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Standard quantum mechanics employs complex Hilbert spaces, but whether complex numbers are fundamental or merely convenient has long been debated. For decades, real-valued equivalents were considered mathematically possible but cumbersome.…
Quantum state tomography, the ability to deduce the density matrix of a quantum system from measured data, is of fundamental importance for the verification of present and future quantum devices. It has been realized in systems with few…
Quantum computers require quantum logic, something fundamentally different to classical Boolean logic. This difference leads to a greater efficiency of quantum computation over its classical counter-part. In this review we explain the basic…
Two models of computer, a quantum and a classical "chemical machine" designed to compute the relevant part of Shor's factoring algorithm are discussed. The comparison shows that the basic quantum features believed to be responsible for the…
Quantum computing is a growing field where the information is processed by two-levels quantum states known as qubits. Current physical realizations of qubits require a careful calibration, composed by different experiments, due to noise and…
Quantum machine learning (QML) seeks to exploit the intrinsic properties of quantum mechanical systems, including superposition, coherence, and quantum entanglement for classical data processing. However, due to the exponential growth of…
The quantum marginal problem consists in deciding whether a given set of marginal reductions is compatible with the existence of a global quantum state or not. In this work, we formulate the problem from the perspective of dynamical systems…
Quantum phase estimation is one of the key algorithms in the field of quantum computing, but up until now, only approximate expressions have been derived for the probability of error. We revisit these derivations, and find that by ensuring…
Recently Shor showed how to perform fault tolerant quantum computation when the error probability is logarithmically small. We improve this bound and describe fault tolerant quantum computation when the error probability is smaller than…
The notion of a macroscopic quantum state must be pinned down in order to assess how well experiments probe the large-scale limits of quantum mechanics. However, the issue of quantifying so-called quantum macroscopicity is fraught with…
Quantum computing promises to solve problems previously deemed infeasible. However, high error rates necessitate quantum error correction for practical applications. Seminal experiments with zoned neutral atom architectures have shown…
We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions…
Quantum algorithms for scientific computing require modules implementing fundamental functions, such as the square root, the logarithm, and others. We require algorithms that have a well-controlled numerical error, that are uniformly…
Almost all public secure communication relies on the inability to factor large numbers. There is no known analytic or classical numeric method to rapidly factor large numbers. Shor[1] has shown that a quantum computer can factor numbers in…
Lectures on quantum computing. Contents: Algorithms. Quantum circuits. Quantum Fourier transform. Elements of number theory. Modular exponentiation. Shor`s algorithm for finding the order. Computational complexity of Schor`s algorithm.…
Quantum theory has evolved from a set of provisional rules to an indispensable framework that underlies much of modern technology and infrastructure. Yet, after a century, Born's probability postulate remains at odds with the theory's…
Since Harrow, Hassidim, and Lloyd (2009) showed that a system of linear equations with $N$ variables and condition number $\kappa$ can be solved on a quantum computer in $\operatorname{poly}(\log(N), \kappa)$ time, exponentially faster than…
We report on the current state of factoring integers on both digital and analog quantum computers. For digital quantum computers, we study the effect of errors for which one can formally prove that Shor's factoring algorithm fails. For…
Properties of Shor's algorithm and the related period-finding algorithm could serve as benchmarks for the operation of a quantum computer. Distinctive universal behaviour is expected for the probability for success of the period-finding…
Quantum computing algorithms require that the quantum register be initially present in a superposition state. To achieve this, we consider the practical problem of creating a coherent superposition state of several qubits. Owing to…