mod02lec12 - Quantum Computing Concepts: Entanglement and Interference​ - Part 2

Summary

This 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

00:00

🔄 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.

05:00

🚀 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.

10:00

🚫 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.

15:01

🌌 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.

Keywords

💡Quantum Teleportation

Quantum teleportation is a process by which the state of a quantum system is transferred from one location to another, without physical travel of the system itself. In the video, this concept is central as it explains how Alice can transmit the state of a qubit to Bob using an entangled pair and classical communication. The script mentions a real-world application where a qubit was teleported from the ground to a satellite, showcasing the practicality of this quantum phenomenon.

💡Entanglement

Entanglement is a quantum phenomenon where pairs or groups of particles interact in such a way that the quantum state of each particle cannot be described independently of the others, even when the particles are separated by large distances. The script discusses entanglement as a prerequisite for quantum teleportation, where Alice and Bob share an entangled state, which is essential for the teleportation process to work.

💡Qubit

A qubit, short for quantum bit, is the basic unit of quantum information. Unlike a classical bit, which can be either 0 or 1, a qubit can be in a superposition of states, representing both 0 and 1 simultaneously. The video script uses qubits to illustrate quantum teleportation, explaining how Alice's measurement of her qubit collapses it to a certain state, which Bob can then recreate using his qubit and a series of quantum gates.

💡Superposition

Superposition is a fundamental principle of quantum mechanics where a quantum system can exist in multiple states simultaneously until it is measured. The script refers to superposition when discussing the initial state of qubits, which are often in a zero state before being transformed into a useful superposition state through quantum gates.

💡Quantum Gates

Quantum gates are the building blocks of quantum circuits, analogous to logic gates in classical computing. They perform operations on qubits and are used to manipulate their states. The script mentions specific quantum gates such as the Pauli-X (bit flip) and Pauli-Z (phase flip) gates, which are used in the teleportation process to correct the state of Bob's qubit.

💡No-Cloning Theorem

The no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state. This theorem is highlighted in the script to emphasize the difference between quantum and classical computing, where information can be copied freely. The script explains this concept by discussing the impossibility of creating a quantum circuit that can copy any arbitrary qubit.

💡Phase Flip

A phase flip is a quantum operation that changes the phase of a qubit, flipping it from one phase to its opposite. In the context of the video, a phase flip is used to correct the state of Bob's qubit during the teleportation process, as explained when discussing the transformation required for the qubit states alpha 0 minus beta 1.

💡Bit Flip

A bit flip is a quantum operation that changes the state of a qubit from |0⟩ to |1⟩ or vice versa. The script describes how a bit flip is applied using a Pauli-X gate during the teleportation process, specifically when Bob needs to correct his qubit state to match Alice's original state.

💡Quantum Circuit

A quantum circuit is a theoretical model consisting of qubits and quantum gates that perform operations on these qubits. The script describes a quantum circuit used in teleportation, where a combination of gates is applied based on the measurement outcomes (m and n) to recover the original qubit state at the receiving end.

💡Interference

Interference is a quantum mechanical phenomenon where the superposition of quantum states can lead to constructive or destructive interference, affecting the probabilities of different outcomes. The script touches on interference as a property that can be harnessed in quantum algorithms, such as Grover's algorithm, to amplify correct answers or diminish incorrect ones.

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