A computer is imagined in which all the parts are simple two-state objects obeying quantum mechanical laws. A time-independent Hamiltonian for such a system is written and analyzed. The computation is done in logically reversible steps. It is used to study possible limitations on computers due to fundamental physical laws. The ideal machine can run ballistically, coasting through the calculation in a short time TB. If there are imperfections due to irregularities in construction or finite temperatures the forward moving calculation is often reversed or scattered. External work must be done to drag the computer forward. The free energy lost in doing a calculation is kT In2 per logical step divided by turn factors. One is the mean free number of steps per scattering, measuring how perfectly the machine is built. The other is the mean time (divided by TB) that the calculation actually takes, which measures how gently you urge it forward.
A computer is imagined in which all the parts are simple two-state objects obeying quantum mechanical laws. A time-independent Hamiltonian for such a system is written and analyzed. The computation is done in logically reversible steps. It is used to study possible limitations on computers due to fundamental physical laws. The ideal machine can run ballistically, coasting through the calculation in a short time TB. If there are imperfections due to irregularities in construction or finite temperatures the forward moving calculation is often reversed or scattered. External work must be done to drag the computer forward. The free energy lost in doing a calculation is kT In2 per logical step divided by turn factors. One is the mean free number of steps per scattering, measuring how perfectly the machine is built. The other is the mean time (divided by TB) that the calculation actually takes, which measures how gently you urge it forward.
In this chapter, the authors discuss the part in an effort to analyze the physical limitations of computers due to the laws of physics. It is a result of computer science that a universal computer can be made by a suitably complex network of interconnected primitive elements. A more elaborate circuit is a full adder which takes a carry, c , and adds it to the two lines a and b and has an additional line, d, with a 0 input. Calculations can be done with reversible machines containing only reversible primitives. There are many sources of imperfections in the machine, but the first one people would like to consider is the possibility that the coefficients in the couplings, along the program line, are not exactly equal. The switches are represented by diagonal lines, and in boxes put the other matrices that operate on registers such as the NOT b .
In this chapter, the authors discuss a part in an effort to analyze the physical limitations of computers due to the laws of physics. It is a result of computer science that a universal computer can be made by a suitably complex network of interconnected primitive elements. There are, however, a number of questions that the people would like to discuss in more detail such as the question of imperfections. It is analogous to the use of transistors in conventional machines where the people don't properly use all the analog continuum of the behavior of transistors, but just try to run them as saturated on or off digital devices so the logical analysis of the system behavior is easier. Of course the large system obeys quantum mechanics, but it is in interaction with the heat baths and other things that could make it effectively irreversible.
Quantum computing is the delicate art of using a quantum-mechanical system to perform some kind of computation. Such a system differs in a number of significant ways from ordinary computers:• A quantum computer actually does a lot of computations at the same time, each solution is represented by a particular possible state. This is inherent to the physical properties of such a system.• Unless you are sure that you have reached the final (unique) solution, you can not "look" at the "values" of the variables in the computer program that is being run: since all solutions are possible at the same time, observing the values will influence the system and put it into one particular, but otherwise arbitrary, state.• All computations must be reversible, that is, a statement like "a = 5" can only appear in the initialization phase of a program, because the previous value will be lost. On the other hand, a statement like "a = a + 5" is reversible, since you can retrieve the original value.The properties of such a quantum computer make it possible to devise very efficient algorithms. One such algorithm, described by L. Grover (see the references), is searching in an unsorted array of items. With ordinary computers this can not be done more efficiently than in O (N) steps (an exhaustive search: each item is examined in the worst case), with this quantum algorithm, however, the number of steps needed is O (N 1/2 ).Devising quantum algorithms requires a thorough understanding of quantum mechanics, there is no doubt about that. Another question is: do we need quantum computers and special programming languages to program (or perhaps emulate) these algorithms? This article tries to show that, at least for the quantum search algorithm, a classical programming language like Fortran is quite fit (yes, the pun is fortunate).
A computer is a physical device that helps us process information by executing algorithms. An algorithm is a well-defined procedure, with finite description, for realizing an information-processing task. An information-processing task can always be translated into a physical task. When designing complex algorithms and protocols for various information-processing tasks, it is very helpful, perhaps essential, to work with some idealized computing model. However, when studying the true limitations of a computing device, especially for some practical reason, it is important not to forget the relationship between computing and physics. Real computing devices are embodied in a larger and often richer physical reality than is represented by the idealized computing model. Quantum information processing is the result of using the physical reality that quantum theory tells us about for the purposes of performing tasks that were previously thought impossible or infeasible. Devices that perform quantum information processing are known as quantum computers. In this book we examine how quantum computers can be used to solve certain problems more efficiently than can be done with classical computers, and also how this can be done reliably even when there is a possibility for errors to occur. In this first chapter we present some fundamental notions of computation theory and quantum physics that will form the basis for much of what follows. After this brief introduction, we will review the necessary tools from linear algebra in Chapter 2, and detail the framework of quantum mechanics, as relevant to our model of quantum computation, in Chapter 3. In the remainder of the book we examine quantum teleportation, quantum algorithms and quantum error correction in detail. We are often interested in the amount of resources used by a computer to solve a problem, and we refer to this as the complexity of the computation. An important resource for a computer is time. Another resource is space, which refers to the amount of memory used by the computer in performing the computation. We measure the amount of a resource used in a computation for solving a given problem as a function of the length of the input of an instance of that problem.
Common introduction to all the records of this series. The content of this record follows below. Mathematical Materiality (MM) in brief. Why are the laws of physics the same everywhere and at every instant, even though nothing forces matter to obey them? MM proposes an answer: the laws are not rules imposed on matter from the outside; rather, matter itself is made of mathematical relations. And mathematics cannot change: a relation binds two distinct terms everywhere and always. In logic, nothing exists on its own: to exist means to be in relation to another logical entity, and outside this web of relations nothing can exist, not even other universes. This reading already meets experimental measurements at two points. First, the ratio of dark to ordinary matter is read as 3√3 = 5.196; when left free in the fits, Planck and DESI DR2 data give 5.21–5.22 (as registered in record 2.44, https://doi.org/10.5281/zenodo.23329080). Second, the acceleration scale of galaxies is read as a₀ = cH₀/6, the same scale that appears in Verlinde's emergent gravity: the computed value is 1.09–1.18 × 10⁻¹⁰ m/s² against the measured value of 1.20 (record 2.46.1, https://doi.org/10.5281/zenodo.23506655). Both readings were formulated with the measurements already known; the tests to check them against future data are registered in the same records. In the first logical nexus between space and time — which became homogeneous with Einstein — there is a non-local phase, made only of a pure logical distinction: space is not time. Between the two there is no relation of quantitative calculation, only the logical entity they have in common, which is needed to make them exist with respect to each other. This level is not distant: it is present in every phase of reality, in every place and at every instant. It shows in entanglement, where two distant systems remain correlated more strongly than any explanation based on separate properties and free of faster-than-light influences allows, without anything physically travelling between them. In this non-local phase the two distinct terms, space and time, share only one logical element, which we shall call their greatest common divisor (GCD), just as the numbers 3 and 5 have only 1 in common. The local world arises when a relation is counted through this GCD: its multiples and its fractions form a matrix. Seen by an observer inside the system like us, the multiples appear as time and the fractions as space. When that same multiple is directly contained in the matrix of the point of observation, it turns into mass — that is, into concrete matter as we usually understand it. There is therefore a close link between time and mass, in full agreement with Einstein's relativity, which ties mass to spacetime. The observed objects, while being logically autonomous entities in their local relations, enter into relation with the observer through a logical diffraction — that is, a decomposition into fractions — of the observer's matrix. This reading also gives a reason for the limit of the speed of light, which in Einstein's relativity is instead taken as a starting principle. The speed of light is the fixed exchange rate between the multiples and the fractions of the same GCD, meaning how much space a single step of time is worth: exceeding it would mean, for some observer, covering space without counting time, and thus leaving the local world. Logic always starts from a logical relative, and this exchange rate has nothing absolute in itself either: it is a necessary relative which, inside the system, becomes an absolute hypothesis, the same for everyone and impossible to exceed. Outside the system, by contrast, it has no justification at all, because outside there is only non-existence: justifying something requires another term to relate it to, and out there there is none. Asking for a reason from outside is like asking what lies north of the North Pole. The same theoretical framework that reads space, time, mass and the observed objects also admits quantum mechanics. The non-local phase and the mirror-like logical reciprocity between the two ends of the nexus give a reason for the probabilistic behaviour of matter. When the observed object is large enough, its mass fixes it in the matrix of the point of observation: for this reason we see it in one precise place and at one precise instant. When instead the object falls below a certain mass threshold, the principal GCD (that of the point of observation) and the GCD of the diffraction that generates the object are forced to become equal, reflecting each other as in a mirror. At that point the object no longer occupies a single point of the spacetime generated by the observer, but spreads over several possible positions, and we observe it as probability. It is like having to describe the number 10 using only the number 1: the only way is to distribute the 1 over ten positions, one for each unit of the 10. Today we know that even objects made of thousands of atoms behave in this way, and no experiment has yet found the boundary. The idea that the passage from quantum to classical behaviour depends on mass is close to that of Penrose and Diósi; in MM the threshold is the point where the two GCDs, the principal one and that of the local diffraction, become equal. For all these reasons the universe behaves like a simulation without really being one. It follows exact rules, made of counts, everywhere and at every instant, but no machine runs them and no programmer wrote them: a structure of mathematical relations does not need to be computed in order to exist. Tegmark (2008) had already noted that such a simulation need not be run, while Vazza (2025) estimated that running it physically on a machine would require resources that the very laws of the universe do not allow. The work is shared openly so that it can be judged by as many readers as possible: anyone who knows someone interested can pass on this single link, valid for the whole framework (with an overview in English and Italian): https://doi.org/10.5281/zenodo.23535916. The entry page, with the complete list of all records, is https://doi.org/10.5281/zenodo.22667516. To cite the framework: Lanciano, U. (2026). The number 1 in front of the mirror: Mathematical Materiality from the nexus to the cosmos. Zenodo. https://doi.org/10.5281/zenodo.23535916. The records have been deposited since 5 August 2026, each with its own date, and are released under CC BY 4.0: anyone may use them by citing the source. Objections and errors found are welcome at riverantares@gmail.com, since Zenodo has no space for comments. This record contains the four-page summary of the framework, in English and in Italian: the question of why physical laws hold everywhere with nothing to enforce them; the Great Reversal; the non-local phase and its empirical trace in Bell-type experiments; the five postulates with the example of 3 and 5; the matrix; the equals sign as a mirror; a clock made of 3 and 5, whose two means give Einstein's slowing of moving clocks (a known identity: Bondi 1964; Knuth and Bahreyni 2014); and the comparison with two measured numbers. The full overview, with the list of all records, is at DOI 10.5281/zenodo.22667516. The author is not a professional physicist; the framework and its premises are the author's own. The texts were written with the help of AI assistants. There has been no review by human referees.
Quantum mechanics is one of the cornerstones of modern physics. It governs the behavior and the properties of matter in a fundamental way, in particular on the microscopic scale of atoms and molecules. Hence, what we may call a classical computer, i.e., those machines on or under the desktops in our offices together with all their potential descendants, are themselves following the rules of quantum mechanics. However, they are no quantum computers in the sense that all the inside information processing can perfectly be described within classical information theory. In fact, we do not need quantum mechanics in order to explain how the zeros and ones – the bits – inside a classical computer evolve. The reason for this is that the architecture of classical computers does not make use of one of the most fundamental features of quantum mechanics, namely the possibility of superpositions. Throughout the entire processing of any program on a classical computer, each of the involved bits takes on either the value zero or one. Quantum mechanics, however, would in addition allow superpositions of zeros one ones, that is bits – now called qubits (quantum-bits) – which are somehow in the state zero and one at the