On an (n+2)-bit quantum network, a gate for conditional NOT operation with n≧5 bits of controls (a C^nNOT gate) can be simulated with 24n-64 gate of conditional two-bit operations, as well as with 32n-4 gates of CNOT and one-bit operations. These small numbers of elementary gates (which are approximately half or two thirds of the number known so far) help toward implementation of the oracle U_f: |x>|y>→|x>|y+f(x)> on quantum computers.
On an (n+2)-bit quantum network, a gate for conditional NOT operation with n≧5 bits of controls (a C^nNOT gate) can be simulated with 24n-64 gate of conditional two-bit operations, as well as with 32n-4 gates of CNOT and one-bit operations. These small numbers of elementary gates (which are approximately half or two thirds of the number known so far) help toward implementation of the oracle U_f: |x>|y>→|x>|y+f(x)> on quantum computers.
Logic gates such as the controlled-NOT (c-NOT) and Toffoli gates play a key role in quantum information processing (QIP) and quantum computing. A natural extension of such gates would necessarily operate on one quantum bit (qubit) conditional on the state of the remaining qubits in the system. We show that such selective gates, termed $(\mathrm{controlled}{)}^{n}\ensuremath{-}\mathrm{NOT}$ gates, or ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates, are convenient in nuclear magnetic resonance (NMR) implementations of QIP and are straightforward to implement. NMR pulse sequences for these gates can be built using classical methods as well as insights from geometric algebra. These methods yield equivalent NMR pulse sequences for the generation of ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates for any number of control spins. In this work, a catalog of ${\mathrm{c}}^{n}\ensuremath{-}\mathrm{NOT}$ gates for systems of as many as 16 spins is provided along with an experimental implementation of a ${\mathrm{c}}^{3}\ensuremath{-}\mathrm{N}\mathrm{O}\mathrm{T}$ gate on a four spin system, 13C alanine.
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.
While quantum algorithms are known to be very powerful, there are also people who think that “quantum computers will never be able to run the if/then/else type of logic”. On the other hand, though many experts know that such a claim is a misconception, in literature there is still lacking of a general description of the procedure for implementing the if/then/else conditional statements in the quantum way. In the current work, we present such a general method in details, explaining how the unitary transformation required for any if/then/else operation can be constructed from simpler unitary transformations and projective operators. Moreover, by using the well-known controlled-NOT gate, Fredkin gate and Toffoli gate as examples, it is elaborated that the structure of the existing conditional quantum operations agrees with our general method. An even more complicated example is also provided, showing that our method is convenient for solving real-world if/then/else problems with quantum computers. Thus the result not only refutes the above claim, but also helps to develop the skill on quantum programming.
A quantum computer is a device for computation that makes direct use of distinctively quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. In a classical (or conventional) computer, the amount of data is measured by bits, in a quantum computer; it is measured by quantum of bits (qubits). This work describes, and implements the universal quantum logic gates, the terminology is introduced with two well known quantum gates, the quantum NOT, and the quantum XOR gates. The NOT and XOR gates have already been described in classical reversible logic. This work gives the implementation of the quantum circuits, such as the Quantum Half-Adder Circuits, which consists of quantum control control not gate (CCNot) and quantum control not gate (CNot). Also, it describes and implements the Quantum Full-Adder Circuits, which consists of two Quantum Half-Adder Circuits and one control not gate (CNot). Depending on the quantum circuits, one can implement the quantum basic arithmetic operations such as (addition, subtraction, multiplication and division). After each implementation, the computational complexity of each step is calculated. Key words:
With the technique of opposite phases, the sequence pulse for exact controlled not gate was designed, and the controlled controlled phase shift gate, which is the key gate in Grovers algorithm, and the double controlled not gate were also constructed by employing only two body interaction. The symmetric operators of quantum states were made up. All the above are the tools of the quantum state engineering, therefore, theyare very useful for both theoretic and experimental investigations of quantum computation.
We provide a method for compiling approximate multicontrolled single qubit gates into quantum circuits. Without ancilla qubits, the total number of elementary gates to decompose an n-qubit multicontrolled gate is proportional to 32n elementary operations. The proposed decomposition depends on an optimization technique that minimizes the CNOT gate count for multitarget and multicontrolled CNOT and SU(2) gates. We also provide an approximate decomposition with ancilla qubits with lower-circuit complexity. Computational experiments show the reduction of CNOT gates when multicontrolled U(2) gates are applied. As multicontrolled single-qubit gates serve as fundamental components of quantum algorithms, the proposed decomposition offers a comprehensive solution that can significantly decrease the count of elementary operations employed in quantum computing applications.