Fractional Quantum Advantage — Stepping Stones to Dramatic Quantum Advantage

  1. A fraction or percentage of dramatic quantum advantage. Such as 50% (half), 90% (almost there), or 10% (a small fraction).
  2. A qualitative degree of quantum advantage. Minimal, substantial or significant, or dramatic.
  3. A symbolic degree. Such as one, two, or three stars, or gold, silver, or bronze.
  4. A fraction or percentage of a qualitative or symbolic degree of quantum advantage. Such as 50% substantial quantum advantage or 75% one-star quantum advantage.
  5. Near quantum advantage. Roughly 90–95% or closer to some qualitative or symbolic degree of quantum advantage. Such as near dramatic quantum advantage, near substantial quantum advantage, or near two-star quantum advantage.
  6. A multiple of classical performance. 2X, 10X, 50X, 100X, 1,000X, 1,000,000X.
  7. Some vague rhetorical characterization. Indicating better than classical, but not the full potential of a quantum speedup.
  1. References
  2. Quantum advantage
  3. Three levels of quantum advantage
  4. Fractional quantum advantage
  5. Broad characterizations of fractional quantum advantage
  6. More modest improvements over classical solutions
  7. Symbolic references to degree of quantum advantage
  8. One, two, and three star quantum advantage
  9. Gold, silver, and bronze levels of quantum advantage
  10. Platinum, stainless steel, and brushed aluminum levels of quantum advantage
  11. Near quantum advantage
  12. How many orders of magnitude faster than a classical computer?
  13. What about factors of 2, 4, 5, 10, 20, 50, 100, 250, 500, and 1,000?
  14. Approximate advantages for various qubit counts
  15. Wall clock problems
  16. Two-hour business process optimization problems
  17. Larger wall clock problems
  18. Some other classifications of relative performance
  19. What is the quantum advantage of your quantum algorithm or application?
  20. What is the net quantum advantage of your quantum algorithm or application?
  21. Be sure to divide net quantum advantage by the number of classical processors used by an application
  22. Raw, net, and final quantum advantage
  23. What fraction of your application performance can really utilize a quantum algorithm effectively?
  24. Full quantum advantage: Generation of true random numbers (TRNG)
  25. Every fraction of quantum advantage counts
  26. Summary and conclusions

References

  1. What Is Quantum Advantage and What Is Quantum Supremacy?
    https://jackkrupansky.medium.com/what-is-quantum-advantage-and-what-is-quantum-supremacy-3e63d7c18f5b
  2. What Is Dramatic Quantum Advantage?
    https://jackkrupansky.medium.com/what-is-dramatic-quantum-advantage-e21b5ffce48c
  3. What Is the Quantum Advantage of Your Quantum Algorithm?
    https://medium.com/@jackkrupansky/what-is-the-quantum-advantage-of-your-quantum-algorithm-8f481aefe8d0
  4. Quantum Advantage Now: Generation of True Random Numbers
    https://medium.com/@jackkrupansky/quantum-advantage-now-generation-of-true-random-numbers-237d89f8a7f2

Quantum advantage

  1. Dramatic quantum advantage. My proposed threshold of a factor of a quadrillion advantage over a classical solution.
  2. Fractional quantum advantage. Something less than dramatic quantum advantage.
  • Unless specified otherwise, quantum advantage implies a dramatic quantum advantage.

Three levels of quantum advantage

  1. Minimal quantum advantage. A 1,000X performance advantage over classical solutions. 2X, 10X, and 100X (among others) are reasonable stepping stones.
  2. Substantial or significant quantum advantage. A 1,000,000X performance advantage over classical solutions. 20,000X, 100,000X, and 500,000X (among others) are reasonable stepping stones.
  3. Dramatic quantum advantage. A one quadrillion X (one million billion times) performance advantage over classical solutions. 100,000,000X, a billion X, and a trillion X (among others) are reasonable stepping stones.

Fractional quantum advantage

  1. 1% minimum quantum advantage would be 0.01 times 1,000X or 10X.
  2. 10% minimum quantum advantage would be 0.10 times 1,000X or 100X.
  3. 50% minimum quantum advantage would be 0.50 times 1,000X or 500X.
  4. 5% substantial or significant quantum advantage would be 0.05 times 1,000,000X or 50,000X.
  5. 25% substantial or significant quantum advantage would be 0.25 times 1,000,000X or 250,000X.
  6. 1% dramatic quantum advantage would be 0.01 times one quadrillion X or ten trillion X.
  7. 0.1% dramatic quantum advantage would be 0.001 times one quadrillion X or one trillion X.
  8. 0.0001% dramatic advantage would be 0.000001 times one quadrillion X or one billion X.

Broad characterizations of fractional quantum advantage

  1. Near quantum advantage. Very close to 100%, maybe high nineties, maybe even 90% or even 80%. But generally simply close enough for many purposes.
  2. 90–100%
  3. 75–90%
  4. 50–75%
  5. 50%. Halfway to full quantum advantage.
  6. 25–50%
  7. 10–25%
  8. 1–10%
  9. 0.1%
  10. 0.01%
  11. 0.001%

More modest improvements over classical solutions

  1. 1,000,000X
  2. 1,000X
  3. 100X
  4. 50X
  5. 20X
  6. 10X
  7. 4X
  8. 3X (200%)
  9. 2X (100%)
  10. 75%
  11. 50%
  12. 25%

Symbolic references to degree of quantum advantage

  1. Stars. One, two, and three star quantum advantage.
  2. Prize medal metals. Gold, silver, and bronze levels of quantum advantage.
  3. Alternative metals. Platinum, stainless steel, and brushed aluminum levels of quantum advantage. To more realistically reflect the relative differences.

One, two, and three star quantum advantage

  1. One star. Minimal quantum advantage — 1,000X.
  2. Two stars. Substantial or significant quantum advantage — 1,000,000X.
  3. Three stars. Dramatic quantum advantage — one quadrillion X.

Gold, silver, and bronze levels of quantum advantage

  1. Gold. Three stars. Dramatic quantum advantage — one quadrillion X.
  2. Silver. Two stars. Substantial or significant quantum advantage — 1,000,000X.
  3. Bronze. One star. Minimal quantum advantage — 1,000X.

Platinum, stainless steel, and brushed aluminum levels of quantum advantage

  1. Platinum. Gold. Three stars. Dramatic quantum advantage — one quadrillion X.
  2. Stainless steel. Silver. Two stars. Substantial or significant quantum advantage — 1,000,000X.
  3. Brushed aluminum. Bronze. One star. Minimal quantum advantage — 1,000X.

Near quantum advantage

  1. Very near 100%. 99.9% or even 99%.
  2. 98%.
  3. 95%.
  4. High 90’s%.
  5. 90% or higher.
  6. Possibly 85%.
  7. Possibly even 80%.
  8. Maybe even 75%. Especially in the early days, when that would be a major accomplishment.
  1. Near dramatic quantum advantage.
  2. Near substantial quantum advantage.
  3. Near minimum quantum advantage.
  4. Near three-star quantum advantage.
  5. Near silver quantum advantage.
  6. Near platinum quantum advantage.
  7. Near stainless steel quantum advantage.
  1. Near dramatic quantum advantage.
  2. Depends on the context. A particular context may make it clear what degree of quantum advantage is being referred to.

How many orders of magnitude faster than a classical computer?

What about factors of 2, 4, 5, 10, 20, 50, 100, 250, 500, and 1,000?

Approximate advantages for various qubit counts

  1. 10 qubits = ~ 1,000 parallel computations.
  2. 20 qubits = ~ 1,000,000 parallel computations.
  3. 30 qubits = ~ one billion parallel computations.
  4. 40 qubits = ~ one trillion parallel computations.
  5. 50 qubits = ~ one quadrillion parallel computations.
  6. 60 qubits = ~ one quintillion parallel computations.
  1. 5 qubits = 32 parallel computations.
  2. 6 qubits = 64 parallel computations.
  3. 7 qubits = 128 parallel computations.
  4. 8 qubits = 256 parallel computations.
  5. 12 qubits = 4K parallel computations.
  6. 16 qubits = 64K parallel computations.
  7. 24 qubits = 16M parallel computations.
  8. 28 qubits = 256M parallel computations.
  1. QV 32 = 5 qubits = 32 parallel computations.
  2. QV 64 = 6 qubits = 64 parallel computations.
  3. QV 128 = 7 qubits = 128 parallel computations.
  4. QV 256 = 8 qubits = 256 parallel computations.
  5. QV 512 = 9 qubits = 512 parallel computations.
  6. QV 1024 = 10 qubits = 1024 parallel computations.
  7. QV 2048 = 11 qubits = 2048 parallel computations.
  8. QV 4096 = 12 qubits = 4096 parallel computations.

Wall clock problems

Two-hour business process optimization problems

Larger wall clock problems

  1. A day or two. Might be meaningful for an application which is run on the weekend, part of a weekly process.
  2. A week or two. Might be meaningful for some design process which occurs over a small number of months.
  3. A month or two. Might be meaningful for some process which occurs over many months or even a year or more.

Some other classifications of relative performance

  1. Below classical. There might in fact be problems which can be more easily solved on a quantum computer, such as simulating physics or chemistry, or generation of true random numbers, but not necessarily faster, just easier.
  2. Near parity with classical. Ditto.
  3. Modestly better than classical. Like 10–50% faster. Generally not of much value, but could still be of value for some wall clock problems, where time is critical.
  4. Moderately better than classical. Like 2X to 20X.
  5. Well above classical. Like 50X to1,000X.
  6. Better than even a massively parallel or distributed classical solution. 10,000X to 1,000,000X. May be easier to manage even if not providing a solution that was not achievable at all using classical methods.
  7. Degrees of advantage over parallel and distributed classical solutions. Comparable to 2, 4, 8, 10, 16, 20, 25, 32, 48, 64, 80, 96, 128, 256, 512, 1K, 2K, 4K, 8K, 16K, 32K, 50K, 64K, 75K, or 100K classical processors in either a massively parallel or distributed classical solution.
  8. Full dramatic quantum advantage. A factor of a quadrillion over the best classical solution.
  9. Amazingly above classical. True quantum supremacy — classical can’t even get the job done with any realistic level of resources and time.

What is the quantum advantage of your quantum algorithm or application?

  1. 10.
  2. 100.
  3. 1,000.
  4. 10,000.
  5. 100,000.
  6. 1,000,000.
  7. 10,000,000.

What is the net quantum advantage of your quantum algorithm or application?

Be sure to divide net quantum advantage by the number of classical processors used by an application

Raw, net, and final quantum advantage

  1. Raw quantum advantage. What the quantum algorithm actually does. n qubits in a Hadamard transform produces 2^n quantum states (product states), implying a raw quantum advantage of 2^n. While the comparable classical code sequence evaluates a single linear computation, the quantum circuit accomplishes 2^n evaluations in a single quantum circuit execution.
  2. Net quantum advantage. Discount raw quantum advantage by the number of circuit repetitions (shot count) to yield the net quantum advantage. The shot count indicates how many times the quantum circuit must be run to generate and process enough results to develop a statistically meaningful result, the expectation value. Repetitions are needed in part because quantum computations are noisy (errors), but also because quantum computations are probabilistic by nature even if there were no noise or errors per se.
  3. Final quantum advantage. Divide the net quantum advantage by the number of classical processors required by the classical solution, to get the final quantum advantage, which is the advantage of the quantum solution over the classical solution, not merely a quantum processor to single classical processor comparison.

What fraction of your application performance can really utilize a quantum algorithm effectively?

Full quantum advantage: Generation of true random numbers (TRNG)

Every fraction of quantum advantage counts

Summary and conclusions

  1. Full, dramatic quantum advantage is a factor of a quadrillion better than the best classical solution.
  2. A fractional quantum advantage could be a relatively minor advantage, a fairly major advantage, a notable fraction of full dramatic quantum advantage, or anything between.
  3. Minimal quantum advantage. A 1,000X performance advantage over classical solutions. 2X, 10X, and 100X (among others) are reasonable stepping stones.
  4. Substantial or significant quantum advantage. A 1,000,000X performance advantage over classical solutions. 20,000X, 100,000X, and 500,000X (among others) are reasonable stepping stones.
  5. Relatively minor advantages such as 2X to 100X could still be of significant value for wall clock problems where time is limited, so if a classical solution cannot complete the task within that critical time interval, then a quantum solution which can will be a meaningful advantage to the organization regardless of how small a fraction of full dramatic quantum advantage it might be.
  6. The raw quantum advantage of a quantum algorithm or application must be discounted (divided) by the shot count or circuit repetitions which are required to achieve a statistically meaningful quantum result to get the net quantum advantage.
  7. Be sure to divide net quantum advantage by the number of classical processors used by an application, so that the comparison is quantum solution to classical solution — at the application level, not simply quantum algorithm to classical algorithm at the processor level.
  8. We should reward all increments of progress towards dramatic quantum advantage. Every fraction counts. Eventually, some day, all quantum solutions will offer dramatic quantum advantage, but we’re not there yet, not even close.

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Freelance Consultant

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Jack Krupansky

Jack Krupansky

Freelance Consultant

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