Quantum Class 3, Tues 20220906
1 Action at a distance
Two qbits can be entangled tho they are far apart.
How can this be?
Newton faced the same conceptual leap.
How can the earth affect the moon's orbit?
Some people posited invisible whirlpools in space that dragged the planets around.
That was falsified by the existence of retrograde orbits.
Now we say that it happens because, in general relativity, the earth's mass bends spacetime, and the moon just follows a geodesic.
My view is that if something is useful but inexplicable, then just use it.

Or, do like Andreas Osiander, Copernicus's editor, "these hypotheses need not be true nor even probable. [I]f they provide a calculus consistent with the observations, that alone is enough." https://en.wikipedia.org/wiki/Nicolaus_Copernicus
If Giordano Bruno had talked evasively like that, maybe he wouldn't have been burnt at the stake in 1600. However he may have been burnt for other reasons.
2 Questions from Quantum Computing 2022 Update
big application.
what is superposition?
why is it powerful?
change in IBM strategy, or, what is circuit knitting?
What is IBM's quantum parallelization?
What is NIST doing re quantum crypto?
How is Intel advancing quantum HW?
What have some other companies done?
3 Hidary, Quantum Computing: An Applied Approach, 2nd ed, chapter 1
Companion site: https://github.com/jackhidary/quantumcomputingbook

Quantum computer: uses properties of quantum mechanics to compute
world is quantum.
compare to classical computer.

quantum properties
superposition
entanglement

state: complete math description of state.
a complex vector.
classical analog: e.g., position of a particle.
Schrodinger's equation computes future state as a function of current state and stuff.
Compare to Newton: future position depends on force etc.
Analogously to Newton, only simple cases have closed form solutions. 2 body not 3 body. hydrogen atom.
Even if there's a closed form solution, it may be chaotic, and so not as useful.
Must simulate when no closed solution. Unfortunately that's all the good cases. College classes use solvable examples, not realistic ones.
See Wolfram, A New Kind of Science.

superposition: linear combo of states is a legal state.
the weights are complex numbers.
everything in quantum mechanics uses complex numbers.
superposition does not work classically.
A qubit $q$ is a quantum analog to a classical bit.
the quantum analog to classical bits 0 and 1 are $0\!\!>$ and $1\!\!>$.

q's state is a superposition (linear combo) of those two basis states:
$q = a0\!\!> + b1\!\!>$ ,
where the weights $a$ and $b$ are complex numbers, and $  a  ^2 +  b  ^2 = 1$.
Note the weird notation (Dirac notation). In $0\!\!>$, $$ is like a left bracket and $>$ like a right one.
It is wrong to think that $q$ is really in one of the two states, but you don't know which one. This is the hidden variable theory. It has been proved experimentally to be false.
$q$ is really in both states simultaneously.
You cannot observe its state, unless it is $0\!\!>$ and $1\!\!>$, in which case you observe $0$ or $1$. This is the classical case.

measurement of a state $\Psi$, and the Born rule (p 5):
Measurement is an operator or matrix, M, applied to a state $\Psi$.
M changes the qbit irreversibly, see the polarization example in the book.
You cannot reclaim the old value.
M has eigenvalues.
Represent $\Psi$ as a linear combo of M's eigenvalues $\psi_i$, considered as a basis.
$\Psi= \alpha\psi_1+\beta\psi_2$, where $\alpha^2+\beta^2=1$.
you can use different basis systems to represent the same vector, and rotate between them.
M changes $\Psi$ state randomly to one of the basis vectors.
the probability of $\Psi$ changing to a particular basis vector is the modulus squared of the weight of that basis vector.
define $z^c$ to be the complex conjugate of $z$.
if $\Psi= \alpha\psi_1+\beta\psi_2$, where $\alpha^c\alpha+\beta^c\beta=1$ then the probability of $\Psi$ changing to $\psi_1$ is $\alpha^c\alpha$. (the Born rule)
$\alpha^c\alpha$ is called the modulus squared.

There are many possible measurement operators available.
You can choose which to apply to $q$. Say, position.
That prevents you from applying the others to $q$, say, momentum, because you don't have $q$ available any more.
Heisenberg uncertainty: measuring, say, position, prevents you from accurately measuring momentum.
$q$, that is, $q$ 's value, can be considered to be a vector of length two: $$\begin{pmatrix} a  0\!\!> \\ b  1\!\!> \end{pmatrix} $$ or simply $$\begin{pmatrix}a\\b\end{pmatrix}$$.
You operate on $q$ with a matrix multiplication: $q_2 = M q$.
Unless $M$ is a measurement operator, it is invertible, so you can go backwards.
Contrast to classical operators like and and or.

Examples of 1qubit gates
not, aka X. page 28.
square root of not
Y, Z
S (rotation by $90^o$), T ($45^o$), phase shift
Hadamark. "it enables us to take a qubit from a definite computational basis state into a superposition of two states"
All operators used in quantum computation other than for measurement must be reversible.  textbook.
No cloning: You cannot copy a qubit, but can move it.

The life cycle of a qubit:
Create a qubit with a classical value, 0 or 1.
Operate on it with matrices, which rotate it to have complex weights.
Measure it by randomly projecting it onto a basis vector.
So far, not very powerful.

a quantum state $\Psi$ usually has many qubits.
compare to a classical byte with 8 classical bits.

However the different qubits in $\Psi$$ might be entangled.
This is very weird and powerful.
Entanglement means that if you measure one qubit then what you observe restricts what would be observed when you measure the other qubit.
Even if the two qubits are 1000 mi apart. This has been experimentally observed.
However that does not let you communicate.
3.1 Entanglement
Crazy counterintuitive idea that's the basis for quantum speedup.

Classical metaphor for entanglement:
Start with a piece of paper.
Tear it into two halves.
Put each half into an envelope, seal them, and mix them up, so that you can't tell which half is in which envelope.
Address and mail one envelope to a friend in Australia, and the other to a friend in Greenland.
When the Australian opens his envelope, he knows what the Greenlander will find in his.
However that doesn't let the Australian send any info to the Greenlander, or vv.
This has been demonstrated with real qubits transported 1000 miles apart.
Entanglement means that if you measure one qubit then what you observe restricts what would be observed when you measure the other qubit.
However that does not let you communicate.
The preceding metaphor is wrong in that it has a hidden variable, the unobserved halfpaper state. That does not happen in quantum physics. With qubits, the states are not fixed until one is observed. I'm trying to get the idea across.
4 Chapter 2: history
Read it on your own, but here are some additions:
The property list on p15 is controversial and seems designed to exclude DWave.
Like for classical computation, the main ideas of quantum computing were proposed before actual machines could be built.
5 Videos to watch for next time
IBM Quantum 2022 Updated Development Roadmap, 18:49, 20220510
Google Quantum AI Update 2022, 25:21, 20220414