A quantum computer is a multi-particle interferometer that comprises beam splitters at both ends and arms, where the n two-level particles undergo the interactions among them. The arms are designed so that relevant functions required to produce a computational result is stored in the phase shifts of the 2^n arms. They can be detected by interferometry that allows us to utilize quantum parallelism. Quantum algorithms are accountable for what interferometers to be constructed to compute particular problems. A standard formalism for constructing the arms has been developed by the extension of classical reversible gate arrays. By its nature of sequential applications of logic operations, the required number of gates increases exponentially as the problem size grows. This may cause a crucial obstacle to perform a quantum computation within a limited decoherence time. We propose a direct and concurrent construction of the interferometer arms by one-time evolution of a physical system with arbitrary multi-particle interactions. It is inherently quantum mechanical and has no classical analogue. Encoding the functions used in Shor's algorithm for prime factoring, Grover's algorithm and Deutsch-Jozsa algorithm requires only one-time evolution of such a system regardless of the problem size n as opposed to its standard sequential counterpart that takes O(n^3), O(n) and O(n2^n).
A quantum computer is a multi-particle interferometer that comprises beam splitters at both ends and arms, where the n two-level particles undergo the interactions among them. The arms are designed so that relevant functions required to produce a computational result is stored in the phase shifts of the 2^n arms. They can be detected by interferometry that allows us to utilize quantum parallelism. Quantum algorithms are accountable for what interferometers to be constructed to compute particular problems. A standard formalism for constructing the arms has been developed by the extension of classical reversible gate arrays. By its nature of sequential applications of logic operations, the required number of gates increases exponentially as the problem size grows. This may cause a crucial obstacle to perform a quantum computation within a limited decoherence time. We propose a direct and concurrent construction of the interferometer arms by one-time evolution of a physical system with arbitrary multi-particle interactions. It is inherently quantum mechanical and has no classical analogue. Encoding the functions used in Shor's algorithm for prime factoring, Grover's algorithm and Deutsch-Jozsa algorithm requires only one-time evolution of such a system regardless of the problem size n as opposed to its standard sequential counterpart that takes O(n^3), O(n) and O(n2^n).
Quantum computing is based on the exploitation of quantum effects to perform some important computations more efficiently than ever possible on classical computers. Well known quantum algorithms are Shors polynomial time factoring algorithm and Grovers data base search scheme. Their potential is due to the parallelism of a quantum computation increasing exponentially with the size of the computing device. IBM has succeeded in building a first working quantum computing device with up to 7 qubits based on NMR technology, however, the construction of large multi-qubit systems appears to become extremely difficult due to the loss of coherence of the quantum states. Like the design process of modern microprocessors which is carried out by means of sophisticated software, modelling and simulation of the entire QC device is even more relevant. Simulation is required in order to understand the functioning of quantum algorithms, to construct novel quantum algorithms, to write quantum programs, to teach quantum computing, to construct real quantum computers, to understand and fight de-coherence and to understand and control the influence of the environment. Since quantum parallelism increases exponentially with the size of the quantum computer, parallel supercomputers are required for realistic simulations. I intend to give an introduction into the simulation of quantum computers using conventional computers. The simulation is based on a generic physical model of quantum computer hardware. Issues like parallel gate level simulators, modelling of QCs, simulators for physical quantum computers are addressed.
After some remarks on the fundamental physical nature of information, Bennett and Fredkin’s ideas reversible computation are introduced. This leads to the suggestions of Benioff and Feynman as to the possibility of a new type of essentially ‘quantum computer’. If we happen to build such devices, Deutsch scientists showed that‘quantum parallelism’ leads to new algorithms and new complexity classes. This is dramatically illustrated by Shor's quantum algorithm for factorization which is polynomial in time in contrast to algorithms for factorization on a classical Turing computer. This discovery has potentially important implications for the security of many modern cryptographic systems. The fundamentals of quantum computing are then introduced - reversible logic gates, qubits and quantum registers. The key quantum property of ‘entanglement’ is described, with due homage to Einstein and Bell. As an illustration of a quantum program, Grover's database search algorithm is described in some detail. After all this theory, the status of experimental attempts to build a quantum computer is reviewed: it will become evident that we have a long way to go before we can factorize even small numbers. Finally, we end with some thoughts about the process of ‘quantum compilation’ - translating a quantum algorithm into actual physical operations on a quantum system - and some comments on prospects for future progress.
Quantum computing exploits quantum mechanical properties to perform computations. It enables quantum parallelism and provides much more powerful data processing capabilities than classical computers. With a quantum computer, one can exponentially accelerate quantum system simulation and accelerate some important classical algorithms. Traditional quantum algorithms use unitary evolution to process information. A quantum computing process is the product of a series of unitary operators. In 1994, Shor invented a quantum prime factorization algorithm that exponentially accelerates the factorization of an integer. In 1996, Grover proposed a quantum search algorithm that accelerates the search of an unsorted database using square-root steps. Afterwards, the development of quantum algorithms slowed down, leading to a subsequent query by Shor in 2003 regarding why no more significant quantum algorithms have been found. Since 2009, many important new quantum algorithms have been proposed, such as a quantum algorithm for solving linear equations with the capability of exponential acceleration, a quantum algorithm for sparse Hamiltonian simulation using linear combinations of unitary operators, and a novel algorithm for Hamiltonian simulation, which provides exponential improvements in precision. In this paper, we first describe the basic principles of quantum computation. We then describe the Shor algorithm and Grover/Long search algorithm. These quantum algorithms are the general quantum algorithms that use unitary operations in their computational processes. Next, we introduce the basic principles of duality quantum computing, which was proposed in 2002. In contrast to traditional quantum computing, duality quantum computing allows the use of linear combinations of unitary operators in the computation process, meaning the multiplication, division, addition, and subtraction of unitary operators are all possible in duality quantum computing. Thus, duality quantum computing provides more flexibility for constructing quantum algorithms and the techniques used in classical algorithm design can be directly used to construct quantum algorithms. We then review recent work regarding newer quantum algorithms that enable the linear combination of unitary operators, which are actually duality quantum algorithms. Additionally, a duality quantum simulation algorithm for open quantum systems is introduced. This algorithm not only reduces computational complexity, but also improves accuracy exponentially. Finally, a summary and the future prospects of quantum algorithms are provided.
This paper introduces a comprehensive quantum calculator using Qiskit that integrates quantum arithmetic and logic functions into a single calculator. The calculator relies on quantum Fourier transforms (QFT), controlled-phase gates, and quantum modular arithmetic to perform basic arithmetic operations such as addition, subtraction, and multiplication. These arithmetic techniques used quantum parallelism and interference to perform calculations more efficiently, particularly for tasks involving large numerical inputs. Numerical values are stored in qubit registers and processed using QFT-based circuits, which enable efficient and reversible mathematical computations. In addition to arithmetic operations, the calculator also implements quantum logic algorithms, including Grover’s Search and the Deutsch-Jozsa algorithm. Grover’s algorithm is an unstructured search algorithm that operates by constructing an oracle and iteratively amplifying the probability amplitude of the target solution. This amplitude amplification significantly increases the likelihood of measuring the desired outcome. Due to this property, Grover’s algorithm has practical relevance in applications such as database searching and solving combinatorial optimization problems. The Deutsch-Jozsa algorithm determines whether or not a certain Boolean function is constant or balanced and offers an exponential speed up compared to classical methods for this class of problem. A key fundamental function of the calculator is the ability to generate hybrid quantum circuits that support both arithmetic and logical functions. That allows the exploration of simple as well as complex problem-solving strategies that take advantage of both quantum arithmetic and decision-making algorithms. For instance, combining Grover’s amplification with function characterizations from the Deutsch-Jozsa algorithm provides not only interesting circuit configurations but also adds value to the possibilities of the algorithms. This research offers a modular idea and allows exploring the fundamental principles of quantum computing generally speaking and will be a valuable resource for educational as well as research purposes in the rapidly growing field of quantum algorithm development.
A quantum machine may solve some complex problems that are intractable for even the most powerful classical computers. By exploiting quantum superposition and entanglement phenomena, quantum algorithms can achieve from polynomial to exponential speed up when compared to their best classical counterparts. A quantum computer will be a part of a heterogeneous, multi-core computer in which a classical processor will interact with several accelerators such as FPGAs, GPUs and also a quantum co-processor. Figure 1 shows the different layers of the quantum computer system stack [1]. Building such a quantum system requires contributions from different fields such as physics, electronics, computer science and computer engineering for addressing the following challenges: i) build scalable quantum chips integrating qubits with long coherence times and high-fidelity operations, ii) develop classical control electronics at possibly cryogenic temperatures and iii) create the microarchitecture as well as the software for quantum computation.
This paper proposes a novel computation architecture centered around the exploitation of quantum coherence. The core concept involves structuring a computational system to leverage the inherent parallelism afforded by quantum coherence, thereby aiming to significantly enhance computational efficiency compared to traditional architectures. We outline a framework where interconnected quantum systems operate in a manner that maximizes the duration of quantum coherence, allowing for parallel processing of multiple computational tasks simultaneously. The architecture is designed to mitigate decoherence effects, a primary challenge in quantum computing, through strategic system design and operational protocols. This approach represents a fundamentally new paradigm in computation, shifting from sequential, deterministic operations to a massively parallel, probabilistic one based on the quantum mechanical principles of superposition and coherence. The potential impact of this architecture extends across various fields, including scientific simulations, optimization problems, and machine learning, where the ability to process vast amounts of data concurrently offers a transformative advantage. ---