All computers are quantum computers but some are more quantum than others. Universal classical computers rely on the transistor, a quantum semiconductor, for their operation but use simple binary digital logic based on bits, 1 and 0. Quantum computers, those we are interested in here, utilize the qbits https://www.w3.org/1998/Math/MathML" display="inline"> | 1 〉 and https://www.w3.org/1998/Math/MathML" display="inline"> | 0 〉 , which are quantum states, to realize their quantum logic. These qbits are used in conjunction with superposition probability amplitudes, such as https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 + , https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 − , https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 + , and https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 − , to interact with optical components such as beam splitters, interferometers, and polarization rotators to perform quantum logical operations. In this chapter, a basic description of quantum logic, and quantum optical configurations, utilized in workable quantum computers is given.
All computers are quantum computers but some are more quantum than others. Universal classical computers rely on the transistor, a quantum semiconductor, for their operation but use simple binary digital logic based on bits, 1 and 0. Quantum computers, those we are interested in here, utilize the qbits https://www.w3.org/1998/Math/MathML" display="inline"> | 1 〉 and https://www.w3.org/1998/Math/MathML" display="inline"> | 0 〉 , which are quantum states, to realize their quantum logic. These qbits are used in conjunction with superposition probability amplitudes, such as https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 + , https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 − , https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 + , and https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 − , to interact with optical components such as beam splitters, interferometers, and polarization rotators to perform quantum logical operations. In this chapter, a basic description of quantum logic, and quantum optical configurations, utilized in workable quantum computers is given.
Computer technology fundamentally works as on/off switches. These switches are digits: 1s and 0s; 1s are on, 0s are off. In computerese these switchable binary digits are called bits. Quantum computing utilizes subatomic bits, or “qubits,” that can exist as a 1 or a 0 or as both simultaneously. Quantum computers use quantum-mechanical interactions that allow amazing calculating power. The chief advantage of a quantum computer is that it can be in multiple states simultaneously, known as “superposition,” while also simultaneously acting on each possible state. A quantum computer could therefore perform innumerable calculations in parallel using only one processing unit.
In quantum mechanics, the physics of probability amplitudes, that exists in the imaginary and complex mathematical realm, is brought to the measurable realm by the iconic Born rule https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 that convert the superposition probability amplitude https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 into a measurable interferometric probability distribution. When Born first introduced his rule, in 1926, he did so in the absence of a physical derivation. In fact, Born never offered a derivation for his most crucial rule. In this chapter a transparent interferometric derivation of Born's rule is given. Furthermore, it is indicated that far from a ‘collapse’ the https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 → | ψ 〉 | ψ 〉 ∗ transition, for interferometric configurations, clearly involves a mathematical expansion . The https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 → | ψ 〉 | ψ 〉 ∗ transition for quantum entanglement is also discussed.
This Chapter introduces the Feynman Path Integral description of quantum transition amplitudes, considering the simplest case of a quantum system described by one quantum variable q and its conjugate momentum p . The transition amplitude from time https://www.w3.org/1998/Math/MathML" display="inline"> t 1 to time https://www.w3.org/1998/Math/MathML" display="inline"> t 2 = t 1 + T is obtained by discretising the time axis in N time steps of amplitude https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , implying T=N https://www.w3.org/1998/Math/MathML" display="inline"> ϵ . For https://www.w3.org/1998/Math/MathML" display="inline"> ϵ → 0 , the Feynman Path Integral formula is derived: https://www.w3.org/1998/Math/MathML" display="inline"> < q N | e - i T H / ℏ | q 0 > = ∫ d [ q ( t ) ] e i S q ( t ) , q ' ( t ) / ℏ , where integration is over all trajectories https://www.w3.org/1998/Math/MathML" display="inline"> q ( t ) which satisfy the boundary conditions at https://www.w3.org/1998/Math/MathML" display="inline"> t 1 and t 2 . https://www.w3.org/1998/Math/MathML" display="inline"> S ( q , q ' ) is the classical Action, a function of https://www.w3.org/1998/Math/MathML" display="inline"> q and of its time derivative. The classical limit, corresponding to https://www.w3.org/1998/Math/MathML" display="inline"> ℏ → 0 , is considered, whereby the solution of the Feynman Path Integral reduces to the classical trajectory, https://www.w3.org/1998/Math/MathML" display="inline"> q c ( t ) , which minimises the classical Action. The lattice approximation , i.e. small, non vanishing https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , with evaluation of the Feynman integral by numerical integration, provides a useful approximation in the case of the fundamental theory of the strong interactions (Quantum Chromodynamics, or QCD), to be considered later; Chapters 17 to19. This chapter is completed by consideration of the extension of gthe formalism to imaginary time, where the Feynman Path Integral becomes the Partition Function of Statistical Mechanics, and by the definition of the Green's functions, ground-state expectation values of products of quantum operators q(t) taken at different times, https://www.w3.org/1998/Math/MathML" display="inline"> t 1 , t 2 … t N .
Today's computers work on bits that exist as either 0 or 1. Quantum computers aren't limited to two states; they encode information as quantum bits, or qubits, which can exist in superposition. Qubits represent atoms, ions, photons or electrons and their respective control devices that are working together to act as computer memory and a processor. Because a quantum computer can contain these multiple states simultaneously, it has the potential to be millions of times more powerful than today's most powerful supercomputers. A processor that can use registers of qubits will be able to perform calculations using all the possible values of the input registers simultaneously. This superposition causes a phenomenon called quantum parallelism, and is the motivating force behind the research being carried out in quantum computing. Due to technical obstacles, till date, a quantum computer has not yet been realized. But the concepts and ideas of quantum computing has been demonstrated using various methods like NMR, Ion Trap, Quantum Dot, Optical Methods, etc. A quantum computer manipulates qubits by executing a series of quantum gates, each a unitary transformation acting on a single qubit or pair of qubits. In applying these gates in succession, a quantum computer can perform a complicated unitary transformation to a set of qubits in some initial state. The qubits can then be measured, with this measurement serving as the final computational result. Research must devise a way to maintain decoherence and other potential sources of error at an acceptable level. Probably the most important idea in this field is the application of error correction in phase coherence as a means to extract information and reduce error in a quantum system without actually measuring that system. Thereby, quantum computers will emerge as the superior computational devices and perhaps one day make today's modern computer obsolete.
Digital computers are based on devices that can take on only two states, one of which is denoted by 0 and the other by 1. By concatenating several 0s and is together, 0-1 combinations can be formed to represent as many different entities as desired. A combination containing a single 0 or 1 is called a bit. In general, n bits can be used to distinguish among 2 n distinct entities and each addition of a bit doubles the number of possible combinations. Computers use strings of bits to represent numbers, letters, punctuation marks, and any other useful pieces of information. In a classical computer, the processing of information is done by logic gate. A logic gate maps the state of its input bits into another state according to a truth table. Quantum computers require quantum logic, something fundamentally different to classical Boolean logic. This difference leads to a greater efficiency of quantum computation over its classical counterpart. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
If we include all three spatial dimensions, the time-dependent wave function https://www.w3.org/1998/Math/MathML" display="inline"> Ψ ( x , y , z , t ) depends on four coordinates. A wave function of this form determines the probability density https://www.w3.org/1998/Math/MathML" display="inline"> ρ t ( x , y , z ) = | Ψ ( x , y , z , t ) | 2 for measurements of the spatial position https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) . The product https://www.w3.org/1998/Math/MathML" display="inline"> | Ψ ( x , y , z , t ) | 2 d x d y d z therefore equals the probability of finding a particle in a small cubic volume with side lengths https://www.w3.org/1998/Math/MathML" display="inline"> ( d x , d y , d z ) around https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) at time t . The normalization condition for this probability density is given by 4.1 https://www.w3.org/1998/Math/MathML" display="block"> ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ | Ψ ( x , y , z , t ) | 2 d x d y d z = 1 for all t .