But, what is this good for?
Clive Cenxin Aw, Zaw Lin Htoo
3 November 2023,
Friday Hacks 745pm
Quantum Theory
rd
My talk
1.
The
How
of Quantum
(oh no, maths)
2.
The
So What
of Quantum
(can eat or not)
label this “0”
label this “0”
label this “1”
label this “1”
What’s weird is that discrete objects can be in superposition and interfere
discrete
mutually exclusive outcomes
We only ever find a photon “
here
” or “
there
” with certain probabilities
input
output
Born Rule
: Probability of obtaining is the square of the number in front of
Generally, beamsplitters can reflect and transmit different amounts of light
This is like having
quantum bits
with the Born rule that determines the output
qubits
Beamsplitters = Single-qubit Transformations
can be ignored because
we’ll be squaring it
Single-bit Transformations = Single-bit Logic Gates
The action of the beamsplitter reminds us of classical logic gates
Single-qubit Logic Gates
Indeed, there are direct quantum analogues of one-bit logic gates
But the beamsplitter itself has no classical counterpart
And general beamsplitters, even less so!
The closer it is to
, the more likely
you would observe 0
Outcome probabilities
according to Born’s rule
These gates compose as we expect
Summary of Some Single-qubit Gates
A Two-qubit Gate
We also have two-qubit controlled-logic gates
This is the quantum version of XOR, also known as the CNOT gate
Generally, gates apply on the last qubit when the first qubit is
A Simple Circuit
(binary representation
of numbers {0,1,2,3})
Similarly, define
Rotates the target times, where is the state of the control qubits
A Slightly Less Simple Circuit
With control qubits, we have possible values of
It can be verified that the circuit does the same thing as before
Another Simple Circuit
Bringing Them Together
We know that , so let’s rewrite everything in terms of
Remember that
Remember that
What are the outcome probabilities?
Remember that
From the Born rule, we have
If we only care about the probability of the target qubit being
This is just the average probability of measuring given
An Inefficient Algorithm
With additional control gates,
we can make depend on
inputs and such that
Let’s take , ,
and find the probability of
measuring the target qubit
Therefore, this algorithm takes
as input and , and returns
might
be a factor of
is not
a factor of
Not efficient: in the worst case,
need to test all
Constructively interferes when is a prime factor of
Observing gaurantees that is not a prime factor!
Building More Efficient Algorithms
But, similar ideas can be used to build more efficient algorithms
Almost looks like a Fourier transform!
With some modification, the quantum Fourier transform
requires one and two-qubit gates
Compare this to the fast Fourier transform:
So, what is this inherent uncertainty good for?
Quantum
Fourier
transform
Interference of
prime-factor-
related terms
Shor’s algorithm [1994]
for prime factorisation
and discrete logarithm
Exponential speedup to break public-key cryptography schemes
Grover’s algorithm [1994]
for unstructured search
and function inversion
Database search
(quadratic speedup)
HHL algorithm [2008]
for estimating a scalar
value of solution vector
Linear systems
(quadratic speedup)
See “
Quantum algorithms: A survey…
” arXiv:2310.03011 [quant-ph]
A Second Look At The Circuit
This time, with just one control, one target, and
Alice
Alpha Centauri
Bob
Beta Capricorni
Give a qubit each to Alice and Bob, who are far away from each other
What are the possible outcomes of Alice & Bob?
Their outcomes are random, but always correlated
Alice’s and Bob’s qubits are
entangled
!
Correlations persist even when light-years apart
This is what Alice and Bob wanted: a box that generates entanglement
But can they really trust that it is truly quantum entanglement?
Fake It (
Classically
) Till You Make It (
Quantum
)
Scammers try to fool Alice and Bob using the internet
Bob’s detector simply sends its outcome to Alice’s detector, which repeats it
But assuming special relativity is correct, nothing can travel faster than light
So, the scammers cannot use this method
The scammers can fool them with “
hidden
” variables and some h4x0r skills
λ := rand([0,1])
a(λ) = λ
b(λ) = λ
Now, the detectors are just revealing the “hidden” variable
What about a more complicated setup with inputs for Alice & Bob?
generate λ (somehow)
a
x
(λ)
b
y
(λ)
and are randomly chosen, and only when the qubit arrives to them
Alice and Bob play a game with the setup:
Them vs. The Machine
Them vs. The Machine
The game
: Measure and for many randomly-chosen and
The score:
What is the maximum score achievable with a fake device?
this is 0
when this is +2 or –2
this is 0
when this is +2 or –2
always +2 or –2
Therefore, the best score the scammers can obtain is
What is a score that is achievable with a
real quantum
device?
Use and
After some calculations using the Born rule, we find
A real quantum device can achieve scores larger than any
score possible with any classical device
Bell’s theorem [1964], CHSH inequality [1969]
: As long as
(1) Faster-than-light communication is impossible
(2) There
exists
some information that allows
someone
to predict Alice’s and Bob’s measurement outcomes
(2) There
exists
some information that allows
an eavesdropper
to predict Alice’s and Bob’s measurement outcomes
Then,
implies impossibility of predicting outcomes
So, how can entanglement be useful for us?
Randomness
with perfect
correlations
Impossibility
of predicting
outcomes
E91 protocol [Eckert 1991]
for distributing private keys with
security guaranteed by physics
A possibility to fix the broken cryptography?
Bell’s theorem brings together the
key pillars of quantum theory
Randomness due to uncertainty is intrinsic to nature:
it does not—
it can not
—come from our lack of knowledge
Quantum correlations at a distance exist, and cannot be
explained by classical physics (unless special relativity is wrong)
What else has quantum physics done for us?
Quantum Technologies Any% Glitchless Speedrun in 3:13
Quantum Tunnelling
Particle on the left of barrier without
enough energy to pass through
Certain about energy of particle
Somewhat certain about kinetic energy,
hence momentum of particle
Heisenberg relation :
can’t know where exactly the particle is
Nonzero chance of finding it across the
barrier—despite not having enough energy!
Barrier could be physical gap between probe and sample
Tunnelling probability highly sensitive to barrier strength
Sample height can be very precisely measured
Scanning Tunnelling Microscope
Barriers could be that of many adjacent atoms
Energy of electrons small, but tunnelling lets them hop around
Resulting collective behaviour of electrons explain
• Why materials are conductors or insulators
• Why semiconductors work better doped
• How some superconductors work
Solid State Physics
The Photoelectric Effect in Reverse
Photoelectric effect: photons with certain frequencies generate electricity
Conversely: applying electricity emits photons with certain frequencies
add energy
introduce
photon
wants to
join friend
deexcites
“
Photons are bosons and like to be with their friends
” —Peter Shor:
Presence of other photons with the same frequencies
stimulates
even
more photons to be
emitted
,
amplifying
the amount of
light radiated
Light Amplification by Stimulated Emission of Radiation (LASER)
Other materials emit photons when energy is applied to the system
Similar to how quantum effects restrict the minimum frequency of
emitted photons, it can also restrict the maximum frequency
Photons with very specific frequencies—i.e., colours—can be emitted
2023 Nobel Prize in Chemistry
Quantum Dots
Another Use of Entangled Photons
Trace amount of positron-emitting material is ingested
As it travels around your blood stream, the positrons interact
with electrons to generate entangled pairs of photons
Photodetectors around you detect these emitted photon pairs
Because they are perfectly (anti)correlated, it is possible to
triangulate the exact location the photons were generated
Position Emission Tomography (PET) scans
Other Applications of Quantum Theory
Nucleus of atoms in your body act like many qubits,
applied magnetic fields act like logical gates
Magnetic Resonance Imaging (MRI)
Fixed, discrete, energy levels of atoms means
fixed transition time between energy levels
Atomic Clocks
stratum 0 of the NTP
Exact mechanism of the radiative process
All of Nuclear Physics
What else has quantum physics done for us?
Quite a bit, actually
But we don’t call them
quantum technology
: we just call them
technology
1.
The
How
of Quantum
Born rule, Bloch sphere, quantum circuit model
2.
The
So What
of Quantum
Uncertainty
: Quantum fourier transform, Shor’s algorithm
Entanglement
: Bell’s theorem, Quantum key distribution
And some others…
Apart from scanning tunneling microscopes, understanding electrical conductivity,
semiconductors, and superconductors, lasers, quantum dots, PET and MRI scans,
atomic clocks, and all of nuclear physics, what has quantum theory ever done for us?
Thank you for your attention!
A First Approach to the Modern Paradigm,
and What It’s Good For
Quantum Theory
Clive Cenxin Aw, Zaw Lin Htoo
3
rd
November 2023,
Friday Hacks 745pm
Outline
Uncertainty:
Interference of discrete objects
Entanglement:
Beyond classical relations
Conclusion
default