Understanding Quantum Computation Part 1

Information Processing
A physical system is in a way corresponding to a computation machine. The computer (physical system) takes and input (initial state), put it through an algorithm (laws of physics) and produces and output (final state). Where that initial and final states are useful, we can call it information.

The Experiment
In classical physics, the simplest information bearing system in the computer is the “bit”, which has 2 “degrees of freedom”, with the label 1 and 0 each. If we do an experiment on the bit, and get an ouput, we measure that output and say whether the the label was just 1 or 0. We can say that the “value of the bit” is an “observable” that can take the possible values of either 1 or 0. In practice we know that the bit is an abstract system of some electrical sub system and often the observable of that sub system is the voltage V across some component, where it can take the values, e.g. 0 <= V <= +5 volts.

In this case, we have identified 2 observables, namely the voltage V and the value B (of the bit). V can take on many labels between 0V and +5V, we say the set of labels that V can possibly be is the “spectrum of the observable V“. Similarly, the spectrum of B is the set delim{lbrace}{0, 1}{rbrace}.

Quantum Multiverse Interpretation
In order to understand why we do the things we do in the rest of this journey, we need to first explain the multi-universe interpretation of quantum physics. (Bear in mind that this is not the only interpretation of the physical phenomenons that we experience, this is only one that we will use for now)

In reality, every object in our universe has counterparts in other universes. Every event that affects an object also affects its counter parts in all other universes. This phenomenon is known as “quantum interference”. Hence if we were to do an experiment on a system and then consider one observable of the system, I would get one label for each of the counterpart systems in all universes. Although any of these labels will be in the spectrum of the observable, it may not be the same throughout all the universes. Furthermore, each isolated quantum system has many observables that may interact with each other.

Specification of Quantum Systems
In order for meaningful information to be carried in the system, change or transformation must occur. It is however arbitary as to where to draw the line between when the system merely changed its configuration or has ceased to exist. For example we could still theorectically measure the angle between a pair of scissors even if it has been disassembled into parts, but if the metal has been melted down into a pool of liquid, then eventhough the atoms still exists, it will no longer be aproperiate to speak of angles. Therefore despite the transformations, there should be some characteristic of the system where it stays invariant throughout the transformation so that it can be said to continue to exist.

Hence forth, in order to fully specify a quantum system that is of interest to us, we need to define its invariant chracteristics, or “constitution”, namely
– static constitution (observable, spectrum of observable)
– dynamic constitution (law of transfomation of the observable)
– state (as in the Heisenberg state)

Static Constitution
The static constitution of a system refers to all the algebraic equations relating the observables of the system that remains invariant. For example, if a system S has N observables X_1, X_2X_n, then if there is a function F of some X, say F(X_a, X_b ...) = Z that remains true at any time, then it follows that F is an algebra of the observables of the system and it is part of the static constitution of the system. For example, a triangle has 3 observables angle A, B and C and F(A, B, C) = A + B + C = 180 remains true regardless of time transformation of the observables. Thus F is part of the static constitution of the triangle.

The Measurement Technique
To discover the algebra of the quantum observables, we need a few steps more, particularly to get around the problem of measuring the observable.

In the case of the classical computer, the observable, voltage, can take on labels or real numbers in the range, e.g. 0 <= V <= 5, which is a continum of infinite real numbers. A commercial voltmeter we have today only gives an approximate of its true value down to a limited number of decimal places. In many cases such as this one, there are no measurement intruments that can literally measure it precisely, and hence prepare it to a desired value. To simplify the problem, we define another observable, the value of the bit, f(V) and then map V to either 0 or 1. The function f in this case is defined as

f(V) = delim{lbrace}{tabular{000}{0000}{{V  data-recalc-dims== 2.5}{}{1}{V < 2.5}{}{0}}}{rbrace}" title="f(V) = delim{lbrace}{tabular{000}{0000}{{V >= 2.5}{}{1}{V < 2.5}{}{0}}}{rbrace}"/>

Things are slightly trickier in quantum systems because we no longer deal with a single real number value x, but a range of real number values x_1, x_2, x_3 ... x_n across all universes i.e. V is now a set of real number values all of which are between 0 and 5. Correspondingly, the pick and use other functions to map our spectrum of the observable to a new spectrum.

g: x right 1, always mapping to the value 1
h: x right (x - x_1)(x - x_2)(x - x_3)...(x - x_n), always map to 0, and so on. Then if a function F which relates g, h, etc. say F(g, h, ... ) = Z is true regarding of time transformation, then F forms the static consitution of the quantum system. Theorectically, there could be many of such F.

Dynamic Constitution
The dynamic consitution would then be the deriavative of the F with respect to time, meaning that they are the equations the govern how the observables will transform with respect to time while keeping true to the relation F.

State or Heisenberg State
A special function that maps an observable to a real number known as its “Expectation Value” is used to specify its state. The expectation value is the average outcome of the quantum observable over a region of the multi-universe. Therefore this special function predicts what the value of the observable will be in each universe that the observable is measured. But then how can we literally traverse the universes and collect all the outcomes? In other words, where can we get this special function that will help we achieve this feat? It turns out that the “Hermitian Matrix”, its properties and its algebra offers such a capability if you interpret its operations in the following manner that we will be discussing.

Matrix Mechanics
The formulation of Hermitian Matrices was first proposed by Heisenberg, Max Born and Pascual Jordan and it was called Matrix Mechanics. If we consider each observable as a Hermitian Matrix, then the eigenvalues of the matrix constitute the spectrum of the observable representated by the hermitian matrix. Subsequently, we can define the expectation function in the using Hermitian Matrix formulation.

Simplest Quantum System
The simplest physical system is one that has only 2 degress of freedom. An observable with only 2 possible values in its spectrum is called a “boolean observable”. The simplest quantum physical system is therefore a system with all its non-trivial observables being boolean observables and we call it a “qubit”. i.e. if all these observables are represented by Hermitian Matrices, then each of these matrix must have 2 eigenvalues each and must be 2 by 2 matrices according to properties of Hermitian Matrix. In fact, the qubit will have as many boolean observables as the number of possible 2 by 2 matrices.

Sharp Observables
Consider a quantum system in a state defined as
{expectation}(delim{[}{matrix{2}{2}{a b b^{*} c}}{]}) = a
And 2 observables represented by the matrix
hat{Z} = delim{[}{matrix{2}{2}{-1 0 0 1}}{]}, eigenvalues = {-1, 1}
hat{X} = delim{[}{matrix{2}{2}{0 1 1 0}}{]}, eigenvalues = {-1, 1}
That is to say, the observables Z and X each has the spectrum of possible values -1 or 1. But when we apply the expectation function, {expectation}(hat{Z}) = -1, while {expectation}(hat{X}) = 0. This means across the multiverse, the outcome of hat{Z} is always -1 while the outcome of hat{X} is -1 in half the universes and 1 in the other half. We say that hat{Z} is sharp in this case, while hat{X} is not sharp. Remember that hat{Z} and hat{X} are observables of the same quantum system. That is to say although hat{Z} turns out to be the same in the multiverse, the quantum system as a whole is different across the same region. In fact the “uncertainty principle” tells us that even if we try to make hat{X} sharp, some other observable will inevitably become unsharp.