In computational molecular science, calculation of electrostatic interactions involving charged atoms - the strongest interactions in condensed phases, is a major bottleneck. We propose a quantum-classical algorithm for fast, yet, accurate computation of the Coulomb electrostatic energy for a system of point charges. The algorithm employs the Ewald method based decomposition of electrostatic energy into several energy terms, of which the "Fourier component" (long-range electrostatics) computed on a quantum device, utilizing the power of Quantum Fourier Transform (QFT). We demonstrate that the algorithm complexity is N log M and that the quantum advantage for a system of point charges in the three-dimensional space is achieved when the number of grid points M^3 exceeds the number of charges N. The numerical error is small <10^-3. The algorithm can be implemented to run the all-atom Molecular Dynamics simulations on a quantum device requiring 15 qubits, thereby expanding the scope of applications of QFT-based methods to computational chemistry and biophysics.

We review recent results on quantum one-way functions, including quantum fingerprinting or quantum hashing (we use these two terms as synonyms even though they have very small difference). This includes the analysis of their properties, different modifications, circuit implementation on an IBM Q platform, as well as on an experimental quantum setup. We discuss computational aspects of quantum hashing, its cryptographic properties and possible usage in communication protocols and algorithms.

In this work, we present a quantum query algorithm for searching a word of length $m$ in an unsorted dictionary of size $n$. The algorithm uses $O(\sqrt{n})$ queries (Grover operators), like previously known algorithms. What is new is that the algorithm is based on the quantum fingerprinting-hashing technique, which (a) provides a first level of amplitude amplification before applying the sequence of Grover amplitude amplification operators and (b) makes the algorithm more efficient in terms of memory use -- it requires $O(\log n + \log m)$ qubits. Note that previously developed algorithms by other researchers without hashing require $O(\log n + m)$ qubits.

The paper considers the problem of finding a given substring in a text. It is known that the complexity of a classical search query in an unordered database is linear in the length of the text and a given substring. At the same time, Grover’s quantum search provides a quadratic speed-up in the complexity of the query and gives the correct result with a high probability. We propose a hybrid classical–quantum algorithm (hybrid random–quantum algorithm, to be more precise) that implements Grover’s search to find a given substring in a text. As expected, the algorithm works (a) with a high probability of obtaining the correct result and (b) with a quadratic query acceleration compared to the classical one. What is new is that our algorithm uses the uniform hash family functions technique. As a result, our algorithm is much more memory efficient (in terms of the number of qubits used) compared to previously known quantum algorithms.

The problem of reliable identification of quantum hash states, which are images of quantum hash functions, is considered. It is shown that this can be done with a high probability for a linear number of O(n/(1-ϵ^2)) experiments when hashing words of length n with ϵ -resistance to collisions. The results of a numerical experiment on the implementation of the procedure are presented, demonstrating the result of the above. We provide such an experiment to simulate quantum hashing for binary words of length 5.

In this article, the properties of quantum hash functions are further explored. Previous findings show that so-called small-bias sets (special subsets of the set of elements of a cyclic group) generate a “phase” quantum hash function. Here, it was proved that they also generate an “amplitude” quantum hash function. Namely, it turned out that constructing small-bias sets while generating amplitude quantum functions yields a well-balanced combination of the cryptographic properties of unidirectionality and collision resistance. As a corollary of the obtained theorem, a general statement about the generation of new amplitude quantum hash functions based on universal hash families and small-bias sets was proved.

The possibilities of generating a contextual query using the quantum Hyperspace Analogue to Language (HAL) model are analyzed. An approach to establishing a semantic connection between words in texts is developed using the generally accepted basic formula for Bell’s parameter as a parameter of connection. A modification of the approach that allows determination of hyponymic isotopy and thus establish a semantic connection between the concepts is proposed. Prospects for using the quantum states of light to implement this approach are discussed.

In the paper, we consider using high-dimensional quantum states (so-called qudits) for the implementation of the quantum hashing technique. The importance of qudits for quantum procedures that implement quantum hashing and protocols based on it follows from two considerations. First of all, d-dimensional qudit describes a d-level quantum system, which can be in a superposition of its d basis states without the need of using highly entangled qubit states. The second consideration is that the d-dimensional qudit can be represented by a single particle. These two properties of qudits are considered in this paper to generalize the notion of quantum hashing. The latter is used to present a variant of a quantum protocol for secret key verification.

A new version of quantum hashing technique is developed wherein a quantum hash is constructed as a sequence of single-photon high-dimensional states (qudits). A proof-of-principle implementation of the high-dimensional quantum hashing protocol using orbital-angular momentum encoding of single photons is implemented. It is shown that the number of qudits decreases with increase of their dimension for an optimal ratio between collision probability and decoding probability of the hash. Thus, increasing dimension of information carriers makes quantum hashing with single photons more efficient.

In this paper, we show the possible development of the technique of quantum hashing, which brings it closer to practical implementation, namely we propose a modified version of the quantum hash function, using the structure of one-photon multidimensional quantum states in basis of the orbital angular momentum.

In the classical hashing theory, collision is a coincidence of the values of a function with different arguments.This paper formulates a quantum analogue of the collision property.A variant of formalization of the concept of quantum function resistant to collisions was proposed.Within the framework of this formalization, the theorem (sufficient condition) on the quantum function that is resistant to collisions was proved.

Quantum hashing is a promising generalization of the cryptographic hashing concept on the quantum domain. In this paper, we construct a quantum hash via a sequence of single-photon states and perform a proof-of-principle experiment using orbital angular momentum (OAM) encoding. We experimentally verify the collision resistance of the quantum hash function depending on the number of qubits in use. Based on these results, we conclude that theoretical estimates are confirmed for different bases of OAM states and the proposed technique can be useful in computational and cryptographic scenarios. The possibility of multiplexing different OAM bases can make this approach even more efficient.

In the paper, we investigate two problems on strings. The first one is the String matching problem, and the second one is the String comparing problem. We provide a quantum algorithm for the String matching problem that uses exponentially less quantum memory than existing ones. The algorithm uses the hashing technique for string matching, quantum parallelism, and ideas of Grover's search algorithm. Using the same ideas, we provide two algorithms for the String comparing problem. These algorithms also use exponentially less quantum memory than existing ones. Additionally, the second algorithm works exponentially faster than the existing one.

Fingerprinting and cryptographic hashing have quite different usages in computer science, but have similar properties. Interpretation of their properties is determined by the area of their usage: fingerprinting methods are methods for constructing efficient randomized and quantum algorithms for computational problems, while hashing methods are one of the central cryptographic primitives. Fingerprinting and hashing methods are being developed from the mid of the previous century, while quantum fingerprinting and quantum hashing have a short history. In the paper we present computational aspects of quantum fingerprinting, discuss cryptographic properties of quantum hashing. We investigate the pre-image resistance of this function and show that it reveals only O(1) bits of information about the input.

In the paper based on the notion of small-biased sets we define the quantum transformation that describes the quantum hash function over the cyclic group. We discuss its similarity to the well-known Quantum Fourier Transform and show possible applications to constructing space-efficient algorithms in various computational scenarios, including two-party quantum communication model, simultaneous message passing model, and the model of quantum online streaming algorithms.

The problem of searching for a given substring in the text was considered. It is known that classical algorithms solve this problem in a linear time depending on the length of the text and the specified template. Quantum algorithms speed up the search by “square root times”. In this paper, we proposed a quantum algorithm that solves the search problem a) with a high probability of getting the correct result and b) with the same acceleration (by “square root times”) as compared with the classical one, but it c) requires much less memory (based on the number of qubits used) than the previously known quantum algorithms.

This is a review of quantum methods for machine learning problems that consists of two parts. The first part, "quantum tools", presents the fundamentals of qubits, quantum registers, and quantum states, introduces important quantum tools based on known quantum search algorithms and SWAP-test, and discusses the basic quantum procedures used for quantum search methods. The second part, "quantum classification algorithms", introduces several classification problems that can be accelerated by using quantum subroutines and discusses the quantum methods used for classification.

We propose a scheme of the universal quantum processing unit based on integrated optic waveguide excitation transfer of qubits between optical micro-resonators and Kerr nonlinear interaction between neighboring cavities. We present the protocols for the implementation of single-qubit gates and a two-qubit controlled phase gate. The optimal regimes of gates operation are studied using input-output formalism.

Modern quantum technologies are NISQ (Noisy Intermediate-Scale Quantum) devices, which are used to create insufficiently accurate quantum computers with low computing power. However, quantum technologies have advanced considerably during the past years. Thus, the issue of demonstrating “quantum supremacy” in the era of NISQ technologies is on the agenda. This study demonstrates that “quantum supremacy” is forthcoming. We propose procedures for constructing a universal family of hash functions based on a quantum hashing process that maps the original sequence w to a quantum hash state and then by random trans-formation to the state | ψ (cid:105) and generating the sequence u , which is an approximate description of the state | ψ (cid:105) . We proved that the proposed procedure generates a family of nondeterministic hash functions F , which allow us to reliably distinguish between different arguments. The F family can be considered an (cid:178) -universal family of nondeterministic hash functions. We assume that the development of this research area will cast light on the effect of “quantum supremacy” and will also have a certain impact on the advance of post-quantum cryptography.

This is a review of quantum methods for machine learning problems that consists of two parts. The first part, "quantum tools", presented some of the fundamentals and introduced several quantum tools based on known quantum search algorithms. This second part of the review presents several classification problems in machine learning that can be accelerated with quantum subroutines. We have chosen supervised learning tasks as typical classification problems to illustrate the use of quantum methods for classification.