mod02lec12 - Quantum Computing Concepts: Entanglement and Interference - Part 2
Summary
TLDRThis educational video script delves into the quantum mechanics principles of superposition, entanglement, and interference, crucial for quantum computing. It explains quantum teleportation, emphasizing its reliance on classical communication channels due to the no-cloning theorem. The script also touches on the no-cloning theorem, which prohibits the duplication of arbitrary qubits, and the potential of algorithms like Grover's, which utilizes quantum interference. The lecture aims to transition from theory to practical application, guiding viewers on creating quantum circuits and programming with Qiskit.
Takeaways
- 🧬 **Quantum Teleportation Basics**: The script explains the concept of quantum teleportation, where a qubit's state is transferred from one location to another using entanglement and classical communication.
- 🔄 **Transformations in Teleportation**: It details how Bob must apply bit flips and phase flips (using Pauli X and Z gates) to recover the original state based on Alice's measurements.
- 🔗 **Entanglement Swapping**: The script introduces entanglement swapping, a process where entanglement is transferred between parties to create a new entangled pair.
- 🚫 **No-Cloning Theorem**: It emphasizes the no-cloning theorem in quantum computing, which states that it's impossible to create an identical copy of an arbitrary unknown quantum state.
- 🌐 **Quantum vs Classical Communication**: The script clarifies that while quantum teleportation is instantaneous at the quantum level, the classical communication of measurement results is limited by the speed of light.
- 🌌 **Experimental Verification**: Quantum teleportation has been experimentally verified, with examples including teleportation from the ground to a satellite and between islands.
- ⚛️ **Quantum Gates and Operations**: The script discusses how quantum gates like the Hadamard gate can create superpositions, and how these gates are used in combination to perform quantum computations.
- 🌟 **Interference in Quantum Computing**: It explains how interference, a wave-like property, is used in quantum algorithms to amplify correct answers and diminish incorrect ones.
- 🛠️ **Building Quantum Algorithms**: The process of constructing quantum algorithms involves determining state space, applying transformations, and using entanglement and interference to solve problems.
- 📚 **Practical Application**: The script concludes by encouraging the application of theoretical knowledge to practical exercises and programming on real quantum computers using frameworks like Qiskit.
Q & A
What is the significance of the state 'phi' in quantum teleportation as described in the script?
-In quantum teleportation, 'phi' represents the qubit state that Alice wants to transmit to Bob. Bob needs to perform certain transformations based on Alice's measurements to recreate the original state 'phi' on his end.
How does Bob recover the original qubit state sent by Alice if he receives the bit '01' from her?
-If Bob receives '01' from Alice, he needs to perform a bit flip on his state to change it from 'alpha_1 + beta_0' to 'alpha_0 + beta_1', which corresponds to the desired state '5'.
What quantum gate is used to perform a phase flip in the context of quantum teleportation?
-A phase flip in quantum teleportation is performed using the Pauli-Z gate, often denoted as Z gate.
What is entanglement swapping and how is it achieved?
-Entanglement swapping is a process that allows two parties who do not share an entangled state to become entangled through a series of quantum operations involving previously shared entangled states. It is achieved by using the teleportation protocol.
What is the no-cloning theorem in quantum computing, and why is it important?
-The no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state. This is important because it ensures the security of quantum information and sets a fundamental difference between quantum and classical computing.
Why can't a CNOT gate be used to copy an arbitrary qubit?
-A CNOT gate cannot be used to copy an arbitrary qubit because it would require the second qubit to be in the same state as the first, which contradicts the principle that the output of the XOR operation (CNOT) must equal the first qubit.
How does interference play a role in quantum computing?
-Interference in quantum computing is used to amplify certain states and diminish others by applying suitable phase changes. This is crucial in algorithms like Grover's algorithm, where interference is used to increase the probability of the correct answer.
What is the purpose of the Hadamard gate in the context of quantum algorithms?
-The Hadamard gate is used in quantum algorithms to create an equal superposition of states. By applying the Hadamard gate to each qubit, the system enters a state of superposition, which is a prerequisite for many quantum algorithms.
How does the measurement process in a quantum system relate to the superposition state?
-In a quantum system, measurement collapses the superposition state to a definite state. To understand the superposition state, one must perform multiple measurements and analyze the probability distribution of the outcomes.
What is the role of the IBM Quantum Experience in learning quantum computing?
-The IBM Quantum Experience provides a platform where users can create and run quantum circuits on real quantum computers. It is an essential tool for learning and experimenting with quantum computing algorithms.
Outlines
🔄 Quantum Teleportation Protocol
The paragraph discusses the quantum teleportation protocol, explaining how Bob can recover a qubit sent by Alice based on her measurements. It covers different scenarios where Alice sends different bit values (mn) and how Bob responds with transformations like bit flip (X gate) or phase flip (Z gate). The explanation includes the use of EPR pairs and how different states require different gate applications. The paragraph also touches on the possibility of using different initial states for the EPR pairs and the generalizability of the protocol.
🚀 Entanglement Swapping and Quantum Teleportation
This section delves into the concept of entanglement swapping, where Alice can help Bob and Carol share an entangled state even if they don't directly share one. It uses the teleportation protocol to achieve this by Alice communicating her part of an entangled state to Carol. The paragraph also addresses the misconception that quantum teleportation allows for faster-than-light communication, clarifying that classical communication channels still apply. Additionally, it introduces the no-cloning theorem, which states that it's impossible to create an identical copy of an arbitrary qubit, contrasting this with classical computing's ability to copy information freely.
🚫 The No-Cloning Theorem
The paragraph explores the no-cloning theorem in quantum computing, which prohibits the creation of identical copies of an arbitrary unknown quantum state. It explains why quantum circuits cannot clone qubits, even with operations like the CNOT gate. The explanation involves a thought experiment where a cloning circuit is assumed to exist, leading to a contradiction when considering the inner product of states. The conclusion is that such a circuit cannot copy non-orthogonal states, thus upholding the no-cloning theorem.
🌌 Quantum Interference and Algorithm Construction
This section introduces the concept of quantum interference, drawing parallels with classical wave interference. It explains how phase changes can be used in quantum circuits to amplify or diminish certain states, which is crucial for quantum algorithms. The paragraph outlines the process of constructing quantum algorithms, starting from determining the state space and inputs, applying transformations, and using entanglement and interference to bias towards correct answers. It mentions Grover's algorithm as an example where interference plays a significant role. The summary also covers the process of measuring quantum states and interpreting the results through probability distributions.
Mindmap
Keywords
💡Quantum Teleportation
💡Entanglement
💡Qubit
💡Superposition
💡Quantum Gates
💡No-Cloning Theorem
💡Phase Flip
💡Bit Flip
💡Quantum Circuit
💡Interference
Highlights
Explanation of quantum teleportation and how Bob can recover Alice's qubit state based on her measurements.
Description of no transformation needed when Alice measures '00' and Bob's state is already aligned with Alice's.
Process of bit flip and phase flip to align Bob's state with Alice's in different measurement scenarios.
Use of Pauli X and Z gates to perform bit and phase flips in quantum teleportation.
Circuit diagram explanation showing how Z and X gates are applied based on Alice's measurements.
Discussion on the flexibility of choosing different initial entangled states for quantum teleportation.
Entanglement swapping as a method to change the partners in an entangled state.
Limitation of quantum teleportation by classical communication speeds due to the need for Alice's measurement results.
Introduction to the no-cloning theorem and its implications for quantum computing.
Explanation of why quantum circuits cannot copy arbitrary qubits, contrasting with classical computing's ability to copy information.
Attempt to refute the no-cloning theorem using the CNOT gate and its failure.
Teleportation not being a counterexample to the no-cloning theorem due to the collapse of the original qubit post-measurement.
Proof of the no-cloning theorem using a thought experiment and inner product calculations.
Introduction to quantum interference and its role in quantum computing algorithms.
Explanation of how interference can amplify or diminish certain states in a quantum system.
Overview of constructing quantum algorithms using superposition, entanglement, and interference.
Discussion on the process of measuring quantum states and interpreting the results through probability distributions.
Summary of the key concepts covered, including qubits, quantum gates, and the properties of superposition, entanglement, and interference.
Introduction to practical applications of quantum computing algorithms on IBM quantum computers using Qiskit.
Transcripts
[Music]
so let's take the each case uh one by
one let's say mn is zero zero okay
in this case bob needs to do nothing
because as you can see alpha he already
has if alice measures zero zero
bob state already must be alpha zero
with beta one which is nothing but five
so
bob needs to do
uh
no transformation on phi dash to create
five
but if bob state is uh if bob receives
the
bit
zero one from alice
there is some change that needs to be
done to bob state
to change it to
uh to file okay
and what does that change
it's a bit flip
let's look at this particular term alpha
one plus beta zero if you do a bit flip
one changes to zero zero changes to one
right
so from alpha one plus beta zero we look
at alpha zero plus beta one which is
nothing but
5
which is what you want to get
similarly if mn is 1 0
then bob state alpha 0 minus beta 1 will
need to go through
a phase flip
okay
and
again in uh
as we saw in the previous these lectures
our face flip can be performed by a poly
z gate again just like a bit flip can be
performed by a poly x gate
and in this case if m
and n are one and zero respectively then
if bob applies
a phase flip or a single z gate then
he'll be able to recover five
if uh alice sends mn as one and one
respectively
then bob is uh
because bob stayed at that point is
alpha one minus beta0 he'll need to
apply both a
bit flip and a phase flow which means
he'll have to apply both an x gate and a
z
okay now let's look at this this circuit
so this looks a bit complicated but it's
actually very simple z to the power n
simply means that we either apply a z
gate or we do not apply a z because uh
remember that m can only assume the
value zero or one okay so z to the power
zero is basically one it means there's
no transformation similarly x to the
power of n can means either an x gate is
applied or not
so when m is zero zero basically we are
applying no gates so that to the power 0
and x to the power 0 means there's no
transformation
with a bit flip
there's no set gate and there's just one
x gate
with the face flip there's one set gate
and no x k xk
and a phase flip we need to apply both
the z gate and the xk8 to bob's state
so bob can recover
uh
the
original qubit that alice poses which is
okay
so
hope this uh was easy to understand uh
now note that
uh we took uh
size zero zero as the best state that
was shared between alice and bob our
priority but we do not have to take that
we can alice and bob can share any of
the four ball states uh
zero zero
psi zero one psi one zero or five one
one and uh
she will be able to communicate
her measurements and bob will be able to
recover uh
state 5 based on those measurements it's
just that this particular table will
look a little bit different which means
you'll have to apply
a different sequence of dead index gates
or a different
combination of zen index gates to
recover the
the qubit
point
so try this out as an exercise it should
be fairly straightforward and you can
the the math that you've been doing for
the past few minutes will uh
be identical to what you will need to uh
[Music]
to infer uh the same file
regardless of what
epr pair you have at the beginning
teleportation is not this theory it's
been experimentally verified
in one experiment in china a qubit was
uh
teleported from the ground to a
satellite 1400 kilometers above the
earth uh based on a an epr pair being
shared between the two endpoints a
priori
a phenomenon known as antagonist
stopping was also demonstrated between
two of the canary islands which are
islands in the atlantic ocean
and let's just look at what entanglement
stopping is it's a slight variation on
the teleportation protocol that
we have witnessed
let's say bob and alice already share
a well state let's call this iig
and let's say alice and carol share a uh
different velocity uh let's call it phi
kl okay
what do you want to achieve here what we
want is
uh to allow
bob and carols
to possess an entangled state
rather than bob and alice or alice and
carol okay
and we can achieve this through
teleportation
what can uh alice do because she knows
her part of the entangled stage she says
is bob
she can communicate that state to carol
using the teleportation protocol because
alice and carol already shared an apr
pair right
and
once she does that
carol then gets to know
the uh alice's part of bob's cupid of of
uh uh bob's of the entangled stage she
shares with bob
okay
so the final result that is achieved is
bob and carol share the state psij which
was originally shared between bob and
alice so
thereby we have
swapped the
epr pair
between alice and carol
so does this mean you can communicate
information infinitely fast
not quite
uh well measurement results that is the
m and n that you saw can only be
communicated over uh classical
communication channels okay through
regular networking channels that uh
we all use today to communicate uh any
piece of information over long distances
and
uh classical communication channels just
cannot exceed the speed of light which
means that
our teleportation protocol even using
quantum entanglement is always going to
be
limited by the capacity of our classical
publication let us look at another
interesting property of quantum
computing this is actually a limitation
of quantum computing
and it says that it's not possible to
build a quantum circuit that can make a
copy of
any arbitrary qubit psi
what does this mean it means that there
is no unitary transformation that can
take qubit sky
which is not already known before and
produce two copies of that
this is a big difference from classical
computing where copying an arbitrary bit
or copy any piece of information from
one record to another that be it
memory or in secondary storage is one of
the most basic operations you can do and
classical computers can make arbitrary
number of copies of any arbitrary
information
but it turns out quantum computing
cannot actually do that
uh what about the c naught that we saw
uh
can we refashion the c naught gate to
make
copies
let us see
if a c naught gate could copy a qubit
let's say it were able to copy uh the
qubit q1
it
what it implies is that
both our first qubit and the second
qubit must assume the value q1 which
means our second qubit q2 dash which is
the xor of q1 and q2 must be equal to q1
but if we uh
solve that equation that xor of q1 and
q2 is equal to q1 it turns out that q1
must be either 0 or 1 okay
so
we cannot actually copy an arbitrary
qubit q1 which is not 0 or 1 which is
any qubit alpha 0 plus beta 1
using a c not k okay so let's think of
other circuits that can do that
uh
did alice not just transfer teleport to
qubit to bob
could he not have used that circuit
but what exactly did alice do
she teleported a cupid to bob but
because she measured her qubit phi
that qubit collapsed to
some state m
and therefore she lost that result after
measurement and finally it's only bob is
left with five
so this is not a copy this was a
transfer of qubit five from alice to bob
okay so
the teleportation is not a counter
argument to what we just
saw
so it turns out that the proof of this
of what we call no cloning theorem that
is
the fact that an arbitrary qubit cannot
be cloned using a quantum circuit
it's actually quite simple so let's
assume for uh
safe argument that there exists such a
circuit okay and
this circuit is able to clone uh
an arbitrary state psi
let's assume that the gate or the
transformation
is represented by u so
you applied to our
qubits
psi s
will produce state size phi that is
we get two copies of five
now if such a circuit existed
it should be able to uh make two copies
of
uh
uh
any possible sign right so let's assume
some value
and let's assume the circuit can make
copy of that okay let's just take the
first value which is represent the first
value by psi so our circuit is able to
produce a copy
state size psi and
similarly for any uh different phi given
as input it can produce two copies of
five okay
now let's compute the inner products of
both sides of these two equations okay
what that means is
we
take the
graph of this side and we take the brass
this side and we apply it to
these two as kept okay what is uh
the synthesized two what is the bra it
is a conjugate transpose right
so u changes to u dagger
uh
get psi turns to brass i and get s turns
to f okay note that s is a our constant
of the circuit but we assume that there
exists a circuit and there is some
constant value that it will our circuit
will always use to make copies of a
different uh
of the other qubit
so
here is the expression we get for the
second term this remains as is
and
uh here for the right hand side
the cat size changes to brass i
and here they get five remains the same
okay now if you actually do the inner
product and this i will leave to you as
an exercise
you'll find
that uh the left hand side just resolves
to the bra
psi applied to
5 and on the right hand side you see
that this just
is
the bra
psi applied to phi
and you take the square okay so we have
a bracket here and you have bracket here
and on the right hand side there's a
is a square of that okay
you can see u dagger and u you multiply
them remember that
their unitary matrices it becomes
identity matrix okay and the rest of the
math actually is also fairly easy uh
let's try it out
but looking at what we got here what do
you see here this term this bracket
psi applied to phi
happens to be equal to its own square
and what does that mean if x equal to x
square
what exactly uh
what is x it simply means that x equal
to either zero or 1 right
one with either sign plus or minus
and
in
the implication of that is that
either psi is equal to 5
or
pi and phi are orthogonal okay
so
our circuit
uh if it can produce a uh a copy of an
input qubit it can only do so
for
two uh for inputs that for for two kinds
of uh inputs which happen to be author
okay the same one doesn't count because
it just means that
there is one particular qubit that it
can make a copy of but
our circuit then uh what we've shown
from this is uh limited in this
application it can only
make copies of a pair of orthogonal
vectors it cannot make a copy of any
arbitrary
site which
can be represented using
any pair of probability amplitudes
so this proves the no cloning theorem
and hope that this was intuitive to
understand
again try out this particular math as an
exercise
so now we move on from entanglement to
the property of interference
remember your
high school physics now
uh
interference is a property that you
probably learned there
and we just come to how exactly that's
applied here uh
what we have in quantum mechanics are
three main properties which also can be
applied in quantum computing we have
already seen superposition we have
dealt with that in length in the first
week and also to some extent in today's
lecture
uh we have seen the property of
entanglement interference is a third
such problem
uh
now any quantum mechanical unit exhibits
wave particle duality that is it can
behave either as a wave or as a particle
and
when it behaves as a wave it produces
interference effects okay if you
remember
uh the young's double state experiment
if you pass a wave through two different
uh
two through two slits that are close by
you end up getting light and dark
fringes or an interference pattern on a
screen at the back
and why does this happen it happens
because
uh
the two waves that go through the two
different slits
uh
whenever the phases get aligned
they constructively interfere and the
amplitude amplitudes add up so you get
the light portion on screen
and when they're out of phase
uh their amplitudes cancel each other
and that's destructive interference and
you see the dark patches on the screen
okay or the dark fringes
now in quantum circuit it turns out we
can apply suitable phase changes i
remember one of the key
components of any
superposition qubit is its phase or its
relative phase right so we can apply
suitable phase changes to
any quantum state through the judicious
use of gates
and
what that can do is it can amplify
certain states in our quantum space
and diminish other states okay so any
general quantum algorithm is all about
trying to find out an answer to a
problem
by
extracting out the
states of the two to the power and state
space
that
correspond to the right answer okay and
you'll see a lot of examples of this
this may sound a bit abstract to you but
just keep in mind that the phenomenal
interference can be
very easily applied and there's one
particular algorithm that you can keep
in mind called grover's algorithm where
the use of interference is
quite star
so finally let us see the intuition
behind uh the construction of any
arbitrary quantum algorithm to solve any
arbitrary problem okay just like you
would
use an algorithm to solve the problem in
classical computers
so in
uh
in our in quantum computing
first thing you have to do is to
determine the state space and inputs
okay in effect determine uh what uh what
the size of the
input state should be that is what n
should be and what each of the input and
values should be usually we begin with
uh states
in zero state so which we can
then transform into something useful and
as you already saw if we apply the
hadamard transform simultaneously to all
of the input states we end up getting an
equal superposition of 2 to the power n
states
and now through some combination of
gates
we can perform various kinds of
transformation of superposition states
and use phenomenon of entanglement and
interference to bias correct answers or
to
amplify the correct states and to
diminish the incorrect ones
and finally
in our n-qubit system we can measure all
of the n-qubits and thereby get a an
n-bit state okay
that's what we get here on the classical
uh
line you see here
and
what we will have to do is then measure
our output using a large number of shots
so
remember that
when you measure something it gives you
a definite value either zero or one
right
so
but the probability of getting a zero or
one for each qubit happens to be
definite and it is based on whatever
calculation we do here so
for our output to reflect
what the probability
what what the superposition state is at
this point we need to run a large number
of outputs and then
do the probability distribution to
figure out what
this superposition state
based on the set of measurements that we
get okay
you will see how this is done using real
examples and in more detail in the next
few weeks
and
some of the algorithms that you'll see
are bhaja
grovers and banshee mazirani and
you'll see that they follow this pattern
uh
gently speaking and
you'll see what kind of clever
combination of gates we can use to
do something useful much faster than we
could on any classical computer using
classical algorithm
let's summarize what you've learned so
far
last week you learned all about qubits
quantum gates and quantum circuits
in this lesson you have learnt how the
quantum mechanical properties of
superposition entanglement and
interference can together be harnessed
to build quantum computing algorithms
now we will move from theory to practice
you will learn how to create real
quantum circuits on real ibm quantum
computers and you will learn how to
write programs using the kiskit
programming free books
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