Survival guide for Quantum Computing a general introduction: I

For why you should (and if you could) read the book please see the previous post — https://luysii.wordpress.com/2024/08/26/another-shot-at-quantum-computing/.

However, it’s clear that Reiffel is preaching to the choir, working out early versions of the book on a group in Palo Alto who were hardly neophytes.

She assumes a lot, but eventually gets around to completely defining things, but not all in one place, making it tough on the true neophyte.

If there is one crucial concept to understand quantum computation it is the qubit. Here is how she introduces it

Example: the crucial definition of qubit
On p. 3 we are told that a qubit can take on a continuum of values

On p. 13 we are told that a qubit is also called a quantum bit

We are also told on that page that for a two dimensional complex vector space to be viewed as a qubit two linearly independent states called |0 > and | 1 > must be distinguished

On p. 14 we are told that qubits can take on not only the values of |0 > and | 1 > but also any superposition of these values (a |0 > + b | 1 >) where a and b are complex numbers and |a|^2 + |b|^2 = 1

So I wrote up some notes for myself to clear the air, and hopefully get you through the first few pages.

It assumes you know a smattering of linear algebra — see the previous post for 9 links to an introduction I wrote about linear algebra for quantum mechanics which should give you all you need to know (it really isn’t very much to get started).

So here come some important definitions

p.5 Quantum State Vector/State Vector (written |v > not v or v within an arrow over it) ::=
A vector in a complex vector space (dimension not specified) representing a state (undefined) in a quantum system (undefined)

p. 14
Given V a complex vector space |v1 >,|v2 >, |v3 > in V, a, b in C, the field of complex numbers

a

stands for the complex conjugate of a

Inner Product/ Dot product (written “” ::=
: V x V –> C
: ( |v1 >, |v2 >) –>
such that
l. is greater than or equal to 0.
2.

=
3. + b|v2 )> = a + b

Length of |v >/ Norm of | v> ( Written | |v> | ) ::= sqrt ( v1 | v1 >)
p. 14 All quantum state vectors have length 1. No reason given, but it is so measurement probabilities on state vectors add to 1.

Given V a complex vector space |basis1 >,|basis2 > in V, a, b in C the field of complex numbers

p. 13 Two state system ::= A quantum system (undefined) which can be modeled by a two dimensional complex vector space (C^2). All states are a linear combination of its two basis states { | basis1 >, | basis2 > } ; the { , } means set

p.13 Superposition ::= A linear combination of two basis vectors of V
(written a | basis1 > + b | basis2 >) where both a and b are nonZero; a and b are called amplitudes

This is why two state system is a terrible term. Since a and b are real numbers, the two state system contains an infinite number of linear combinations hence states, not just two

Now with enough background under your belt you can understand the final definition of qubit given on p. 13

Qubit ::= a Two state system with two linearly independent vectors chosen in advance and fixed in subsequent discussions which form a basis. The elements of the qubit are superpositions of the form a | basis1 > + b | basis2 > such that
|a|^2 + |b|^2b = 1
No explanation is given for this, but it follows from postulate 3 of quantum mechanics found in Neilsen and Chuang (aka Mike and Ike) p. 18.

Standard basis of a qubit ::= { |0>, |1> } in that order. Convention when ‘basis’ is used without any sort of modifier, you are to assume that the standard basis is being used.