Variational Quantum Algorithms

let's dig into the subject of

variational quantum algorithms

so variational circuits are the

practical embodiment

of the idea that we want to train

quantum computers the same way we train

neural networks and the basic way a

variational quantum circuit is is that

if there is some quantum circuit that

forms the

basic subroutine of a larger algorithm

the quantum subroutine takes in the

state preparation or

kind of input data x and it also has

some circuit parameters

theta and then it outputs some

measurement statistics

these measurement statistics go through

some classical processing

and then you use some optimizer or some

update rule to

update the parameters in some outer

classical optimization

loop now variational circuits are also

called parametrized quantum circuits or

even quantum neural networks

so a variation of circuits consists of

the following ingredients

so first thing we want to do is prepare

some initial state psi

and as usual this is often a ground

state or a zero state or some fixed

reference state

and then we want to execute some

parametrized unitary transformation

which

breaks down to a sequence of gates and

the parametrized

is important here because that's that's

where the variational parameters come in

those are the things that we're going to

vary are the parameters of the gates

so the architecture of the circuit is

fixed but the parameters fed to the

gates are not fixed

and then to convert quantum information

back to classical information we want to

measure some particular observable

which we'll call b

so a particular variational algorithm

will contain

a few fundamental ingredients so

first you need to decide on a circuit

and sats

so an ansat is the

structure the architecture of the

circuits and

this is sometimes fixed in place for a

particular algorithm

or it's sometimes up to the user to

decide what this can be

we also need a problem-specific cost

function so this is something that

codifies a particular objective of

interest that we want to minimize or

maximize

and relates it back to the outputs of

the quantum circuits

and then we need a training algorithm

and for me i would like to really focus

on gradient descent but you could use

other training procedures if you'd like

and what the training algorithm should

do is it should take

some function computed from the output

measurements of the quantum circuit and

then

update this circuit's parameters based

on that information

so a famous example of a variational

circuit is called the variational

quantum eigenself it's one of the very

first variational algorithms

and it's in particular focused on

quantum chemistry problems so simulating

chemicals using quantum computers

so the three ingredients i said you need

are the ansats the cost function the

training

so the ansats could be something that's

very much

heavily tied to the intuition or the

chemical

or physical nature of the problem in

this case something called unitary

coupled clusters singles and doubles

it's a particular answer that's related

to quantum chemistry

we need some problem-specific cost

function and in the case of eqe

this is actually a energy measurement so

you have some

hamiltonian observable some some

output that you observe that measures

the energy of your circus

and then the training procedure and

again in this particular example i've

chosen gradient descent and i'm going to

use the perimeter shift rule

to compute the gradients and the goal is

to minimize

the cost function which is the energy so

it's going to find the minimum energy

state of a particular hamiltonian or a

particular physical system

another example is qaoa i've heard also

called kwawa but i don't really like

that's

that name it stands for quantum

approximate optimization algorithm

so in this particular case the ansats is

a very particular structure it's it's

something that comes from the

initial statement of the problem itself

and it consists of a repeated series or

a repeated layering of different

circuit sub-circuits so there's a cost

circuit which implements something

related to a cost function and then

there's a mixer

circuit or sub-circuit which implements

something which kind of jumps

or coherently moves into different

configurations

so in the case of qaoa the cost function

is something that is encoding a

optimization problem so you might have

an optimization problem

a discrete optimization problem with

many clauses that have to be satisfied

and you can encode this

into an ising type model a spin chain

type model which

can then can be converted to some

observables that you would measure on a

quantum circuit

and this particular example i'm going to

use gradient descent but instead of

using

standard optimizer i'm going to use a

shots frugal optimizer

for instance something called rosalind

so there's there's lots of different

ingredients that play here that you can

pick and choose or might be specifically

chosen by the variational algorithm

itself

so there's a number of different uh

variational algorithms out there in the

literature there's actually quite a few

and you can break them down into

different subject

areas so there's ones related to

chemistry or physics so these are

preparing quantum states that emulate

physical systems

or tell you interesting properties of

physical systems

there are variation algorithms related

to

mathematical problems such as factoring

or solving linear equations

and these can be seen as near-term

candidates to

replace things like shore's algorithm or

hhl algorithm for solving linear

systems of equations there's also

a number of variational algorithms tied

to machine learning and this is not too

surprising because variational

algorithms are inheriting a lot of

structure from machine learning so

machine learning is one of the natural

application areas

so there's quantum generative

adversarial networks there's quantum

classifiers there's

different kinds of quantum neural

networks for instance recurrent networks

or graph networks or optical

implementations or convolutional quantum

versions of neural networks

so it's quite an interesting research

area right now and there's lots of ideas

out there

and i encourage you to check out this

website at the bottom of the slide here

if you want to see some more examples

so a little bit more about the ansats

nsats is a german word

it's basically in physics it means

something like an educated guess

or an additional assumption that's kind

of chosen at the start but which is kind

of

verified as being correct throughout the

course of the problem

so in the case of variational circuits

the ansats is

the particular structure of the quantum

circuit

and as i said sometimes the structure is

completely fixed by the problem

and other times the structure is more

flexible

so some might say it's completely

selectable by the user

so in vqe in particular there's no

requirement that you have to use any

particular nsats

the only thing that's fixed is the cost

function whereas in qaoa

some of the cost function influences

the actual circuit that you're using

so the ansats is an important ingredient

and there's lots of ways to choose it

and it's still very much an open

question about

what are the best ends and states in

invasional circuits for different

problems

so again the reason for choosing a

particular and sats

uh it could be many so there could be

some intuition or some

some like logical or physical or

mathematical basis for choosing a

particular and sats

so i did mention that vqe doesn't force

you to select an assets but you might

want to choose one that we know

is likely to be similar to how actual

chemicals or chemistry systems work

again the answers can be dictated by the

structure of the problem itself

the nsats can come from intuition board

from other fields

like machine learning or the nsats can

be something that's taken in order to

make something more trainable or the

ancesto can

really be arbitrary you can use your

imagination and there's no reason to

favor one handset's over another one but

the choice of answers

will affect the quality of the model

that you're able to

learn or the quality of the answer

you're able to achieve from your vaginal

circuit

and one general piece of advice is the

deeper

you can make your own sets typically the

more

expressive it can be and the better

results you'll get

so another important thing to take into

account is

the input data so circuits don't just

have free parameters but sometimes you

also need to input data

into them in some problems it's not

necessary but other problems is

necessary

so how do we actually input classical

data

into a quantum variational circuit

there's actually a number of different

strategies you could take here

and it's still very much an open

research question

of how to embed classical data into a

quantum

circuit

so one of the simplest choices you can

make is say well

the easiest way for a parameter to enter

a circuit is through a rotation of a

single qubit so what i could do is i

could just rotate a single qubit

in proportion to the value of a of a

single data point

so a single scalar value so that's very

common

but i really want to warn people that

this is not sufficient

if you do that then the only thing your

circuit will ever

produce as a function of this input data

will be a simple sine function

so it's it's much more complicated story

and there there needs to be

still a lot of exploration done in order

to find the optimal or

reasonable ways to embed data

so just as a kind of mentioning of some

strategies that are already available

that go beyond this very simple

initial strategy is something called

data re-uploading

and the idea is not to up not to embed

data using a single rotation but

actually a sequence of repeated

rotations and maybe there's free

parameters in between those as well

and this can make a more complex

function available to you in your

circuit than you would have if you just

did a single

rotation the other idea is to have

actually a trainable

embedding layer so don't worry so much

about training the

the unitary of the circuit worry about

training the embedding and then use

standard quantum information

metrics and tricks to to classify the

data for instance so

learnable embeddings is also a very

viable strategy

so when we're using variational circuits

if you can compute the gradient using

the premiership rule then that opens up

every possible flavor of grady descent

that you could want so there's standard

grading descent but also in

deep learning there's all sorts of other

great instant optimizers that are

available to you

ones that you see most common are

momentum and atom

probably but there's also a number of

quantum aware optimizers that you could

use and these are things that

inherently take into account that you're

optimizing a quantum circuit

and not just a black box so for instance

i've put three different examples here

of

quantum aware optimizers so one is

it's actually a pair of optimizers

they're called rotosolve and rotoselect

they're in the same family

these actually don't use gradients at

all so instead of using the gradient

they recognize that there's this

sinusoidal structure to quantum circuits

and if you're only ever looking in one

direction in parameter space you can

actually find a minimum quite easily

just by going to the minimum of that

particular sinusoid

and then you can iterate through that

process many times and hope that you can

find

via a sequence of individual jumps to

local minima

eventually end up in a global minimum

another cool quantum ware optimizer is

called quantum natural gradient

and the basic idea here is that the

inherent geometry of quantum circuits of

quantum

computing circuits and quantum physical

systems

is not euclidean so it doesn't look like

the world around us is not like this

rectilinear structure it's more of a

it's more of a sinusoidal structure so

we should take that into account

we should adjust for the inherent

geometry of the

space that we're optimizing in and then

there's a family of optimizers that are

called

shots frugal and what these do is

recognize that in current day

quantum computers the number of circuit

executions is actually a very precious

commodity and if you're having to wait

in a queue online is this can really

slow down your optimization so

these optimizers are much more frugal in

how they use optimize

how they optimize using shots and they

they rely on a lot

smaller number of shots especially

earlier on in training to get

estimates that you need to train the

circuit

final thing i want to mention and this

is a topic that people have probably

heard of before is there's this notion

of baron plateaus

for variational circuits the basic idea

here and it's it's similar to something

that happened in deep learning is that

there are parts of the optimization

landscape where the gradient is zero

and anywhere you go around it the

gradient is also zero

so they're very flat and so it's hard to

use a gradient descent strategy because

everything just looks like you're

completely flat in every direction so

baron plateaus actually come from a

number of different effects there's not

just one effect but there's multiple

ways to do it they can come from the

choice of your circuit hand stats they

can come from the choice of

parameterization

or parameter values or they could come

from your cost function

and there's a number of different

proposals for overcoming these

uh but it's still an open question i

would say to how to avoid these

in general so you could use a

specialized initialization strategy

you could go with a layer-wise expansion

of your circuit bit by bit and

try to avoid it by making the shortcut a

circuit short at the earlier

stages of training or you can go with

adiabatic type approaches where you'll

have a very

kind of slow evolution towards a

targeted goal

and these are nice pictures that

illustrate the bearing plateau

phenomenon where

if you're sitting anywhere in this

landscape except right at this

center you really can't look in any

direction and see anything with flatness

so brother-in-laws are an interesting

barrier that we'll have to overcome in

order to train

variational quantum circuits

you