I am having to consider fundamental principles of mathematics, namely the graphing of quadratic equations. The preferred form for solving using the DWave system relies heavily upon quadratic equations.
I believe the EI (either) logic gate and balanced AND gate have logical functions that can be graphed. If I identify the relationship between the logic gate and mathematical function which describes it visually in the graphically plotted form...
..a QUANTIFIABLE system of logic gates, a most fundamental unit (EI) and mathematics to accompany them into the quadratic implementations of quantum computing logic.
I am not sure the logic has been graphed before. I know the electrical properties for the components of which the gate is comprised are graphed frequently ..but the logic or function of the gate should have a distinct mathematical representation.
..I have found a publication which details a method for graphically representing the ordering and interconnections of localities within a decision structure but no quadrant based graphing of function.
The mapping of logic gate function to the quadrants of a graph might yield further insight into the radial aspects of logic which could help us to pinpoint the neglected science to explain the four wasteful steps contained within each XOR and NAND gate.
I need to explain the "Hokey Pokey" dance and turn around that happens in all digital electronics.
So far, I know that the unidirectional nature of direct current made the design of the most fundamental logic gate difficult and as of yet impossible.
Quantum bits have a bi-directional function that makes the simple unit EI (either) immediately possible.
It took a generation of transistorized logic for us to literally "screw" our way to perceiving another dimension in four ninety degree phases. This now yields a smaller dimension of digital logic that allows us to express more accurately and efficiently logical decision structures. These should translate from logic structure design to circuit to new methods of quantifying logic and graphical/symbolic representation (they are going to love this, it looks like Tron from here).
The more fun logic design tool can wait till the two-dimensional quadrant based radial graphing is expressed.
At the center and origin of a two dimensional quadrant based graph, a screw can be turned from -1 to +1 in both dimensions simultaneously.
This should allow the EI (either) logic gate to interface DWave's QUBO (Quadratic Unconstrained Binary Optimization).
Coffee break.
The two decision related variables can move a point radially through the four quadrants, giving access to π (pi) related functions further enhancing our ability to model logic.
..if a circle contained 1000 degrees radially, a right angle would be 250 degrees and one vertex of a triangle 333.33 degrees.
A 1 kilo-radian metric circle
Interesting, pi 3.14-> an infinitely irrational number and so would be the vertex of a triangle at 333.33->
We needed OR (three) to choose in a singular way by negation of AND (two). EI (one, either) accomplishes the exception operation without negation or wasted function.
If a circle had 3π radians (9.42477796-> degrees), one vertex of a triangle would contain pi radians (3.1415-> degrees).
What relationship does this exercise express by mapping radians?
3π maps π to the vertices of a triangle.
4π maps π to the vertices of a square.
5π maps π to the vertices of a pentagon.
6π maps π to the vertices of a hexagon.
Multiplication and division of radians by factors can also map a chosen value to the corresponding shape.
The number of radians per vertex is a variable while the shape or factor is the coefficient. Mapping relative radian values across scaled dimensions.
This is so useful! ..rate of change can be plotted in relation to the third dimension that designates time scaling. With the basic mathematical relationship, one can scale a function to fine detail for sample rates and frequency analysis.
Technically it is possible with such mathematics to model a sine wave that is as true to the analogue form as causality, computation and our quantum mechanics will allow.
A model of one cycle per second, a sine wave of smooth transition yet artificial ..or not? ..yes but a fake somehow too perfect ..it needs a probability of radiant interference. There is another challenge, modeling white noise that is exactly as random as real noise.
The sun is not up and I am on this one again, I love this build.
EI (either) logic gate as applied to the DWave quantum annealing process.
Possible states of binary logic gates (balanced) for the quadratic form:
1: +1, +1
2: -1, -1
3: +1, -1
4: -1, +1
This can translate by representing the classic system of logic gates more efficiently while development of new logic systems and ways of representing them can include newly confirmed quantum mechanics.
The mapping necessary to bring into existence a system of balanced logic gates that allow the most fundamental operation EI (the either exception) to be declared.
The EI (either) logic gate
1: +1, +1 false
2: -1, -1 false
3: +1, -1 true
4: -1, +1 true
The AND logic gate in the balanced form:
1: +1, +1 true
2: -1, -1 true
3: +1, -1 false
4: -1, +1 false
Two 45 degree vectors which can also be mapped to a four radian circle. A 4 degree circle, the four states mapped using radial elements of mathematics.
Plotted at the center on a four quadrant graph, the balanced logic gates (EI/ AND) are represented by a two diagonal vector X.
The logic fits neatly, time for sun up at the cafe. I cannot share what I build with others (no one would understand and agree) so I enjoy my coffee and smile. :)
Using this method we can interface quantum mechanical binary logic with existing transistorized binary logic, coldware to hardware.
Omg ..we can map any decision structure to another point or transpose it elsewhere relative three given scaled dimensions.
X is the graphical representation for the quantum superposition of EI and AND logic.
We need at least two neurons in communication, the same is true of a logic gate. A system with a minimum of two channels allowing choice. One gate for exception (EI), the other for inclusion (AND). For a four quadrant graph, these plot a two vector X shape over the center point.
This suggests that part of each decision structure exists on the imaginary plane (negative and i numbers). The other part of the decision exists in the plane of real numbers (positive or "real").
There is a third dimension of time for quantification in samples decided by mathematical scaling for the dimension.
What this finally yields are states of one qbit channel that can be represented as a waveform of state changes over time. Any qbit can be graphed and scaled over a sample rate, this includes the perpendicular "superimposed" quantum state that can graph/scale zero amplitude.
The state of two qbits acting as one logical gate can also be represented as a frequency scaled over time. The superimposed quantum state is quantified as zero, radially perpendicular to the amplitudes (-1, +1) of the EI and AND logical states.
Developing a theoretical implementation for balanced logic gates EI (either) and AND. This includes a decision structure design method.
Decision structure ordering can be represented by locality among steps. For a more defined solution or information handling, sides connecting vertices are variables which present locality by quantifiable scale rather than whole steps.
The vertices of a decision structure shape are also called nodes, connected by decision paths (sides) which can themselves contain quantifiable and variable information.
So far the EI (either) operator seems to function properly, tested by mathematical and logical probe.
EI (either) logic interfaced well with:
Binary
Boolean logic
Transistorized decision structures
Quadratic functions, 2D, two variables and coefficients.
Quartic functions, 4D, four variables and coefficients.
Schrodinger's equation, waveform mathematics with relation to quantum physics and general relativity.
Superconducting Coldware, quantum superposition IC's (integrated circuit). For cyberpunk fans, a new type of "ICE" to foreshadow the incoming tech level changes.
The four variables of the quartic function suit the EI (either) logic well, the balanced truth table must have four elements. So far the circuits which make such behavior possible must be in a superconducting environment.
It just so happens that the state of two entangled quantum bits are described by quantum physics using quartic functions.
This is not the last of the quartic function and EI cooperating, there is interesting development to be tested and confirmed.
The final theoretical test, can the EI (either) logic gate represent and manipulate quantum states as well as the Hadamard gate which is the current standard for the quantum annealing processor.
I am so interested to know if the EI (either) logic gate might be an alternative to the current Hadamard gate system. That would create a page for quantum textbooks.
It looks bad for Hadamard, I see unbalanced four element logic tables. More lopsided 3 - 1 logic, this time implemented in coldware. That was exactly the problem with truth tables resulting from the limitations of transistorized circuits.
The case for implementation of EI (either) logic is more affirmative and defined by the inquiry.