Chapter 3 - Physics of Quantum Information

It was a great realization that information is physical and that a (classical) Turing machine is not the end of the story of computation. The physical system in which the information is stored and manipulated is important and qubits are quite different from bits.

In this chapter, some background in quantum mechanics is provided. Not all of this chapter will be directly relevant to our discussion, but it is included for the sake of completeness of our understanding of how quantum mechanics from a textbook is related to quantum computing. The connection is, as of yet, clear but the story seems incomplete from a physicists perspective and for the subject of error prevention methods, some of this chapter will be vital. In particular, the section(s) concerning the density matrix. Not only is this vital, but not usually covered in most quantum mechanics classes, either undergraduate or graduate.

It is also worth emphasizing that this chapter is primarily aimed at physicists and for those others which are interested in the background physics. It is not necessary for much of what follows.

Schrodinger's Equation

A common starting point in quantum mechanics is Schrodinger's equation. This equation is not derived, or justified here, but is given in a general form:

	$H\left\vert \Psi \right\rangle =i\hbar {\frac {\partial }{\partial t}}\left\vert \Psi \right\rangle ,$	(3.1)

where $H\,\!$ is the Hamiltonian, $\hbar \,\!$ is Planck's constant (divided by $2\pi \,\!$), and $t\,\!$ is time. The Hamiltonian contains what is known about the system's evolution. Most of the time in these notes, we let $\hbar =1\,\!$.

This equation is (formally) solved by taking the time derivative to be an ordinary derivative (we assume no explicit time dependence for $H\,\!$), so

	$H\left\vert \Psi \right\rangle =i{\frac {d\left\vert \Psi \right\rangle }{dt}}.$	(3.2)

This means that

	$-iHdt={\frac {d\left\vert \Psi \right\rangle }{\left\vert \Psi \right\rangle }},$	(3.3)

so

	${\begin{aligned}\ln \left\vert \Psi \right\rangle &=-iHt+C,\\\Rightarrow \left\vert \Psi (t)\right\rangle &=e^{-iHt}\left\vert \Psi (0)\right\rangle .\end{aligned}}$	(3.4)

Now if $H\,\!$ is Hermitian, and it is, then the matrix

	$U=e^{-iHt}$	(3.5)

is unitary. (See Appendix {app:alg}, Sec.{sec:linalg}.) Any transformation on a closed system can be described by a unitary transformation and any unitary transformation can be obtained by the exponentiation of Hermitian matrix.

Aside, for inclusion later in an appendix about Group theory.

It may seem strange to exponentiate a matrix. However, you can define a function of a matrix according to its Taylor expansion. The details of this are primarily unimportant here, but just to show how it goes, it is written out.

The Taylor expansion of an exponential is the following:

	$e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}$	(3.6)

and this can be used to exponentiate a matrix by letting the matrix replace $x/,/!$ in the equation. This can also be used to prove that

	$e^{ix}=\cos x+i\sin x.$	(3.7)

End Aside

The end result and important point is that the evolution of a quantum state is, in general, given by a unitary matrix

	$\left\vert \Psi (t)\right\rangle =U\left\vert \Psi (0)\right\rangle .$	(3.8)

So our objective in quantum information processing is to create a unitary evolution, and eventual measurement, which will produce a particular outcome.