Fundamentals of Quantum Computation

Fundamentals of Quantum Computation

Copyright: © 2021 |Pages: 30
DOI: 10.4018/978-1-7998-3799-2.ch001
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Abstract

This chapter briefly describes the basic concepts and principles of quantum computing. Firstly, the concepts of qubit, quantum coherence, quantum decoherence, quantum entanglement, quantum density operators, linear operators, inner products, outer products, tensor products, Hermite operators, and unitary operators are described. Then, the four basic assumptions of quantum mechanics are introduced, focusing on the measurement assumptions of quantum mechanics. Finally, the definition of commonly used quantum logic gates is given including single qubit gates, double qubit gates, and multiple qubit gates. These contents provide the necessary theoretical basis for subsequent chapters.
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Basic Concepts Of Quantum Computation

Basic concepts of quantum computation are described, such as qubit, tensor products, inner products, and outer products.

Qubit

A classical bit has a state: either 0 or 1. In quantum computation, information elements are represented by qubits (Schumacher, 1995), which have two possible states |0〉 and |1〉, which are called basis states. Any two-level quantum system can be used to implement qubits. For instance, the ground and excited states of electrons in a hydrogen atom, and the two different polarizations of a photon, can be called as |0〉 and |1〉, respectively. Dirac uses the symbol |∙〉 to represent a quantum state, also known as a ket, which is equivalent to a column vector. 〈∙| called a bra, is equivalent to a row vector and is a dual vector of |∙〉, which can be simply understood as a conjugate transpose vector of a vector (Dirac, 1947). A qubit can also be a linear combination of two basis states, often called superposition states:

978-1-7998-3799-2.ch001.m01
,(2.1) where the numbers α and β are complex numbers, satisfying 978-1-7998-3799-2.ch001.m02. Therefore, they are called the probability amplitude. When the quantum state 978-1-7998-3799-2.ch001.m03 is measured, it collapses to |0〉 with a probability of |α|2, and collapses to |1〉 with a probability of |β|2. Hence a qubit can contain both |0〉 and |1〉, which is quite different from classical bits.

The special basis states |0〉 and |1〉 are also called computation basis states. Their vector forms are

978-1-7998-3799-2.ch001.m04
,(2.2) and their dual vectors are

978-1-7998-3799-2.ch001.m05
(2.3)

It is known from (2.2) that the states of a qubit are unit vectors in a two-dimensional vector space (Rudin, 1999), which can also be in states 978-1-7998-3799-2.ch001.m06 and 978-1-7998-3799-2.ch001.m07. The two states are also called basis states, denoted as |+〉 and |‑〉, respectively.

A qubit also has a useful geometric representation, i.e., Bloch sphere representation (Nielsen & Chuang, 2000)

978-1-7998-3799-2.ch001.m08
,(2.4) where numbers θ and φ define a point on the unit three-dimensional sphere. It provides a useful means of visualizing the state of a single qubit, and often serves as an excellent testbed for ideas about quantum computation and quantum information.

Figure 1.

Bloch spherical representation of quantum bits

978-1-7998-3799-2.ch001.f01

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