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Quantum Computation: Overview
Quantum Computation: Overview
In the simplest physical terms, quantum computation is an implementation of an arbitrary unitary operation on a finite collection of two-level quantum systems that are called quantum bits or qubits for short. It is typically depicted in a quantum circuit diagram as in Figure 1.
Figure 1. The quantum circuit model of quantum computation. (left) The input state from the left is processed by a unitary operator , and then the output state is measured in the computational basis on the right. (right) A quantum computer is programmable, and the unitary gate in the left panel is decomposed into elementary gates that act on one qubit or two at each time.
U
U
Each qubit is associated with a line that indicates the time evolution of the state specified on the left, and time flows from left to right. The quantum logic gate operations (or gates for short) on single or multiple qubits are denoted by a rectangular box often with labels indicating the types of the gates. Measurements are denoted by square boxes with needles. After a measurement, time-evolution is represented by dashed lines to remind that the information is classical, that is, there is no superposition.
The input state is prepared in one of the states in the logical basis, typically . After an overall unitary operation, the resulting state is measured in the logical basis, and the readouts are supposed to be the result of the computation.
In order for a quantum computer to be programmable, a given unitary operator Uˆ must be implemented as a combination of other more elementary unitary operators
|0〉⊗|0〉⊗⋯⊗|0〉
In order for a quantum computer to be programmable, a given unitary operator Uˆ must be implemented as a combination of other more elementary unitary operators
U=⋯
U
1
U
2
U
L
where each is chosen from a small fixed set of elementary gate operations. The latter operations are the elementary quantum logic gates of the quantum computer.
U
k
In this collection of tutorial documents, we will examine widely-used choices for elementary gates and illustrate how a set of elementary gates achieve an arbitrary unitary operation to realize the so-called universal quantum computation.
Single-Qubit Gates
The Pauli Gates
The Hadamard Gate
Rotations
Euler Rotations
A Discrete Set of Universal Rotations
Two-Qubit Gates
Controlled-NOT Gate (CNOT)
Controlled-Z Gate (CZ)
SWAP Gate
Controlled-Unitary Gates
General Unitary Gates
Multi-Control NOT Gate
The Toffoli Gate
The Fredkin Gate
Implementations
Multi-Control Unitary Gates
Gray-Code Method
Quadratic Implementations
Universal Quantum Computation
Decomposition into Two-Level Unitary Matrices
Implementation of Two-Level Unitary Matrices: Idea
Implementation of Two-Level Unitary Matrices: Gray Code Sequence
Universal Quantum Computation Theorem
Computational Model of Measurement
Measurement in an Arbitrary Basis
Pauli Measurements
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