We consider a quantum circuit in which shift and rotation operations on qubits are performed by swap gates and controlled swap gates. These operations can be useful for quantum computers performing elementary arithmetic operations such as multiplication and a bit-wise comparison of qubits.
Quantum computing is a process that incorporates interacting physical systems that represent quantum bits and quantum gates. We present the quantum bit (qubit), the quantum register and the quantum gates. The qubit is described as a vector in a two-dimensional Hilbert space and the quantum register, which comprises a number of qubits, as a vector in a multidimensional Hilbert space. Quantum gates are Hilbert space operators that rotate the qubit or the quantum register vectors. Quantum computations are modeled and described using a quantum circuit model. We also present a quantum computer simulator based on the circuit model of quantum computation. In this model quantum computations and quantum algorithms are represented by circuits, which comprise quantum gates and quantum registers. The well-known Deutsch's algorithm is described and the corresponding quantum circuit is presented. Possible applications of quantum computers are be presented and discussed.
A quantum gate is the quantum analog of the classical logic gate of digital computers. It is a basic structural unit of a quantum circuit which operates on a small number of qubits to deliver a predefined output. A quantum circuit is a graphical representation of the sequence of the quantum gates used and the measurements performed in a quantum computation. Various quantum logic gates, namely, the Hadamard, T phase, NOT, Pauli-X, controlled NOT (CNOT), controlled-Z, SWAP and controlled phase gates as well as higher-level Toffoli and Fredkin gates are described. Conventions of reading quantum circuit diagrams are explained. Operation of a quantum full adder consisting of registers and gates is discussed, elaborating the role of the NOT gate, the sum calculation with CNOTgates and the carry C-out calculation with Toffoli gates. A primary tool used for performing arithmetic operations on a quantum computer is the quantum Fourier transform (QFT). The QFT adder and multiplier circuits are presented.
Abstract A circular shift operator (or cyclic rotation gate) ROTk ROT k applies a rightward (or leftward) shift to an input register of n qubits o by as many positions as encoded by an additional input k∈N k ∈ N . Specifically, the qubit at position x is moved to position (x+k)modn ( x + k ) mod n . While it is known that there exists a quantum rotation operator that can be implemented in O(log(n)) O ( log ( n ) ) -time, through the repeated parallel application of the elementary Swap Swap operators, there is no systematic procedure that concretely constructs the quantum operator ROT ROT for variable size n of the quantum register and a variable parameter k. We fill the gap, providing a systematic implementation of the cyclic rotation operator (denoted ROT ROT ) in a quantum circuit model of computation whose depth is O(log(n)) O ( log ( n ) ) . We show how the circular shift operator can be utilized in quantum approaches to text processing, focusing on the problem of getting all possible cyclic rotations of a string in O(log2(n)) O ( log 2 ( n ) ) depth.
This chapter represents our first step into quantum information theory (QIT). The key to operating such a transition is to become familiar with the concept of the quantum bit, or qubit , which is a probabilistic superposition of the classical 0 and 1 bits. In the quantum world, the classical 0 and 1 bits become the pure states |0〉 and |1〉, respectively. It is as if a coin can be classically in either heads or tails states, but is now allowed to exist in a superposition of both! Then I show that qubits can be physically transformed by the action of unitary matrices , which are also called operators . I show that such qubit transformations, resulting from any qubit manipulation, can be described by rotations on a 2D surface, which is referred to as the Bloch sphere . The Pauli matrices are shown to generate all such unitary transformations. These transformations are reversible , because they are characterized by unitary matrices ; this property always makes it possible to trace the input information carried by qubits. I will then describe different types of elementary quantum computations performed by elementary quantum gates, forming a veritable “zoo” of unitary operators, called I , X , Y , Z , H , CNOT, CCNOT, CROSSOVER or SWAP, controlled- U , and controlled-controlled- U . These gates can be used to form quantum circuits , involving any number of qubits, and of which several examples and tools for analysis are provided. Finally, the concept of tensor product , as progressively introduced through the above description, is eventually formalized.
Abstract Advances in quantum information science are closely related to our ability to prepare, manipulate, and measure the states of multiple quantum bits (qubits). To this end, quantum gates are the key components by which specific operations over individual and multiple qubits are implemented. In the context of quantum optics, qubits can be encoded in any degree of freedom of a multi-photon state, e.g. polarization, frequency or spatial configuration. The faithful reciprocal exchange of information between two qubits, a so-called SWAP, is one of the fundamental operations that underpin modern quantum architectures. Here, we present a novel method realizing multiple-qubit SWAP operations over several qubits, without relying on any other gates, in an integrated-photonic setting. In principle, our approach allows for the implementation of SWAP operations with unity efficiency regardless of the number of involved qubits. Further, we discuss how this approach can be applied to qudits. Our findings provide a powerful new functional element for the design of compact quantum-photonic circuitry.
Quantum computing (Q-computing) is a revolutionary technology that applies quantum mechanics (QM) to implement calculations significantly quicker than traditional computers. Q-computing employs Q-bits (qubits) rather than classical bits, which can take on a superposition state, making it possible to represent both 0 and 1 at once. This distinctive property allows qubits to perform multiple calculations simultaneously, making Q computers exponentially more powerful as more qubits are added, compared to the linear scaling of classical computers. However, challenges such as qubit instability and error rates remain significant obstacles to fully realizing this potential. A vital aspect of this technology is the development of Q arithmetic circuits, which act as the foundational blocks for efficiently executing Q algorithms. Q half-adder and full-adder circuits are integral to performing fundamental arithmetic operations within Q computers. In contrast to classical adders, Q adders rely on Q gates such as Controlled NOT (CNOT) and Toffoli gates. These gates exploit Q phenomena like entanglement and superposition, allowing for more efficient computations. The Q half-adder consists of two qubits and adds two single bits, producing a carry and sum output. It utilizes the CNOT gate to compute the sum and the Toffoli gate (controlled-controlled-NOT [CCNOT]) to compute the carry. The Q full-adder builds on this, adding three qubits to produce a sum and a carry output, further increasing computational capacity. Through the entanglement of qubits, these circuits allow for more parallelism and speed compared to classical adders. Q carry-look-ahead adders (QCLAs) and Q ripple-carry adders (QRCAs) are used to perform more complex arithmetic operations based on these half-adder and full-adder structures. Q arithmetic circuits have broad applications, boosting secure key generation in cryptography, speeding up matrix operations for faster model training in machine learning, and enabling more precise, efficient image transformations in Q image processing, benefiting fields like medical imaging and computer vision.