We generalize the binary quantum counting algorithm of Lesovik, Suslov, and Blatter [Phys. Rev. A 82, 012316 (2010)] to higher counting bases. The algorithm makes use of qubits, qutrits, and qudits to count numbers in a base-2, base-3, or base-
d representation. In operating the algorithm, the number
n<N=dK is read into a
K-qudit register through its interaction with a stream of
n particles passing in a nearby wire; this step corresponds to a quantum Fourier transformation from the Hilbert space of particles to the Hilbert space of qudit states. An inverse quantum Fourier transformation provides the number
n in the base-
d representation; the inverse transformation is fully quantum at the level of individual qudits, while a simpler semiclassical version can be used on the level of qudit registers. Combining registers of qubits, qutrits, and qudits, where
d is a prime number, with a simpler single-shot measurement allows us to find the powers of 2, 3, and other primes
d in the number
n. We show that the counting task naturally leads to the shift operation and an algorithm based on the quantum Fourier transformation. We discuss possible implementations of the algorithm using quantum spin-
d systems,
d-well systems, and their emulation with spin-1/2 or double-well systems. We establish the analogy between our counting algorithm and the phase estimation algorithm and make use of the latter's performance analysis in stabilizing our scheme. Applications embrace a quantum metrological scheme to measure voltage (an analog to digital converter) and a simple procedure to entangle multiparticle states.
Abstract Efficient encoding of classical data into quantum state—currently referred to as quantum encoding—holds crucial significance in quantum computation. For finite-size databases and qubit registers, a common strategy of the quantum encoding entails establishing a classical mapping that correlates machine-recognizable data addresses with qubit indices that are subsequently superposed. Herein, the most imperative lies in casting an algorithm for generating the superposition of any given number of qubit indices. This algorithm is formally known as quantum superposing algorithm. In this work, we present an efficient quantum superposing algorithm, affirming its effectiveness and superior computational performance in a practical quantum encoding scenario. Our theoretical and numerical analyses demonstrate a substantial enhancement in computational efficiency compared to existing algorithms. Notably, our algorithm has a maximum of 2n − 3 controlled-not (CNOT) counts, representing the most optimized result to date.
At the time of this writing more than 60 (sixty) companies in the world are building quantum computers. These computers, based on quantum physics principles, are radically different from those that operate according to the more familiar principles of classical physics. A quantum algorithm takes a number of classical bits as its input, manipulates them so as to create a superposition of all their possible states, further manipulates this exponentially large superposition to obtain the final quantum result, and then measures the result to get (with the appropriate probability distribution) the same number of output bits as in its input. For the middle phase, there are elementary operations which count as one step and yet manipulate all the exponentially many amplitudes of the superposition. The natural language of these quantum gates is that of linear algebra in a complex (Hilbert) vector space. Since 2017 it is known that it is possible to replace the linear algebra with some string-rewriting rules which are no more complicated than the basic rules of arithmetic. The original system was introduced by Terry Rudolph and has been promoted and disseminated in large-scale outreach projects (among others) by Diana Franklin (University of Chicago) and Sofia Economou and Ed Barnes (Virginia Tech) as well as several other educators at the high-school level. In this paper we show how a slightly modified (though still very elementary) system can be used to communicate a visual and entirely operational understanding of key quantum computation concepts such as: superposition, probability, entanglement, phase, interference and unitary state evolution, as they occur in well-known quantum algorithms. We give concrete examples of proving properties for quantum gates and quantum circuits without resorting at all to complex numbers or matrix multiplication. Only simple, abacus-like operations are used-hence the title of the paper. The system we present allows a novice learner to actually trace a quantum algorithm as if it were a classical computation, which is a rare (and, frankly, borderline incredible) luxury in the area of quantum computation, where traditional debugging is impossible. Examples include the phase kickback phenomenon and the famous Deutsch-Josza algorithm. We end with a discussion (and more examples) of how this approach can create a genuine bridge to the mathematics of quantum computation, that is, of vector and tensor algebras in complex spaces for students who may have little or no proper mathematical background.
The use of quantum processing units (QPUs) promises speed-ups for solving computational problems, in particular for discrete optimisation. While a few groundbreaking algorithmic approaches are known that can provably outperform classical computers, we observe a scarcity of programming abstractions for constructing efficient quantum algorithms. A good fraction of the literature that addresses solving concrete problems related to database management concentrates on casting them as quadratic unconstrained binary optimisation problems (QUBOs), which can then, among others, be processed on gate-based machines (using the quantum approximate optimisation algorithm), or quantum annealers. A critical aspect that affects efficiency and scalability of either of these approaches is how classical data are loaded into qubits, respectively how problems are encoded into QUBO representation. The effectiveness of encodings is known to be of crucial importance for quantum computers, especially since the amount of available qubits is strongly limited in the era of noisy, intermediate-size quantum computers.
Encoding classical data into quantum systems is a foundational step in the execution of nearly all quantum algorithms, and a critical bottleneck in realizing practical quantum advantage. This review provides a comprehensive account of the concepts, algorithms, and practical considerations associated with quantum data encoding. We trace the development from its early conceptual foundations to recent advances, considering commonly used access models, such as quantum state preparation, unitary synthesis, QRAM and block encoding. We survey the circuit size, depth, space-time tradeoffs, as well as non-Clifford resources required for fault-tolerant implementation. We also discuss the roles of different access models in quantum algorithms. Special attention is given to structured data, such as sparse data, Boolean functions and data represented by tensor networks. This review bridges theory and applications, serving both as a pedagogical guide for newcomers and as a reference for active researchers. We also highlight the pivotal role of quantum data encoding in quantum computing and provide insights into future directions that will enable quantum advantage.