mod04lec21 - Variational Quantum Eigensolver

Summary

This video script delves into the Variational Quantum Eigensolver (VQE), a quantum algorithm pivotal for quantum chemistry simulations. It elucidates the VQE's three-part name—variational, quantum, and eigensolver—each integral to its function. The script outlines the quantum-classical hybrid optimization loop, detailing the role of the Hamiltonian mapping, trial state preparation, and the optimizer's selection. It underscores challenges in fermionic-to-qubit mapping, initial state preparation, and optimizer efficiency. The script also discusses the importance of the variational principle in ensuring the cost function's faithfulness and the implementation's efficiency, providing insights into quantum chemistry's quest for minimum energy states and inter-atomic distances.

Takeaways

• 🌟 The Variational Quantum Eigensolver (VQE) is a quantum algorithm used primarily for quantum chemistry simulations, focusing on finding the ground state energy of molecules.
• 🔍 VQE is composed of three parts: 'variational' from the variational principle in quantum mechanics, 'quantum' referring to the quantum computing aspect, and 'eigensolver' which is about diagonalizing matrices to find eigenvalues and eigenstates.
• 📈 The algorithm involves a hybrid quantum-classical optimization loop where a cost function, typically the energy of a system, is implemented on the quantum side and parameters are tuned by a classical optimizer.
• 🧬 Quantum chemistry simulations aim to efficiently simulate the behavior of molecules, treating electrons as fermions described by a fermionic Hamiltonian.
• 🔄 A key challenge in VQE is the transformation from a fermionic problem to a qubit Hamiltonian, which requires translating a complex system into a form understandable by quantum hardware.
• 📊 The trial state preparation and the choice of an appropriate classical optimizer are crucial for the efficiency of VQE, as they directly impact the algorithm's ability to find the optimal solution.
• 💡 The variational principle ensures that the energy of any trial state in the VQE will be greater than or equal to the ground state energy, making the cost function faithful to the goal of finding the ground state.
• 🛠️ The implementation of VQE involves creating parameterized quantum circuits with layers of single and entangling gates, which are varied to explore the state space and find the minimum energy state.
• 📉 The measurement part of the VQE process involves computing expectation values for different terms in the Hamiltonian, which requires careful consideration to minimize the number of calls to the quantum hardware.
• 🔧 The choice of classical optimizer is critical, with techniques like the Simultaneous Perturbation Stochastic Approximation (SPSA) being used for their efficiency in making only a few calls per iteration.

Q & A

What is Variational Quantum Eigensolver (VQE)?

VQE is a quantum algorithm used for simulating quantum chemistry problems. It is a type of variational quantum algorithm designed to find the ground state energy of molecules by minimizing the expectation value of a Hamiltonian operator.

What are the three parts of the VQE name?

The three parts of the VQE name are 'variational', 'quantum', and 'eigensolver'. 'Variational' comes from the variational principle in quantum mechanics, 'quantum' refers to the application of quantum physics or computing, and 'eigensolver' is about diagonalizing a matrix to find its eigenvalues and eigenstates.

How does VQE apply to quantum chemistry simulation?

VQE applies to quantum chemistry simulation by allowing the simulation of molecular systems. It uses a quantum-classical hybrid optimization loop to minimize the energy of a trial state, which represents the state of the molecules being simulated.

What is the role of the quantum circuit in VQE?

In VQE, the quantum circuit is a parameterized circuit that prepares the trial state based on the parameters tuned by the classical optimizer. The quantum computer then measures the expectation value of the energy for that trial state.

What is the Hamiltonian mapping portion in VQE?

The Hamiltonian mapping portion in VQE is the process of transforming the fermionic Hamiltonian, which describes the system of molecules, into a qubit Hamiltonian that can be understood and processed by the quantum computer.

Why is the initial trial state preparation important in VQE?

The initial trial state preparation is important because it sets the starting point for the optimization process. The choice of the initial state can significantly affect the efficiency and success of finding the global minimum energy state.

What challenges are there in the mapping from fermionic to qubit Hamiltonian?

The challenges in mapping from fermionic to qubit Hamiltonian include reducing the size and complexity of the Hamiltonian for efficient simulation on quantum hardware, while maintaining accuracy. This is an active area of research with various techniques being explored.

How does the variational principle relate to the VQE algorithm?

The variational principle in VQE ensures that the energy of any trial state will always be greater than or equal to the ground state energy. This principle guarantees that the cost function used in the optimization process is faithful, meaning that minimizing the trial state energy will lead to the ground state energy.

What is the significance of the entangling gates in the VQE quantum circuit?

Entangling gates in the VQE quantum circuit are significant because they create entanglement between qubits, allowing the quantum system to explore a more extensive portion of the state space. This increased exploration helps in finding a state closer to the true ground state.

Why is the choice of classical optimizer important in VQE?

The choice of classical optimizer is important in VQE because it directly affects the efficiency and effectiveness of finding the minimum energy state. The optimizer must be capable of navigating the complex energy landscape with a limited number of quantum hardware calls.

What is the role of the Born-Oppenheimer approximation in VQE?

The Born-Oppenheimer approximation in VQE allows for the simplification of the quantum chemistry problem by treating the nuclei as fixed and focusing on the electrons and their interactions. This approximation is useful for systems with lower energy and where the mass of the nucleus is significantly different from that of the electrons.

Outlines

00:00

🔬 Introduction to Variational Quantum Eigensolver (VQE)

The paragraph introduces the Variational Quantum Eigensolver (VQE), a quantum algorithm designed for quantum chemistry simulations. It explains the three parts of the name: 'variational' from the variational principle in quantum mechanics, 'quantum' referring to the quantum computing aspect, and 'eigensolver' which is about diagonalizing matrices to find eigenvalues and eigenstates. The VQE algorithm is part of a quantum-classical hybrid optimization loop, where a quantum circuit with tunable parameters is adjusted by a classical optimizer to minimize the cost function, often representing the energy of a molecular system. The quantum computer provides the cost function values, while the classical computer updates the parameters. The paragraph also discusses the importance of choosing the right optimizer and the challenges in mapping fermionic problems to qubit representations.

05:01

🌐 Challenges in Quantum Chemistry Simulation

This paragraph delves into the challenges faced in quantum chemistry simulations using VQE. It highlights the need to map complex fermionic Hamiltonians to qubit Hamiltonians, which is non-trivial and an active area of research. The goal is to reduce the size of the Hamiltonian for better simulation on quantum hardware, possibly with trade-offs in approximations. The paragraph also discusses the importance of preparing an effective initial trial state and choosing a classical optimizer that can work efficiently within the limitations of current quantum hardware. It uses the example of a simple two-electron molecule to illustrate the concept of finding the minimum energy state, which is a key objective in quantum chemistry.

10:03

🔍 Detailed Explanation of VQE Components

The paragraph provides a detailed look at the components of the VQE algorithm. It explains the process of translating a fermionic Hamiltonian into a qubit Hamiltonian using techniques like the Jordan-Wigner or Bravyi-Kitaev transformations. It also discusses the variational principle, which ensures that the energy of any trial state is at least as high as the ground state energy, thus guiding the optimization process. The paragraph further describes the implementation of the quantum circuit, which involves alternating layers of single-qubit gates and entangling gates, and how the number of layers affects the exploration of the state space. It concludes with a discussion on the measurement process, which involves computing expectation values for different terms in the Hamiltonian.

15:05

🛠️ Quantum Circuit Design and Measurement

This paragraph focuses on the design of the quantum circuit used in VQE and the measurement process. It describes how the trial state is prepared using a series of quantum gates, including single-qubit rotations and entangling operations. The paragraph explains the trade-off between the depth of the quantum circuit and the hardware's ability to maintain coherence and accuracy. It also discusses the measurement process, where expectation values for different Hamiltonian terms are calculated. The paragraph emphasizes the importance of reducing the number of terms in the Hamiltonian to minimize the number of calls to the quantum hardware, which is crucial for efficient computation.

20:07

📉 Classical Optimization in VQE

The paragraph discusses the role of the classical optimizer in the VQE algorithm. It describes the optimizer's task of navigating the complex cost function landscape to find the global minimum, which corresponds to the ground state energy of the system. The paragraph introduces the SPSA (Simultaneous Perturbation Stochastic Approximation) technique, which is used for its efficiency in making only two calls per iteration to the quantum hardware. The paragraph also touches on the importance of the initial parameters provided to the optimizer, as they can significantly affect the likelihood of finding the global minimum amidst potential local minima.

25:08

🧪 Practical Application of VQE in Quantum Chemistry

The final paragraph highlights the practical application of VQE in quantum chemistry by referencing an experiment where the algorithm was used to calculate the energy of known molecules on actual quantum hardware. It implies the translation of theoretical concepts into experimental practice, showcasing the potential of VQE for real-world quantum chemistry simulations.

Keywords

💡Variational Quantum Eigensolver (VQE)

VQE is a quantum algorithm used for finding the ground state energy of molecules, which is crucial in quantum chemistry. It operates by preparing trial quantum states and measuring their energies through a hybrid quantum-classical optimization loop. In the script, VQE is highlighted as one of the first variational algorithms, tailored for quantum chemistry applications, emphasizing its importance in simulating molecular behavior.

💡Quantum Chemistry Simulation

Quantum chemistry simulation refers to the use of quantum computing to model the behavior of molecules and their chemical reactions. It is central to the video's theme as it illustrates the practical application of VQE. The script discusses how VQE can be used to simulate the behavior of molecules efficiently, focusing on finding the minimum energy state of a system.

💡Hamiltonian

In the context of the video, the Hamiltonian represents the total energy of a quantum system, often used in quantum chemistry to describe molecular systems. It is a key component in VQE as the algorithm aims to find the ground state energy of the Hamiltonian. The script mentions the need to map the fermionic Hamiltonian, which describes electrons, to a qubit Hamiltonian that can be processed by quantum hardware.

💡Parameterized Circuit

A parameterized circuit in quantum computing refers to a quantum circuit where certain parameters can be adjusted to change the circuit's behavior. In VQE, these circuits are used to prepare trial states, with parameters tuned by a classical optimizer to minimize the energy expectation value. The script explains that the quantum circuit implemented is parameterized, and the parameter is what is tuned by the classical optimizer.

💡Eigensolver

An eigensolver is a method for finding eigenvalues and eigenvectors (eigenstates) of a matrix, which is essential in diagonalizing a matrix. In quantum computing, it is used to find the energy levels and states of a quantum system. The script describes eigensolvers as a way to diagonalize a matrix to know the eigenvalue along with the eigenstate, which is a fundamental process in VQE.

💡Quantum-Classical Hybrid Loop

This refers to the iterative process where a quantum computer prepares and measures quantum states, and a classical computer processes the results and sends back optimized parameters. It is a core aspect of VQE, as highlighted in the script, where the quantum computer returns the energy of a trial state, and the classical optimizer adjusts the parameters to find the optimal value.

💡Fermionic Hamiltonian

A fermionic Hamiltonian is a mathematical description of a system of fermions, like electrons in a molecule. The script discusses the transformation from a fermionic problem, which is the generic form of the system, to a qubit Hamiltonian that can be processed by quantum hardware, emphasizing the complexity of this translation.

💡Qubit Representation

Qubit representation is the method of encoding information in a quantum computer using qubits, which can be in a state of 0, 1, or a superposition of both. The script mentions the need for a transformation from a fermionic problem to a qubit representation, which is the binary-like system understood by quantum hardware.

💡Entanglement

Entanglement is a quantum phenomenon where the state of one particle becomes dependent on the state of another, even when separated by large distances. In the script, entanglement is discussed as a crucial part of the quantum circuit in VQE, where layers of entangling gates are used to create complex, entangled states that can represent the molecular system more effectively.

💡Classical Optimizer

A classical optimizer is an algorithm running on classical computers to find the minimum or maximum of a function. In VQE, it is used to tune the parameters of the quantum circuit to minimize the energy expectation value. The script mentions the importance of choosing the right optimizer that can work efficiently with the limitations of quantum hardware, such as the number of circuit layers and measurements.

💡Born-Oppenheimer Approximation

The Born-Oppenheimer approximation is a simplification used in quantum chemistry where the motion of electrons is considered independently of the motion of the nuclei. The script refers to this approximation when discussing the electronic Hamiltonian, explaining that it allows for the focus on electron interactions while treating the nucleus as fixed, which simplifies calculations.

Highlights

Introduction to the Variational Quantum Eigensolver (VQE), a quantum algorithm designed for quantum chemistry simulations.

Explanation of the three components in VQE: variational, quantum, and eigensolver, each playing a crucial role in the algorithm.

Description of the quantum-classical hybrid optimization loop central to VQE, where the quantum computer and classical optimizer work in tandem.

Importance of parameterized quantum circuits in VQE and their tuning by classical optimizers.

Challenge of Hamiltonian mapping in quantum chemistry, translating fermionic problems into qubit Hamiltonians.

Discussion on the active research area of reducing Hamiltonian complexity in quantum chemistry simulations.

The role of initial state preparation in VQE and its impact on the efficiency of quantum chemistry simulations.

Overview of the quantum chemistry simulation process, including the simulation of molecular behavior and identification of minimum energy states.

Insight into the fermionic Hamiltonian and its components: kinetic, potential, and interaction energies.

The Born-Oppenheimer approximation in quantum chemistry, simplifying calculations by treating the nucleus as fixed.

The goal of quantum chemistry simulations: finding the ground state energy and the corresponding molecular distance.

Details on the implementation of VQE, including the structure of the quantum circuit and the role of entangling gates.

Trade-offs in the design of quantum circuits, balancing depth and entanglement for optimal results.

Measurement process in VQE and how it contributes to calculating the expectation value of the Hamiltonian.

The classical optimizer's role in VQE, selecting the best parameters to minimize the energy of the trial state.

The use of the SPSA (Simultaneous Perturbation Stochastic Approximation) technique in the classical optimizer for efficient parameter tuning.

Practical demonstration of VQE in quantum chemistry, calculating the energy of known molecules using actual quantum hardware.

Transcripts

[Music]

in this section we're going to

talk about variational quantum

eigensolver it's a particular

application particular version of

quantum variational quantum algorithm

and it's we're going to see its

application in quantum chemistry

simulation

remember that

vqe

was one of the first algorithms that

came on the variational side

and it was tuned to quantum chemistry

application there are three parts in

this name as you can see variational

quantum and eigensolver variational

comes from variational principle and

quantum mechanics we're going to see

that a little bit uh about that in

detail

quantum naturally stands for quantum

physics or quantum computing as you

would want to see it eigen solver is

essentially a way

to diagonalize the matrix to know the

eigen value along with the eigen state

of a particular matrix so when you take

a generic matrix and you want to

diagonalize it you can transform it and

finally get into a form where it is

diagonal and that

diagonalization process is what is as

called as an eigen solver at the end of

it you're going to get the eigen values

and the corresponding eigen states okay

so we saw this picture at least at the

lower half of the picture from the

variational quantum algorithm discussion

so you have a quantum classical hybrid

optimization loop

where you have some aspect of the cost

function implemented in quantum and the

classical optimization tunes the

parameter that goes into this

parameterized circuit remember the

quantum

circuit that we implement is a

parameterized circuit and the parameter

is what is tuned by the classical

optimizer running in a classical

hardware the quantum computer returns

the cost function or in this case the

energy

of that particular

trial state and the quantum classical

optimization tunes and the job of it is

to find the optimal value

remember in the previous discussion we

talked about input going into this um

loop uh in this particular case of vqe

there is what is called as a hamiltonian

mapping portion we are going to talk

about that and naturally the trial state

the initial triad state the and sets

and then subsequently the optimization

loops takes over and the parameters are

tuned

and then the handsets actually sets the

quantum circuit that gets implemented so

it is very important and then the

optimizer what are what is the kind of

optimizer you want to choose and set it

up so that it works efficiently in the

quantum hardware system that we have

so here is the solution framework

remember what we are talking about here

is a quantum chemistry simulation which

means that we have system of molecules

we want to be able to simulate its

behavior efficiently

molecules

we can look at it as a fermionic problem

so the electrons are fermions

the description of it is in a particular

form factor called a fermionic

hamiltonian and it's typically

generically called as a fermionic

problem

remember the quantum computer

is a binary system it has two levels

only zero and stage zero and state one

so it is what we call as a qubit

hamiltonian so it has a qubit

representation uh for intuitive purposes

think of it this way our general world

works on decimal numbers however when

you run it in the actual classical

hardware it runs in binary so there

needs to be a way to transform

the decimal number system into a binary

number system the binary number system

is what is understood by the classical

hardware

much like that

in the quantum space you have a generic

system called a fermionic problem but

our hamiltonian or what runs in the

quantum hardware is a qubit hamiltonian

so there needs a transformation from

this generic into a specific form factor

that we have in a quantum computer so we

need to be able to translate this and

this is a non-trivial problem

and that's why we highlight this we need

a classical cost function remember we

talked about tuning of the parameters

and that prepare and then we prepare the

trial state the using the quantum

circuit in this picture the classical is

shown in the left and quantum is shown

in the right but the idea is the same so

you have the

quantum state prepared psi of theta

theta being the parameter and then you

get the measurement measure the

expectation values and that's what goes

to the classical and the classical

pieces together that and computes the

total energy and then it tunes the

parameters so that it finds some optimal

things

what are the challenges here

each step of the way there is a

challenge and it's an active area of

research

the mapping from homeonic to cubic

uh this is important because while there

are different mechanisms already

existing that can do this there are

certain tuning that we can potentially

do

the lesser the size of the hamiltonian

that we need to simulate the better it

is in the nest hardware so any

optimization any learning that we can

have any anything that can reduce the

complexity of the hamiltonian on the

cubit side as part of this

transformation is always helpful so

there is an active area of research to

figuring out how to reduce

this hamiltonian even more

with in some cases with trade-offs in

terms of approximations and in some

cases um

exact ones so there are various

techniques that are used uh to do this

and it's an active area of research the

other active area of research you would

have guessed by now is the initial state

preparation we have talked about this a

lot in the variational quantum algorithm

section um so what is the trial state

that we're going to prepare that answers

the quantum circuit that we're going to

implement the parameterization of that

quantum circuit

that becomes important

and then you have the classical

optimizer so what kind of optimizer

can leverage the system that we

currently have and the limitation of the

hardware for example it can't be too

deep a circuit and the number of

measurements number of runs of that

quantum has to be reduced any optimizer

that can do that kind of

tricks and get benefit from it will

always be beneficial so there is a host

of optimizers that is available out

there we can pick and choose based on

the problem at hand and see which works

the best

so coming to the problem at hand

so

quantum chemistry involves simulation of

molecules

so here is an example of a simple

molecule think of it as a two electron

system and your

job is to find the minimum energy of

that particular molecule

so look at the x-axis think of it as the

inter-atomic distance the distance

between them the two molecules um and

then um the energy so as we move the

distance from it the energy landscape

changes so if you are very close to the

the two systems are close together it's

in the unstable regime the energy is

very high there is a push to move

together and if you are in the regime

where they are too far off they are not

interacting much it's called as a

dissociation regime but there is a space

or the distance where there is certain

equilibrium back and forth and that is

their system that is the energy where

you have the minimum energy and then it

increases as the distance increases

so your job or most of quantum chemistry

are important problems in quantum

chemistries um is to identify what

distance

do you get this minimum energy and what

is that minimum energy these are the two

problems that are of interest and

finding this particular minimum energy

is a very important aspect to quantum

chemistry analysis

so what is now a fermionic or an

electronic hamiltonian so here is a bit

of a math but i will give an intuition

to this particular thing

in high school physics that you may have

done

remember the

the simple pendulum example so it

oscillates

between this and this we know that

the total energy of the system is the

sum of the potential energy and the

kinetic energy you also know that in the

pendulum case

this point the highest point is where

the it is all potential and when it gets

to the

equilibrium point at the middle is where

it's all kinetic and it's back to

potential so you know that there are two

energies to it one is potential and

kinetic much like that you have the same

thing represented here you have the

kinetic and you have the potential

energy and then here is the interaction

part remember even in the classical

physics

when you go from here to here there is a

change from potential to kinetic right

so there is a way to go from potential

to kinetic here there is an interaction

between the different

electrons so electrons are there there

is an energy of that electron visa with

the nucleus and the rotation etcetera

that it has all this is captured in the

kinetic and potential but the

interactions between the two electrons

is what this term is let's think of this

as the total energy of this system note

that we have the subscript e which means

that it's a hamiltonian of the

electronic system only doesn't include

the nucleus part um this is in what is

called as a born-oppenheimer

approximation regime this is generally a

good approximation where you can think

of nucleus as fixed and you can sort of

move that away and just focus on the

electrons and its interaction um much

like

for us

in the broader scheme of things we treat

sun as something physically fixed and

everything else we do it with respect to

the sun central

sun is the center and everything is

around it and you have relatively

relative to this position of this sun we

do the calculations and that works out

quite well uh we know that

the sun and the solar system is not

fixed in space uh it roams around in the

galaxy and indeed in the universe so

it's not something absolutely fixed but

that approximation works out because

what we are interested in is the

relative

positions of these planets and their

interactions much like that

this approximation sort of takes the

nucleus part of it away and focuses just

on the

electron part and that works out well

for

systems that have lower energy

primarily because nucleus

the mass of the nucleus and electron is

so different so that approximation works

out quite well so what are we solving

for uh as you as described earlier what

we want to identify are two things what

is the ground state

uh in the previous page

the location the distance

think of it as an example so what is the

state and what is the value of the

energy of that ground state that is what

we are trying to solve

now uh these are details and this is the

second order approximation you can

expand that um particular electronic

habitonian in the second order

much like a taylor series type expansion

you can expand that out

and there are different approaches to go

from the fermionic system to the qubit

system so jordan wigner is a popular one

driving is another one there are

multiple of these available

that takes this as input this

hamiltonian and translates that into

what is called as a qubit hamiltonian

again remember we are going for example

an intuition is going from a decimal

number system to a binary number system

for example now think of it that way

um so you're translating that and there

are

systematic way of doing it these are

non-trivial

things but

the system exists to go from this

hamiltonian to this hamiltonian which is

what

the qubit system understands

so there are many tradeoffs involved as

i mentioned so it's an active area of

research to reduce

these

number of

number of components of this hamiltonian

there is active area of research there

the lesser it is the better it is

this is an important

actually very intuitive but very

important principle the variational part

in the variational algorithms is from

the variational principle

so what is the idea so remember in this

hybrid loop where you have the classical

talking to the quantum

you pass the parameters to the quantum

it prepares the trial state and computes

the energy of that particular trial

state and gives that system back the

trial state is represented by the psi of

theta

h is the hamiltonian as we discussed in

the previous chart that is the energy of

the system and this is the expectation

value so this one on the numerator is

telling you what is the energy of the

system and this is the normalization

factor so let's ignore this for the

present let's just look at the numerator

um so

eg is the actual ground state energy of

the system what this variational

principle basically tells you is that

energy of the trial state will always be

greater than equal to the ground state

energy which seems straightforward in

the sense that

any

any particular theta value that is not

the ground state by definition ground

state is the lowest energy of a

particular system which means that that

is the minimum of that particular system

so any state of that system

should necessarily be greater than the

energy of any state of the system should

necessarily be greater than that of the

ground state that's what this one is

telling you

and this

ties in with

the

the principles of the cost function from

before remember um

we we we had that faithfulness and

easily estimatable as two important

qualities of the cost function here this

inequality uh demo

tells us that it is faithful that is if

i minimize this i will necessarily be

getting to the ground state there is no

uh difference in here and therefore this

cost function of energy of that

hamiltonian calculating the energy of

the hallmate onion is um is faithful to

what we want to solve that is to find

the

eg or the energy of the ground state and

this particular inequality

ensures that the faithfulness of the

cost function is determined the

implementation part we will see it

subsequently in terms of the two second

part of it was can i implement this

efficiently and that part we will see in

the next portions

so the actual implementation now we have

little bit more detail the same

classical quantum hybrid uh

structure that we saw before is what is

shown here

so what you're seeing is on the left is

the quantum part on the right is the

classical optimizer giving the parameter

value theta going into the string

the qubit hamiltonian is represented by

this in generic form

so psi g of theta this is the trial

state is the is at this point in time so

we will get into this details shortly

but at this point after

all these

gates are performed you get into this

trial state corresponding to the

parameter theta and then this portion

measure does the measurement part and

get the results out so what do we have

here in this particular thing

um so all the initial the initial state

of all the qubits in this case there are

six qubits shown um is set to zero by

default they are all reset to zero

then what you have is an alternate um

operations so first you have a layer of

single qubit gates then u and n stands

for entangling it this is uh two qubit

gate it could be a c naught it could be

a control z or any of those two

multi-qubit gate operations basically it

is entangling uh the system

and then you have another single qubit

gate so

what you have is sandwich of

entanglement between two cubicles and

you can do this n number of times you

see the d here is stacking of this

in d number of times

the more layers they are the more likely

the department gets which is bad from

the hardware standpoint we don't want

too deep a circuit too but the plus for

the reason for doing that if at all is

more deeper it gets the more entangled

state it gets the more state it explores

in the state space as you know the

quantum system has exponential state

space a significant majority of them are

entangled states so the more entangled

they are you go into those more

exponential parts of the states and you

are likely to find better answers so

there is always a trade-off involved in

terms of

how many how much of entanglement do you

do in each layer and how many of these

layers do you stack

this is a function of what we have in

the hardware the lifetime the noise

profile of that particular hardware it

will be a combination of this to choose

the right settings

for these particular parameters the the

kind of entangling you want to do that

is you end and the number of layers that

you want to stack one after the other

this portion is the measurement part

we'll cut to in detail so at the end of

it you will get the expectation value so

what you get here this is um

i've shown it little bit

in a generic sense but really what you

get from a single execution of this is

the inner portion of this summation

which is the expectation of value of

sigma alpha remember hamiltonian is

summation over the sigma alpha the sigma

alpha

is a set of poly strings applied on each

of these qubits the poly gates applied

on each of this qubit what you get by

single execution of this particular

quantum circuit is the expectation value

of

this particular energy we need to do n

number of sampling the number of shots

for each of them to get the

right value expectation value and then

you have a pre-factor and then the

summation actually e of theta is

actually done in a classical hardware

inside um what i have shown for

simplicity is the e of theta is what is

get from quantum but that's not actually

true the summation of a portion of this

equation is done in classical hardware

before going to the classical optimizer

but for simplicity i've shown it as e of

theta here but what you actually get out

of a particular quantum

execution is this expectation value of

sigma alpha

okay so that is the energy measure and

the classical optimizer then goes into

this classical optimization part

so let's go to the trial state

preparation

so

as you saw you have the single dates so

if you see the notation of it

so the first index here is stands for

the qubit number

and

the second index stands for the layer so

for example this is qubit number one

this is layer 0 this is layer 1 and you

have so on so forth so this is a one

time operation the single qubit

layer here and this forms one group that

is entangling followed by single qubit

so if you have say two of this you have

the first layer which is

this as indicated then you have the u

entangling and the single qubit layer

and then you have one more of it if it

is d equal to 2 you will have 2 of this

that's what gets done

so um each

the parameter theta k is shown as a

vector here but actually

think of each of them as the parameter

value theta

for

that particular

location in a way so you will get all

these values

the theta vector coming from the

classical optimizer

so the entangling gates as i mentioned

is two qubit operations sandwiched

between the single qubit rotations right

d is the number of layers that we

do the choice the circuit that we end up

doing this the quantum circuit here

is what is the colon and sats now this

is as you can variable see is in a

parameterized circuit what kind of

handsets

do we do a traditional or a hardware

sensitive ones earlier we did discuss in

detail about different ways of doing

hardware and assets

problem inspired problem agnostic and so

forth problem inspired is what's very

useful at least uh in this particular

paper and the subsequent hardware

implementation of this that was

demonstrated by ibm um it was a problem

inspired handsets uh coming from quantum

chemistry uh that was used and uh also

there are other

more advanced and such more recent ad

and such that was derived that are

sensitive to hardware noise and those

are the hardware sensitive ones that we

have so this one

is a craft to find the right handsets

and how to use them and also make it

sensitive to the hardware that we have

the second part is the measurement part

remember the hamiltonian that the qubit

hamiltonian i've just expanded out the

summation here is a sample hamiltonian

for example for this q six uh in this

example i'm showing it for four qubits

one two three four this example on the

plot here shows six qubit but

nonetheless for simplicity there's a

four qubit example

so what this one is showing is that uh

we have um we have to do uh the

expectation value so the number of calls

uh to the quantum circuit is 4 because

you have to compute

the expectation value for each of these

strings

for example x y z x z 0 x and so on and

so forth so each call

needs to be made so each of them

corresponds to a call to the quantum

circuit

there is typically one layer of rotation

needed for this qubit hamiltonian

followed by the measurement in the basis

state which is the c basis so this

rotation is based on the particular

value be it x y or z if it is z or z

there is no rotation needed you apply an

identity but in other cases you may have

to do a single qubit rotation before

measuring

uh note that the number of calls is

proportional to the summation the number

of summation terms that we have each

term needs needs to be averaged to get

the result and then finally to calculate

the expectation value the lesser the

number of strings the number of these

terms the better it is because we don't

have to go to the quantum each step um

so there are a lot of opportunities to

tune this um for example in the

radiation quantum algorithm section we

talked about uh communicative property

non-commuting portions using the

commuting property to eliminate some

aspects of

these strings so there are many

techniques that are there to reduce the

number of terms in this hamiltonian

finally the classical optimizer

so

here is this is um classical optimizer

what you are going to see here is not

unique to quantum this is true of

classical machine learning techniques as

well

this is an imagined uh cost function

space so this is the say the space that

you're looking at what you want for

example let's say is the global minima

which is indicated by this portion right

here

let's say you have no clue about this

state this is highly non-linear as you

can see there is a barren plateau the

big barren plateau is right here

if you land up here you're less likely

to find any answer

but there are also local minima as you

can see there are many places where you

can get stuck and not be able to get out

of it

let's say you have a trial state initial

state the initial parameters that's why

initial parameter again as i described

before is very critical if you land up

here for example the odds are that you

will get to the global minima but if you

land up somewhere here

the odds are that you're going to get

stuck somewhere in the local minima here

then the question of one is the

calculation part how many calls to the

quantum hardware needs to be made that

is important the second part is

how do i traverse how do i move around

in this space and that's what a

classical optimizer does there are

various approaches that are used

the one that is used by ibm in this

particular paper where this was used to

demonstrate a chemical simulation was

the spss simultaneous perturbation

stochastic approximation technique um

the key is because

um

the value proposition of this particular

technique and it seems to do well with

particularly the quantum chemistry space

is that it makes only two calls per

iteration

irrespective of the optimization so for

example what it does is you have the

current let's say theta happens to be

the current point seems to be here you

get two points here based on the

the slope of the different things it

moves in certain direction it tries to

find a

technique it's it's a technique to move

in different directions and takes it to

it but the important point is each

iteration requires only two calls per

iteration which is what is very

important we want to reduce the number

of calls that we make now the lesser it

is the better it is so um this is what

is used uh in the actual hardware when

the the experiment that was done in this

paper

where they run these experiments in the

actual hard way to calculate the energy

of

some of the molecules that we know