mod04lec21 - Variational Quantum Eigensolver
Summary
TLDRThis 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
🔬 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.
🌐 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.
🔍 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.
🛠️ 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.
📉 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.
🧪 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.
Mindmap
Keywords
💡Variational Quantum Eigensolver (VQE)
💡Quantum Chemistry Simulation
💡Hamiltonian
💡Parameterized Circuit
💡Eigensolver
💡Quantum-Classical Hybrid Loop
💡Fermionic Hamiltonian
💡Qubit Representation
💡Entanglement
💡Classical Optimizer
💡Born-Oppenheimer Approximation
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
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