Cbits and Qbits

Unlike classical digital computers in which a bit can represent either 1 or 0 at any time, quantum computers use a two-level system, i.e., a qubit, to implement logical operations based on quantum mechanical laws, which can represent both values at once. Owing to the superposition property of qubits, quantum computers have natural parallel processing advantages and thus have potential to exceed the computational efficiency of classical computers for particular tasks. Quantum logic gates are the generalization of classical logic gates in computational networks. It has been proved that two-qubit quantum gates together with one-qubit quantum gates are adequate for constructing networks with any possible quantum computational property. Directional couplers are the most critical elementsfor constructing the quantum gates. In recent years, photonic quantum technologies have emerged as a promising experimental platform for quantum computing. Single photons have robust noise resistance, long coherence time, high transmission speed and great compatibility with other systems. They can be easily manipulated and encoded in any of several degrees of freedom, for example, polarization, path, spatial mode or time bin. Optical waveguide technology enables the realizing of complex optical schemes comprised of many elements with desired scalability, stability and miniaturization. Femtosecond laser direct writing of waveguide has been adopted as a powerful tool for integrated quantum photonics with characteristics of rapidness, cost-effectiveness, mask-less and single-step process. In particular, it has the ability to build arbitrary three-dimensional circuits directly inside bulk materials, which is impossible to achieve with conventional lithography. In this article we review the femtosecond laser writing and quantum characterization of directional coupler and important one-qubit and two-qubit optical quantum logic gates, such as Hadamard gate, Pauli-X gate, controlled-NOT gate, and controlled-Phase gate. The qubits in these gates are usually encoded through optical paths or polarizations of photons. The key to the realization of polarization-encoded one-qubit gates is to achieve flexible wave-plate operations, which is described in detail. Controlled-NOT gate and controlled-phase gate are the most crucial two-qubit gates in the linear optics computation and sometimes they can be converted into each other by adding some one-qubit gates or special superposition states. Many different kinds of waveguide circuits have been used to implement these two-qubit gates. The outlook and challenges for the femtosecond laser writing of three-qubit gates, such as Toffoli gate and Fredkin gate, are briefly introduced.

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).

Security of QKD is guaranteed by the quantum mechanics laws rather than unproven assumptions employed in computational cryptography. Unfortunately, the secret-key rates are way too low and transmission distances are limited. The post-quantum cryptography (PQC) is proposed as an alternative to QKD. However, the PQC protocols are based on conjecture that there are no polynomial time algorithms to break the PQC protocols. To overcome key challenges of both post-quantum cryptography and QKD, we propose to use the QKD only in initialization stage to set-up corresponding cybersecurity protocols. The proposed concept is applied to both computational security and PQC protocols. The proposed QKD-enhanced cybersecurity protocols are tolerant to attacks initiated by quantum computers.

Quantum computing is aninterdisciplinary field that lies at the intersection of mathematics, quantum physics, and computer science, and finds applications in areas including optimization, machine learning, and simulation of chemical, physical, and biological systems. It has the potential to help solve problems that so far have no satisfying method solving them, and to provide significant speedup to solutions when compared with their best classical approaches. In turn, quantum computing may allow us to solve problems for inputs that so far are deemed practically intractable. With the computational power of quantum computers and the proliferation of quantum development kits, quantum computing is anticipated to become mainstream, and the demand for a skilled workforce in quantum computing is expected to increase significantly. Therefore, quantum computing education is ramping up. This article describes our experiences in designing and delivering quantum computing workshops for youth (Grades 9–12). We introduce students to the world of quantum computing in innovative ways, such as newly designed unplugged activities for teaching basic quantum computing concepts. We also take a programmatic approach and introduce students to the IBM Quantum Experience using Qiskit and Jupyter notebooks. Our contributions are as follows. First, we present creative ways to teach quantum computing to youth with little or no experience in science, technology, engineering, and mathematics areas; second, we discuss diversity and highlight various pathways into quantum computing from quantum software to quantum hardware; and third, we discuss the design and delivery of online and in-person motivational, introductory, and advanced workshops for youth.

Quantum mechanical systems exhibit particle wave duality property. This duality property has been exploited for information processing. A duality quantum computer is a quantum computer on the move and passing through a multi‐slits. It offers quantum wave divider and quantum wave combiner operations in addition to those allowed in an ordinary quantum computer. It has been shown that all linear bounded operators can be realized in a duality quantum computer, and a duality quantum computer with n qubits and d‐slits can be realized in an ordinary quantum computer with n qubits and a qudit in the so‐called duality quantum computing mode. The quantum particle‐wave duality can be used in providing secure communication. In this paper, we will review duality quantum computing and duality quantum key distribution.

The purpose of the work is to increase the chi-square test power by analyzing the spectrum of output states of this test discrete form. The refusal from a hypothesis of a spectrum continuity of a chi-square molecule output states allows you to observe 22 spectral lines of different probability amplitudes on a sample of 11 experiments. It is proposed to obtain a mode of support for the quantum superposition of the output states of the chi-square molecule through the random selection of 11 experiments from the general population of 21 experiments. This makes it possible to obtain about 200,000 different states of the molecule, which is quite sufficient to estimate the 8-level states of the probability amplitude of 22 output spectral lines. In the case of the histogram with 6 intervals, the hypothesis of a continuous spectrum of the chi-square test for a sample of 11 experiments does not allow us to distinguish between normal and uniform laws of data distribution with a confidence probability acceptable for practice. The situation changes dramatically if the spectrum discrete nature of the chi-square molecule states is taken into account. It is possible to obtain significantly different digital portraits of the normal and uniform data distribution. Their differences are about 40 bits as per the Hamming distance. The chi-square molecule can be interpreted as a certain square neuron with a Gammerstein structure. Simulation of Schrödinger equations on an ordinary computer has exponential computational complexity in relation to the number of electrons (degrees of freedom or experiments). For this reason, the reproduction of quantum computations on 22 qubits and more within the framework of quantum mechanics on an ordinary computer is technically difficult to perform. A completely different situation arises with the support of quantum superposition of the chi-square molecule. The support of quantum superposition of the chi-square molecule has linear computational complexity. As a consequence, it is possible to implement 22-qubit, 222-qubit, 2222-qubits, and higher quantum computations on an ordinary computer.

Reversibility is a fundamental property of microscopic physical systems, implied by the laws of quantum mechanics, which seems to be at odds with the Second Law of Thermodynamics (Schiff 2008; Toffoli and Margolus 1990). Nonreversibility always implies energy dissipation, in practice, in the form of heat. Using reversible cellular automata (CAs) to simulate such systems has caused wide attention since the early days of the investigation of CAs (Toffoli and Margolus 1990; Kari 2005). On the other hand, if a CA is not reversible but reversible over an invariant closed subset, e.g., the limit set (Taaki 2007), it can also be used to describe physical systems locally. In this chapter, (Theorems 11.1, 11.2, and 11.4 were reproduced from Zhang and Zhang (2015) with permission @ 2015 Old City Publishing Inc. Theorems 11.3 and 11.5 were reproduced from Taaki (2007) with permission @ 2007 Old City Publishing Inc.) we present a formal definition to represent this class of generalized reversible CAs, and investigate some of their topological properties. We refer the reader to Zhang and Zhang (2015), Taaki (2007) for further reading. Other variants of generalized reversibility can be found in Castillo-Ramirez and Gadouleau (2017).

In this paper, we present the experimental realization of multi-qubit gates $% \Lambda_n(not) $ in macroscopic ensemble of three-qubit and four-qubit molecules. Instead of depending heavily on the two-bit universal gate, which served as the basic quantum operation in quantum computing, we use pulses of well-defined frequency and length that simultaneously apply to all qubits in a quantum register. It appears that this method is experimentally convenient when this procedure is extended to more qubits on some quantum computation, and it can also be used in other physical systems.

Adiabatic quantum computing (AQC) is a model of computation that uses quantum mechanical processes operating under adiabatic conditions. As a form of universal quantum computation, AQC employs the principles of superposition, tunneling, and entanglement that manifest in quantum physical systems. The AQC model of quantum computing is distinguished by the use of dynamical evolution that is slow with respect to the time and energy scales of the underlying physical systems. This adiabatic condition enforces the promise that the quantum computational state will remain well-defined and controllable thus enabling the development of new algorithmic approaches. Several notable algorithms developed within the AQC model include methods for solving unstructured search and combinatorial optimization problems. In an idealized setting, the asymptotic complexity analyses of these algorithms indicate computational speed-ups may be possible relative to state-of-the-art conventional methods. However, the presence of non-ideal conditions, including non-adiabatic dynamics, residual thermal excitations, and physical noise complicate the assessment of the potential computational performance. A relaxation of the adiabatic condition is captured in the complementary computational heuristic of quantum annealing, which accommodates physical systems operating at finite temperature and in open environments. While quantum annealing (QA) provides a more accurate model for the behavior of actual quantum physical systems, the possibility of non-adiabatic effects obscures a clear separation with conventional computing complexity. A series of technological advances in the control of quantum physical systems have enabled experimental AQC and QA. Prominent examples include demonstrations using superconducting electronics, which encode quantum information in the magnetic flux induced by a weak current operating at cryogenic temperatures. A family of devices developed specifically for unconstrained optimization problems has been applied to solve problems in specific domains including logistics, finance, material science, machine learning, and numerical analysis. An accompanying infrastructure has also developed to support these experimental demonstrations and to enable access of a broader community of users. Although AQC is most commonly applied in superconducting technologies, alternative approaches include optically trapped neutral atoms and ion-trap systems. The significant progress in the understanding of AQC has revealed several open topics that continue to motivate research into this model of quantum computation. Foremost is the development of methods for fault-tolerant operation that will ensure the scalability of AQC for solving large-scale problems. In addition, unequivocal experimental demonstrations that differentiate the computational power of AQC and its variants from conventional computing approaches are needed. This will also require advances in the fabrication and control of quantum physical systems under the adiabatic restrictions.

Quantum information theory has provided the formal framework for describing how information is stored, transmitted, and transformed in physical quantum systems [Renes, 2022; Tomamichel, 2015; Wilde, 2013]. Its entropic formulations underpin our understanding of quantum computation, communication, and cryptography. Yet this theory traditionally treats all quantum operations as freely available, ignoring computational restrictions. In practice, however, any manipulation of quantum information must be performed by devices of bounded complexity and runtime. Capturing such realistic constraints requires extending quantum information theory to include computational efficiency as a fundamental component. This work takes a first step toward building a computational version of quantum information theory, one that treats efficiency as part of the theory itself. The goal is to understand how the behavior of quantum information changes when the parties involved can only perform computationally efficient operations. This approach bridges the abstract, ideal setting of quantum information theory with the practical limitations of real quantum devices, offering a means to study information processing under realistic resource constraints. At the center of this work are two new quantities: the quantum computational min-entropy and the quantum computational max-entropy. These entropies extend standard quantum entropies by explicitly limiting the computational power of the observer or adversary. The quantum computational min-entropy captures how unpredictable a quantum system A remains to an observer holding system B, when that observer is restricted to quantum circuits of bounded size. Formally, for a bipartite state ρ_{AB}, we define {H^c}^s_{min}(A|B)_{ρ} ≔ -log d_A max_{ℰ^s_{B→A'}} F((𝕀_A ⊗ ℰ^s)(ρ_{AB}),|Φ_{AA'}⟩⟨Φ_{AA'}|) , where the maximization is over quantum channels that can be implemented by circuits of size at most s, and F denotes fidelity with a maximally entangled state. In the classical setting, the min-entropy can be expressed through the maximal probability of correctly guessing a random variable given some side-information. In the fully quantum setting, this idea extends to uncertainty about quantum information [König et al., 2009], quantifying how well one system can be inferred from another using local quantum operations. Our definition generalizes this operational viewpoint by restricting the computational power of the observer to efficient quantum circuits. This definition extends the operational meaning of the information-theoretic quantum min-entropy [König et al., 2009] by incorporating computational constraints, and it provides the fully quantum counterpart of the classical unpredictability entropy [Hsiao et al., 2007]. We establish fundamental properties for the computational min-entropy, including monotonicity in the circuit size and smoothing parameters, efficient data-processing inequalities, and fully quantum leakage and purification chain rules, which were left as open questions in earlier definitions of quantum computational entropies [Yi-Hsiu Chen et al., 2017; Munson et al., 2025]. For classical–quantum states, it coincides with the previously defined quantum computational unpredictability entropy [Noam Avidan and Rotem Arnon, 2025], showing that the new definition correctly generalizes known results. We also introduce the quantum computational max-entropy through a duality relation [Tomamichel et al., 2010] with the min-entropy using a fixed purification. Finally, we prove unconditional separations between the computational and information-theoretic entropies, demonstrating that computational restrictions can fundamentally alter entropic behavior even for simple states. These results establish the fundamental mathematical framework for studying quantum information within realistic computational constraints. By integrating efficiency directly into entropic quantities, they open the door to a fully developed computational quantum information theory that parallels its information-theoretic counterpart. Such a framework provides the foundation for analyzing cryptographic security against computationally bounded quantum adversaries [Noam Avidan and Rotem Arnon, 2025] and the limits of efficient quantum state manipulation. More broadly, it suggests that many core notions in quantum information theory may have refined computational analogues yet to be explored.

Certain physical systems that one might consider for fault-tolerant quantum computing where qubits do not readily interact, for instance photons, are better suited for measurement-based quantum-computational protocols. Here we propose a measurement-based model for universal quantum computation that simulates the braiding and fusion of Majorana modes. To derive our model we develop a general framework that maps any scheme of fault-tolerant quantum computation with stabilizer codes into the measurement-based picture. As such, our framework gives an explicit way of producing fault-tolerant models of universal quantum computation with linear optics using protocols developed using the stabilizer formalism. Given the remarkable fault-tolerant properties that Majorana modes promise, the main example we present offers a robust and resource efficient proposal for photonic quantum computation.

The miracle of quantum error correction Correcting errors might sound like a dreary practical problem, of little aesthetic or conceptual interest. But aside from being of crucial importance for the feasibility of quantum computation, it is also one of the most beautiful and surprising parts of the subject. The surprise is that error correction is possible at all, since the only way to detect errors is to make measurements, but measurement gates disruptively alter the states of the measured Qbits, apparently making things even worse. “Quantum error correction” would seem to be an oxymoron. The beauty lies in the ingenious ways that people have found to get around this apparently insuperable obstacle. The discovery in 1995 of quantum error correction by Peter Shor and, independently, Andrew Steane had an enormous impact on the prospects for actual quantum computation. It changed the dream of building a quantum computer capable of useful computation from a clearly unattainable vision to a program that poses an enormous but not necessarily insuperable technological challenge. Error correction is not a major issue for classical computation. In a classical computer the physical systems that embody individual bits – the Cbits – are immense on the atomic scale. The two states of a Cbit representing 0 and 1 are so grossly different that the probability is infinitesimal for flipping from one to the other as a result of thermal fluctuations, mechanical vibrations, or other irrelevant extraneous interactions.

Quantum computers have made extraordinary progress over the past decade, and significant milestones have been achieved along the path of pursuing universal fault-tolerant quantum computers. Quantum advantage, the tipping point heralding the quantum era, has been accomplished along with several waves of breakthroughs. Quantum hardware has become more integrated and architectural compared to its toddler days. The controlling precision of various physical systems is pushed beyond the fault-tolerant threshold. Meanwhile, quantum computation research has established a new norm by embracing industrialization and commercialization. The joint power of governments, private investors, and tech companies has significantly shaped a new vibrant environment that accelerates the development of this field, now at the beginning of the noisy intermediate-scale quantum era. Here, we first discuss the progress achieved in the field of quantum computation by reviewing the most important algorithms and advances in the most promising technical routes, and then summarizing the next-stage challenges. Furthermore, we illustrate our confidence that solid foundations have been built for the fault-tolerant quantum computer and our optimism that the emergence of quantum killer applications essential for human society shall happen in the future.

A quantum machine is a human-made device whose collective motion follows the laws of quantum mechanics. Quantum machine learning (QML) is machine learning for quantum computers. The availability of quantum processors has led to practical applications of QML algorithms in the remote sensing field. Quantum machines can learn from fewer data than non-quantum machines, but because of their low processing speed, quantum machines cannot be applied to an image that has hundreds of thousands of pixels. Researchers around the world are exploring applications for QML and in this work, it is applied for pseudo-labelling of samples. Here, a PRISMA (PRecursore IperSpettrale della Missione Applicativa) hyperspectral dataset is prepared by quantum-based pseudo-labelling and 11 different machine learning algorithms viz., support vector machine (SVM), K-nearest neighbour (KNN), random forest (RF), light gradient boosting machine (LGBM), XGBoost, support vector classifier (SVC) + decision tree (DT), RF + SVC, RF + DT, XGBoost + SVC, XGBoost + DT, and XGBoost + RF with this dataset are evaluated. An accuracy of 86% was obtained for the classification of pine trees using the hybrid XGBoost + decision tree technique.

Since the pioneering works of Feynman 1) and Deutsch 2) using elements following the laws of quantum mechanics, interest has been growing in the area of so-called quantum computation. Among other things, a quantum teleportation scheme of the type proposed by Bennett et a1. 3) is of great interest from the measurement theoretical point of view. The original teleportation scheme of an unknown state iq'», which consists of the story of Alice and Bob, however, is insufficient due to the unclear treatment of the EPR 4) problem and the assumption of von Neumann's projection postulate. At the same time, the former unclear treatment follows from the latter assumption. This teleportation scheme, based on the existence of non­ local long range correlations between EPR pairs of particles relevant to the notion of wave function collapse, necessarily leads to epistemological discussion on the nature of physical reality. Unfortunately, tentative solutions to this problem always depend on what the authors decide to define as realism. 5 ) Therefore, such philosophical arguments concerning the conceptual aspects of EPR phenomenon are beyond the realm of physics. We shall show that a quantum teleportation can be achieved within the framework of present-day quantum mechanics without resorting to von Neumann's projection postulate. Today, the naive Copenhagen interpretation is criticized for various reasons. For example, in Machida and Namiki's theory of measurement, 5),6) which does not rely upon von Neumann's projection postulate, making use of the statistical operator as the description of the quantum state and introducing the state of the apparatus system, they showed that the resolution of the difficulty concerning the wave function collapse does not consist of a demonstration of the disappearance of other branch waves, but follows from the concept of decoherence. Their theory is widely applicable, even to incomplete measurements. 5) According to these lines of thought, one of the present authors (A. M.) attempted to analyze the process of decoherence. 7) In this paper, reviewing what is a correct understanding of the EPR problem, we intend to show how we can obtain a correct teleportation scheme without resorting to von Neumann's projection postulate. A reasonable explanation of the EPR problem was given by Machida. 8) In order