A quantum computer simulator is presented. This simulator is an engineering work and no deep understanding of quantum mechanics is required from the user. The simulator is based on the circuit model of quantum computation in which quantum gates act on quantum registers which comprise a number of quantum bits (qubits). The inputs to the simulator are the initial states of the qubits that form a quantum register and the quantum gates applied at each computation step. The inputs are entered through a graphical user interface. The outputs of the simulator are the matrices that represent the quantum register state at each quantum computation step and graphical outputs that show the probability of measuring each one of the possible quantum register base states and the phase of each state at each computation step. The well-known Deutsch's algorithm and the quantum Fourier transform, which is the base of many quantum algorithms, are presented using this simulator. Furthermore, the generation and variation of entanglement during quantum computations can be calculated using this simulator. The quantum computer simulator is a useful tool for the study of quantum computer circuits, quantum computing, and the development of new quantum algorithms.
It has been proven that a quantum computer is superior to an electronic com puter in solving some NP problems. Based on quantum operations, this article proposes a new quantum sum and quantum multiplication, and then the floating point quantum multiplier and quantum sum are created on the basis of fixed point number operations. These studies lay the foundation for the quantum implementation of digital filters. This article provides a new way to calculate the summator on a quantum computer. This method uses the quantum Fourier transform (QFT) and reduces the number of qubits needed to be added by eliminating the need to temporarily transfer bits. This approach also allows you to add a classical number to a quantum superposition without encoding a classical number in a quantum register. This method also allows for mass parallelization during its execution. Adding and multiplying capabilities based on QFT are improved with some changes. The proposed operations are compared with the operations of close quantum arithmetic.
We discuss a new approach to simulate quantum algorithms using classical probabilistic bits and circuits. Each qubit (a two-level quantum system) is initially mapped to a vector in an eight dimensional probability space (equivalently, to a classical random variable with eight probabilistic outcomes). The key idea in this mapping is to store both the amplitude and phase information of the complex coefficients that describe the qubit state in the probabilities. Due to the identical tensor product structure of combining multiple quantum systems as well as multiple probability spaces,
n qubits are then mapped to a tensor product of
n 8-dimensional probabilistic vectors (i.e., the Hilbert space of dimension
2n is mapped to a probability space of dimension
8n). After this initial mapping, we show how to implement the analogs of single-qubit and two-qubit gates in the probability space using correlation-inducing operations on these classical random variables. The key defining feature of both the mapping to the probability space and the transformations in this space (i.e., operations on the random variables) is that they are not linear, but instead affine. Using this architecture, the evolution of the
2n complex coefficients of the quantum system can be tracked in the joint fully-correlated probabilities of the polynomial number of random variables. We then give specific procedures for implementing (1) the Deutsch-Jozsa algorithm, and (2) the Quantum Fourier Transform in the probability space. Identical to the Quantum case, simulating the Quantum Fourier Transform in the probability space requires
O(n) probabilistic bits and
O(n2) (i.e., quadratic in the number of quantum bits) operations.
The Quantum Fourier Transform (QFT)andQuantumPhaseEstimation (QPE)are twofundamental algorithmic frameworks in quantum computation, and they provide the mathematical basis of many exponential-speedup quan tum algorithms, such as Shor’s factoring algorithm and quantum simulation techniques. In this work we present a detailed theoretical and computational study of the Quantum Fourier Transform and its integration in the Quan tum Phase Estimation. The discussion begins with the linear algebraic structure of the Hilbert spaces, the unitary transformations, and the complex vector representations that enable quantum interference and superposition. The QFT is studied as the unitary transform of computational basis states in the amplitudes in the frequency domain by phase rotations and tensor-product decompositions. The efficient circuit implementation in terms of Hadamard and controlled phase gates of this transform is discussed. The advantage of the transform in terms of polynomial time complexity over the classical discrete Fourier transform in some quantum scenarios is discussed.The QFT is implemented and the Quantum Phase Estimation algorithm is proposed to obtain the eigenphase information of the unitary operators. In the report the mathematical representation of QPE is derived by using the controlled unitary evolution, the inverse Fourier transforms and the measurement theory in finite dimensional Hilbert spaces. We discuss the probabilistic interpretation of phase extraction in detail and the convergence properties of the bi nary phase approximation. We also discuss the relation of QPE to eigenvalue problems in quantum mechanics and its applications to Hamiltonian simulation, quantum chemistry, cryptography, and hidden subgroup prob lems.Implementation aspects of these algorithms are studied using quantum circuit representations and oracle based computation models. We also discuss error sensitivity, qubit precision requirements, decoherence limita tions and practical constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The work synthesizes the mathematical foundations, computational complexity analysis and physical realizations of the QFT and the QPE, demonstrating how the Fourier-based quantum transformations provide the essential mechanism underlying modern quantum algorithmic speedups and scalable quantum information processing.
The quantum computer is emerging technology since it can resolve complex tasks such as factorization or search problem faster than classical computers. However, simulation of the quantum computer is difficult because computational load increases exponentially as the number of qubits used in the circuit is increased. This paper proposes a hardware architecture for quantum circuit simulation. To implement simulator smaller, our architecture operates based on a 1-input single gate regardless of the number of qubits used in the circuit In addition, our architecture helps to reduce simulation time by skipping calculation and data transfer. The proposed architecture is implemented on FPGA and the experimental results show that our architecture reduces 79.2% of the multiplication process when we simulate 20-qubit quantum Fourier transform with a small hardware size.