Beware of Quantum Algorithms Dependent on Fine Granularity of Phase

In a nutshell

What is phase?

  • The quantum state of a quantum system has wave-like or cyclical behavior. A two-state quantum system such as a qubit has two waves which are not necessarily in sync. Their degree of synchronization — actually the difference between them — is their relative phase or relative phase angle. That relative phase angle is known as the phase of the two-state quantum system (qubit.)
  • If the 0 and 1 states are the two main components of the state of a qubit, phase is the third component of the state of a qubit.
  • Granted, it’s a vague and ambiguous term, but fine granularity of phase generally means more gradations than an S or T gate (90 or 45-degree gradations.) Say 12 or 16 or more gradations. The open question is whether dozens, hundreds, thousands, or even billions or more gradations of phase are supported by both the theoretical physics of qubits and the technology and engineering of real qubits.
  1. Qubit values have a third component beyond superposition of binary 0 and 1 — phase.
  2. Using phase as a simple binary flag such as for constructive and destructive interference is probably reasonably safe.
  3. Two or three bits of precision of phase (three to eight gradations of phase) are probably reasonably safe as well. Such as the S and T gates (90 and 45-degree rotations.)
  4. But all bets are off beyond a few bits of precision — exactly where the breakpoint occurs is indeterminate at this stage of the technology.
  5. One would hope that 16 gradations of phase would work reasonably well, but even that is a question mark.
  6. Maybe 32 to 64 gradations, or even 128 to 256 gradations of phase might be workable for more advanced qubits.
  7. But one could be skeptical of hundreds or thousands, let alone millions or billions of gradations of phase.
  8. In any case, vendors of quantum computers (and service providers as well) need to fully document any limitations on phase for their qubits.
  9. Vendors of quantum computers should annotate their product roadmaps with number of gradations of phase supported at each stage of the roadmap, as well as the number of bits of precision supported for quantum Fourier transform (QFT) and quantum phase estimation (QPE) at each stage of the roadmap.
  10. Vendors also need to perform validation testing of phase limits to compare reality to theory for documented limitations. Confirm theory and documentation, or note any discrepancies.
  11. Even classical quantum simulators need clear technical documentation of precision and granularity of phase, including the degree to which phase limits can be configured.
  12. It should be easy to configure classical quantum simulators to precisely match the limitations of real quantum computers.
  13. The main characteristics of phase which need careful documentation are the minimum and maximum phases, the minimum increment between phases, and the expected uncertainty for phase.
  14. No, “Just try it and see what happens!” is not a reasonable or professional approach. Trial and error is not a plan.
  15. First things first, get the theory straight. Folklore won’t cut it.
  16. We need more information on implementation of phase angle rotations. Hardware and software — how the theory meets the road.
  17. Portability of algorithms and applications is problematic. Different machines have different capabilities and different limitations when it comes to phase. That said, there may be a least common denominator of capabilities and limitations for a particular collection of machines. General portability is problematic — for fine granularity of phase.
  18. As hardware evolves, the rules may change, sometimes incrementally, and sometimes in quantum leaps.
  19. Sector is too young for most people to run into phase granularity issues. Wait a few years as algorithms and applications become more ambitious.
  20. Do probability amplitudes have the same characteristics, limitations, and issues as phase? Possibly. Maybe. I would expect so. But I just don’t know and haven’t been able to verify one way or the other. Vendors need to supply detailed technical guidance.
  21. Unfortunately, there may be different stories and details on phase granularity depending on whether we’re talking about the present, current offerings, the near future, next year, the medium-term future, and the much longer-term future. Timing is everything. What’s true for today may no longer be true tomorrow, next week, next month, next year, in two years, in five years, in seven years, in ten years, or in twenty years.

Topics covered in this paper

This informal paper will cover:

  1. In a nutshell
  2. Caveats
  3. What is phase?
  4. Cycles and waves
  5. Period or wavelength
  6. Four stages or phases of a single cycle of a wave
  7. Circles and sine waves
  8. Phase — quadrants of a cycle vs. fraction of a cycle
  9. Phase angle
  10. Phi for phase angle
  11. Units for phase (phase angle)
  12. Gradations of phase vs. granularity
  13. Theoretical gradations of phase vs. granularity
  14. Phase as a decimal fraction
  15. Why is two pi used everywhere when phase is used?
  16. Euler’s formula
  17. Range of phase
  18. Phase is a continuous value
  19. Is phase an infinitely-valued continuous value?
  20. Is phase a discrete continuous value?
  21. How can phase be discrete if two pi is irrational?
  22. What does a decimal phase fraction between 0.0 and 1.0 mean if two pi is irrational?
  23. Is phase multivalued?
  24. Are phase angles bright lines or fuzzy and uncertain?
  25. Are phase angles black and white or shades of gray?
  26. Is phase digital or analog?
  27. What is the resolution or precision of a phase angle?
  28. Aspects of phase in quantum computing
  29. How close can phase come to 0.0 and pi and not be treated as not having any phase?
  30. Phase granularity may be fine for smaller circuits
  31. Bloch sphere
  32. Rotations of the Bloch sphere
  33. Phase is rotation about the Z axis
  34. Phase is real in the range 0.0 to 1.0
  35. Theta and phi
  36. Phase is relative
  37. S and T gates should be reliable
  38. R-phi gate (or PHASE gate, phase shift) may or may not be reliable
  39. Phase cannot be measured directly
  40. Measuring phase — indirectly
  41. Phase as a binary flag or for constructive and destructive interference
  42. Degree of granularity
  43. Where is the “break” point for phase?
  44. Limitations due to theory, engineering, or portability
  45. Theoretical limitations
  46. Is there a Planck-type unit for phase?
  47. Levels of translation
  48. Precision of entries in unitary matrices
  49. Precision of exponentials in unitary matrices
  50. Precision for exponentials in a quantum Fourier transform (QFT)
  51. Precision of pi and sqrt(2) when used in unitary matrices
  52. Quantum vs. classical hardware issues
  53. Uses of phase in quantum algorithms
  54. Quantum algorithms which rely on granularity of phase
  55. QFT and QPE are not currently viable at-scale anyway
  56. Algorithms which use phase kickback
  57. Extreme granularity of phase for quantum Fourier transform (QFT)
  58. No, banding won’t facilitate precision of phase in quantum Fourier transforms
  59. Extreme granularity of phase for quantum phase estimation (QPE)
  60. When will quantum phase estimation (QPE) be viable?
  61. When will quantum Fourier transform (QFT) be viable?
  62. How does phase granularity scale as an algorithm is scaled?
  63. Shor’s algorithm as worst-case example so far
  64. Is Shor’s really the worst-case limit for quantum algorithms, or just the starting point for serious, heavy-duty quantum algorithms?
  65. Hardware vendors (and service providers) need to fully document any limitations on phase for their qubits
  66. What does current vendor documentation say about granularity of phase?
  67. Product roadmaps should detail the number of gradations of phase supported at each stage of the roadmap
  68. Product roadmaps should detail the number of bits of precision supported for quantum Fourier transform (QFT) and quantum phase estimation (QPE)
  69. Providers of quantum simulators need to fully document the behavior of phase
  70. No, “Just try it and see what happens!” is not a reasonable or professional approach
  71. Trial and error is not a plan
  72. Vendor validation testing of phase limits — compare to theory
  73. Publish reports for validation testing of phase limits
  74. Accumulated error after performing n x PHASE(1/n)
  75. Model categories of algorithm and application phase granularity as tee-shirt sizes — S, M, L, XL, XXL, XXXL
  76. What multiples of phase granularity are supported?
  77. Do probability amplitudes have the same characteristics, limitations, and issues as phase?
  78. Still at laboratory curiosity stage, so all bets are off
  79. Sector is too young for most people to run into phase granularity issues
  80. Need for hardware and architecture advances
  81. As hardware evolves, the rules may change
  82. First things first, get the theory straight
  83. What does the theory say about phase granularity?
  84. What are the actual physics of phase granularity?
  85. Impact of granularity and uncertainty on constructive and destructive interference
  86. How will phase interact with quantum error correction (QEC)?
  87. How stable is physical phase?
  88. We need more information on implementation of phase angle rotations
  89. Are trapped-ion qubits better for high-precision phase?
  90. Timeframe
  91. No specific granularity guidance I can offer — it’s all up in the air
  92. Conclusion

Caveats

  1. This paper won’t delve deep into the all of the actual physics of phase, just deep enough to address the headline issue of precision and granularity of phase.
  2. No graphics, no diagrams, no Greek symbols, no math, no matrices. Just addressing the headline issue in as plain language as possible.
  3. Quantum computing technology is a moving target, constantly evolving, so what might be true today — or at any other time — may not necessarily be true at some arbitrary time in the future.

What is phase?

In the subatomic world of quantum mechanics, everything simultaneously has both particle-like behavior and wave-like behavior.

Cycles and waves

The concept of phase relates to the cyclical or periodic wave-like behavior of a quantum system.

Period or wavelength

A wave has a period or wavelength. Wavelength corresponds to one complete cycle of the wave.

Four stages or phases of a single cycle of a wave

One complete cycle of a wave has our stages or phases:

  1. Rising from zero to a maximum amplitude peak.
  2. Falling from that maximum amplitude peak back to and through zero.
  3. Falling from zero to a maximum negative amplitude trough.
  4. Rising from that maximum negative amplitude trough back to and through zero.
  5. Rinse and repeat — periodic.

Circles and sine waves

The two most common forms of cycles or waves are:

  1. Circle. Four quadrants, each comparable to one phase of a sine wave.
  2. Sine wave. The path a point on a circle takes if the circle is rolled.

Phase — quadrants of a cycle vs. fraction of a cycle

There are two distinct ways to look at phase:

  1. The four phases of a single cycle of a wave.
  2. A specific fraction of a single cycle of a wave. A specific point or relative distance along the path of a wave in terms of a fraction of the complete path for a single cycle.

Phase angle

Generally, phase corresponds to an angle — phase angle — as in a fraction of a full circle.

Phi for phase angle

The greek letter phi is commonly used to refer to a phase angle.

Units for phase (phase angle)

There are three distinct units for measuring phase, as an angle:

  1. Radians. The language of formal mathematics. Two pi radians in a circle. Each of the four phases of a cycle or the four quadrants of a circle correspond to pi over 2 radians.
  2. Degrees. Common, vernacular unit for measuring angles. Simple, direct numeric conversion factor — two pi radians equals 360 degrees. Pi over two radians corresponds to 90 degrees.
  3. Decimal fraction. Fraction of a full circle. From 0.0 to 1.0. 1.0 corresponds to two pi radians or 360 degrees. 0.5 corresponds to pi radians or 180 degrees. 0.25 corresponds to pi over two radians or 90 degrees.
  1. exp(i phi) — presume phi is an angle in radians.
  2. exp(two pi i phi) — presume phi is an angle as a decimal fraction.
  1. 0.0 radians = 0 degrees = 0.0 decimal fraction.
  2. pi over four radians = 45 degrees = 0.125 decimal fraction.
  3. pi over two radians = 90 degrees = 0.25 decimal fraction.
  4. pi radians = 180 degrees = 0.5 decimal fraction.
  5. Three pi over two radians = 270 degrees = 0.75 decimal fraction.
  6. Two pi radians = 360 degrees = 1.0 decimal fraction.

Gradations of phase vs. granularity

Granularity of phase and gradations of phase are approximately the same thing.

Theoretical gradations of phase vs. granularity

The only significant difference between granularity of phase and gradations of phase may be that gradations are more theoretical or ideal while granularity is more focused on the practical aspects of distinguishing two phase angles.

Phase as a decimal fraction

Generally in quantum computing, it makes more sense to conceptualize operations on phase in terms of decimal fractions, such as 1.0, 0.5, 0.25, etc. After all, pi, pi over two, and pi over four are irrational numbers, which cannot be easily conceptualized from a numeric computational perspective.

Why is two pi used everywhere when phase is used?

Just to highlight, as already mentioned, that two pi is the conversion factor from phase angle as a decimal fraction (0.0 to 1.0) to radians (zero to two pi).

  • exp(two pi i phi) — phi is presumed to be an angle as a decimal fraction.

Euler’s formula

Euler’s formula converts exp(i phi) to and from cos(phi) + i sin(phi).

  1. Even multiples of pi, including 0.0. cos(0) + i sin(0) == 1.0.
  2. Odd multiples of pi, including pi. cos(pi) + i sin(pi) == -1.0.
  3. Even multiples of pi plus pi over two. cos(pi over two) + i sin(pi over two) == i.
  4. Odd multiples of pi plus pi over two. cos(pi plus pi over two) + i sin(pi plus pi over two) == -i.

Range of phase

There are four distinct ways to look at the range of phase (or phase angle):

  1. A full circle or single full wave (one cycle). Zero to two pi radians, 0 to 360 degrees, or 0.0 to 1.0 as a decimal fraction.
  2. Half a circle. The other half of the circle merely retraces the first half, back to where it started. Zero to pi and pi to two pi radians, 0 to 180 degrees and then 180 degrees to 360 degrees, and 0.0 to 0.5 and then 0.5 to 1.0 as a decimal fraction.
  3. Two half circles which are mirror images, but with the sign flipped. Zero to pi and zero to minus pi radians, 0 to 180 degrees and 0 to minus 180 degrees, and 0.0 to 0.5 and 0.0 to minus 0.5 as a decimal fraction.
  4. Infinite circles or waves (cycles), but periodic based on the size of one cycle. Four pi radians is the same as two pi radians, 720 degrees is the same as 360 degrees, and 2.0 is the same as 1.0 as a decimal fraction.

Phase is a continuous value

Although the 0 and 1 values of a qubit are clearly very discrete — there are two of them, a binary pair of values, phase at least superficially seems to be a continuous value, between zero and two pi radians or between 0.0 and 1.0 as a decimal fraction.

Is phase an infinitely-valued continuous value?

Okay, phase appears to be a continuous value, but are there really an infinity of phase angles between zero and two pi?

Is phase a discrete continuous value?

It seems reasonable to presume that there is some limit as to how fine a granularity there can be between any two adjacent phase angles.

How can phase be discrete if two pi is irrational?

There are two pi radians in a circle or single cycle of a wave, so if phase is discrete at some level, any level, how can there be any discrete value for the smallest difference between two phase angles which divides two pi, an irrational number? It doesn’t seem possible.

What does a decimal phase fraction between 0.0 and 1.0 mean if two pi is irrational?

How exact, precise, or meaningful can phase expressed as a decimal fraction between 0.0 and 1.0 be if two pi is irrational? If a phase fraction of 0.5 corresponds to pi radians, what does that really mean, precisely? Or a fraction of 0.25, 0.125, 0.1, 0.2, etc.

Is phase multivalued?

Regardless of whether phase is a true continuous value or discrete, at a minimum it supports multiple values.

Are phase angles bright lines or fuzzy and uncertain?

Separate from the question of actual precision or granularity of phase angles, there is the question of whether there is some degree of fuzziness or uncertainty between adjacent phase angles.

Are phase angles black and white or shades of gray?

Maybe the better question is how distinctive the adjacent shades of gray really are.

Is phase digital or analog?

A much more basic question is whether phase should be treated as an analog metric rather than a digital metric — is phase continuous or discrete?

What is the resolution or precision of a phase angle?

It is probably common for the API interfaces for quantum circuits to use double precision floating point for all real numbers, but that begs the question of what resolution or precision is actually used down at the hardware interface level where a unitary matrix is converted to the analog electrical signals to control qubit operations.

Aspects of phase in quantum computing

When contemplating the use of phase in quantum computing, here are the considerations which must be kept in mind:

  1. Minimum phase. Greater than zero. Or zero if no phase.
  2. Maximum phase. Less than two pi radians or 1.0 as a decimal fraction.
  3. Smallest increment or delta. Between two adjacent phases.
  4. Error or uncertainty. In the quantum mechanical sense.
  5. Measurement resolution. Can’t directly measure phase, but can use phase estimation techniques — quantum phase estimation (QPE).
  1. Smallest phase greater than zero.
  2. Smallest phase greater than pi radians or 0.5 as a decimal fraction.
  1. Greatest value less that two pi radians or 1.0 as a decimal fraction.
  2. Greatest value less than pi radians or 0.5 as a decimal fraction.

How close can phase come to 0.0 and pi and not be treated as not having any phase?

That’s part of the question about minimum and maximum values of phase — being able to distinguish from zero and pi radians.

Phase granularity may be fine for smaller circuits

For quantum circuits using only relatively few qubits and relatively few gates, phase may work just fine, such as 4–8 qubits or maybe even 12–16 qubits and a comparable number of gates. Exactly where the breakpoint is would need to be determined for each distinct quantum computer.

Bloch sphere

This paper won’t delve into the full details of the Bloch sphere for visualizing the quantum state of a qubit, other than simply to give some context for visualizing precision and granularity of qubit phase angles.

Rotations of the Bloch sphere

The Bloch sphere can be rotated around all three axes, X, Y, and Z.

Phase is rotation about the Z axis

Changes to phase of a qubit are achieved by rotations about the Z or X axis of the Bloch sphere.

Phase is real in the range 0.0 to 1.0

As a decimal fraction, phase (phase angle) is a real number in the range 0.0 to 1.0, representing a rotation about the Z axis of the Bloch sphere.

Theta and phi

The Greek symbol theta is commonly used to refer to rotation about the Y axis, which has the effect of changing the probabilities for the 0 and 1 states, but leaving the phase unchanged.

Phase is relative

Although each of the two probability amplitudes for a qubit state is a complex number with its own imaginary part — the phase, there is only one phase for the qubit state and it is relative — the relative phase between the phase angle of the 0 state and the phase angle of the 1 state.

S and T gates should be reliable

The S and T quantum logic gates rotate the Bloch sphere about the Z axis by 90 degrees and 45 degrees, respectively (pi over two and pi over four radians, or 0.25 and 0.125 as a decimal fraction.) This is rather coarse granularity — four and eight gradations, respectively, so these gates should be reasonably reliable on most quantum computers.

R-phi gate (or PHASE gate, phase shift) may or may not be reliable

The R-phi quantum logic gate, sometimes called the PHASE (shift) gate, rotates the Bloch sphere about the Z axis by an arbitrary phase angle, expressed in radians. Whether this gate is reliable or problematic depends on whether the phase angle is a clean multiple of a low-granularity angle, such as two, four, or eight gradations. It is only when the number of gradations — and the granularity of the phase angle — get greater than eight that issues might begin to appear.

Phase cannot be measured directly

Note that the phase of a qubit value cannot be measured directly — it collapses when the qubit is measured.

Measuring phase — indirectly

Although the phase of a qubit cannot be directly measured, there are clever tricks to indirectly infer the value of the phase of a qubit, notably quantum phase estimation (QPE). This paper won’t delve into either this technique or others since the focus here is on resolution of phase — how many unique values it can have.

Phase as a binary flag or for constructive and destructive interference

The concept of interference makes use of the phase of a qubit to perform clever tricks used in quantum computations. This paper won’t delve into the details of interference or how it is used, other than the narrow aspect of the granularity or resolution of phase — how many values of phase can be used in the quantum computations using a single qubit.

Degree of granularity

Just to inspire thinking about the utility or value of various categories of fine granularity, here are some possible ranges:

  1. Binary or flag — two states of phase (actually four — pi over two and minus pi over two radians, plus the two zeroes of the Bloch sphere, 0.0 and pi radians)
  2. Two or three bits (4–8 states)
  3. 8–12
  4. 16
  5. 32
  6. 50–64
  7. 80–100
  8. 128–150
  9. 192–256
  10. 512
  11. 1024
  12. 2K
  13. 4K
  14. Thousands
  15. Millions
  16. Billions
  17. Trillions
  18. Quadrillions
  19. 10²⁰ to 10⁵⁰
  20. Beyond 10⁵⁰

Where is the “break” point for phase?

Where might the transition point or break point be between phase granularity which works reliably and phase granularity which begins to break down? Great question, but at present the answer is unknown — and can vary from machine to machine.

  1. Anything more than just a binary flag.
  2. Anything more than a handful, such as 3 to 5 gradations.
  3. 8 gradations.
  4. 12
  5. 16
  6. 24
  7. 32
  8. 50
  9. 64
  10. 100
  11. 128
  12. 256
  13. 1024
  14. 4K
  15. 16K
  16. 64K
  17. 256K
  18. 1M
  19. Millions
  20. Billions
  21. 10⁴⁰
  22. 10⁵⁰
  23. 10¹⁰⁰

Limitations due to theory, engineering, or portability

Phase could be limited or constrained in three areas:

  1. Theory. Raw physics. Physical limitations, regardless of particular implementations.
  2. Engineering of real machines. Not all theory can readily, practically, or economically be turned into a real machine.
  3. Portability. Consistency between machines, vendors, and technologies and architectures.

Theoretical limitations

Theoretical limitations could arise in two ways:

  1. Limitations of quantum mechanics in general.
  2. Specific physics of particular qubit designs.

Is there a Planck-type unit for phase?

I imagine that phase is not infinitely continuous between zero and two pi. Relatively continuous, yes, but not absolutely and infinitely continuous.

Levels of translation

One of the overall issues with precision and granularity of phase — or any aspect of quantum computing, for that matter — is the significant number of levels of interfaces which must be transitioned to translate from a high-level algorithm and application at the software level to actual physics at the hardware level.

  1. Abstract phase — e.g., pi as an abstraction vs. approximation of pi to so many digits.
  2. Representation in source programming language — e.g., Python.
  3. High-level API representation.
  4. Network interface representation.
  5. Software driver representation. Where exactly do the entries in unitary matrices get turned into physical hardware control?
  6. FPGA representation.
  7. Digital electronics representation.
  8. Analog electronics representation.
  9. Cryogenic electronics and photonics (e.g., microwaves or lasers) representation.
  10. Physical control of qubits implementation.
  11. Physical effect on qubits. The actual quantum mechanics.

Precision of entries in unitary matrices

Unitary matrices are the method used to express the fine details of the operation to be performed by each quantum logic gate. Each entry of the matrix is a complex number.

  1. Single-precision floating point.
  2. Double-precision.
  3. Quad precision.

Precision of exponentials in unitary matrices

In addition to the raw precision of the entries of unitary matrices, there is the question of precision for the exponents for exponentials. Generally, a rotation will be specified by one or more entries in the unitary matrix of the form:

  • exp(two pi i phi) — if phi is a decimal fraction (0.0 to 1.0)
  • exp(i phi) — if phi is in radians (0.0 to two pi)

Precision for exponentials in a quantum Fourier transform (QFT)

When implementing a quantum Fourier transform (QFT), phi, as a decimal fraction will be 1.0 over N, where N is 2^n. If n is fairly large, such as in Shor’s algorithm for factoring large semiprimes, even a very modest value of n = 50 means N = 2⁵⁰, which is roughly 10¹⁵, so that 1.0 over N (1 over 10¹⁵) is a very tiny number. And that’s the motivation for this paper — how small N (or how large n) can be and still produce valid results in a quantum computation.

Precision of pi and sqrt(2) when used in unitary matrices

pi and the square root of 2 are irrational numbers, which means that they cannot be expressed exactly with a finite number of digits or bits of precision. That raises the question of exactly how many digits or bits of precision are needed to use pi or sqrt(2) or 1 over sqrt(2) in the entries of a unitary matrix.

Quantum vs. classical hardware issues

Besides theoretical limitations, practical aspects of designing and engineering hardware may cause limitations on use of phase.

  1. Quantum aspects of a qubit.
  2. Classical control of qubit gate operations.
  1. Classical digital electronics.
  2. Classical analog electronics.

Uses of phase in quantum algorithms

There are many potential uses of phase in quantum algorithms, but they can be divided into two broad categories:

  1. Those which rely on phase merely as a binary flag. Such as constructive and destructive interference.
  2. Those which rely on some degree of granularity of phase.

Quantum algorithms which rely on granularity of phase

The major examples of quantum algorithms which rely on granularity of phase are:

  1. Quantum Fourier transform (QFT).
  2. Quantum phase estimation (QPE).
  3. Algorithms which use the technique of phase kickback.

QFT and QPE are not currently viable at-scale anyway

Although quantum Fourier transform (QFT) and quantum phase estimation (QPE) are considered essential algorithmic building blocks for quantum computing, at least in the long-term, it is widely accepted that they are not viable at-scale in the near-term, being currently suitable only for only the tiniest of demonstration cases. This non-viability may be due more to coherence, gate errors, and circuit-depth requirements, so we don’t even get to the stage of testing the limits of phase angle granularity.

Algorithms which use phase kickback

Phase kickback is a general technique which can be used in a very wide variety of algorithms. Whether phase kickback relies on granularity of phase will depend on the particular usage.

  1. Phase is used merely as a binary flag.
  2. Phase has some degree of granularity, more than a mere two-state flag.

Extreme granularity of phase for quantum Fourier transform (QFT)

This informal paper won’t provide a deep description of quantum Fourier transform (QFT) — for more background, read the Wikipedia Quantum Fourier transform article.

  • exp(two pi i j k over 2^n)

No, banding won’t facilitate precision of phase in quantum Fourier transforms

A large quantum Fourier transform can quickly result in very small values of phase, even zero or very close to zero. Banding (also known as approximate quantum Fourier transform (AQFT)) is a technique to skip over all of the zero values of phase and even most of the very small values of phase which are so close to zero as to not having a significant effect on the computation of the quantum Fourier transform. Adding a significant number of zeros or very small values is unlikely to significantly impact the net result significantly, so banding simply skips over them — any matrix entries outside of a relatively narrow band.

Extreme granularity of phase for quantum phase estimation (QPE)

Quantum phase estimation (QPE) is essentially a specialized application of quantum Fourier transform (QFT), so the comments for QFT apply to QPE.

When will quantum phase estimation (QPE) be viable?

Although quantum phase estimation (QPE) is not viable today, the open question is when or at what stage of evolution of the quantum computing sector will QPE in fact become technically viable? The short answer is that it is unknown, but not in the very near future. And by viable, it is presumed that more than a small number of bits of precision is supported, such as eight to sixteen or more bits or 256 to 64K or more gradations.

  1. 18 months? Not likely
  2. 2–3 years? Unlikely.
  3. 5 years? Possibly.
  4. 7 years? A distinct possibility.
  5. 10 years? One would hope so.

When will quantum Fourier transform (QFT) be viable?

Quantum phase estimation (QPE) is essentially a quantum Fourier transform (QFT), so the answer to the question of when or at what stage quantum Fourier transform (QFT) will be technically viable is essentially the same as the answer to the same question about quantum phase estimation (QPE).

How does phase granularity scale as an algorithm is scaled?

Scaling of algorithms is a great unknown. Scaling of phase (granularity) is a key one of those unknowns.

Shor’s algorithm as worst-case example so far

To the best of my knowledge, Shor’s algorithm for factoring very large semiprime numbers is the worst-case algorithm so far in terms of overall complexity, raw qubit requirements, raw gate count, and size of quantum Fourier transform (QFT). The latter implying that Shor’s is the worst-case so far in terms of need for very fine granularity of phase.

  • exp (two pi i j k over 2^n)

Is Shor’s really the worst-case limit for quantum algorithms, or just the starting point for serious, heavy-duty quantum algorithms?

As quantum algorithm designers struggle with even very modest-sized algorithms and Shor’s algorithm is far beyond the size of algorithms realizable today, it’s rather difficult to contemplate algorithms much more complex than Shor’s, but that shouldn’t stop us from doing so.

Hardware vendors (and service providers) need to fully document any limitations on phase for their qubits

Granularity of phase must be fully documented for each quantum computer. Designers of algorithms and developers of applications need to know exactly what they are working worth — they need to know the rules of the road and where the lines are on the road.

What does current vendor documentation say about granularity of phase?

As far as I have seen, current vendor documentation and technical specifications are completely silent (i.e., useless) with regard to the granularity of phase.

Product roadmaps should detail the number of gradations of phase supported at each stage of the roadmap

Product roadmaps from hardware vendors should document in detail the number of gradations or granularity of phase supported at each stage of the product roadmap. The roadmap should clearly answer questions such as:

  1. When will even four bits (16 gradations) of precision be supported?
  2. Ditto for five bits (32 gradations)
  3. 6 bits (64 gradations)
  4. 8 bits (256 gradations)
  5. 10 bits (1024 gradations)
  6. 12 bits (4K gradations)
  7. 16 bits (64K gradations)
  8. 20 bits (one million gradations)
  9. 24 bits (24M gradations)
  10. 32 bits (4G gradations)
  11. And beyond.

Product roadmaps should detail the number of bits of precision supported for quantum Fourier transform (QFT) and quantum phase estimation (QPE)

Product roadmaps from hardware vendors should document in detail the number of bits of precision supported for quantum Fourier transform (QFT) and quantum phase estimation (QPE) at each stage of the roadmap. The roadmap should clearly answer questions such as:

  1. When will even four bits of precision be supported?
  2. Ditto for five bits.
  3. 6 bits.
  4. 8 bits.
  5. 10 bits.
  6. 12 bits.
  7. 16 bits.
  8. 20 bits.
  9. 24 bits.
  10. 32 bits.
  11. And beyond.

Providers of quantum simulators need to fully document the behavior of phase

Even classical quantum simulators need clear technical documentation of precision and granularity of phase, including the degree to which any limitations on phase can be configured.

No, “Just try it and see what happens!” is not a reasonable or professional approach

Algorithm designers and application developers should work from documentation and specifications for the quantum computers they will be using. It’s inappropriate and outright unprofessional to throw up your hands and say “Well, let’s just try it and see what happens!” Good grief! We can, should, and must do better, much better.

Trial and error is not a plan

Same point. With a lot of immature technologies there is a great temptation to use trial and error as if it were a plan, when it isn’t.

Vendor validation testing of phase limits — compare to theory

Theory is great and documentation and specifications are great, but there also needs to be technical validation testing to confirm that theory, documentation, and reality really are in sync.

Publish reports for validation testing of phase limits

Every quantum computer — or service offering access to a quantum computer should come with a report documenting the validation testing for phase limits, either confirming theory or noting any discrepancies.

Accumulated error after performing n x PHASE(1/n)

The basic question for the uncertainty of a phase rotation is if you pick a granularity, n, such as 16, 50, 64, 100, 256, 500, 1024, 4K, 16K, 64K, etc., and execute that many PHASE gates each with a phase rotation angle of 1 over n, do you really get back to zero. What is the residual error?

Model categories of algorithm and application phase granularity as tee-shirt sizes — S, M, L, XL, XXL, XXXL

There may be roughly six categories of algorithm and application phase granularity requirements which can be distinguished using the classic tee-shirt size metaphor:

  1. S — Small. Requires no more or little more than S and T gate phase granularity.
  2. M — Medium. Dozens of gradations of phase, such as 24 to 64.
  3. L — Large. Hundreds of gradations of phase, such as 256 to 1024.
  4. XL — Extra Large. Thousands of gradations of phase.
  5. XXL — Extra-extra Large. Millions or billions of gradations of phase.
  6. XXXL — Extra-extra-extra Large. Far beyond billions of gradations of phase.

What multiples of phase granularity are supported?

Vendors need to clearly document what multiples of phase granularity are supported, such as:

  1. Powers of 2?
  2. Multiples of 2?
  3. Multiples of 4? Since there are four quadrants in the Bloch sphere.
  4. Multiples of 3 or other odd numbers? 3, 6, 9, 12, 15? 5, 7, 11, 13, 23? What happens if the multiple isn’t divisible by 2 or 4 to get 0 at the midpoint and maximum magnitude at 90 and 270 degrees?

Do probability amplitudes have the same characteristics, limitations, and issues as phase?

It’s hard to say whether probability amplitudes of qubit state have the same characteristics, limitations, and issues as phase angles. Possibly. Maybe. I would expect so since they’re all just rotations about the Bloch sphere. But I just don’t know for sure and haven’t been able to verify one way or the other. Vendors need to supply detailed technical guidance.

Still at laboratory curiosity stage, so all bets are off

Quantum computers and qubits are still a very active area of research and not even close to being ready for commercialization. As such, quantum computers remain laboratory curiosities and will remain so for years to come.

Sector is too young for most people to run into phase granularity issues

Most people are struggling to get basic, simple quantum algorithms to work. Anything beyond an S or T gate is generally beyond most small projects.

Need for hardware and architecture advances

Regardless of where quantum computer hardware is today, there’s no question that there will be much research and many waves of incremental and quantum-leap advances in quantum computing hardware in the coming years.

As hardware evolves, the rules may change

It is also important to note that as hardware evolves, the rules may change, sometimes incrementally, and sometimes in quantum leaps.

First things first, get the theory straight

There is too much hearsay and folklore about phase angles floating around. Folklore won’t cut it.

What does the theory say about phase granularity?

Just to reiterate, as far as I can tell, the theory of qubits doesn’t say anything at all about what the granularity or precision of phase might be.

What are the actual physics of phase granularity?

Same question, but focused on the actual physics of whatever physical phenomenon actually underlies phase. There’s nothing that I have been able to find.

Impact of granularity and uncertainty on constructive and destructive interference

The concepts of constructive and destructive interference depend on the addition and subtraction of two phase angles of identical magnitude but either identical or opposite signs. This raises the question of the impact of granularity or precision of phase when two phase angles are interfered.

How will phase interact with quantum error correction (QEC)?

Will quantum error correction (QEC) have any impact on phase granularity or limits on phase in general? Hard to say.

How stable is physical phase?

Does the phase of a qubit have jitter?

We need more information on implementation of phase angle rotations

Even when we finally get all of the theory of phase angles worked out, there is then the question of how the theory is implemented in real, physical hardware — and the software which controls the hardware. How the theory meets the road.

Are trapped-ion qubits better for high-precision phase?

Trapped-ion qubits may be better for higher-precision phase.

Timeframe

When discussing future capabilities of quantum computers it is useful to distinguish the timeframe:

  1. Now — currently available.
  2. Coming soon — over the next few months.
  3. Coming over the remainder of the year.
  4. Next year.
  5. 1 year — must wait a whole year.
  6. 2 years.
  7. 5 years.
  8. 7 years.
  9. 10 years.
  10. 15 years.
  11. 20 years.
  12. Ultimate, maximum capabilities.

No specific granularity guidance I can offer — it’s all up in the air

After all of this, the bottom line is that there simply isn’t any specific guidance I can offer on granularity of phase and what exactly algorithms and applications can or can’t depend on.

  1. They need to get the theory straight.
  2. They need to deeply understand exactly what they are building.
  3. They need to fully characterize what they are building.
  4. They need to fully test and validate what they have built.
  5. They need to fully document what they have built.

Conclusion

  1. We are still at the toy and experimentation stage of quantum computing — these hard issues are beyond current imagination.
  2. There are many unknowns which vendors need to clarify, such as any limitations or issues with phase of qubits.
  3. Algorithm designers should only rely on a few bits of precision for phase, for now.
  4. Algorithms depending on modest granularity of phase (dozens) should tread very carefully.
  5. Depending on fine granularity of phase (hundreds or thousands of gradations) is likely beyond the limits of current technologies over the next few to five years.
  6. Phase granularity in the millions, billions, and beyond should not be considered as practical for the indefinite future.
  7. Whether extremely fine granularity of phase is practical at all even in the long term remains to be seen.
  8. Portability of algorithms and applications is problematic when different quantum computers have different limits on phase.

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Freelance Consultant

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