Quantum Computing. Hafiz Md. Hasan Babu

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      IOP Publishing

      Quantum Computing

      A pathway to quantum logic design

      Hafiz Md Hasan Babu

      Chapter 2

      Basic definitions of quantum logic

      Quantum computing was advanced as a technological attempt to build a propositional structure that would permit relating the events of interest in quantum mechanics. Quantum computing seeks to replace the Boolean structure which, although appropriate for addressing classical physics, is insufficient for representing atomic elements. The scientific structure of the propositional linguistics of classical systems is a power set, partly ordered by set enclosure, with a pair of operations that denote conjunction and disjunction.

      During the progress of quantum computing, numerous lines of study have tried to address quantum mechanics from a logical viewpoint. This book offers a map of these multiple methods in order to familiarize the reader with the very different approaches and problems deliberated in the quantum computing literature. When possible, redundant theories, algorithms, and examples, are avoided in order to provide an instinctive understanding of the ideas before designing or presenting the associated mathematics.

      The procedure of a two-bit ‘controlled-NOT’ quantum logic gate has been designed which, in concurrence with simple single-bit operations, forms a common quantum logic gate for quantum computation. The two quantum bits are placed in the internal and external degrees of choice of a single trapped element which is first laser cooled to the zero-point energy. Decoherence properties are acknowledged for the operation, and the prospect of expanding the system to more qubits seems favorable.

2.1 The quantum bit

      A quantum bit (qubit) is typically derived from the state of a two-level quantum system, such as the ground and excited states of an atom or the vertical and horizontal polarizations of a single photon. A qubit, represented by ∣A〉, is the basic unit of information in a quantum computer, which can hold two states, 0 or 1, simultaneously or at different times. A qubit can also be a superposition (both states at the same time) of these two states, i.e. a linear combination of the binary values ∣1〉 and ∣0〉 (α∣0〉i+β∣1〉i, where α and β are the probabilities of being the ∣1〉 and ∣0〉 states, respectively), whereas classical bits or binary bits are in one of two possible states, labeled 1 and 0.

2.2 The quantum gate

      A quantum gate is a basic quantum circuit operating on a small number of qubits. Previously, various quantum gates with different functionalities have been designed. Among them, the NOT, CNOT, controlled-V, and controlled-V⁺ gates represent an important class of quantum gates. These gates are shown in figure 2.1. In this figure, the control, target, and contact qubits are represented by the ·, ⊗, and ∥ symbols, respectively. In a quantum gate the number of outputs must be equal to the number of inputs.

      Figure 2.1. Basic quantum gates. (a) Quantum NOT gate. (b) Quantum CNOT gate. (c) Controlled-V gate. (d) Controlled-V⁺ gate.

2.2.1 The quantum Feynman gate

      The quantum Feynman gate is a 2 × 2 quantum gate which implements the logical functions of P = A and Q=A⊕B, and is illustrated in figure 2.2. The quantum Feynman gate can be used for copying a bit. When B is set to zero, then the output vector will be P = A and Q = A, which ensures copying of the input A.

      Figure 2.2. The quantum Feynman gate.

2.2.2 The quantum Tofolli gate

      The quantum Tofolli gate is a 3 × 3 gate in which A, B, and C are input vectors and the output vectors are P = A, Q = B, and R=AB⊕C logical functions. The quantum Tofolli gate can implement the ‘AND’ operation when C is set to zero. The quantum Tofolli gate is presented in figure 2.3. The Tofolli gate is an important quantum gate and is widely applied in the construction of quantum circuits.

      Figure 2.3. The quantum Tofolli gate.

2.2.3 The quantum Fredkin gate

      The quantum Fredkin gate is illustrated in figure 2.4, and is a 3 × 3 quantum gate in which A, B, and C are input vectors and the output vectors are the P = A, Q=A¯B⊕AC, and R=A¯B⊕AB logical functions. One of the important applications of the quantum Fredkin gate is swapping. The input bit A is used as a control bit