Author: workhouse123
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Reversible Computation
We are studying classical gates to help us develop quantum gates. Quantum gates are unitary. This means they are reversible: they can be “run backward”. More practically, the meaning is that the inputs can be deduced from the outputs. Most classical gates however, are irreversible, and cannot as such be extended to quantum gates. For…
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Universal gates
It is well known that AND, NOT and OR form a universal set of gates. Consider for example the case n = 1. We have four distinct functions imple- mented as in Table 6.2. TABLE 6.2: The four 1-bit functions. Function Action Form Gate f 1 : 0 → 0 1 → 0 f(x) =…
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The Circuit Model and Universal Gates
Classical computation using binary variables works on Boolean logic, and implementation of basic logical operations are done through logic gates that are well known. We will revise their behaviour and notation and express their action as matrix operators. We will think of a computation as effected by a circuit evaluating some Boolean function whose input…
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Computability and Models for Computation
For a long time historically, computation was a matter of actually solving, or finding algorithms to solve, various mathematical problems using mechan- ical or other algorithms. It was only in the early twentieth century that the process of computation was modelled in mathematical terms, largely in the works of Alan Turing, Alonso Church, Kurt G¨odel,…
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Introduction
Now that we have the laws for qubits, we need to develop a system for meaning- fully manipulating them. Much of the current paradigm for quantum comput- ing is motivated by classical computation theory, especially the circuit model for computation. In this chapter, we will briefly overview the classical model of Boolean circuit theory, and…
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Purification
The notion of Schmidt decomposition immediately leads to a converse construction known as purification: given a density matrix ρ A for a mixed state of a system A, one can construct a supersystem AB of which it is a subsystem, such that |ψ AB i is a pure state, and ρ A = Tr B…
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Composite Systems
There is another sense in which density operators are a useful way to describe nature. In general, it is impossible to isolate the system of interest from some parts of its environment. We then have to regard our system as a subsystem of a larger system: “system + environment”. If the large system in a…
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Quantum Mechanics with Density Operators
We now have an alternate formulation of quantum mechanics, in terms of density operators instead of state vectors, that is good for open systems as well. Let’s go through the axioms of quantum mechanics framed in this language. 5.2.1 States and observables Postulate 1. Quantum State: The state of a quantum system is described by…
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Decomposition of the density operator
Often the density operator is the primary descriptor of a state. The de- composition in terms of component states ρ = X i p i |iihi|, is not always unique. For a pure state, it must be obvious that there is only one such decompo- sition, and this can be proved from the definitions: Theorem…
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The Bloch ball and the density operator
The representation of a single qubit state on the Bloch sphere can be ex- tended to the density operator. The Bloch sphere is parametrized by spherical angles or in terms of the Bloch vector of Equation 4.2, which characterizes the polarization of the state. Can we use this kind of description for a mixed state?…