The ENIAC Moment will require relatively high nines of qubit fidelity, it is expected that various quantum error correction schemes will be able to utilize relatively noisy qubits — a low number of nines — to achieve the perfect fidelity of logical qubits.

Relevance of qubit fidelity to The FORTRAN moment of quantum computing

The FORTRAN Moment of quantum computing is predicated on full support for quantum error core — so that non-elite technical staff can develop relatively sophisticated quantum algorithms and applications without the need to worry about manual error mitigation or even how many nines of qubit fidelity are needed to achieve required application result accuracy.

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

Quantum Fourier transform (QFT) and quantum phase estimation (QPE) are two of the most powerful algorithmic building blocks for quantum algorithms, but unfortunately they are not practical at present due to the very low qubit fidelity of current and near-term NISQ quantum computers. The question is how many nines of qubit fidelity would be needed for QFT and QPE to become practical for the kind of production-scale applications needed to achieve dramatic quantum advantage. Quick answer: unknown, at present.

Relevance of qubit fidelity to achieving dramatic quantum advantage

The real bottom line for qubit fidelity is whether it is sufficient to enable a quantum computer to achieve a dramatic quantum advantage over classical computing.

  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)

One of the key parameters for execution of a quantum circuit is shot count or circuit repetitions, which is the number of times the execution of a quantum circuit must be repeated. The application can then perform a statistical analysis of the distribution of the quantum results, in order to determine which particular result is the most likely result, the so-called expected value.

  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

Part of the appeal of quantum error correction (QEC) is that it is theoretically possible to construct perfect logical qubits using physical qubits which are of relatively low fidelity. But there is a tradeoff between lower physical qubit fidelity and logical qubit capacity since it will be quite some time before quantum computers have more than fairly limited capacities of physical qubits, let alone an interesting capacity of logical qubits.

Regular calibration needed for qubit fidelity

Unfortunately, you can’t just assemble a quantum computer, turn on the power and, presto, qubits magically have the expected fidelity. Even in normal and best operation qubit fidelity can decay or fluctuate over the course of a day. This necessitates periodic calibration testing and adjustment to assure that qubits are able to achieve their best 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

Coherence time is not an absolute 100% or nothing — there is no perfect coherence until time expires and then coherence falls off a cliff. Rather, it’s a curve, so that coherence decays gradually until it’s so bad that it’s not worth continuing to execute the circuit. Exactly how far out on that curve you can go before execution of gates is problematic is unclear and will vary for different qubit technologies, different implementations, and different algorithms and applications.

  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

It would be very helpful if classical quantum simulators could be easily configured with noise models which very closely match the nines of qubit fidelity of real quantum computers. This would allow classical quantum simulators to be used as debugging aids even if real quantum computers are available.

Where should vendors report error rates and qubit fidelity?

In my own model, each machine produced by a vendor would have two key documents:

  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

It’s not related to qubit fidelity, but the notion of quantum computer system availability would seem to make sense, comparable to system availability for classical computer systems.

Nines of reliability

Nines of reliability in the context of quantum computing is simply a reference to either:

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

Need a roadmap for nines

It sure would be nice to have a roadmap of milestones for achieving each increment of nines for qubit fidelity, but that’s not practical at this time.

  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

The IBM Quantum Volume metric measures overall performance for the largest square circuit which meets some threshold of accuracy of the final result, while the focus of this paper is simply the fidelity of a single qubit. These are two distinct metrics.

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

The single metric approach of IBM’s Quantum Volume metric masks much of the underlying complexity of adequately characterizing a wide variety of quantum algorithms. Providing the user with both qubit count and qubit fidelity enables the user to characterize a wider variety of algorithm topologies, especially those which are much more shallow or much deeper than strictly square circuits.

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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Freelance Consultant

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