All aboard! Can you find a mansion on the hill? ... let's see what particles and waves can do, and what makes quantum computing so different to classical computing...
The web version for mobile and desk is here
(- the references will pop up when selected - not available on the email version.
- note also, the email version does not include updates)
The Fragile Sea Quantum Series
In Q01 we looked at the standard model of particle physics and went for a brief walk through the establishment of scientific theories.
In Q02 we travelled into the wonderful, strange, world of particles, waves, and fields, on the road to quantum computing.
Welcome to Q03. In this post the conceptual outlines of classical and quantum computing are introduced.
This post is a little longer to encompass the areas, please bear with me. Let’s start by defining a few aids.
1. Knowledge domains
In quantum, I've found that experts will jump from one knowledge domain to another in the same sentence, a lingual transition that can be hard to follow, for example, between the physics, quantum math, and quantum computing. In a way, these knowledge spaces are discrete domains with their own complexities.
There’s no formal convention here for what constitutes a knowledge domain, definitions may change as familiarity with the subject matter increases. Being aware of when a subject transitions between different domains has helped me to break down difficult subjects and better understand them.
2. ‘Imputed’.
Knowledge can be imputed from one domain to another but remain discrete. Here’s an example: the quantum physical states of particles (which we’ll meet in this post), have no correlation to the computing logic imputed as binary 0 or 1 in a quantum bit (a qubit) [1]. A qubit is the basic unit of information in quantum computing which serves as the quantum equivalent of a classical bit [2].
The convention of “imputing” means assigning a property, meaning, or value to something by agreement or decision, rather than by necessity or inference [3]. The property is not connected from one system to the other – by convention, we elect a chosen representation between two discrete systems.
3. ‘Analogue’ (or ‘analog’).
An analogue means a counterpart or equivalent in another system. It’s subtly different to imputation – an analogue is a similar feature, whereas an imputation is a chosen representation of a feature in one domain, applied to another.
When we say something has “no analogue in xyz” it means there is no equivalent concept or phenomenon in that other system that directly matches or operates like the feature in this system. As we’ll see, quantum computing has features that have no analogue in classical computing.
OK, hope that helps. Let’s go!
Classical and quantum conceptual outlines
Our classical computational history is built on sequences of binary 0 and 1 digits. All the applications, graphics, videos, and classical computing we use every day arise from imputing a logical 0 with a physical electrical state, typically a low voltage or ‘off’, and a logical 1 with a high voltage or ‘on’. Some implementations swap this convention (a high voltage is imputed as off), but that doesn’t matter for our purposes.
The point is, there’s no direct physical connection between the electrical property, for example, ‘off’ and the logical computational entity called zero, we impute it and build our classical computers from there.
1s and 0s are all we use
A Turing machine is an abstract, mathematical model of computation invented by Alan Turing in 1936, to express how a function or problem can be computable by any mechanical process [4].
It’s a very powerful abstract model that has enabled both classical and quantum computing, further developed with Alonzo Church (Turing’s doctoral adviser at Princeton in the 1930s [5], [6]).
Essentially, the Church–Turing thesis holds that any process that can be described algorithmically can be carried out on a Turing machine, which in practice can always be encoded using binary symbols. Informally, this means that anything computable can be computed using only 1s and 0s.
Chris Ferrie explains the breakthroughs and how it developed theoretically and historically [7], see also [8].
In classical computing, when we say that an electrical property such as 'on' has no direct connection to a binary bit, similarly in quantum computing, physical entities (particles, waves, fields), and their properties (momentum, spin, etc), also have no direct correlation with qubits. We select unique states of quantum physical entities, for example electron spin-up, spin-down, or photon linear, horizontal or circular polarization states (known as ‘basis states’), and impute them with ‘computational meaning’ – that is, for example, spin up = 0 and spin down = 1, in binary terminology.
A basis state is a selected property (e.g., spin-up) of an underlying quantum physical entity (an electron, ion, or photon). As we’ll see, there can be many basis states [9], but normally two are selected, such as particle momentum and location, or spin-up and spin-down, tied together by certain properties as 'conjugate variables', which will come up again later.
Superposition
This section is the most difficult in the post, because the underlying quantum reality is so unlike our everyday reality.
The most confounding quantum reality is that a single particle can be put into a superposition of selected basis states (like position and momentum etc). This means that before measurement, the particle does not have definite, measurable properties of these basis states, that is, we cannot measure a particle in superposition, and the selected basis states are indeterminate. Instead, it has a wavefunction encoding probabilities of finding it in one or the other basis states after measurement [10]. (A further property called entanglement with more than one particle will be discussed later).
Its ‘quantum state’ exists as a superposition of the selected basis states prior to measurement [11]. Its quantum state refers to the state before measurement as a combination of possibilities—not physical coexistence in a classical sense. Until measurement 'collapses' the wave function, the states are not available to be measured.
This demonstrates how easy it is to get lost in multiple knowledge domains: a ‘quantum state’ primarily refers to the mathematical representation of a quantum system, but one which fully encodes all knowledge about its physical properties and behaviors. The connection is indirect - quantum states mathematically encapsulate the physical properties (position, momentum, spin, etc.), allowing physicists to calculate probabilities for measurement outcomes corresponding to those physical properties.
A quantum system can be a single particle, a composite of particles (e.g., a molecule), a computational and/or experimental structure, whatever is in current focus [12].
How can the quantum system encode all possibilities for multiple basis states at once? We met wave functions in Q02: The wave function for a particle encodes probability amplitudes for each basis state; when squared, these give the probability (before measurement) of finding the system in one or the other of each possible state after measurement.
As discussed in Q02, The Heisenberg Uncertainty Principle asserts that two basis states (that are tied conjugate variables) cannot be measured precisely at the same time. These variables could, for example, be position and momentum, or energy and time (e.g., for energy transitions of electrons).
A variable (basis state) is conjugate with another and the uncertainty principle applies only when the precise measurement of one increases the uncertainty of the other, and where they carry the mathematical property that the order in which measurements are performed affects the outcome (non-commutative) [13].
The result cannot be predicted in advance - only the overall probability distribution. At the point of measurement, the wave function collapses to one definite state. Note here also that some interpretations refute the Copenghagen Interpretation of ‘collapse’ and instead propose other processes. We’ll discuss this in Q05 [14].
Superposition is a physical property intrinsic to quantum systems. By exploiting superposition, quantum algorithms can compute with multiple possible inputs in parallel, which is fundamentally different from classical bits that carry only a single value at a time.
Unlike a classical bit, which must be either 0 or 1, a qubit in superposition can be expressed as a linear combination (a sum) of both, as imputed from the selected physical basis states, each having a complex amplitude (see below) in superposition. The squared magnitudes of these amplitudes represent the probabilities of measuring the system in each basis state, and the probabilities always sum to 1 (one) [15], [16], [17].
A quantum logic circuit gate (for example a Hadamard gate) can place a particle into a superposition through a variety of means, for example a laser acting on a trapped ion.
Notice in that sentence the jump between computational (the Hadamard gate) and physical (the laser and the trapped ion), two separate knowledge domains. Note the preceding paragraphs also. I won’t keep pointing this out, but I’ve found it useful to be aware of such lingual transitions.
A potential objection
Here we should cover off a subtle nuance. Chris Ferrie’s book ‘What You Shouldn’t Know About Quantum Computers’ debunks many ‘pop science’ myths around quantum physics, myth 2 being ‘qubits are zero and one at the same time’. He notes, “a qubit is represented not by ‘0 and 1 at the same time’ but by two complex numbers. These numbers take on a continuum of values so they are indeed much more versatile than the binary option afforded to a single bit” ([18], p.50).
This is correct. A superposition is an indeterminate state and cannot be defined as 0 or 1 or both, only probabilities; when the indeterminate state collapses, either 0 or 1 will output.
Complex numbers involve real numbers and imaginary numbers that allow quantum states to represent magnitude and phase, rotations, and other quantum phenomena that cannot be described by real numbers alone. [19], [20]. This enables, as Ferrie notes, ‘a continuum of values’, which are indeed much more versatile than a single bit.
Imaginary numbers are used in many areas of math and logic, including complex analysis, signal processing, control theory, and engineering, to name a few [21] [22], [23]. They enable the joint encoding of multiple complex variables, such as spin and momentum, and including probability functions.
A ‘complex amplitude’ is the complex number itself, encoding both the “size” (probability/magnitude) and phase (direction/angle), which are behaviours of the quantum states of trapped particle (s). A complex amplitude is a mathematical representation corresponding directly to the experimentally observed probabilities and quantum behaviors in a quantum system, e.g., in trapped particles [24].
Fundamentally, interference is used in quantum systems to control and amplify the probabilities of certain outcomes. Quantum interference is the phenomenon where the probability amplitudes of different basis states combine, either reinforcing (constructive interference) or canceling out (destructive interference), due to the wave-like nature of quantum states. Interference will be discussed later in this post.
Do we know what we’ve achieved already?
Sometimes I wonder if we fully appreciate what great minds have achieved over the last 150 years.
If a superposition of all possible probabilities for selected basis states in quantum physical systems can be represented in computational logic, then they can be computed. But both physics and math were needed - observed physical behaviour, experiment, theoretical constructs, quantum logic, linear algebra, and math, to capture all probabilities and properties, and make them calculable.
This is where the formulation and representation of complex amplitudes enable quantum computation, imputed from discrete physical quantum properties. Once quantum behaviour, that is, rotation, phase, amplitude (etc), could be modelled mathematically, quantum computation became possible [25].
That's some achievement mathematically. It’s no wonder that physicists like Max Tegmark, president of the Future of Life Institute, MIT, and author of many books and over 200 papers, can write Our Mathematical Universe[26], suggesting that reality is mathematics. We might not agree, but as Feynman said (noted in Q02), we need to make constructs even when there is no experimental proof. The contructs have been spectacularly successful.
In our time, we see so much negativity daily about humanity, that some positivity is worth hailing. I don’t deny that the daily news is often awful and beyond comprehension, but great minds have come before us, and we live among very clever people still. The achievements in physics and computing are truly outstanding, and humanity should acknowledge them as something special.
The results have produced historic achievements that are both great and terrible, and I've no words to make those two outcomes better or worse than they are. There are no words. We must make the future better, that’s all, it’s enough. We live under the constant cloud of sub- ten minute annihilation, a better future is desirable, even necessary, or we face something we'll likely never know happened.
I for one, am simply amazed at the achievements of bright minds, and long may we appreciate how difficult it is to find true success in these areas. Think about it for a moment. Imagine the intellectual leaps and brainwork in coming up with the equations and constructs to model the reality that is there, and then to discover (often later) that the models are accurate to billions of decimal places, in scales both unimaginably large and small. The physics and the math agree to a great extent in these major achievements. Not complete, and perhaps they never will be, but incredible enough to have got us this far. The many humans who achieved these intellectual feats have enabled the life we know today. And now we must take control of the future.
Superposition is not all - entanglement
If superposition is suitably stupefying, there’s more. Entanglement is connected to superposition. A particle can be singularly placed in a superposition state, or two (or more) particles can become entangled, which immediately places them in superposition [27]. The exponential possibility of quantum advantage only becomes available with both superposition and entanglement [28].
Quantum entanglement is a phenomenon where two particles become linked so that the state of one instantly determines the state of the other, no matter how far apart they are [29], [30]. The phrase “spooky action at a distance” was famously written by Albert Einstein to describe quantum entanglement, in a letter to physicist Max Born in March 1947. The full quote is:
"I cannot seriously believe in [quantum mechanics] because the theory is incompatible with the requirement that physics should represent reality in space and time without spooky action at a distance" [31].
Nevertheless, without violating the generally accepted laws of physics, entanglement is a quantum reality, linked to superposition. Superposition is a mathematical and physical requirement to enable entanglement to be possible; two or more particles can then only be described, in all their basis state possibilities, by one wave function as though they were one particle, a unified whole [32]. We’ll dig into this more deeply in Q04 and Q05.
Many reams of paper and discussions have issued on ‘spooky action at a distance’ and it’s no surprise that theories have emerged around quantum entanglement in our brains generating consciousness [33].
A minor diversion into quantum, AI, and consciousness
In my view, this year and next will see many breakthroughs in quantum physics, quantum computing and AI combined [34], [35], [36], along with neuromorphic computing [37], and perhaps, continued fanciful connections to consciousness theories. What a wonderful world.
Regarding AI and consciousness, I currently tend to sit in the Yann LeCun camp, i.e., We Won't Reach AGI By Scaling Up LLMS [38], but I’m open to what the future may unfold. The paper, Neuroscience-Inspired Artificial Intelligence (Hassabis et al 2017 [39]), remains for me the gold standard in the alignment and the gaps between neuroscience and AI.
To that should be added the neurobiological discovery, reported in August 2024, of the ‘glue’ that “makes memories stick for a lifetime” [40]. The discovery, written up in Tsokas et al 2024 [41], details an extraordinary chemical pathway via an interplay of protein molecules, with a more ancient fallback pathway in the background, where both assist to retain persistent memory even with cellular regeneration.
While it is true that AI has copied much from neuroscience, these pathways, to date, have no analogue in AI, and so the long-term persistence of memory is one impediment to artificial general intelligence (AGI), that remains to be solved, if it can. It’s not just a storage and recall issue, dynamic and persistent connection, interconnection, dynamic linkages, and other factors are involved.
An AI may temporally appear to be human, or like Rachel (Sean Young) in Blade Runner, to have implanted memories, but there’s a chasm to cross yet, in the long-term memory persistence pathways and recall evident in nature.
What about our brains?
To this should be added the extraordinary bi-hemispheric nature of our neurobiology, for example see 'Split brain: divided perception but undivided consciousness' Pinto et al 2017 [42]. I wrote about this in more detail in a short story on a personal computer becoming sentient, originally in 1981, and reproduced in this link without any textual modifications. (The commentary on dual brain hierarchies at the end of the story was added recently).
The appearance of AGI seems to be enough to convince many that what we are seeing is true intelligence, and there’s quite a divide between those who think we are almost there, and those (like me) who don’t [43], [44], [45]. I guess it depends on whether it matters or not – if it’s good enough, why should we care? I think it matters.
It may be that we’ll come to a point where tests of some characteristics and capabilities that do exceed human ability will indeed be good enough, certainly for commercial exploitation, but I think we’ll come to regret handing over the farm so soon in ways only dimly foreseen now [46], [47], [48]. There are many human capabilities still beyond generative AI.
Even so, the intersection of quantum, AI, and consciousness, is producing some fascinating research and papers [49], [50], [51]. It is indeed a wonderful world, a haven of new possibilities and ‘many paths to the future’.
Hurdles to quantum computing
In future, as Feynman suggested, perhaps all computers will be quantum. Because the technology is dealing in probabilities in fragile environments, subject potentially to quantum fragility, noise, heat, (etc), the path to durable quantum computing at scale remains ahead of us. In advance of new materials and other breakthroughs yet to come, the physical environment in which to trap particles is supremely complex, and error correction is a critical ongoing focus.
The computations are run several times, called shots, 1,000 times or more. Some algorithms can run perfectly well only once and deliver a result as discussed below, but mostly, complex error correction is necessary in a variety of ways. Probability and linear algebra figure prominently in quantum computation, presented mathematically through powerful and elegant Dirac notation [52], [53].
Maintaining a stable quantum environment is difficult
Many coherent environments last only in the micro-seconds before decohering [54], the longest stable state so far for a single qubit exceeded one hour in 2021 [55], and at room temperature, 39 minutes in 2013 [56]. For more than one qubit, the record times are considerably shorter and more complex to maintain – the current record achieved in 2023 in a solid state device is 0.1 milliseconds (100 microseconds) and in 2024 [57], in a room temperature ‘metal-organic framework’, 100 nanoseconds [58].
Beware however, coherence times can be reported as average or best, and some vendors have reported 400 microsecond coherence times as best, but across an entire chip with up to 1000 qubits or more, the average coherence is lower, nearer the 100 microseconds noted above.
Each week brings reports of improvements in error correction, and the number of concurrent qubits that reach coherence. Barrera et al (May 2025) provides details of applying a protocol to “24 quantum processors from six vendors, testing problems with up to 156 qubits and 10,000 layers… This constitutes the most extensive cross-platform quantum benchmarking effort to date, with circuits reaching a million two-qubit gates” [59]. Matt Swayne from Quantum Insider summarised the results, with appropriate vendor comments, and the limitations and challenges that remain [60].
One of the issues here is that quantum decoherence increases dramatically as more qubits are coupled and the circuit depth grows [61], [62]. It is obvious immediately that progress proceeds in many firms worldwide but there’s some way to go yet to achieve stability and durability for multi-qubit quantum systems.
IBM quantum computers and tools
IBM has an excellent training series that explains how their quantum chips work (not all quantum devices work the same way). They offer access to their quantum computers for education and research on several tiers (the free tier allows 10 minutes per month): "IBM has dozens of quantum computers around the world, and we've recently upgraded our fleet to exclusively have processors larger than 100 qubits…" [63].
The IBM Qiskit SDK (software development kit) is a powerful enabler akin to an integrated development environment (IDE), with excellent tutorials and documentation [64], [65], plus a responsive support desk. The drag and drop tools to create a quantum circuit are very similar to scripting a piece of music, which makes me smile and reminds me of the deep relationship between math and music [66], [67].
Access to IBM quantum computers
In another link, IBM explains how their quantum chips are fabricated and the almost zero degrees Kelvin (−273.15 °C or -459.67 °F) at which they work. They explain how a user instructs the chip to compute: “Microwave transmission lines are fabricated on the chip to deliver microwave signals to the qubits. When we apply highly calibrated microwave pulses - with specific frequency, amplitude, shape, and duration - to the lines, we can make the qubits do specific things. This forms the basis of our quantum gates… The instructions for the microwave pulse go from your computer, through the cloud, and to room-temperature control electronics, which interpret those instructions and physically generate the pulses. After the room-temperature control boxes create the pulses, they travel through cables into a dilution refrigerator and eventually to the quantum chip. The signal goes into the resonators, through a wirebond, and then flows down the transmission line into our qubits” [68].
Quantum advantage – is it achievable?
News articles of systems achieving quantum advantage have been met with some skepticism, but the industry has generally adopted a cautious path [69]. This is an area where things can get a bit sticky. No, quantum computers aren’t the solution to everything, no, they won’t readily replace classical computers, no, they are not suitable for many types of computations, and yes, there’s still some distance to go to achieve stable environments, manage multiple qubits, and really leverage the benefits we can see in the mansion on the hill over there.
There’s no doubt, however, that combining qubits with quantum properties, such as superposition, entanglement, and interference, provides an exponential increase in computational power over classical computing for certain types of applications. Lately, verifiable quantum advantage over classical computing has started to emerge in papers [70] and workshops [71].
Linear quantum advantage
I’ve seen linear quantum advantage work directly through programming a Bernstein-Vazirani algorithm in IBM Qiskit to find a hidden string of numbers, requiring only one query. [72].
Linear quantum advantage reduces the number of steps by a constant or proportional factor compared to the classical method, whereas exponential quantum advantage solves problems where the gap between classical and quantum approaches expands significantly with problem size, and the problems quickly become impossible to solve on a classical computer [73].
Programming quantum advantage - use case
The Bernstein-Vazirani algorithm has specific use-cases, mainly in cryptography to find keys for certain symmetric algorithms, error correction, optimization, and in machine learning to determine hidden weights [74], [75].
Taking a very long total bit string, and a smaller hidden string, the classical method for this task on a binary computer is sequential, requiring multiple queries to the function to deduce the hidden string. For a string of length 'n', a classical computer might need up to 'n' queries in the worst case. Even if it was divided into parallel paths, up to ‘n’ queries would potentially be required and as explained below, there's a law that limits speedup from parallel processing.
In contrast, the quantum approach, by leveraging superposition, can determine the hidden string with just one query. This means that regardless of how long the total bit string is, the hidden bit string can be found in a single run.
Quantum parallelism is not ‘all solutions at once’.
Another of Chris Ferrie’s debunks is that ‘quantum computers try all solutions at once’. And he’s correct, that’s not what is happening. This is the most important part: quantum mechanics describes physical systems using wave functions (or state vectors), which are expressed as probability amplitudes. These contain all possible outcomes as a weighted superposition, calculated using wave (or matrix) math. We can’t access all possible outcomes in superposition, and the quantum computer is not calculating all solutions at once. It’s doing something way stranger.
In the Bernstein-Vazirani algorithm, the quantum circuit prepares a single superposition of all possible input states, and through constructive interference, when measured 'on collapse of the superposition’, only one result will output with certainty. There’s no general “try all answers at once” ability - superposition encodes all possible inputs at once, but they’re unavailable to access until measurement collapses the superposition and yields only one outcome.
Even then, a brute-force quantum search and measurement would only give a random outcome - not the best one, not ‘all solutions at once’, and not the one that is desired. To get the desired result, the setup requires ‘choreographing interference’ [76], [77], [78].
Interference?
OK, so what is interference? In quantum mechanics, interference occurs because each possible path a particle or quantum system can take to get from A to B is described in math by a complex number as noted above, called a probability amplitude. All possible paths or states contribute to the final probability, not one alone. The genius is in the discovery of the right math to encompass ‘all possible paths or states’.
If we drop a stone into a still water lake, then drop a second stone a few seconds later, the expanding and colliding rings will have (in some measure) both magnitude (height between troughs), and phase, where two colliding rings add together (constructive) or cancel each other out (destructive).
In quantum systems, each basis state corresponds to a classical binary bit value, but it is the superposition of basis states, each weighted by a complex probability amplitude (a complex number), that encodes both magnitude (the probability of finding the system in that state after measurement) and phase (which determines how amplitudes for different paths/states can add and interfere).
The formal math derives the magnitude squared of the wave function, which gives the probability of finding a particle at a given location if measured [79].
Particles or waves?
But wait a minute. Are we talking about particles or waves? This reveals the oddities of the quantum realm. The superposition principle and the use of amplitudes come from one of the foundations of quantum theory, first formalized by Louis de Broglie in 1924, known as wave particle duality [80]. Superposition is a wave-like behaviour, quantum states can be added together like adding waves. The probability amplitudes enable these ‘quantum waves’ or wavefunctions to interfere, but when measured, the output is ‘particle-like’. This can be characterised as a quantum system with waves of possibilities until measured; measurement produces a single, observable result, compatible with classical computers. No matter how strange it is, it works.
How to ‘favour’ the correct result
In quantum algorithms like Bernstein-Vazirani, logic gates set up constructive and destructive interference so that only the correct answer’s amplitude remains, and the others cancel out. When measured, the right answer has certainty, because interference has made all other possibilities unlikely or impossible. When quantum computing logic such as the algorithm above, is applied to the underlying quantum physics, the right answer can be imputed – but it only works for specific problems where interference can be controlled to advantage.
I understand it’s difficult to believe but it works. For other more complex algorithms and processes, the math and the encoding are far deeper.
Better than classical parallel computing
Quantum computing is beyond classical parallelism in binary computers. Classical parallelism has its limits - Armdahl’s law posits that there is a limit to speedup from parallel processing on a classical computer, basically, the sequential processes, for example, at the start and at the end of a computation, exhibit limitations and bottlenecks that cannot be overcome by using parallelism alone. Marin Ivezic explains well how quantum computing overcomes Armdahl’s law. [81].
Classical computing parallelism scales, within limits, by adding more processing and threads. Quantum parallelism is completely different [82].
But that’s not all
That’s all good, but there’s more; we’re almost at the end for this post.
A qubit is a two-state (or two-level) quantum system [83]. To be clear, classical computing is also a two-level system – the difference is that classical bits can only encode one definite state - 1 or 0, but never both, or a combination [84].
But there are other, more exotic quantum systems with more available states, such as qutrits (three levels) or even higher dimensions [85]. Beyond qubits and two selected basis states (e.g. spin-up, spin-down), many more basis states exist and can be selected, perhaps an infinite number [86], which is quite exciting to adherents of the multiverse point of view [87].
Research is being conducted on qutrits (0,1,2), ququarts, and beyond, collectively qudits. A qudit is a quantum information unit with d states*, where d is any integer equal to or greater than 2, for example, qubits are 2-level (d=2), qutrits are 3-level (d=3 = 0,1,2), ququarts are 4-level (d=4), and so on. These are imputed by using more than two basis states in the underlying quantum world, such as spin-left, spin-right, spin-up, spin-down (etc).
(* The states must be ‘orthogonal’ which is defined differently in telecommunications and physics, but in quantum mechanics typically refers to the relationship between two quantum states indicating they are independent [88]).
Beyond qubits, multiple basis states and entanglements in superposition exhibit properties far exceeding even the exponential advances in qubit quantum computing, and with potential benefits in error correction, and computational efficiencies [89], [90], [91]. There are, however, daunting challenges ahead to achieve stability.
How far can we go?
If quantum computing with qubits is hard to grasp, qutrids and beyond are much further out on the unbelievable scale, way out there. The potential exponential gains lie far beyond, perhaps, our wildest imaginations, and far beyond even qubit quantum computing.
We are proceeding down the path of a vast open-ended quantum journey. Potentially, a qudit system could achieve or exceed the trillions of synapses in our magnificent brains, but stable systems are still, conceivably, many years away, even as qudit research proceeds [92], [93].
It's not equal to the human brain
We should be cautious though. It’s true, for example, that 40 qudits of dimension 10 is equal to 1040 possible pure quantum states, way beyond the number of synapses in a human brain [94].
In contrast, however, biological synapses are not just ‘bits’, but systems that exhibit complex biochemical and bioelectrical dynamics. Perhaps AI will find exotic new materials and ways to achieve these sorts of magnitude leaps in quantum computing, but I cautiously don’t think we’ll get there for a good while yet [95].
A journey well travelled...
This has been some journey! The understanding I’ve come to is this: when quantum physicists and educators speak or write, they transition, without thinking or being unduly aware of it, through multiple domains or ‘spaces’, and often in the same sentence.
The lingual transitions between knowledge domains are seamless and a normal part of their daily activities, but for me, it took a long time to understand ‘jumping between spaces’, and how to follow the course of an argument between discrete knowledge domains.
Finding computable logic to express physical behaviour
Physicists and logicians observe quantum behaviour, and use mathematical constructs (such as a Hilbert state [96]) in which they can model data and physical behaviour, to create complex algorithms using quantum math, then combine knowledge domains in potential solutions that, if successful, carry a very high degree of accuracy.
The goal is to find and test and refine those formulas that best fit the experimental data in the physical world, across a great many scales and magnitudes. From that comes explanatory power. That’s why it can be said that in many ways the standard model of particle physics is accurate to a very great degree, even if incomplete.
Standing on the shoulders...
Imagine if you will, Einstein, prior to 1905, and up to 1907, spending months grappling with concepts, math, and logic, to find the formulas that, when tested, have remained accurate across all known dimensions from the infinitesimally small, to the macro-large, even when it took some years for tests to confirm some aspects of theory at different scales [97].
Or in June 1925, twenty-three-year-old Werner Heisenberg, suffering from hay fever, walking all night on the Island of Helgoland in the North Sea, and coming up with a radically new mathematics. This became matrix mechanics, the first complete formulation of quantum mechanics [98], as memorably retold in David Lindley’s book ‘Uncertainty: Einstein, Heisenberg, Bohr, and the Struggle for the Soul of Science’ [99], and very briefly, in Carlo Rovelli’s book Helgoland [100]. (Incidentally, Rovelli’s book has crystal clear expositions on his own views around ‘collapse’, gravity, and the Copenhagen Interpretation.)
Or in 1933, when Leo Szilard was waiting at the lights to cross Southampton Row, London, when the lights turned green and he stepped off the sidewalk, coming up with the idea of a sustained chain reaction. His key insights were mathematical and became the backbone of modern chain reaction math [101], a foundation for nuclear physics both unimaginably terrible and incredibly useful for humankind (think of medical devices, and other uses in industry, energy, telecommunications, the internet, and computing) [102].
Wrapping up
I deliberately raised superposition in this post before entanglement in the next, a beast of a subject. In Q04 we’ll meet entanglement, superdense coding, decoherence, and teleportation, plus a few other topics besides, such as realism, localism, and the no-cloning theorem. They may become monstrous, so I might split Q04 into a and b – separate posts, and deal with entanglement first. There’ll be a short pause before they come out.
As we’ll see, if some concepts seem unbelievable, sometimes they just are. They exist, they work, and the best is yet to come…
Almost every discussion on entanglement notes Einstein’s ‘spooky action at a distance’ term, in context to a famous paper, Einstein-Podolsky-Rosen (EPR) 1935 [103], a terrific paper to read before Q04 if you are inclined. Einstein was unhappy with entanglement, and the subject has generated decades of papers, discussions, and new physics.
The term has unfortunately appeared to make entanglement mysterious. At times, I’ve wondered if this is just to make the quantum world seem unobtainable by any mere mortals. Entanglement is mysterious, but not completely obscure. I want to show details that, to the best of my knowledge, are rarely mentioned, or at the least, rarely gathered in one place. They’ve made it easier for me to understand, and maybe they will for you too.
Bye for now
Thanks for making it this far!, it’s time to go.
As summer’s light wends its way towards autumn, a poem in the Japanese classical waka form [104]:
Was it only yesterday that
Summer reached its end?
For stepping out my door this morning
My sleeves are chilled
By autumn’s first breeze…
Kinkai wakashū 180 [105]
Till next time, then. Take care, Brent.
P.S. Please feel free to subscribe here, it’s free. My posts go out by email as well as online when released. They’re not regular.
I release notifications and links on LinkedIn and Facebook and
welcome comments there. I regret I’m unable to respond readily at the
present time. If there’s something particularly egregious you wish to contest,
or that would help my own understanding, or if you do wish to make contact,
please use the contact page on this web.
I own all the books referenced in this post and have mostly read them all cover to cover. All the written words, apart from acknowledged quotes, are my own. Some phrases and definitions are common to many sources.
For around 35% of primary research, I have used perplexity.ai to check multiple sources and gain a sense of the accuracy of my own writing, but the words written (unless direct quotes) come from developing my own understandings and concepts over many years. All the papers and most of the web sources were already known to me or found before or during writing, without using AI.
© 2025 The Fragile Sea - all rights reserved
[1]: T. Siegfried, ‘A quarter century ago, the qubit was born’, July 05, 2017. https://www.sciencenews.org/article/quarter-century-ago-qubit-was-born
[2]: J. Schneider and I. Smalley, ‘What is a qubit? | IBM’, 2024. https://www.ibm.com/think/topics/qubit
[3]: B. van Tiel and B. Geurts, ‘Conventions, coordination, and arbitrariness’, Aug. 2025, https://www.doi.org/10/g9xmkc
[4]: V. Hadzilacos, ‘Church’s thesis and the universal Turing machine’, 2025, https://www.cs.toronto.edu/~vassos/teaching/c63/handouts/ChurchsThesisUniversalTM.pdf.
[5]: B. J. Copeland, ‘The Church-Turing Thesis’, in The Stanford Encyclopedia of Philosophy, Winter 2024., E. N. Zalta and U. Nodelman, Eds. Metaphysics Research Lab, Stanford University, 2024. https://plato.stanford.edu/archives/win2024/entries/church-turing/.
[6]: L. De Mol, ‘Turing Machines’, in The Stanford Encyclopedia of Philosophy, Summer 2025., E. N. Zalta and U. Nodelman, Eds. Metaphysics Research Lab, Stanford University, 2025. https://plato.stanford.edu/archives/sum2025/entriesuring-machine/.
[7]: C. Ferrie, ‘Whence Quantum Computing?’, Medium, May 27, 2024. https://csferrie.medium.com/whence-quantum-computing-ac5fb1efa642
[8]: Wikipedia, ‘Church–Turing thesis - Wikipedia’, Aug. 21, 2025. https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis
[9]: QuEra, ‘What is Physical Qubit’, 2025. https://www.quera.com/glossary/physical-qubit
[10]: Wikipedia, ‘Quantum superposition’, Wikipedia. Sept. 01, 2025. https://en.wikipedia.org/w/index.php?title=Quantum_superposition&oldid=1309002183.
[11]: M. Ivezic, ‘Quantum Superposition: How Qubits Live in Many States at Once’, PostQuantum - Quantum Computing, Quantum Security, PQC, May 10, 2017. https://postquantum.com/quantum-computing/quantum-superposition/
[12]: Wikipedia, ‘Quantum state’, Wikipedia. Aug. 29, 2025. https://en.wikipedia.org/w/index.php?title=Quantum_state&oldid=1308355504.
[13]: Wikipedia, ‘Conjugate variables’, Wikipedia. May 24, 2025. https://en.wikipedia.org/w/index.php?title=Conjugate_variables&oldid=1291971958.
[14]: P. Ball, ‘Physics Experiments Spell Doom for Quantum ‘Collapse’ Theory’, Quanta Magazine, Oct. 20, 2022. https://www.quantamagazine.org/physics-experiments-spell-doom-for-quantum-collapse-theory-20221020/
[15]: study.com, ‘Quantum Superposition | Definition, Principle & Examples’, study.com, May 10, 2025. https://study.com/academy/lesson/quantum-superposition-definition-principle-examples.html
[16]: NIST, ‘Quantum Logic Gates’, NIST, Mar. 2018, https://www.nist.gov/physics/introduction-new-quantum-revolution/quantum-logic-gates.
[17]: Q. News, ‘Quantum Computing Basics: Understanding Qubits And Superposition’, Sept. 08, 2024. https://quantumzeitgeist.com/quantum-computing-basics-understanding-qubits-and-superposition/
[18]: C. Ferrie, ‘What You Shouldn’t Know About Quantum Computers’, Medium, May 27, 2024. https://csferrie.medium.com/what-you-shouldnt-know-about-quantum-computers-0a13863cc0c2
[19]: Wikipedia, ‘Probability amplitude’, Wikipedia. Feb. 24, 2025. https://en.wikipedia.org/w/index.php?title=Probability_amplitude&oldid=1277356219.
[20]: Quantum Grad, ‘Understanding Complex Numbers: A Quantum Mechanics Primer #1’, Apr. 29, 2025. https://www.quantumgrad.com/article/446
[21]: The Math Doctors, ‘Making Sense of Imaginary Numbers – The Math Doctors’, Apr. 20, 2025. https://www.themathdoctors.org/making-sense-of-imaginary-numbers/
[22]: Wikipedia, ‘Imaginary unit’, Wikipedia. Aug. 08, 2025. https://en.wikipedia.org/w/index.php?title=Imaginary_unit&oldid=1304764780.
[23]: M.-O. Renou, A. Acin, and M. Navascues, ‘Quantum Physics Falls Apart without Imaginary Numbers’, Scientific American, Apr. 01, 2023. https://www.scientificamerican.com/article/quantum-physics-falls-apart-without-imaginary-numbers/
[24]: Wikipedia, ‘Complex number’, Wikipedia. Aug. 08, 2025. https://en.wikipedia.org/w/index.php?title=Complex_number&oldid=1304899041.
[25]: F. D. Giovanni, ‘From Bits to Qubits: Mathematical Representation of Quantum Gates’, EE Times Europe, Jan. 15, 2024. https://www.eetimes.eu/from-bits-to-qubits-mathematical-representation-of-quantum-gates/
[26]: M. Tegmark, Our mathematical universe: my quest for the ultimate nature of reality. London: Penguin Books, 2015.
[27]: A. Migdall, ‘Quantum Entanglement: Unlocking the mysteries of particle connections’, Space, Mar. 16, 2022. https://www.space.com/31933-quantum-entanglement-action-at-a-distance.html
[28]: T. Garlinghouse, 2023, and 4:39 P.m, ‘Physicists ‘entangle’ individual molecules for the first time, bringing about a new platform for quantum science’, Sept. 05, 2025. https://www.princeton.edu/news/2023/12/08/physicists-entangle-individual-molecules-first-time-hastening-possibilities-quantum
[29]: Caltech Science Exchange, ‘What Is Quantum Entanglement? Quantum Entanglement Explained in Simple Terms - Caltech Science Exchange’, 2025. https://scienceexchange.caltech.edu/topics/quantum-science-explained/entanglement
[30]: A. Muller, ‘What is quantum entanglement? A physicist explains Einstein’s ‘spooky action at a distance,’’ Astronomy Magazine, Oct. 07, 2022. https://www.astronomy.com/science/what-is-quantum-entanglement-a-physicist-explains-einsteins-spooky-action-at-a-distance/
[31]: Wikipedia, ‘Quantum entanglement’, Wikipedia. Aug. 10, 2025. https://en.wikipedia.org/w/index.php?title=Quantum_entanglement&oldid=1305202659.
[32]: L. Tse, ‘Superposition and entanglement flee the quantum nest’, Physics World, May 30, 2022. https://physicsworld.com/a/superposition-and-entanglement-flee-the-quantum-nest/
[33]: D. Orf, ‘Quantum Entanglement in Your Brain Is What Generates Consciousness, Radical Study Suggests’, Yahoo News, July 10, 2025. https://www.yahoo.com/news/quantum-entanglement-brain-generates-consciousness-171000903.html
[34]: M. Swayne, ‘2025 Expert Quantum Predictions — Quantum Computing’, The Quantum Insider, Dec. 31, 2024. https://thequantuminsider.com/2024/12/31/2025-expert-quantum-predictions-quantum-computing/
[35]: IQM, ‘Your Guide to Quantum AI - The future of computing?’, Apr. 16, 2025. https://meetiqm.com/blog/quantum-ai-the-future-of-computing-or-just-hype/
[36]: Quantinuum, ‘Quantum Computers Will Make AI Better’, Jan. 22, 2025. https://www.quantinuum.com/blog/quantum-computers-will-make-ai-better
[37]: R. D. Caballar, ‘What Is Neuromorphic Computing? | IBM’, 2025. https://www.ibm.com/think/topics/neuromorphic-computing
[38]: Yann LeCun: We Won’t Reach AGI By Scaling Up LLMS, (May 30, 2025). [Video]:. https://www.youtube.com/watch?v=4__gg83s_Do.
[39]: D. Hassabis, D. Kumaran, C. Summerfield, and M. Botvinick, ‘Neuroscience-Inspired Artificial Intelligence’, Neuron, vol. 95, no. 2, pp. 245–258, July 2017, https://www.doi.org/10/gbp987
[40]: S. Makin, ‘Brain Scientists Finally Discover the Glue that Makes Memories Stick for a Lifetime | Scientific American’, Aug. 28, 2024. https://www.scientificamerican.com/article/brain-scientists-finally-discover-the-glue-that-makes-memories-stick-for-a/
[41]: P. Tsokas et al., ‘KIBRA anchoring the action of PKMζ maintains the persistence of memory’, Science Advances, vol. 10, no. 26, p. eadl0030, June 2024, https://www.doi.org/10/g8p9ns
[42]: Y. Pinto et al., ‘Split brain: divided perception but undivided consciousness’, Brain, vol. 140, no. 5, pp. 1231–1237, May 2017, https://www.doi.org/10/f947x3
[43]: M. Masi, ‘No Consciousness? No Meaning (and no AGI)!’, Qeios, July 2025, https://www.doi.org/10/g9wtbq
[44]: R. L. Kuhn, ‘A landscape of consciousness: Toward a taxonomy of explanations and implications’, Progress in Biophysics and Molecular Biology, vol. 190, pp. 28–169, Aug. 2024, https://www.doi.org/10/g85cqw
[45]: R. Qin, C. Zhou, and M. He, ‘A comprehensive taxonomy of machine consciousness’, Information Fusion, vol. 119, p. 102994, July 2025, https://www.doi.org/10/g9wtbw
[46]: X. Luo et al., ‘Large language models surpass human experts in predicting neuroscience results’, Nat Hum Behav, vol. 9, no. 2, pp. 305–315, Feb. 2025, https://www.doi.org/10/g8r7n5
[47]: K. Grace, J. Salvatier, A. Dafoe, B. Zhang, and O. Evans, ‘When Will AI Exceed Human Performance? Evidence from AI Experts.’ arXiv, May 03, 2018. https://www.doi.org/10.48550/arXiv.1705.08807
[48]: M. Shin, J. Kim, B. van Opheusden, and T. L. Griffiths, ‘Superhuman Artificial Intelligence Can Improve Human Decision Making by Increasing Novelty’, Proc. Natl. Acad. Sci. U.S.A., vol. 120, no. 12, p. e2214840120, Mar. 2023, https://www.doi.org/10/grxdg4
[49]: R. Marozau, ‘The Role of Quantum Information in Artificial Cognitive Consciousness’, May 2025, https://www.authorea.com/users/875987/articles/1273633-the-role-of-quantum-information-in-artificial-cognitive-consciousness.
[50]: M. Swayne, ‘Is Consciousness Research The Next Big Quantum Use Case?’, The Quantum Insider, Jan. 11, 2025. https://thequantuminsider.com/2025/01/11/is-consciousness-research-the-next-big-quantum-use-case/
[51]: T. Goncalves, ‘Consciousness as a Measuring Interface: Hilbert Space, AI, and the Reframing of Reality.’ Social Science Research Network, Rochester, NY, May 26, 2025. https://www.doi.org/10.2139/ssrn.5269521
[52]: Microsoft, ‘Dirac Notation in Quantum Computing - Azure Quantum’, Mar. 19, 2025. https://learn.microsoft.com/en-us/azure/quantum/concepts-dirac-notation
[53]: Wikipedia, ‘Bra–ket notation’, Wikipedia. May 10, 2025. https://en.wikipedia.org/w/index.php?title=Bra%E2%80%93ket_notation&oldid=1289704087.
[54]: D. Tordera, ‘What is the longest time a qubit has survived with 0.9999 fidelity?’, Quantum Computing Stack Exchange, Dec. 23, 2018. https://quantumcomputing.stackexchange.com/q/1687
[55]: P. Wang et al., ‘Single ion qubit with estimated coherence time exceeding one hour’, Nat Commun, vol. 12, no. 1, p. 233, Jan. 2021, https://www.doi.org/10/gkrbm5
[56]: J. Morgan, ‘Quantum memory ‘world record’ smashed’, BBC News, Nov. 15, 2013. https://www.bbc.com/news/science-environment-24934786.
[57]: X. Zhou et al., ‘Electron charge qubits with 0.1 millisecond coherence time.’ arXiv, Feb. 19, 2023. https://www.doi.org/10.48550/arXiv.2210.12337
[58]: A. Yamauchi et al., ‘Room-temperature quantum coherence of entangled multiexcitons in a metal-organic framework’, Science Advances, vol. 10, no. 1, p. eadi3147, Jan. 2024, https://www.doi.org/10/g9wtcf
[59]: J. A. Montanez-Barrera, K. Michielsen, and D. E. B. Neira, ‘Evaluating the performance of quantum processing units at large width and depth.’ arXiv, May 28, 2025. https://www.doi.org/10.48550/arXiv.2502.06471
[60]: M. Swayne, ‘Quantum Computer Benchmark Shows Quantinuum Retains Coherence at Record Scale’, The Quantum Insider, Mar. 25, 2025. https://thequantuminsider.com/2025/03/25/quantum-computer-benchmark-shows-quantinuum-retains-coherence-at-record-scale/
[61]: SpinQ, ‘What Is Qubit Coherence Time and Why It Matters | SpinQ’, May 03, 2025. https://www.spinquanta.com/news-detail/what-is-qubit-coherence-time-and-why-it-matters
[62]: SpinQ, ‘Qubit Coherence Time: A Critical Factor in Quantum Computing | SpinQ’, July 22, 2025. https://www.spinquanta.com/news-detail/qubit-coherence-time-a-critical-factor-in-quantum-computing
[63]: IBM, ‘IBM Quantum Business Foundations - Overview’, IBM Quantum Learning, 2025. https://quantum.cloud.ibm.com/learning/en/courses/quantum.cloud.ibm.com/learning/en/courses/quantum-business-foundations/quantum-technology
[64]: Introduction to Qiskit | Coding with Qiskit 1.x | Programming on Quantum Computers, (2025). [Video]:. https://www.youtube.com/watch?v=Tk9LOL9--Y4.
[65]: IBM, ‘IBM Quantum Documentation’, IBM Quantum Documentation, 2025. https://quantum.cloud.ibm.com/docs/quantum.cloud.ibm.com/docs/en
[66]: A. Freireich, ‘Does Music Really Relate to Math?’, Teachers Who Tutor | LA, Mar. 06, 2023. https://lateacherswhotutor.com/2023/03/06/does-music-really-relate-to-math/
[67]: L. Azaryahu, I. Ariel, and R. Leikin, ‘Interplay between music and mathematics in the eyes of the beholder: focusing on differing types of expertise’, Humanit Soc Sci Commun, vol. 11, no. 1, p. 1153, Sept. 2024, https://www.doi.org/10/g9tdfp
[68]: IBM, ‘IBM - Running quantum circuits’, IBM Quantum Learning, 2025. https://quantum.cloud.ibm.com/learning/en/courses/quantum-computing-in-practice/quantum.cloud.ibm.com/learning/en/courses/quantum-computing-in-practice/running-quantum-circuits
[69]: M. Swayne, ‘Taking Advantage: Researchers Offer a Measured Path Toward Quantum Advantage’, The Quantum Insider, July 23, 2025. https://thequantuminsider.com/2025/07/23/taking-advantage-researchers-offer-a-measured-path-toward-quantum-advantage/
[70]: Z. Liu et al., ‘Efficiently Verifiable Quantum Advantage on Near-Term Analog Quantum Simulators’, PRX Quantum, vol. 6, no. 1, p. 010341, Mar. 2025, https://www.doi.org/10/g9tdd7
[71]: Simons Institute, ‘Verifiable quantum advantage: old and new ideas’, July 08, 2025. https://simons.berkeley.edu/talks/alexandru-gheorghiu-ibm-quantum-2025-07-08
[72]: Wikipedia, ‘Bernstein–Vazirani algorithm’, Wikipedia. July 21, 2025. https://en.wikipedia.org/w/index.php?title=Bernstein%E2%80%93Vazirani_algorithm&oldid=1301706493.
[73]: H.-Y. Huang, S. Choi, J. R. McClean, and J. Preskill, ‘The vast world of quantum advantage.’ arXiv, Aug. 07, 2025. https://www.doi.org/10.48550/arXiv.2508.05720
[74]: H. Hannan, ‘The Bernstein-Vazirani Algorithm: A Clear Explanation of How It Works - Quantum Positioned’, Sept. 06, 2023. https://quantumpositioned.com/bernstein-vazirani-algorithm/, https://quantumpositioned.com/bernstein-vazirani-algorithm/
[75]: A. R. Khan, B. Rizwan, and F. Hassan, ‘Bernstein-Vazirani Algorithm’, May 2023, https://physlab.org/wp-content/uploads/2023/05/BernsteinVazirani_23100071_Fin.pdf.
[76]: M. Ivezic, ‘Quantum Parallelism in Quantum Computing: Demystifying the ‘All-at-Once’ Myth’, PostQuantum - Quantum Computing, Quantum Security, PQC, Feb. 05, 2019. https://postquantum.com/quantum-computing/quantum-parallelism/
[77]: M. Ivezic, ‘Quantum Superposition: How Qubits Live in Many States at Once’, PostQuantum - Quantum Computing, Quantum Security, PQC, May 10, 2017. https://postquantum.com/quantum-computing/quantum-superposition/
[78]: K. Groenland, ‘Four myths about quantum computing’, Introduction to Quantum Computing for Business, June 24, 2025. https://koengr.github.io/essentials/myths/
[79]: Wikipedia, ‘Wave function’, Wikipedia. Aug. 15, 2025. https://en.wikipedia.org/w/index.php?title=Wave_function&oldid=1306000910.
[80]: Wikipedia, ‘Wave–particle duality’, Wikipedia. Aug. 06, 2025. https://en.wikipedia.org/w/index.php?title=Wave%E2%80%93particle_duality&oldid=1304485903.
[81]: M. Ivezic, ‘How Quantum Could Break Through Amdahl’s Law and Computing’s Limits’, PostQuantum - Quantum Computing, Quantum Security, PQC, Mar. 15, 2025. https://postquantum.com/quantum-computing/quantum-amdahls-law/
[82]: SpinQ, ‘Quantum Computing News: ICQE 2025 & Latest Quantum Research | SpinQ’, June 17, 2025. https://www.spinquanta.com/news-detail/latest-quantum-computing-news-and-quantum-research
[83]: Wikipedia, ‘Two-state quantum system’, Wikipedia. Aug. 14, 2025. https://en.wikipedia.org/w/index.php?title=Two-state_quantum_system&oldid=1305879242.
[84]: CELLFRAME, ‘Q-Bits Explained — Part 1’, Aug. 13, 2018. https://cellframe.net/blog/q-bits-explained-part-1/
[85]: Wikipedia, ‘Qutrit’, Wikipedia. Sept. 07, 2025. https://en.wikipedia.org/w/index.php?title=Qutrit&oldid=1309976877.
[86]: JOE, ‘Is there a finite number to the possible state of electron in an atom?’, Physics Stack Exchange, Aug. 09, 2018. https://physics.stackexchange.com/q/421838
[87]: S. Carroll, ‘Why the Many-Worlds Formulation of Quantum Mechanics Is Probably Correct – Sean Carroll’, June 30, 2014. https://www.preposterousuniverse.com/blog/2014/06/30/why-the-many-worlds-formulation-of-quantum-mechanics-is-probably-correct/
[88]: EBSCO, ‘Orthogonal Functions And Expansions | EBSCO Research Starters’, 2022. https://www.ebsco.com/research-starters/mathematics/orthogonal-functions-and-expansions
[89]: Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, ‘Qudits and High-Dimensional Quantum Computing’, Front. Phys., vol. 8, Nov. 2020, https://www.doi.org/10/ghptsj
[90]: N. Goss et al., ‘Extending the computational reach of a superconducting qutrit processor’, npj Quantum Inf, vol. 10, no. 1, p. 101, Oct. 2024, https://www.doi.org/10/g9tc69
[91]: Quantum News, ‘Exploring Qudit-Based Quantum Gates: Advantages Over Qubit Systems’, Apr. 25, 2025. https://quantumzeitgeist.com/exploring-qudit-based-quantum-gates-advantages-over-qubit-systems/
[92]: B. L. Brock et al., ‘Quantum error correction of qudits beyond break-even’, Nature, vol. 641, no. 8063, pp. 612–618, 2025, https://www.doi.org/10/g9kgzp
[93]: M. Oliveau, ‘Groundbreaking Qudits simulate complex interactions between matter and force fields’, The Brighter Side of News, May 27, 2025. https://www.thebrighterside.news/post/groundbreaking-qudits-simulate-complex-interactions-between-matter-and-force-fields/
[94]: T. Nguyen, ‘Total Number of Synapses in the Adult Human Neocortex’, UJMM: One + Two, vol. 3, no. 1, May 2013, https://www.doi.org/10.5038/2326-3652.3.1.26
[95]: R. Tompa PhD, ‘Why is the human brain so difficult to understand? We asked 4 neuroscientists.’, Allen Institute, Apr. 21, 2022. https://alleninstitute.org/news/why-is-the-human-brain-so-difficult-to-understand-we-asked-4-neuroscientists/
[96]: Mathematics for Quantum Physics, ‘Vector spaces in quantum mechanics - Mathematics for Quantum Physics’, Sept. 07, 2025. https://mathforquantum.quantumtinkerer.tudelft.nl/4_vector_spaces_QM/#41-dirac-notation-and-hilbert-spaces
[97]: BBC, ‘Questions and answers about E=mc2 and the atomic bomb.’, Dec. 02, 2014. https://www.bbc.co.uk/sn/tvradio/programmes/horizon/einstein_equation_qa.shtml
[98]: K. Bourzac, ‘June/July 1925: Werner Heisenberg pioneers quantum mechanics’, July 01, 2025. https://www.aps.org/apsnews/2025/07/werner-heisenberg-pioneers-quantum-mechanics
[99]: D. Lindley, Uncertainty - einstein, heisenberg, bohr, and the struggle for the soul of. 2008.
[100]: C. Rovelli, Helgoland: The Sunday Times bestseller. Dublin: Penguin, 2021.
[101]: Priceonomics, ‘Leó Szilárd: A Forgotten Father of the Atomic Bomb’, Priceonomics, Aug. 06, 2015. https://priceonomics.com/leo-szilard-a-forgotten-father-of-the-atomic-bomb/
[102]: A. Jogalekar, ‘Leo Szil rd, a traffic light and a slice of nuclear history’, Scientific American, Feb. 12, 2013. https://www.scientificamerican.com/blog/the-curious-wavefunction/leo-szilard-a-traffic-light-and-a-slice-of-nuclear-history/
[103]: A. Einstein, B. Podolsky, and N. Rosen, ‘Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?’ July 24, 2009. https://cds.cern.ch/record/405662/files/PhysRev.47.777.pdf.
[104]: Vers Libre, ‘What is Waka Poetry? - Vers Libre’, May 23, 2024. https://verslibre.co.uk/haiku/what-is-waka/
[105]: Temca, ‘Kinkai wakashū 180 | Waka Poetry’, Apr. 05, 2025. https://www.wakapoetry.net/kinkai-wakashu-180/
Comments ()