Ciro Santilli @cirosantilli 34

 Incoming links: Complex number

Algebra over a field  Updated 2024-12-15  +Created 1970-01-01

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A vector field with a bilinear map into itself, which we can also call a "vector product".

Note that the vector product does not have to be neither associative nor commutative.

Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples

complex numbers, i.e. $R^{2}$ with complex number multiplication
$R^{3}$ with the cross product
quaternions, i.e. $R^{4}$ with the quaternion multiplication

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Bra-ket notation  Updated 2024-12-15  +Created 1970-01-01

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Notation used in quantum mechanics.

Ket is just a vector. Though generally in the context of quantum mechanics, this is an infinite dimensional vector in a Hilbert space like

L^{2}

Bra is just the dual vector corresponding to a ket, or in other words projection linear operator, i.e. a linear function which can act on a given vector and returns a single complex number. Also known as... dot product.

For example:

(a x) ∣ y ⟩

(1)

is basically a fancy way of saying:

x \cdot y

(2)

that is: we are taking the projection of

y

along the

x

direction. Note that in the ordinary dot product notation however, we don't differentiate as clearly what is a vector and what is an operator, while the bra-ket notation makes it clear.

The projection operator is completely specified by the vector that we are projecting it on. This is why the bracket notation makes sense.

It also has the merit of clearly differentiating vectors from operators. E.g. it is not very clear in

x \cdot y

that

x

is an operator and

y

is a vector, except due to the relative position to the dot. This is especially bad when we start manipulating operators by themselves without vectors.

This notation is widely used in quantum mechanics because calculating the probability of getting a certain outcome for an experiment is calculated by taking the projection of a state on one an eigenvalue basis vector as explained at: Section "Mathematical formulation of quantum mechanics".

Making the projection operator "look like a thing" (the bra) is nice because we can add and multiply them much like we can for vectors (they also form a vector space), e.g.:

a x + a y

(3)

just means taking the projection along the

x + y

direction.

Ciro Santilli thinks that this notation is a bit over-engineered. Notably the bra's are just vectors, which we should just write as usual with

v

... the bra thing makes it look scarier than it needs to be. And then we should just find a different notation for the projection part.

Maybe Dirac chose it because of the appeal of the women's piece of clothing: bra, in an irresistible call from British humour.

But in any case, alas, we are now stuck with it.

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Cayley-Dickson construction  Updated 2024-12-15  +Created 1970-01-01

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Constructs the quaternions from complex numbers, octonions from quaternions, and keeps doubling like this indefinitely.

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CNOT gate  Updated 2024-12-15  +Created 1970-01-01

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The CNOT gate is a controlled quantum gate that operates on two qubits, flipping the second (operand) qubit if the first (control) qubit is set.

This gate is the first example of a controlled quantum gate that you should study.

⎣ ⎢ ⎢ ⎢ ⎡ 1000010000010010 ⎦ ⎥ ⎥ ⎥ ⎤

(1)

Equation 1.

CNOT gate matrix

Figure 1.
CNOT gate symbol
. Source. The symbol follow the generic symbol convention for controlled quantum gates shown at Figure "Generic controlled quantum gate symbol", but replacing the generic "U" with the Figure "Quantum NOT gate symbol".

To understand why the gate is called a CNOT gate, you should think as follows.

First let's produce a generic quantum state vector where the control qubit is certain to be 0.

On the standard basis:

∣ 00 ⟩ ∣ 01 ⟩ ∣ 10 ⟩ ∣ 11 ⟩

(2)

we see that this means that only

∣ 00 ⟩

and

∣ 01 ⟩

should be possible. Therefore, the state must be of the form:

⎣ ⎢ ⎢ ⎢ ⎡ x y 00 ⎦ ⎥ ⎥ ⎥ ⎤

(3)

where

x

and

y

are two complex numbers such that

∣ x ∣ + ∣ y ∣ = 1.0

If we operate the CNOT gate on that state, we obtain:

⎣ ⎢ ⎢ ⎢ ⎡ 1000010000010010 ⎦ ⎥ ⎥ ⎥ ⎤ \times ⎣ ⎢ ⎢ ⎢ ⎡ x y 00 ⎦ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎡ x y 00 ⎦ ⎥ ⎥ ⎥ ⎤

(4)

and so the input is unchanged as desired, because the control qubit is 0.

If however we take only states where the control qubit is for sure 1:

⎣ ⎢ ⎢ ⎢ ⎡ 1000010000010010 ⎦ ⎥ ⎥ ⎥ ⎤ \times ⎣ ⎢ ⎢ ⎢ ⎡ 00 x y ⎦ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎡ 00 y x ⎦ ⎥ ⎥ ⎥ ⎤

(5)

Therefore, in that case, what happened is that the probabilities of

∣ 10 ⟩

and

∣ 11 ⟩

were swapped from

x

and

y

y

and

x

respectively, which is exactly what the quantum NOT gate does.

So from this we understand more concretely what "the gate only operates if the first qubit is set to one" means.

Now go and study the Bell state and understand intuitively how this gate is used to produce it.

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Dirac equation  Updated 2024-12-15  +Created 1970-01-01

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Adds special relativity to the Schrödinger equation, and the following conclusions come basically as a direct consequence of this!

Experiments explained:

spontaneous emission coefficients.
fine structure, notably for example Dirac equation solution for the hydrogen atom
antimatter
particle creation and annihilation

Experiments not explained: those that quantum electrodynamics explains like:

Lamb shift
TODO: quantization of the electromagnetic field as photons?

The Dirac equation is a set of 4 partial differential equations on 4 complex valued wave functions. The full explicit form in Planck units is shown e.g. in Video 1. "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=1010:

i \partial_{t} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{1} ψ_{2} ψ_{3} ψ_{4} ⎦ ⎥ ⎥ ⎥ ⎤ = - i \partial_{x} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{4} ψ_{3} ψ_{2} ψ_{1} ⎦ ⎥ ⎥ ⎥ ⎤ + \partial_{y} ⎣ ⎢ ⎢ ⎢ ⎡ - ψ_{4} ψ_{3} - ψ_{2} ψ_{1} ⎦ ⎥ ⎥ ⎥ ⎤ - i \partial_{z} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{3} - ψ_{4} ψ_{1} - ψ_{2} ⎦ ⎥ ⎥ ⎥ ⎤ + m ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{1} ψ_{2} - ψ_{3} - ψ_{4} ⎦ ⎥ ⎥ ⎥ ⎤

(1)

Equation 1.

Expanded Dirac equation in Planck units

Then as done at physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600 from why are complex numbers used in the Schrodinger equation?, we could further split those equations up into a system of 8 equations on 8 real-valued functions.

Video 1.

Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)

Source.

Video 2.

PHYS 485 Lecture 14: The Dirac Equation by Roger Moore (2016)

Source.

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Discrete Fourier transform  Updated 2024-12-15  +Created 1970-01-01

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Input: a sequence of

N

complex numbers

x_{k}

Output: another sequence of

N

complex numbers

X_{k}

such that:

x_{n} = \frac{1}{N} \sum_{k = 0}^{N - 1} X_{k} e^{i 2 π \frac{k n}{N}}

(1)

Intuitively, this means that we are braking up the complex signal into

N

sinusoidal frequencies:

$X_{0}$ : is kind of magic and ends up being a constant added to the signal because $e^{i 2 π \frac{k n}{N}} = e^{0} = 1$
$X_{1}$ : sinusoidal that completes one cycle over the signal. The larger the $N$ , the larger the resolution of that sinusoidal. But it completes one cycle regardless.
$X_{2}$ : sinusoidal that completes two cycles over the signal
...
$X_{N - 1}$ : sinusoidal that completes $N - 1$ cycles over the signal

and is the amplitude of each sine.

We use Zero-based numbering in our definitions because it just makes every formula simpler.

Motivation: similar to the Fourier transform:

compression: a sine would use N points in the time domain, but in the frequency domain just one, so we can throw the rest away. A sum of two sines, only two. So if your signal has periodicity, in general you can compress it with the transform
noise removal: many systems add noise only at certain frequencies, which are hopefully different from the main frequencies of the actual signal. By doing the transform, we can remove those frequencies to attain a better signal-to-noise

In particular, the discrete Fourier transform is used in signal processing after a analog-to-digital converter. Digital signal processing historically likely grew more and more over analog processing as digital processors got faster and faster as it gives more flexibility in algorithm design.

Sample software implementations:

numpy.fft, notably see the example: numpy/fft.py

Figure 1.
DFT of $2 sin (t) + cos (4 t)$ with 25 points
. This is a simple example of a discrete Fourier transform for a real input signal. It illustrates how the DFT takes N complex numbers as input, and produces N complex numbers as output. It also illustrates how the discrete Fourier transform of a real signal is symmetric around the center point.

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Frobenius theorem (real division algebras)  Updated 2024-12-15  +Created 1970-01-01

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There are 3: real numbers, complex numbers and quaternions.

Notably, the octonions are not associative.

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Hermitian form  Updated 2024-12-15  +Created 1970-01-01

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The complex number analogue of a symmetric bilinear form.

The prototypical example of it is the complex dot product.

Note that this form is neither strictly symmetric, it satisfies:

< x, y > = \overline{< y, x >}

(1)

where the over bar indicates the complex conjugate, nor is it linear for complex scalar multiplication on the second argument.

Bibliography:

mathworld.wolfram.com/HermitianForm.html

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Polynomial over a field  Updated 2024-12-15  +Created 1970-01-01

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By default, we think of polynomials over the real numbers or complex numbers.

However, a polynomial can be defined over any other field just as well, the most notable example being that of a polynomial over a finite field.

For example, given the finite field of order 9,

G P (3)

and with elements

{0, 1, 2}

, we can denote polynomials over that ring as

G P (3) [x]

(1)

where

x

is the variable name.

For example, one such polynomial could be:

P (x) = 2 x^{4} + x^{2} + 2

(2)

and another one:

Q (X) = x^{3} + 2 x^{2} + 2

(3)

Note how all the coefficients are members of the finite field we chose.

Given this, we could evaluate the polynomial for any element of the field, e.g.:

P (0) = 2 (0 \times 0 \times 0 \times 0) + (0 \times 0) + 2 = 2 P (1) = 2 (1 \times 1 \times 1 \times 1) + (1 \times 1) + 2 = 2 (1) + 1 + 2 = 2 P (2) = 2 (2 \times 2 \times 2 \times 2) + (2 \times 2) + 2 = 2 (16

(4)

and so on.

We can also add polynomials as usual over the field:

P (x) + Q (x) = 2 x^{4} + x^{3} + (1 + 2) x^{2} + (2 + 2) = 2 x^{4} + x^{3} + (0) x^{2} + 1 = 2 x^{4} + x^{3} + 1

(5)

and multiplication works analogously.

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Programmer's model of quantum computers  Updated 2024-12-15  +Created 1970-01-01

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This is a quick tutorial on how a quantum computer programmer thinks about how a quantum computer works. If you know:

what a complex number is
how to do matrix multiplication
what is a probability

a concrete and precise hello world operation can be understood in 30 minutes.

Although there are several types of quantum computer under development, there exists a single high level model that represents what most of those computers can do, and we are going to explain that model here. This model is the is the digital quantum computer model, which uses a quantum circuit, that is made up of many quantum gates.

Beyond that basic model, programmers only may have to consider the imperfections of their hardware, but the starting point will almost always be this basic model, and tooling that automates mapping the high level model to real hardware considering those imperfections (i.e. quantum compilers) is already getting better and better.

This model is very simple to understand, being only marginally more complex than that of a classical computer, see also: quantumcomputing.stackexchange.com/questions/6639/is-my-background-sufficient-to-start-quantum-computing/14317#14317

The situation of quantum computers today in the 2020's is somewhat analogous to that of the early days of classical circuits and computers in the 1950's and 1960's, before CPU came along and software ate the world. Even though the exact physics of a classical computer might be hard to understand and vary across different types of integrated circuits, those early hardware pioneers (and to this day modern CPU designers), can usefully view circuits from a higher level point of view, thinking only about concepts such as:

logic gates like AND, NOR and NOT
a clock + registers

as modelled at the register transfer level, and only in a separate compilation step translated into actual chips. This high level understanding of how a classical computer works is what we can call "the programmer's model of a classical computer". So we are now going to describe the quantum analogue of it.

The way quantum programmers think about a quantum computer in order to program can be described as follows:

the input of a N qubit quantum computer is a vector of dimension N containing classic bits 0 and 1
the quantum program, also known as circuit, is a $2^{n} \times 2^{n}$ unitary matrix of complex numbers $Q \in C^{2^{n}} \times C^{2^{n}}$ that operates on the input to generate the output
the output of a N qubit computer is also a vector of dimension N containing classic bits 0 and 1

To operate a quantum computer, you follow the step of operation of a quantum computer:

set the input qubits to classic input bits (state initialization)
press a big red "RUN" button
read the classic output bits (readout)

Each time you do this, you are literally conducting a physical experiment of the specific physical implementation of the computer:

setup your physical system to represent the classical 0/1 inputs
let the state evolve for long enough
measure the classical output back out

and each run as the above can is simply called "an experiment" or "a measurement".

The output comes out "instantly" in the sense that it is physically impossible to observe any intermediate state of the system, i.e. there are no clocks like in classical computers, further discussion at: quantum circuits vs classical circuits. Setting up, running the experiment and taking the does take some time however, and this is important because you have to run the same experiment multiple times because results are probabilistic as mentioned below.

Unlike in a classical computer, the output of a quantum computer is not deterministic however.

But the each output is not equally likely either, otherwise the computer would be useless except as random number generator!

This is because the probabilities of each output for a given input depends on the program (unitary matrix) it went through.

Therefore, what we have to do is to design the quantum circuit in a way that the right or better answers will come out more likely than the bad answers.

We then calculate the error bound for our circuit based on its design, and then determine how many times we have to run the experiment to reach the desired accuracy.

The probability of each output of a quantum computer is derived from the input and the circuit as follows.

First we take the classic input vector of dimension N of 0's and 1's and convert it to a "quantum state vector"

q_{i n}

of dimension

2^{n}

q_{i n} \in C^{2^{n}}

(1)

We are after all going to multiply it by the program matrix, as you would expect, and that has dimension

2^{n} \times 2^{n}

Note that this initial transformation also transforms the discrete zeroes and ones into complex numbers.

For example, in a 3 qubit computer, the quantum state vector has dimension

2^{3} = 8

and the following shows all 8 possible conversions from the classic input to the quantum state vector:

000 -> 1000 0000 == (1.0, 0.0, 0.0, 0.0,  0.0, 0.0, 0.0, 0.0)
001 -> 0100 0000 == (0.0, 1.0, 0.0, 0.0,  0.0, 0.0, 0.0, 0.0)
010 -> 0010 0000 == (0.0, 0.0, 1.0, 0.0,  0.0, 0.0, 0.0, 0.0)
011 -> 0001 0000 == (0.0, 0.0, 0.0, 1.0,  0.0, 0.0, 0.0, 0.0)
100 -> 0000 1000 == (0.0, 0.0, 0.0, 0.0,  1.0, 0.0, 0.0, 0.0)
101 -> 0000 0100 == (0.0, 0.0, 0.0, 0.0,  0.0, 1.0, 0.0, 0.0)
110 -> 0000 0010 == (0.0, 0.0, 0.0, 0.0,  0.0, 0.0, 1.0, 0.0)
111 -> 0000 0001 == (0.0, 0.0, 0.0, 0.0,  0.0, 0.0, 0.0, 1.0)

This can be intuitively interpreted as:

if the classic input is 000, then we are certain that all three bits are 0.
Therefore, the probability of all three 0's is 1.0, and all other possible combinations have 0 probability.
if the classic input is 001, then we are certain that bit one and two are 0, and bit three is 1. The probability of that is 1.0, and all others are zero.
and so on

Now that we finally have our quantum state vector, we just multiply it by the unitary matrix

Q

of the quantum circuit, and obtain the

2^{n}

dimensional output quantum state vector

q_{o u t}

q_{o u t} = Q q_{i n}

(2)

And at long last, the probability of each classical outcome of the measurement is proportional to the square of the length of each entry in the quantum vector, analogously to what is done in the Schrödinger equation.

For example, suppose that the 3 qubit output were:

q_{o u t} = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ \frac{3}{2} 0.0 \frac{1}{2} 0.0 0.0 0.0 0.0 0.0 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤

(3)

Then, the probability of each possible outcomes would be the length of each component squared:

P (000) P (001) P (010) P (011) P (100) P (101) P (110) P (111) = ∣ ∣ ∣ ∣ ∣ ∣ \frac{3}{2} ∣ ∣ ∣ ∣ ∣ ∣^{2} = ∣ 0 ∣^{2} = ∣ ∣ ∣ ∣ ∣ ∣ \frac{1}{2} ∣ ∣ ∣ ∣ ∣ ∣^{2} = ∣ 0 ∣^{2} = ∣ 0 ∣^{2} = ∣ 0 ∣^{2} = ∣ 0 ∣^{2} = ∣ 0 ∣^{2} = \frac{3}{2}^{2} = 0^{2} = \frac{1}{2}^{2} = 0^{2} = 0^{2} = 0^{2} = 0^{2} = 0^{2} = \frac{3}{4} = 0 = \frac{1}{4} = 0 = 0 = 0 = 0 = 0

(4)

i.e. 75% for the first, and 25% for the third outcomes, where just like for the input:

first outcome means 000: all output bits are zero
third outcome means 010: the first and third bits are zero, but the second one is 1

All other outcomes have probability 0 and cannot occur, e.g.: 001 is impossible.

Keep in mind that the quantum state vector can also contain complex numbers because we are doing quantum mechanics, but we just take their magnitude in that case, e.g. the following quantum state would lead to the same probabilities as the previous one:

∣ ∣ ∣ ∣ ∣ ∣ \frac{1 + 2 i}{2} ∣ ∣ ∣ ∣ ∣ ∣^{2} ∣ ∣ ∣ ∣ ∣ \frac{i}{2} ∣ ∣ ∣ ∣ ∣^{2} = \frac{1 ^{2} + 2 ^{2}}{2 ^{2}} = \frac{1 ^{2}}{2 ^{2}} = \frac{3}{4} = \frac{1}{4}

(5)

This interpretation of the quantum state vector clarifies a few things:

the input quantum state is just a simple state where we are certain of the value of each classic input bit
the matrix has to be unitary because the total probability of all possible outcomes must be 1.0
This is true for the input matrix, and unitary matrices have the probability of maintaining that property after multiplication.
Unitary matrices are a bit analogous to self-adjoint operators in general quantum mechanics (self-adjoint in finite dimensions implies is stronger)
This also allows us to understand intuitively why quantum computers may be capable of accelerating certain algorithms exponentially: that is because the quantum computer is able to quickly do an unitary matrix multiplication of a humongous $2^{N}$ sized matrix.
If we are able to encode our algorithm in that matrix multiplication, considering the probabilistic interpretation of the output, then we stand a chance of getting that speedup.

Bibliography:

arxiv.org/pdf/1804.03719.pdf Quantum Algorithm Implementations for Beginners by Abhijith et al. 2020

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Quadratic form  Updated 2024-12-15  +Created 1970-01-01

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Multivariate polynomial where each term has degree 2, e.g.:

f (x, y) = 2 y^{2} + 10 y x + x^{2}

(1)

is a quadratic form because each term has degree 2:

$y^{2}$
$x y$
$x^{2}$

but e.g.:

f (x, y) = 2 y^{2} + 10 y x + x^{3}

(2)

is not because the term

x^{3}

has degree 3.

More generally for any number of variables it can be written as:

f (x_{1}, x_{2}, \dots, x_{n}) = \sum_{i, j} a_{i} a_{j} x_{i} x_{j}

(3)

There is a 1-to-1 relationship between quadratic forms and symmetric bilinear forms. In matrix representation, this can be written as:

x^{T} B x

(4)

where

x

contains each of the variabes of the form, e.g. for 2 variables:

x = [x, y]

(5)

Strictly speaking, the associated bilinear form would not need to be a symmetric bilinear form, at least for the real numbers or complex numbers which are commutative. E.g.:

[x y] [0210] [x y] = [x y] [y 2 x] = x y + 2 y x = 3 x y

(6)

But that same matrix could also be written in symmetric form as:

[0 1.5 1.5 0]

(7)

so why not I guess, its simpler/more restricted.

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Quaternion  Updated 2024-12-15  +Created 1970-01-01

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Kind of extends the complex numbers.

Some facts that make them stand out:

one of the only three real associative division algebras in addition to the real numbers and complex numbers, according to the classification of associative real division algebras
the simplest non-commutative division algebra. Contrast for example with complex numbers where multiplication is commutative

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Underlying field of a vector space  Updated 2024-12-15  +Created 1970-01-01

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Every vector space is defined over a field.

E.g. in

R^{3}

, the underlying field is

R

, the real numbers. And in

C^{2}

the underlying field is

C

, the complex numbers.

Any field can be used, including finite field. But the underlying thing has to be a field, because the definitions of a vector need all field properties to hold to make sense.

Elements of the underlying field of a vector space are known as scalar.

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