mod01lec09 - Quantum Gates and Circuits - Part 2

Summary

This lecture delves into the intricacies of quantum computing, focusing on multiple qubit gates that transform multipartite quantum states, akin to classical logic gates but with reversibility. It introduces the CNOT gate, pivotal for conditional transformations, and explores other gates like Control Y, Control Z, and the Phase gate. The Hadamard gate's role in creating superpositions is highlighted, essential for exponential state space computations. The lecture also covers universal gates, circuit identities, and the ability to synthesize unsupported gates using available ones, concluding with a foundation for further exploration into entanglement and quantum algorithms.

Takeaways

• 🧑‍🔬 A multiple qubit gate is a generalization of a single qubit gate, capable of transforming multipartite quantum states, similar to how a single qubit gate transforms individual qubits.
• 🔄 Multiple qubit gates can be represented as taking n qubits as inputs and producing n qubits as outputs, which aids in simplifying computations.
• 🔄 The concept of reversibility in quantum gates is crucial; given the output qubits, the input qubits can be recovered, a property stemming from the unitary nature of quantum mechanics.
• 🧬 The CNOT (Control NOT) gate is a significant two-qubit gate, analogous to the XOR gate in classical computing, flipping the state of one qubit based on the state of another.
• 🔄 The CNOT gate, along with other controlled gates like the Control Y and Control Z, manipulates qubits based on the state of control qubits, showcasing conditional transformations.
• 🔄 The Hadamard gate is instrumental in creating superpositions, allowing for the generation of a 2^n-dimensional state space from an n-qubit state, which is vital for quantum algorithms.
• 🔄 Universality in quantum computing refers to the ability of a set of gates to perform any possible quantum transformation, with the Hadamard and T gates, or the Hadamard and phase gates, being examples of universal sets.
• 🔄 Circuit identities allow for the creation of complex gates from simpler ones, leveraging the properties of gates like the Hadamard to simulate other transformations, such as the CZ gate from CX gates.
• 🔄 The Swap gate is an example of a simple circuit identity that can be decomposed into a sequence of CNOT gates, demonstrating how to manipulate qubits to achieve specific state transformations.
• 🔄 The lecture series progresses from basic quantum computing concepts to more advanced topics like entanglement and interference, eventually leading to practical programming on quantum computers.

Q & A

What is a multiple qubit gate and how does it differ from a single qubit gate?

A multiple qubit gate is a generalization of a single qubit gate, capable of transforming a multipartite quantum state. Unlike a single qubit gate, which operates on one qubit, a multiple qubit gate takes multiple qubits as inputs and produces multiple qubits as outputs, enabling more complex transformations.

What is the significance of the reversibility property in multi-qubit gates?

Reversibility in multi-qubit gates is significant because it allows the recovery of input qubits from the output qubits, given the transformation function. This is possible due to the unitary nature of quantum mechanics, where the conjugate transpose (dagger) of a matrix representing the gate is its inverse, ensuring that the gate and its inverse can be used to transform states back and forth.

Can you explain the CNOT gate and its function in quantum computing?

The CNOT gate, also known as the control-NOT or CX gate, is a two-qubit gate that performs a conditional bit flip on the second qubit based on the state of the first qubit (control qubit). If the control qubit is 1, it applies an X gate (bit flip) to the second qubit; if the control qubit is 0, it leaves the second qubit unchanged. This gate is fundamental in quantum computing for creating entanglement and performing conditional operations.

How does the Hadamard gate contribute to creating a superposition state?

The Hadamard gate is instrumental in creating superposition states by applying a transformation to qubits that puts them into a state where each possible state is equally probable. When applied to a multi-qubit state, it results in a superposition of all possible combinations of the qubits, allowing for operations on an exponentially large state space.

What is the role of the universal gate in quantum computing?

In quantum computing, a universal gate or a set of gates is capable of performing any possible quantum transformation, meaning they can convert any input quantum state into any desired output state. This is analogous to universal gates in classical computing, such as NAND or NOR gates, which can be combined to perform any Boolean operation. Examples of universal sets in quantum computing include the Hadamard gate and the T gate, or the Hadamard gate and the phase gate.

How can the control-Z (CZ) gate be synthesized using gates available in typical quantum hardware?

The control-Z (CZ) gate can be synthesized using a combination of a Hadamard gate and a CNOT gate. By applying a Hadamard gate to the target qubit, followed by a CNOT gate with the same qubit as both control and target, and then another Hadamard gate to the target qubit, one can effectively implement a CZ gate, which applies a phase shift conditioned on the control qubit being in state 1.

What is the significance of circuit identities in quantum computing?

Circuit identities in quantum computing are crucial for deriving complex gate operations using simpler, natively supported gates. They allow for the construction of any desired quantum circuit using a limited set of basic gates, which is particularly important given hardware limitations in the types of transformations that can be directly applied.

Can you provide an example of how the SWAP gate can be decomposed into simpler gates?

The SWAP gate, which exchanges the states of two qubits, can be decomposed into three CNOT gates. By applying CNOT gates in a specific sequence, one can effectively swap the states of two qubits without needing a dedicated SWAP gate, which may not be natively supported by all quantum hardware.

What is the purpose of applying a Hadamard gate to each qubit in a multi-qubit state?

Applying a Hadamard gate to each qubit in a multi-qubit state is done to create a superposition of all possible states. This results in a state where each combination of the qubits (from all zeros to all ones) has an equal probability amplitude, allowing for parallelism in quantum computation and enabling algorithms to explore a large state space efficiently.

How does the concept of entanglement relate to the operations performed by multi-qubit gates?

Entanglement is a quantum phenomenon where the state of one qubit becomes dependent on the state of another, even when separated by large distances. Multi-qubit gates, particularly those that perform conditional operations like the CNOT gate, are instrumental in creating and manipulating entangled states, which are essential for quantum algorithms and quantum computing's advantage over classical computing.

Outlines

00:00

🧲 Introduction to Multi-Qubit Gates

This paragraph introduces the concept of multi-qubit gates as an extension of single-qubit gates, which are crucial in transforming multipartite quantum states. It explains how these gates take multiple qubits as inputs and produce the same number of qubits as outputs, highlighting the ease of computation with this approach. The paragraph also delves into the property of reversibility inherent in multi-qubit gates due to the unitary nature of quantum mechanics, where the input state can be recovered from the output state, given the gate's function. The CNOT gate is introduced as an example, analogous to the XOR gate in classical computing, which flips the state of one qubit based on the state of another. The explanation includes the matrix representation of the CNOT gate and its action on qubit states, emphasizing its role in quantum computing circuits.

05:01

🔄 Reversibility and Other Multi-Qubit Gates

The second paragraph expands on the reversibility of quantum gates, emphasizing that given the output qubits, the input qubits can be recovered, a principle stemming from the unitary nature of quantum mechanics. It then discusses other types of multi-qubit gates, such as the control-Y and control-Z gates, which behave similarly to the CNOT gate but act on different bases. The paragraph also introduces the phase gate, which shifts the phase of a two-qubit state based on the state of the control qubit, and the swap gate, which exchanges the states of two qubits. A three-qubit gate, the Toffoli gate (CCNOT), is also mentioned, which flips the third qubit if the first two qubits are in the state |11>. The Hadamard gate is highlighted for its ability to create a superposition of states, a fundamental operation in quantum algorithms, and its matrix representation is discussed.

10:01

🔄 Universality in Quantum Gates

This section discusses the concept of universality in quantum computing, where a set of gates can be combined to perform any possible quantum transformation. It mentions the Hadamard and T gates, along with the CNOT gate, as examples of a universal set that can achieve any two-qubit transformation. The paragraph also touches on the idea that not all necessary gates may be natively supported by quantum hardware, leading to the need for circuit identities that allow the synthesis of non-native gates using native ones. The example of deriving the CZ gate from the CX gate using Hadamard gates is provided, showcasing how different gates can be combined to achieve the desired quantum operations.

15:03

🔄 Circuit Identities and Quantum Computing Basics Conclusion

The final paragraph of the script covers circuit identities, which are essential for creating gates that may not be directly supported by quantum hardware. It uses the example of the swap gate to demonstrate how a sequence of CNOT gates can achieve the desired transformation. The paragraph concludes by summarizing the week's learnings, which include understanding qubits, single and multi-qubit states, quantum gates, and their applications in building quantum circuits and algorithms. It also provides a preview of upcoming topics, such as entanglement and interference, which are key to harnessing the full power of quantum computing, and hints at practical programming on quantum computers in future lectures.

Keywords

💡Multiple qubit gate

A multiple qubit gate is a quantum gate that operates on two or more qubits simultaneously. It is a generalization of the single qubit gate, capable of transforming a multipartite quantum state. In the context of the video, this concept is central to understanding how quantum information can be manipulated in a quantum computer. The script mentions that applying a multiple qubit gate to a state q_1 to q_n results in a different state q'_1 to q'_n, highlighting the gate's role in quantum computation.

💡Reversibility

Reversibility in quantum computing refers to the property of quantum gates that allows the input state to be recovered from the output state. This is a key difference from classical gates, where knowing the output does not necessarily allow one to determine the input. The script explains that quantum mechanics is unitary, meaning that the matrix representing a gate is invertible, which is why the input qubits can be recovered from the output qubits, given the transformation function.

💡CNOT gate

The CNOT gate, also known as the control-NOT gate, is a two-qubit gate that is fundamental in quantum computing. It performs a NOT operation on the second qubit if the first qubit (control qubit) is in the state |1⟩. The script uses the CNOT gate as an example to illustrate how a two-qubit state is transformed, where it flips the state of the second qubit based on the state of the first qubit. This gate is crucial for creating entanglement and performing conditional logic operations in quantum circuits.

💡Unitary

In quantum mechanics, a unitary operation is one that preserves the inner product of quantum states, meaning it is reversible. The script emphasizes that quantum gates are unitary, which is why they can transform qubits in a reversible manner. This property is essential for quantum error correction and the reversibility principle discussed in the video.

💡Dirac notation

Dirac notation, also known as bra-ket notation, is a standard notation used in quantum mechanics to describe quantum states. The script mentions that the CNOT gate can be represented in Dirac notation, which is a way to express the transformation of quantum states in a compact form. This notation is particularly useful for representing the action of quantum gates on quantum states.

💡Hadamard gate

The Hadamard gate is a single-qubit gate that creates a superposition of states. It is often used to put qubits into a superposition of |0⟩ and |1⟩. The script describes the Hadamard gate's role in generating a superposition of a multi-qubit state, which is essential for exploiting the exponential state space of quantum systems. This gate is a building block in many quantum algorithms.

💡Superposition

Superposition is a fundamental principle of quantum mechanics where a quantum system can exist in multiple states simultaneously. The script explains how the Hadamard gate is used to create a superposition of all possible states for an n-qubit system. This property is what gives quantum computers their potential computational advantage over classical computers.

💡Universal gate set

A universal gate set in quantum computing is a set of gates that can be combined to perform any unitary operation on a quantum system. The script mentions that the Hadamard and T gates, along with the CNOT gate, form a universal set. This means that any quantum algorithm can be constructed using these gates, making them essential tools in quantum computing.

💡Circuit identities

Circuit identities in quantum computing are relationships between quantum gates that allow one gate to be synthesized using other gates. The script discusses how certain gates, like the CZ gate, can be created using combinations of other gates, such as the Hadamard and CNOT gates. This is important for practical quantum computing, as hardware may not support all gates natively.

💡Entanglement

Entanglement is a quantum phenomenon where the state of one qubit is dependent on the state of another, even when the qubits are separated by large distances. The script hints at the upcoming discussion of entanglement in subsequent lectures, which is a critical concept for understanding quantum correlations and their use in quantum algorithms.

Highlights

A multiple qubit gate is a generalization of a single qubit gate, capable of transforming multipartite quantum states.

Multi-qubit gates can be viewed as taking n qubits as inputs and producing n qubits as outputs, facilitating computations.

Reversibility is a key property of multi-qubit gates, allowing input states to be recovered from outputs due to quantum mechanics' unitary nature.

The CNOT gate is introduced as the quantum equivalent of the classical XOR gate, with the ability to flip the state of one qubit based on another.

The CNOT gate's matrix representation and its action on the computational basis states are explained.

Controlled-Y and Controlled-Z gates are mentioned as variations of the CNOT gate, affecting the target qubit's state conditionally.

Phase gates are introduced, which shift the phase of a two-qubit state based on the state of the qubits.

Swap gates are discussed, which exchange the states of two qubits.

The Toffoli gate, or CCNOT gate, is explained as a three-qubit gate that flips the third qubit if the first two are in the state |11⟩.

The Hadamard gate is highlighted for its role in creating superpositions of states, a fundamental operation in quantum algorithms.

The concept of universal gates in quantum computing is introduced, paralleling the role of universal gates in classical computing.

Circuit identities are discussed, showing how to derive non-native gates using combinations of native gates supported by hardware.

An example is given on how to synthesize a Controlled-Z gate using Hadamard and CNOT gates.

The Swap gate is also decomposed into a sequence of CNOT gates, demonstrating the flexibility in quantum circuit design.

The lecture concludes with a summary of the week's topics, including qubits, quantum gates, and their applications in circuits and algorithms.

An outlook for the next week's topics is provided, focusing on entanglement and interference, which are crucial for advanced quantum computing concepts.

Transcripts

[Music]

a multiple qubit gate is a

generalization of single qubit gate

it transforms a multiparted quantum

state just like a single qubit k

transforms a single qubit state so the

state q1 to qn

when you apply a multipartite uh

multiple qubit gate to it

converts it to a different and cubic

state q dash

1 to q dash

and this can be represented in a

different way

if you uh

take each part or each qubit of our

multicube state and

verify them individually and you feed

them to this particular gate

so we can

view this as a gate that takes n qubits

as inputs and produce n qubit as qubits

as outputs and more useful to look at it

this way because then we can do

computations

more easily as you see

in subsequent

slides and

there's another important problem uh

property that we get from a multi-qubit

gate which is called reversibility

and

remember we talked right in the

beginning of this week that this one of

the differences between quantum gauge

and

classical logic gates is that uh any

gate if when it takes in n inputs it

actually produces n outputs and not just

one output

and

the property of any such gate is that if

we know this function that's

transforming these n qubits

and we know the

output qubits

then the input qubits q1 up to qn can

actually be recovered from q1 dash to q

and dash so given the output we can

actually recover the input and that's

the reversibility principle

and why is this

this is because quantum theory and

quantum mechanics itself is unitary

which means that any matrix that

represents the gate is unitary so as you

have seen

it means that the conjugate transpose or

a dagger of any matrix must be its

inverse so a times a dagger must be the

identity matrix and what this means is

that any gate

that can be used to transform one

multipartite qubit state into another

the inverse of the gate can be used to

transform the output of uh the output

back to the input

so that comes intuitively from the

notion that

these gates are unitary let's take the

most uh prominent or the most

interesting two qubit uh

gate

called which is which which you call the

c naught gate

and this is the equivalent of the xor

gate in casual computing so if you

remember the xor gate or if you're

familiar with it what it means is you

take in two bits a and b and the output

is represented by a x or b what that

means is that

if both a and b are

different than the output

is one otherwise the output is

zero

so

the equivalent of the xor gate is the c

naught data as you just said

and what this will produce is given

qubits q1 and q2

it will produce outputs q1 and the

second output will be q1 absorbed with

q2

and the circuit representation of this

is as follows

uh we'll come back to the circuit again

but

the matrix that is going to transform

our two qubit state q1 q2 to this

particular qubit state is represented by

this matrix and because it's uh we have

a two qubit state uh the get that is

represented uh by the tensor product of

q1 and q2 is going to be four by one

matrix therefore we need a four by four

matrix to transform one four by one

matrix into another okay let's see how

that happens uh the c naught gate by the

way can also be represented in dirac

notation in in this form if we take the

states uh

0 0 they get 0 0 apply it to the bra 0 0

and then we add other other uh get brass

or outer products uh so this is uh can

be represented as a sum of various uh

outer products

so

let's apply the c naught to our zero

zero sheet okay let's assume q one is

zero q two zero

so our zero zero state

uh and we computed this earlier as the

tensor product of the state zero and

zero that ends up being one zero zero

zero so we apply this c naught matrix to

that which means you do a matrix

multiplication

and what we end up with is one zero zero

zero okay which means this is again

nothing but the zero zero so we apply

the c naught gate to zero zero it does

nothing to it it uh there's no

transformation

now

generally uh what will happen is

if we take the c naught gate uh

if you take if our inputs are of any of

these forms and this is the truth table

uh here are the outputs so for state 0 0

output is going to remain 0 0

if state 0 1 output is going to remain 0

1

for 1 0 it changes to 1 1 1 1 change to

1 0. what does this mean it means that

if our qubit q 1

is 0 then output remains the same

whereas for qubit q1 is 1 then the

output is going to flip 0 flips to 1 1

flips to 0.

and that's why we call

this a c naught gate or in longer form a

control not g or

in there's another term for it called

control x which means that

we are going to be uh applying an x gate

to the qubit q2 but controlled by what

the state of the qubit q1 is

okay

so

here we have a two qubit state and the

first qubit ends up being a control

qubit for the computation that is

happening on the second cube

that's how you should visualize it

so

based on the value of the control qubit

we are going to be computing a not or a

poly x transformation of qubit q2

and

the if the cube if the control cube it

happens to be zero

then

this does not do anything to change the

state of q two the controlled qubit is q

one is going to be flipping the state of

q two

we can also see that this gate is

reversible

if we have both q1 as well as q1 and xor

of q2

that means that we can get back q1 and

q2 if we are given q1 and q1 of q2 so

the reversibility property is maintained

as you can see here

let us look at a few other examples of

multicubic gates

now the c naught gate as you've seen is

also called the control x gate or the

control poly x gate similarly there are

also two cubic gates called control y

and control z

which preserves the strain of a qubit if

the control qubit is zero and transform

it in particular base if the control

qubit is one

another example as you can see here is

the phase gate and

it's represented by this two by two

matrix so it acts on a

bipartite state and it's going to shift

the phase of any uh two qubit state by

phi

uh only if that particular state uh

happens to be uh one one otherwise it

leaves the

the particular state intact

a swap state is another interesting one

which uh swaps two qubits what that

means is if you give the input state q

one q two what you get up what you get

as output is q to q1

and

here is an example of a three qubit

state therefore it requires an eight by

eight matrix for transformation and

this is related to the c naught and as

you can see it's also called the c c

naught that's the control control knot

so it flips our third qubit

only if both the first two bits are

both one so it uses uh the first two

cubes as control qubits and

transforms the third qubit

and the profile gate is a gate with

interesting properties and something

that is used

extensively in quantum computing

now let us see how we can generate a

superposition of a multiplied state and

here's where you see how the power of

quantum computing

emerges and this is the

gate that you see here is one that we'll

end up using in

almost all of the algorithms

in subsequent lectures

so

what we want to achieve here is given an

nq with state we want to produce a 2 by

n 2 2 to the power n state space okay

now we already saw that any n qubit

state can produce a 2 to the power n

state space but how do you produce a 2

to the power and state space that has

certain properties that then we can do

some computation of okay

so take the first qubit of our

multi-qubit state uh q1 and let's say we

apply add-on gate to that okay similarly

we apply a hadamard gate to the second

qubit or the second part of our enquiry

state and likewise we apply hadamard

gate up to the nth qubit okay on each

qubit of our

multiplied qubit state q1 up to qn we

apply a hadamard gate on each of the

qubits so naturally the outputs you're

going to get are the hadamard applied by

quite q one etc and uh for any qubit you

can use a hadamard transform to compute

what the output is

now

this can be taken

together as one gate and we can

represent this in this form uh this is

the direct form of the uh multi-state

hadamard gate and it can be represented

with this exponent and the tensor

product sine and exponent

so

what this gate is if in matrix or in

mathematical form it ends up being a 2

to the power n by 2 to the power n

matrix t

and we can get the actual matrix value

here by doing a tensor product of the

hadamard matrix with itself

uh n times and we are not going to go

into the details of the tensor product

of of how to do the tensor product two

different matrices but it is just a

generalization of the tensor product

that you saw

in in earlier slides where

we

computed the the tensor product of two

uh

single cats to produce another get

the hadamard uh our

end state hadamard gate can then be

applied to the

input multipatter state to produce the

final output

so let's take one example

so our hadamard or multi-state hadamard

gate can let's try to apply it to our

get

where each qubit is in the state 0.

so

the state 0

0 0

n times

when we take the tensor product we get a

matrix with 2 to the power n

rows and 1 column and

only the first row will have the value

zero remaining rows are going to have

the value or the first row will have the

value one and the remaining rows will

all have the value zero so if we apply

our

h let's call it s to the power n gate to

this particular uh

n qubit state

we end up getting and if we if you uh

uh

try this with

a 2 to the power n pi to the power n

matrix you can try it for smaller values

of n

you'll find that we get this particular

matrix you get a

matrix which also has 2 to the power and

rows but now the value in each row is

going to be 1 but the probability

amplitude is

what you can see here 1 by root over 2

to the power n which means that

what we end up with is a

is a superposition state

of all possible

where each

uh combination of uh our n qubits that

is if we set any of the qubits to their

one or the zero so we are going to get

we can get any of one of two to the

power n states right from zero zero zero

zero up to one one one one and

everything in between different

combinations zeros and ones so the state

we end up getting here is a state where

we get all of we have all of the total

power and states in superposition and

they all have uh equal probability

that's what we get here so from a state

that uh has uh just n possibilities or

uh because or rather it has just one

possibility because

all of our qubits are in state 0 we end

up getting a state which can uh which

when measured can lie in any of one of

two to the power n states and now we can

do a lot of user computation on this and

this is a property or that this sort of

hadamard transform you see is very

useful in algorithms where we want to do

fast computations on uh exponential size

state spaces and

get results

much faster than we would in an

equivalent algorithm that we could run

the bar on a classical computer the

property of universality may be already

familiar to you if you uh know how to

program on classical computers uh

what a universal gate is is a

single gate or a set of gates that

compute

any possible function that we would like

that is it can transform

any input state into any output state

through some combination of

those gates

in passive computing

we have if we take the set of case and

or not through any combinations of and

or not we can express any uh boolean

transformation that is we can convert

any boolean state to any other billing

state

the nand and the nor gates are by

themselves universal gates which means

that we can use we can apply uh

one or more nand gates to produce any

possible boolean transformation

similarly we can apply one or more nor

gauge to produce any boolean

transformation

this

the boolean transformation that we're

talking about here are converting one

logical expression

into into another

in quantum computing similarly

what we need

is the ability to convert

one point quantum state to another but

the the notion of universality stays

requires

the ability to convert any possible and

cubic state to any other possible and

and cubic state so

it's very useful to

have a set of gates that can allow

from which we can just pick and

use any particular combination of gates

to produce any transformation so uh

the sets that you see here are uh

candidate universal uh sets so if we

take the h gate and we take a gate

called the t gate which you are not

encountered here but

you may encounter it in subsequent

lectures and the c naught

the using any uh

one or more of these gates it turns out

that we can

affect any uh two qubit transformation

similarly uh the

the state hadamard the the set of gays

hadamard and toefl end up being another

uh universal set uh what this means in

uh quantum computing terms or

in uh linear algebraic terms is that

uh we want a set of gates that

from which we can pick some combination

of gates and implement any possible

unitary

matrix function

moving to circuit identities this is the

last topic we will cover in this lecture

note that not all important gates

that we need that are and that are going

to be useful to us in a quantum circuit

can be directly applied by hardware that

is hardware is limited by what kind of

transformations it can natively support

so if our hardware is limited to

supporting particular kind of

transformations or particular kinds of

gates

if we need a different kind of gate what

we would need is the ability to derive

those gates using combination of gates

that the hardware actually supports okay

so take for example the c naught gate

now

c not gate does a

conditional bit flip on the second qubit

based on the first control cable right

uh now consider the another gear that we

discussed briefly a couple of slides ago

the control z

now what this does is it applies the

polyz gate to the second qubit based on

what the state of the control qubit is

but does the condition face

in other words

now the c naught gate is something that

is often present in most commercial

hardware in commercial quantum computers

so for example in the ibm quantum

computer we can natively apply a c

naught or a csk but

uh the same uh quantum computer does not

natively support a control z gate it's

not directly available so what we need

is we need to derive an identity whereby

we can uh use uh

some combination of c naught gates and

other gates that are supposed supported

by the hardware to produce uh a c z gate

and we can it turns out we can actually

do this fairly simply at least for this

particular example uh

you know that we can use hadamard gauge

to switch between the x and the z basis

right from the x to z basis and back

what this means is that uh the hadamard

applied to state zero produces state phi

one ah produced state minus hadamard

applied to plus produces zero

and had multiplied to minus produces one

so we have learned this

earlier it turns out using matrix

multiplication

these two conditions that you these two

expressions that you see here they hold

what does this mean it means that if you

apply multiply the hadamard by the x and

again multiply the hadamard you end up

getting the z

and likewise if you multiply h by z by h

you get x okay and that's something you

can actually try out as an exercise if

you take the hadamard matrices and x

matrices and the z matrices that you've

already encountered in previous slides

and do the matrix multiplications to

verify for yourself

how can we

realize the c dead gate out of ex gates

using this particular identity

let's take a look at the circuit

uh the z gate

is the application of a hadamard gate

followed by an x-gate followed by a

hadamard right i'm going in the reverse

direction remember that

what this means is that

a z transformation can be produced by

applying a hadamard followed by an x

followed by another hadamard and

because it's a two qubit state we simply

going to be applying this transformation

to our second qubit which is what the

transformation uh usually works on

because our first qubit is just a

control cube okay

so the

uh

just by applying uh an edge followed by

c naught followed by an edge we can

realize uh zero key so if our hardware

supports both hadamard as well as c

naught we can synthesize a

uh or uh we can compose a c that state

another example let's take the swap gate

okay

state q and q two is uh converted to

state q to q1 okay

how can this be

a decomposition of

individual states

it turns out this is also fairly simple

and all we need is we need to apply

three c note gates in sequence to

produce this transformation okay and

we're not going to go on the map but you

can verify this yourself using uh the

tools that we already discussed in

previous slides

all right this brings us to the end of

uh this particular lecture as well as

this particular week so this is the

final module of the quantum computing

basics

uh you've learned all about qubits and

single and multiple qubit states you

learn about quantum gates and their wave

properties both for

single state transformations and

multiplier state transformations and

you've seen how to use them as building

blocks for

quantum circuits or algorithms

next week you will learn about the

concepts of

entanglement interference which are

slightly advanced topics but which are

which still follow from quantum

mechanical integrations and once we uh

learn how we can harness the power of

entanglement interference we will then

move on to practical programming where

you will learn

cascade as well as

how to program on

ibm frontend computers