The Power of Qubits

I’ve been working my way through The Principles of Quantum Mechanics (1930), by Paul Dirac. (It’s available as a Kindle eBook for only 6.49 USD.) It’s perhaps best known for being where he defines and describes his 〈bra|ket〉 notation (which I posted about in QM 101: Bra-Ket Notation). More significantly, Dirac shows how to build a mathematical quantum theory from the ground up.

This is not a pop-science book. Common wisdom is that including even a single equation in a science book greatly reduces reader interest. Dirac’s book, in its 82 chapters, has 785 equations! (And no diagrams, which is a pity. I like diagrams.)

What I wanted to post about is something he mentioned about qubits.

He doesn’t call them “qubits” of course. That term is said to have been coined by theoretical physicist Benjamin Schumacher in 1995.

Nor is Dirac talking about quantum computing. That wasn’t a thing until the Paul Benioff paper, The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines, in May of 1980.

Dirac uses the concrete example of photon polarization (aka photon quantum spin), which is a two-state quantum system, which is exactly what a qubit is, a two-state quantum system. (See: QM 101: Photon Spin) In general, Dirac is describing the basic superposition I’ve shown in many previous posts:

\displaystyle|\Psi\rangle=\alpha|0\rangle+\beta|1\rangle

Although Dirac writes it as:

\displaystyle|R\rangle={c}_{1}|A\rangle+{c}_{2}|B\rangle

Which is the same thing wearing different clothes. Both describe a quantum state consisting of a superposition of two other states, each of which has a complex coefficient (either of which, but not both, may be zero).

What Dirac says about this, and which caught my eye, is:

Thus only the ratio of the two coefficients is effective in determining the state R. Hence this state is determined by one complex number, or by two real parameters. Thus from two given states, a twofold infinity of states may be obtained by superposition.

It’s that last sentence (emphasis mine) that reaffirmed what I’ve been thinking all along about the power of quantum computing. And about why it’s so different from classical computation.

I’ve read that David Deutsch attributes the power of quantum computing to the Many Worlds Interpretation of quantum mechanics, but I believe it’s much simpler than that. The way I see it, it’s roughly for the same reason that a note played on a saxophone is instantly recognizable as such and completely different from that same note played on a piano, guitar, or flute (which are themselves instantly recognizable for what they are).

Put simply, it’s due to, as Dirac puts it, the “twofold infinity” of degrees of freedom two-state quantum systems have compared to classical bits, which are not only discrete but binary. It’s the difference between a coin flip and the entire surface of the Earth.

Recall that a two-state quantum system can be represented by the Bloch sphere:

Figure 1. The twofold infinity of a qubit.

Where we have two quantum eigenstates |0〉 and |1〉 that form the basis of a Hilbert space. (See QM 101: Bloch Sphere for details.) The actual state of the system is represented by a unit-length vector that can point to any point on the surface of the sphere.

As with the surface of the Earth, any location of which can be specified by a longitude and latitude (two real numbers), any state on the Bloch sphere surface can be specified by the two angles, θ (theta) and φ (phi), which are also real numbers.

I’ve posted before about the vast gap between the countably infinite numbers versus the uncountably infinite numbers (see Infinity is Funny for instance). When it comes to classical computing versus quantum computing, that gap is magnified on both ends. On the classical side, there is no infinity, just the two states, 0 and 1. On the quantum side, however, there are two uncountably infinite values (Dirac’s “twofold infinity”).

The same musical note played on different instruments sounds different (and instantly recognizable) because of content of their respective higher harmonics. The Fourier transforms of each, although they’d all show the same fundamental frequency, would have completely different frequency spectrums in terms of those higher harmonics.

In fact, musical notes have many more degrees of freedom than qubits do because musical notes are an infinite series of frequencies, each with their own coefficient defining the amplitude and phase that frequency contributes to the whole.

Even so, once we cross from countable to uncountable, the ballgame completely changes. Those two degrees of uncountable infinity, compared to the binary bit, are what gives quantum computing the power it has.

§

On the other hand, it’s also why quantum computing isn’t much good for the sorts of tasks we routinely give classical computers. For instance, a quantum computer would be incapable of running Tetris (or any game). Or a spreadsheet program. Or pretty much any classical software.

Given that quantum computing doesn’t provide precise (that is, countable and determined) answers, you really wouldn’t want it doing spreadsheets or your banking.

The problem is that, unless a qubit is already in a known eigenstate, any measurement of its value is subject to the Born rule and is therefore probabilistic. (In fact, the Born rule still applies to a qubit in a known eigenstate. It’s just that in such a case the Born rule returns 100% probability for that eigenstate.)

Figure 2. A quantum computing output measuring two entangled bits.

This is why a given quantum computing algorithm is run many times (typically thousands). The results it returns are always probabilistic, and it’s the average of those many runs that provides the answer.

Figure 2 shows the output of a simple quantum computation. I created a trivial example that entangles two qubits and then performs a measurement of both. In theory, the only results should be either |00〉 or |11〉 with equal probability because the entangled pair is in the superposition:

\displaystyle|\Psi\rangle=\tfrac{1}{\sqrt{2}}|00\rangle+\tfrac{1}{\sqrt{2}}|11\rangle 

But, as you see in Figure 2, the incorrect results |01〉 or |10〉 sometimes occurred. So, QC is not something you’d want to risk your bank account with! It might add money, but it also might vanish it.

On the other hand, quantum computers, once we figure out how to scale them to useful capability, would be great at simulating quantum systems. They should be excellent at simulating atomic and molecular systems, for instance.

And, of course, they famously will break RSA encryption (and other public key cryptosystems) because of Shor’s algorithm for factoring numbers. This is another place where that twofold infinity expresses its power. As a very rough analogy, think of water seeking the lowest level of a landscape defined by the inputs to the algorithm (the number to be factored). Where the water ends up is the result (which occasionally includes small puddles — incorrect results — left behind).

§ §

While we’re on the topic of quantum systems, this recent video from Dr. Sabine Hossenfelder:

Earlier this year I posted a five-part series about the measurement problem in quantum mechanics (see Quantum Measurement, Wavefunction Collapse, Quantum Decoherence, Measurement Specifics, and Objective Collapse).

In that series, I talked about how quantum mechanics is linear in contrast to the nonlinearity of classical mechanics. (For a simple example, the parabola of any ballistic object. Or, for that matter, Newton’s second law, F=ma.)

Per Dr. Hossenfelder, this is a well-known (but apparently largely ignored) issue in quantum mechanics. She ultimately presents it as yet another aspect of the measurement problem, the nonlinear change in the linear evolution of a quantum system under the Schrödinger equation.

She also touches on how, although we say “measurement” or “observation”, the better term is interaction. When two quantum systems interact, the result is also a quantum system, but when a quantum system interacts with a system already decohered to a classical state, the result, if any, is also classical.

Typically, such interactions are essentially undetectable unless the classical device is primed with its own energy so that the minute interaction causes a chain reaction amplified by the device’s energy to a classical level. It’s not unlike triggering a mousetrap. A tiny twitch releases the energy stored in the spring.

§ §

The point here is simply that, once again, we should really question our faith that quantum mechanics is complete. (It may even be downright incorrect.) I’ve long thought there was a strong whiff of epicycles to QM, and we should remember that Ptomey’s view of the solar system worked just fine for over 1000 years.

Quantum mechanics is only 100 years old and, despite its stunning success both in practice and in theoretical prediction, and (as with any scientific theory) we should leave the door open.

And speaking of the Schrödinger equation, we’ve known for a long time that it’s a simplified answer that only applies to non-relativistic frames. It’s replaced by the relativistic version, the Dirac equation, which apparently Dirac introduces near the end of the very book I’m reading!

Stay Lorentz invariant, my friends! Go forth and spread beauty and light.

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

14 responses to “The Power of Qubits

  • Wyrd Smythe

    I’ve read a few authors speculating negatively about the difficulty of scaling quantum computing to a useful size. Peter Shor developed his algorithm for factoring numbers in 1994, but so far the record is factoring the number 21 in 2012. An attempt to factor 35 in 2019 failed due to accumulating errors.

    That’s a long way from factoring any RSA public key (which are integers with thousands of bits)!

    The problem is the engineering difficulty of maintain the quantum state of thousands, or even hundreds, of bits. Quantum error correction will be hugely important here, and some are wondering if it’ll ever meet the need.

    We can take some comfort in how humans are pretty incredible engineers, but we might also look to another engineering problem, fusion, which has been “20 years away” for over 50 years.

    We likewise, at least for now, seem stymied by the engineering problems of replicating the human brain, so I think quantum computing could go either way.

  • Wyrd Smythe

    BTW: Reading a technical book from 1930 is … interesting.

  • Lee Roetcisoender

    Interesting stuff Wyrd….. I think Rovelli’s RQM offers a credible paradigm shift from our current model of QM. As you already know, RQM rejects the notion of wave function in its entirety, a move which is revolutionary in itself.

    I think the very notion of a wave function is the biggest mental stumbling block amongst intellectuals for entertaining the idea that mind itself is a quantum system (except Penrose of course). But if one rejects the wave function as RQM does, then all motion resulting in form be it mental, emotional, psychological, classical or quantum reduces to a flash ontology based entirely upon relationships grounded in “the now”.

    Of course, a “flash ontology” leaves time out of the equation because according to RQM, there is only a continuum of “now states” that are constantly changing. Time and space are dominant intuitions for us no doubt about that fact; but I think those intuitions are based entirely upon the usefulness of those intellectual constructs because they help us to navigate, even though ontologically neither time nor space may literally exist. Interesting…………..

    • Wyrd Smythe

      Hey Lee. I haven’t read Rovelli’s Helgoland but are you sure you don’t mean there is no wavefunction collapse in RQM? I’m pretty sure all interpretations of QM feature some version of a wavefunction governing the evolution of the system. For instance, the Wikipedia article for RQM says, “here is no true wave collapse, in the sense in which it occurs in some interpretations.” And shortly thereafter, “Thus, the system will evolve entirely unitarily (without any form of collapse) relative to O’, if O measures S.”

      Why do you think the notion of a wavefunction a stumbling block? Among those who deny quantum effects in the brain, most do so because there is so far no evidence for such. If mind was quantum, it would certainly have a wavefunction. I do agree with Penrose that mind cannot be (classically) computational (in large part because Gödel), but it’s less obvious to me that it must be quantum. That said, I do find the Hameroff-Penrose idea interesting.

      Rovelli does think time is emergent, but I side with Smolin who considers time fundamental. To be honest, I see Rovelli as a bit of a Space Cadet, and I haven’t been greatly impressed by his ideas. (FWIW, I’ve posted about some of his books.)

      As I’m sure I’ve said before, fantastic ideas are fantastic and necessary, but experimental evidence is where the rubber meets the road. Otherwise, they are just science fiction.

  • Lee Roetcisoender

    “Why do you think the notion of a wavefunction (is) a stumbling block?”

    For the same reason you just cited: “If mind was quantum, it would certainly have a wavefunction.”

    I also get it that if one’s own metaphysical position is dualism, then it’s relatively easy to dismiss any notions that do not correspond with that original assumption as fantastic or science fiction.

    Being a pragmatic materialist like Rovelli; for me, the evidence is in the logic an idea where there are no contradictions or paradoxes built into a given model. Mind is the quintessential example of a paradox for materialism because mind is a subjective system whereas the brain is a 100%, unadulterated objective system. It is a contradiction that a single system can be both objective and subjective at the same time. At least a dualist has a quasi legitimate reason for this paradox because a dualist can point to the mind and say it’s not physical.

    In contrast, an idealist like Bernardo Kastrup had nothing but positive accolades for Rovelli’s work until Helgoland came out. He wrote a scathing essay criticizing Rovelli because the book refused to address first cause or the notion of an ontological primitive.

    • Wyrd Smythe

      The notion of a wavefunction is the biggest stumbling block for seeing the mind as a quantum system because, … if mind was a quantum system it would have a wavefunction? I can’t parse the logic there.

      “Being a pragmatic materialist like Rovelli; for me, the evidence is in the logic [of] an idea where there are no contradictions or paradoxes built into a given model.”

      As any pragmatic materialist ought to know, evidence is the result of physical observation. I can point to any number of science fiction and fantasy stories that are entirely self-consistent given their axioms.

      “It is a contradiction that a single system can be both objective and subjective at the same time.”

      We only know of one such system, and that system, in terms of its size and complexity, is utterly beyond any other system we’ve encountered. There is nothing inconsistent with the notion that a system of such size and complexity might have the proverbial “something it is like” to be that system. Indeed, that is exactly what any materialist needs to believe.

      And please stop thinking I’m dismissing anything because of my personal suspicions of dualism. You don’t get to write me off that easily, Amigo. I am entirely comfortable thinking within the bounds of physicalism because, so far, that’s what all the evidence says.

      • Lee Roetcisoender

        “As any pragmatic materialist ought to know, evidence is the result of physical observation.”

        I think we need to put this assessment in its proper context. All of the data that we observe is sovereign; and that data only looses its sovereignty when we adjudicate that data and call the conclusion of that process “evidence”. And once again, all of that so-called evidence is subjective because it is subordinate to another sovereign system called the mind.

        Penrose cites Gödel’s incompleteness theorem as justification for mind being quantum. His rationale is justified and I agree with his conclusion however, a more compelling argument is the accepted paradoxical contradiction that the brain is “the only system in the known universe” that is both objective and subjective at the same time.

        I don’t think so, that conclusion defies the very tenets of logical consistency. There is a reason why human beings have always intuited that there is proverbial ghost in the machine and that is because this so-called ghost is:
        A). a separate and distinct system.
        B). a system that has its own unique properties (quantum).
        C). a system that emerges from and is instantiated by the classical brain.
        D). a system that operates on the substrate of that brain.
        E.) a system that uses the substrate of that brain for its own purposes such as imagination, creativity and the arts to name just a few..

        And finally: Ok, I’ll cut you some slack……….. 😎

      • Wyrd Smythe

        “And once again, all of that so-called evidence is subjective because it is subordinate to another sovereign system called the mind.”

        I think the “it’s all subjective” argument loses its power when dealing with the accumulation and averaging of experience from myriad contributing minds. It is the consistency of information and the consensus over time that gives us the right to claim justified true beliefs.

        More to the original point, an argument, no matter how logical or self-consistent, is not and never will be “evidence”.

        “Penrose cites Gödel’s incompleteness theorem as justification for mind being quantum.”

        There are two parts to his argument, though. In his 1989 book, The Emperor’s New Mind (which took me three years and multiple readings back in the 1990s to fully appreciate), Penrose uses Gödel to argue that the mind cannot be (classically) computational. Such computation is nothing more than arithmetic, so Gödel does apply, and the argument is a strong one. Penrose then speculated that quantum behavior might then account for consciousness. It was after the book came out that he hooked up with anesthesiologist Stuart Hameroff, who had a theory about quantum behavior in the microtubules in the cells of the brain.

        I’m a huge fan of Penrose and his work (I consider him a modern Einstein), and I absolutely agree with the premise the mind cannot be a classical computation.

        But it is as yet unknown whether a physical system of the right structure, size, and complexity, would have the emergent behavior we call consciousness and subjectivity. It is entirely possible it would. We have no basis for comparison and only a dim understanding of the brain.

        Look at it this way: Because of chaotic nonlinear dynamics, orbits involving more than two bodies are not computationally tractable, but the physical system of stars and planets and moons solves a problem involving thousands of orbiting bodies every instant. Or think about how water seeks the lowest level and, when still, always forms a completely level surface. Physical systems routinely solve problems computation cannot (because it works with finite numbers and arithmetic).

        So, the brain may be an analog physical system that routinely solves the problem of consciousness in virtue of its complexity and structure. As I said before, that’s exactly what materialists believe.

        “I don’t think so, that conclusion defies the very tenets of logical consistency.”

        Where do you see logical inconsistency in the premise that the brain is the only system we know of with subjective experience? To me it seems a matter of fact.

        Humans have long attributed personality to the wind, trees, Sun, etc. It’s because we are both pattern-matching machines and fundamentally narcissistic. We tend to see everything in terms of ourselves. (And that which isn’t we often try to kill.) We see agency because we have agency, but ultimately it’s just a form of animism.

      • Lee Roetcisoender

        “It is the consistency of information and the consensus over time that gives us the right to claim justified true beliefs.”

        Yeah, justified true beliefs is a slippery slope because we don’t possess the innate capacity to discern between an intellectual construction that is “useful” and one that represents the “true nature” of reality; we conflate them.

        “Where do you see logical inconsistency in the premise that the brain is the only system we know of with subjective experience?”

        I didn’t state that, what I said was the inconsistency is the assumption that as a single unified system, the brain can be both objective and subjective at the same time. If this is the case, where is that definitive line of demarcation; what part of the brain is objective and what part of the brain is subjective and why?

        The materialist’s position is one based upon naive realism and they readily accepts contradictions and paradoxes as a part of the that world view. Very few will admit it; and being a “pragmatic” materialist myself who also believes there is an underlying reality that is neither mind nor matter, I do not accept paradoxes nor do I accept contradictions, they are things to be resolved not dismissed.

      • Wyrd Smythe

        All our intellectual constructions are merely useful. The utility of science is that it uses consistency and consensus over time to self-correct and converge on better and better models. As a friend of mine once brilliantly quipped, “Science proceeds despite scientists.”

        “[T]he inconsistency is the assumption that as a single unified system, the brain can be both objective and subjective at the same time.”

        That only seems inconsistent in comparison with systems that aren’t brains, but brains are unique and beyond any other system we know. They are also the only system that we’re both outside and inside of.

        And that’s the demarcation: inside versus outside. There is “something it is like” to be a brain.

      • Lee Roetcisoender

        “And that’s the demarcation: inside versus outside. There is “something it is like” to be a brain.”

        I don’t buy it and I don’t believe that you do either; it sounds like a nilly willy whitewash job to me. Dualism is a more plausible explanation than the smoke and mirrors tale of “inside versus outside”.

        We will have to agree once more to disagree. “Pay no attention to the man behind the curtain…”” You take care now and have a good weekend.

      • Wyrd Smythe

        But we don’t disagree about the ideas. I’m totally down with Penrose’s arguments against computation. It’s not just Gödel — nature uses transcendental math (like pi and e), which is also not algorithmic. And I agree some sort of quantum behavior might underlie consciousness.

        But my notions of consciousness are a superposition of that and other possibilities, including material explanations such as I’ve mentioned, various dualist possibilities, and some even more speculative ideas. So, I don’t have a definite answer. I can just agree that, yeah, it might be like that. But maybe not, also.

  • Wyrd Smythe

    It occurs to me this is also a good demonstration of why global phase doesn’t matter but the relative phase between parts of a superposition does.

    The short version is that, at the North and South poles, your longitude is just as irrelevant as a quantum system’s global phase, and for exactly the same reason.

    A longer version is that, in terms of the Bloch sphere, the phase of the system is the “longitude” of the vector in the sphere. That longitude comes about in consequence of the superposition between states. The interference of the two complex coefficients results in that phase. But in the measured states, |0⟩ and |1⟩, that phase degenerates to irrelevancy.

    Metaphorically, it doesn’t matter which direction you face at the North pole, you’re always looking exactly south (and vice versa at the South pole).

  • Matter Waves | Logos con carne

    […] In the last quantum-related post, I wrote about why qubits are so much more powerful than classical computing bits. Basically, a bit has only two states, but a qubit has a two-fold uncountable infinity of states. […]

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