Quantum bits or qubits

The basic element of a quantum computer is the quantum bit or qubit. It is the QC counterpart of the Boolean bit, a classical physical system with two well-defined states. A material realization of a qubit is a quantum two-level system, with energy eigenstates, 0) and 1), and an energy gap AE, which can be in any arbitrary superposition cp) = cos(d/2) 0) + exp(i0)sin(0/2) l).These pure superposition states can be visualized by using a Bloch sphere representation (see Figure 7.1). 

Such an entity represents the basic unit of information in a quantum computer—a quantum bit or qubit. Unlike a classical bit, which can store only a single value—a 0 or 1—a qubit can store both 0 and 1 at the same time. The state of a two-qubit register could be written... 

The individual unit of classical information is the bit an object that can take either one of two values, say 0 or 1. The corresponding unit of quantum information is the quantum bit or qubit. It describes a state in the simplest possible quantum system [1,2]. The smallest nontrivial Hilbert space is two-dimensional, and we may denote an orthonormal basis for the vector space as 0> and 11 >. A single bit or qubit can represent at most two numbers, but qubits can be put into infinitely many other states by a superposition ... 

Analogously, the unit of information in Quantum Information and Quantum Computation is the quantum bit, or qubit, for short. A qubit can assume the logical values 0 or 1. However, it can also be in a logical state containing any linear combination of them, thanks to laws of quantum mechanics [8], Physically, qubits can be represented by any quantum object with two well defined and distinct eigenstates. Examples of qubits are the photon polarization states, electrons in two-level atoms (as an approximation) and nuclear spins under the influence of a magnetic field. 

Fig. 1.7 shows how two different states of an atom can be used to encode a bit [14]. But quantum bits - or qubits - behave differently from classical bits because the state of such a quantum system is characterized by a complex wave function ) which represents a superposition of the states. The atom cannot only exist in the pure states 0) or l) but also in the mixed state [ ) = ( 0) + 1)) Since... 

The quantum analogue to the classical bit is the quantum bit or qubit. In order not to overwhelm the reader with abstract concepts I will introduce the notion of a qubit by an explicit example. In particular, I will discuss the realization of qubits with atoms. This example will be relevant throughout this chapter. 

In this equation, C andT refer to control and target qubits, respectively. The resulting state (output of the qugate) is said to be an entangled state of the two qubits, that is, a state that cannot be written as a product of states for each qubit [30]. The occurrence of such entangled states is another characteristic trait of QC, at the basis of secure quantum communication or cryptography. It also implies that, as opposed to what happens with a classical bit, an arbitrary quantum bit cannot be copied (the COPY classical operation is, in fact, based on the application of a succession of classical CNOT gates) [4]. 

Within the past decade much progress has also been made in experimental realizations of quantum computing hardware. Many architectures have been proposed based on a variety of physical hardware. On a small scale, quantum information has been stored and manipulated in superconducting quantum bits (qubits) [4,5], trapped ions [6,7], electron spins [8-11], nuclear spins in the liquid or solid state [12], and other systems. On the theoretical side, new quantum algorithms have recently been found, exhibiting significant pol momial speedups on quantum computers for solution of sparse linear equations or differential equations [13,14], quantum Monte Carlo problems [15], and classical simulated annealing problems [16]. 

The interaction process of the QC with the environment can be understood as follows. We consider a single qubit interacting with its environment. The environment can be viewed as a high-dimensional quantum system. Initially the quantum bit and the environment are in a product (or non-entangled) state... 

In this section we introduce the space of spin states of a spin-1/2 particle, such as an electron. In quantum computation (the investigation of computers whose basic states are quantum, not deterministic), this space of states is called a qubit, pronounced cue-bit. Just as a bit (a choice of 0 or 1) is the smallest unit of information in a deterministic computer, a qubit is the smallest unit of information in a quantum computer. 

Richard Feynman pointed out that contemporary computers are based on the all or nothing philosophy (two bits 0) or 11)), while in quantum mechanics one may also use a linear combination (superposition) of these two states with arbitrary coefficients a and b a 0> -I- fc l), a qubit. Would a quantum computer based on such superpositions be better than traditional one The hope associated with quantum... 

Why is quantum computing more powerful than its classical counterpart A classical bit can have the logical values 0 or 1. The qubit, its quantum counterpart, can have the values 0 and 1 at the same time. This is directly related to other seemingly paradoxical statements often heard in connection with quantum mechanics, such as, Schrodinger s cat is both dead and alive. This example can be easily associated with a qubit by assigning dead to 0 and alive to 1. 

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