Hamiltonian Dynamics: Application and Simulation with Mario Motta - Qiskit Summer School 2024

hello uh my name is Mario MTA I'm a senior research staff member at IBM Yorktown and I work on the quantum simulation of many body systems today I'll be telling you a bit about the concept of aonian Dynamics which is very important in Quantum computation I'll give you an overview of some applications of aonian Dynamics and of some techniques to simulate it on Quantum devices and I will suggest you some references for some more in-depth Explorations so let's start so to bring everybody on board let us first start by defining what a miltonian Dynamics means

and what it means to simulate it so for that we have to go back to the postulates of quantum mechanics which tell us that the state of an isolated Quantum system evolves in time according to a shinger equation where the time derivative of the state is proportional to a state that is obtained applying the hamiltonian of the system the operator associated with its energy to the state at the current time te the formal solution of the shinger equation is given by the action of a unitary operator the time Evolution operator to the state at the

initial time for a Time dependent hamiltonian a hamiltonian that depends on time this operator is the time ordered exponential that you see in the second equation for a Time independent aonian hamiltonian that remains constant with time it is the exponential of the hamiltonian times minus i t simulating aonian Dynamics for example on a quantum computer means to map exactly or approximate the state of the system at time t on a state of the quantum computer through the application of a Quantum circuit and once we've done that for example we can calculate the expectation value of

an observable at time T so the problem of simulating aonian Dynamics is one of the real challenges of the simulation of quantum systems along with some others for example we have the simulation of aonian igen States like the ground the lowest energy and the excited the higher energy states of aonian which are solutions of a Time independent Shing equation and the simulation of thermal aages are thermal properties which requires us to find the GBS State the the equilibrium state of a canonical system at inverse temperature beta these three problems are deeply intertwined uh first uh

they often determine together the physical properties of some Quantum system we will see an example later when we talk about scattering experiments and in second as we will see the simulation of hamiltonian igen States may use the simulation of hamiltonian Dynamics as a sub routine or vice versa so there are very interesting algorithmic connections between these problems which are fairly important for Quantum Computing so aonian Dynamics is very important because it determines many physical properties let let's take a look together at some applications of aonian Dynamics in the simulation of many body systems so one

of the ways uh to probe material properties and to understand material properties is routinely are laboratory experiments among laboratory experiments there's an important class of experiments which are the scattering experiments and as an example in the chart we can consider ultraviolet visible absorption spectroscopy in this kind of experiment some sample that's the yellow um thing um at equilibrium at temperature T for example at temperature zero when it is in the ground state is heat with monochromatic light light that has a well defined frequency Omega um the intensity of the light emerging out of the sample

is recorded as a function of the frequency given as an absorption Spectrum some examples from astrophysics are shown on the on the right um ultraviolet absorption spectra often present dark lines corresponding to frequencies that the system absorbs the pattern of lines is a little bit like a footprint that allows us to understand some of the properties of the material that is responsible for the absorption for example the chemical composition um but photons are not the only particles that we can throw at a Quantum system in condensed matter for example it's very frequent to use Neutron

scattering experiments in a neutron scattering experiment some Quantum system it can be a Quantum liquid like super fluidum 4 or a spin system is heat with neutrons that have a well- defined momentum K and energy e and what we measure is the intensity of the scattered neutrons with a different momentum K plus q and with a different energy e plus Delta e a higher or a lower intensity of scattered neutrons we see an example on the right means that the interaction between the neutrons and the system has changed the momentum of the neutron by an

amount q and its energy by an amount Delta e so there are many different kinds of scattering experiments but they all share some uh common structure um in many of these SK in experiments a Quantum system is prepared at equilibrium in the ground or thermal state of some hamiltonian h z and then some perturbation is applied the perturbation represents the interaction with the impinging radiation or matter in ultraviolet visible spectroscopy the perturbation operator is a component of the dipo moment to represent interaction with an electric field in the case of Newton scale in an external

potential representing the interaction with the impinging neutrons um the absorption of a photon in uvv spectroscopy with energy H bar Omega is due to some excitation if we imagine to be at zero temperature from the ground state P0 to an excited state P me with an energy difference e minus e0 that is equal to H bar Omega and this excitation is induced by the perturbation um these um Notions are captured in terms more mathematically rigorous by ferm Golden Rule According to which the the absorption of uvite is described by a linear combination of the r

Deltas in the frequency picked at the transition energies of the system with transition rates that are metric elements of the perturbation operator between the ground and the excited state in UV visible spectroscopy these quantity are called transition dipole moments and they are related with the oscillator strengths this uh form is given in frequency domain but with the Fier transform and with the resolution of the identity we can represent it in terms of a Time dependent response function a ground state expectation value of the product between the time evolve the perturbation operator V and the perturbation

operator V response functions are a very fundamental aspect of physics they link experimentally observable properties with some underlying many body quantum mechanical behavior and they are also an important application for a miltonian Dynamics because we have a Time Evolution operator inside the response function uh if you're interested in this kind of uh uh problems I do recommend the book by feter and valka quantum theory of many particle systems it's a fantastic reading fmy Golden Rule that we saw in the previous chart captures essentially a linear response approximation to a more General setup where a system

is prepared at equilibrium for example at 0 temperature in the ground state of an unperturbed aonian h z and then a perturbation Poss time dependent is applied letting the system evolve and according to a Time dependent shinger equation with the unperturbed the miltonian and the perturbation after that a Time dependent expectation value a of T is measured at some time T linear response approximation essentially corresponds to approximating the time Evolution operator to first order in the perturbation operator V and according L the time dependent expectation value differs by the value at the initial time to

an amount that is controlled by a correlation function between the observable a and the perturbation operator V uh the setup of points one to three um is also the setup of quantum quenching experiments which are used to study aspects of non-e equilibrium physics such as thermalization and Universal aspects of critical Dynamics in this kind of setup one doesn't require that the perturbation be weak in other words uh we don't need to approximate the time Evolution operator in any way in principle um so now we have seen some quite important applications of aonian Dynamics which for

example defines the time dependent response functions and time dependent expectation vales which are probed in in various experiments along of course with the problem of preparing a miltonian i State such as the ground state which also was in all these quantities we briefly discussed two kinds of spectroscopy experiments but there's a great variety of them for different systems and situations um but uh how can we simulate miltonian Dynamics on a quantum computer that's the topic of the next part um um so first the first thing we have to do is to map um the degrees

of freedom of the problem to the degrees of freedom of the cubits and for that I remind you to Kevin Sun's lecture this essentially means that we can represent the aonian of a physical system with a cubit operator and an impr principle the time Evolution operator with the unitary uh Cubit operator as well in in practice the simulation of the time evolution is is performed by some Quantum circuit which we will call V of T and this Quantum circuit approximates the time Evolution operator U of T E to the minus th H for a Time

independent aonian with some accuracy Epsilon this referring to the Chart means that time Evolution starting from a given point p z would lead you to some State P of T if you have an approximation if a Quantum circuit implements an approximation V oft to the time Evolution operator U of T you're going to land into a different state c bar of T for example and the accuracy of the simulation can be for example Quantified by the distance between the two states Epsilon which of course you would want to be small the simulation is efficient if

the number of Elementary Gates one and two Cubit Gates in the circuit V oft scales at most polom with respect to the number of cubits system size the desired Precision or accuracy and the evolution time a great discussion about these topics is inside Andrew child's PhD thesis which is referenced down here so before going into something more um um Rich let us stop for a second and make one example hamiltonian that is comprising only one poly operator here at the top the exponential of this hamiltonan we can compute it efficiently and exactly for any value

of the time uh first we can transform the PO operator into a tensor product of Z operators using layers of single Cubit operations for example for the PO X operator we can use the adamar to convert into Z form after that our goal is that of implementing the exponential of a tensor product of Z operators and it is known that that can be done with two letterers of C not Gates surrounding a single Cubit Z rotation of an angle that depends on the evolution time but also the coefficient in front of the PO operator so

this is a simple example of a hamiltonian for which the Dynamics can be implemented exactly no approximations with a linear number of Elementary Gates and for a generic time no matter how long it's of course a special case another example is that of onebody fermionic operators um I refer you again to uh Kevin Sun's lecture which explains very well how to map these fermionic operators onto cubits on the top here we have a one body operator with its creation and distraction operators acting on M spatial orbitals even for this operator the exponential e to the

minus it can be computed efficiently and exactly for any time and the reason is that a onebody fermionic operator can be diagonalized exactly by a the exponential of a onebody operator an operation also called an orbital rotation or a Buu of transformation so we can write our hamiltonian as the product of a bolu of transformation another bolu of transformation and in the middle a diagonal one body operator so linear combination of number operators multiplying the igen values of the operator the orbital rotation can be implemented with a circuit that has a linear number of Z

rotations and a quadratic number of 2 Cubit XX plus y y Gates you can see an example down here here for four orbitals the exponential of the diagonalized one body operator because number operators in Jordan vigner representations are single Cubit Z operators essentially can be implemented with a single layer of Z rotations and then we have again another bolu of circuit to implement the inverse of the B of transformation so with this kind of scheme one achieves an exct solution with a polinomial number of gates for a generic time once again a consequence of the

fact that we can exact diagonalize one body operators it is often the case that the hamiltonian is not an operator for which can we can efficiently simulate the Dynamics exactly we have to introduce approximations however if our hamiltonian is a linear combination as it is often the case of terms hi Each of which can be simulated efficiently then also the aonian itself can be simulated efficiently a way to prove constructively this fact are product formulas product formula product formula are a class of formula that approximate the time Evolution operator for the aonian using products of

the time evolution erators of the individual terms of the hamiltonian hi hence the name product formulas of course before showing some product formulas let us make some general observations uh the first ingredient of a product formula is always to break down the time interval 0 to T into some number of steps NS uh typically of equal duration D over NS the time Evolution operator for a single step is then approximated using products of the exponentials of the hi terms with an error that scales like some power of the time step um and the error of

the product formula decreases to zero when the number of Step increases this is informally speaking because the error per step decreases very quickly with step length compensating the accumulation of Errors for NS steps for interested readers there's a very thorough discussion and presentation of the errors of product formulas in this physical review X paper um the arguably simplest product formula is the sometimes called primitive approximation in the Primitive approximation the time Evolution operator for a single step U Delta T is approximated simply by the product of the exponentials of the individual operators times delta T

which we assume we can compute exactly or with very accurate approximations we saw it being the case for po operators and one body fermionic operators those are examples of individual terms for which we can simulate the Dynamics exactly the Primitive approximation is an approximation because if we expand both U of delta T the exact single step time Evolution operator and V of delta T it's approximation in powers of the time step we find that the two tailor series differ and they differ at the second order the distance between the two unitary operators which bounce the

distance between the final States the accuracy of the simulation for a single time step grows quadr ically with the time step times a coefficient which is essentially the norm of a sum of commutators the simulation for a Time T broken down into NS intervals reaches accuracy Epsilon when the number of steps scales like CT squ over Epsilon um the cost grows quadratically with time and is proportional to 1 / Epsilon however the simulation is efficient ently is efficient so long as the number of terms L grows polom and so does c there are many more

refined product formula that use more gates per steps but achieve a better error than the Primitive one thereby requiring fewer steps and therefore fewer Gates than the Primitive approximation on a whole uh for example the second order Suzuki approximation uh is a product formula that uses the exponentials of the individual terms from one to L time delta T / 2 followed by the product of the same exponentials from L to T if we compare the tailor series of the exact time Evolution operator and of the Suzuki approximation we find that the error scales like a

third power of the time step which is better for a small time step than the second power so the second order Suzuki approximation uses essentially twice as many gates per step compared against the Primitive approximation but on the other hand the number of steps required to achieve the same accuracy Epsilon now scales like T to the 1.5 lower than t^2 times the square root of one / Epsilon which is more favorable than one / Epsilon and the simulation is efficient under the same assumptions as for the Primitive approximation so before um moving on to something

different uh let's stop for a second and make a couple of examples the first one is a two site ising aonian which we can see here in the top portion of the chart this is a linear combination of three poop erators and we can simulate its Dynamics for example with a primitive approximation where we have the exponentials of the individual POI operators the exponentials of the X operators um can be uh executed in parallel so to speak meaning at the same time on the two different cubits with X rotations and the exponentials of the ZZ

interaction with the ZZ rot which as we have seen uh in a chart before requires two c Nots and a single Cubit Z rotation so that's a example of a primitive approximation another example again connected to Kevin Sun's lecture is that of the electronic structure aonian the aonian of electrons in a molecules um the electronic structure in miltonian is the sum of a one body operator and of a two body operator now of course we could represent this aonian as a linear combination of Po operators and then simulate its Dynamics with a product formula for

a linear combination of Po operators that would be possible but the cost per Trotter step would be roughly order a fifth power of the number M of orbitals this is because there is going to be m to the 4th po operators and the exponential of each requires roughly order M um to Cubit Gates a lower scaling comes from low rank decomposition for example it has been known for a long time that the electron repulsion integral the to part of the electronic structure in miltonian can be written as a contraction of rank three tensors um the

number of terms in the summation very importantly grossly larly with the number of orbitals in the system a pictorial representation is here on the right instead of having a dense rank four tensor we have a contraction of rank three tensors if we plug in this decomposition of the two body operator inside the electronic structure hamiltonian and we rearrange uh the elements of the two body part we find that the electronic structure in miltonian can be written as the linear combination between a one body fermionic operator uh which does contain the electron repulsion integral and and

then a sum of squares of one body operators constructed with the tensors of the low rank decomposition these sorts of decompositions are very well known in classical electronic structure and they have been used to obtain lower scaling simulations of electronic structure on quantum computer but also to obtain efficient and noise resilient measurements a few references are here we have essentially written the electronic structure aonian as a linear combination of operators that can be individually diagonalized by means of bogolub of Transformations and so this is the starting point for example for a primitive approximation for each

of the terms of the Onan we apply the bolu of transformation then we apply the exponential of the diagonal part with a single layer of Z rotations for the one body part and with something more complicated ZZ rotations will see it in a second for squares of one body operators then we apply the inverse of the B transformation and we Loop through all the terms in the aonian the computational bottleneck are the exponentials of the square of the one body operators which require a quadratic number of two Cubit Gates ZZ rotations arranged in a circuit

of linear depth an important aspect is that the ZZ rotations uh require all to all connectivity if our device doesn't have all to all connectivity meaning we can apply physically to Cubit gate among all possible pairs of cubits in the device then we we have to use a swap Network to move around the cubits until they become physically adjacent then we can apply a zz rotation and then we can separate them back I show you an example of a swap Network fairly dense one in in this chart the development of swap networks is a very

important aspect of the compilation of quantum circuits and a very fascinating one um so one thing that we may have noticed looking at product formulas is that the cost in as a function of time uh grows more than linearly we have a square for the Primitive approximation and three half for the second order Suzuki a question now arises can one further lower this scaling make it linear or even lower for a generic aonian the answer is no there's a very important result no fast forwarding theorem which is presented in these papers by baral and childes

and and Kari um According to which for generic aonian the simulation of aonian Dynamics scales at Best linearly in time we have seen exceptions single po operators one body fermionic operators which have this property of a cost independent of time coming from their very simple diagonalization in general we can expect a cost as per the product formulas that grows at best linearly with time a question is whether one can achieve a simulation time that is strictly linear um in the in the time itself there are some very fascinating rather Advanced approaches to a miltonian dynamics

that I will flash out for you one is the approximation of the time Evolution op Ator with a tranca tailor series introduced by barrier tal which reaches a computational cost linear in the time multiplied by the one Norm of the aonian with some logarithmic Corrections that have together the the time and the accuracy there is also the so-called cubation approach that approximates the time Evolution operator with a truncated yakobi anger series using um a sophisticated intermediate the construction of a special transformation called the cub iterator and cuz achieves linear cost in time and an additive

an additive uh correction logarithmic in the accuracy this is from guanga and Isaac Chang I invite you to examine these references these are very fascinating uh Advanced algorithms for miltonian Dynamics um an extra comment we have seen how to approximately map the time Evolution operator of a Quantum system onto a Quantum circuit and that makes it very easy to compute time dependent expectation values what about correlation functions that we have seen at the beginning for the scattering experiments these are less straightforward these are not ordinary normal expectation values uh of operators they are expectation values

of products of operators one of which is time evolved Computing a response function requires first of all a ground state preparation we have to um to have the system prepared in the ground state to compute the response function a Time Evolution and then a particular Quantum circuit called the modified adamar test which involves controlled application of the operators in the correlation function which we are assuming to be unitary in this setting um controlled by the state of an auxiliary cubits for which the sigma minus the D exitation operator is then measured this algorithm was introduced

by so metal and since then there has been a great deal of research in trying to economize it some references are here in particular there's one by mitay and Fuji that makes use of um um an ensemble a collection of unitary Transformations and mid circuit measurements to bypass the need for the auxiliary Cubit and the controlled operations so we have seen that aonian Dynamics is important for applications like scattering experiments and that there are very nice algorithms like product formulas to simulate it efficiently and accurately um one could even argue that aonian Dynamics is a

quintessential application for a quantum computer in more formal terms this idea is captured by the bqp completeness of the aonian Dynamics uh bqp is short for bounded error Quantum polinomial time it's a concept that is defined formally in the framework of the theory of quantum computational complexity it's a very reach Advanced topic beyond the scope of this presentation so I will give you an informal discussion but refer you if you're interested to this very good book by Alex kayv Shen viali classical and Quantum computation informally speaking bqp is the class of problems that the quantum

computer can solve with polinomial resources and a high success probability since we have seen that for for many aonian there are algorithms like product formulas to simulate aonian Dynamics with polinomial resources and a controllable accuracy we can informally say that miltonian Dynamics is a problem in the bqp class but there's something more a miltonian Dynamics is bqp complete this means that any other computational problem in the bqp class can be reformulated as the problem of simulating the Dynamics of a suitable aonian the first to realize this circumstance and to offer a proof was Fineman in

his seminal 1980s papers two of them are are here and it is for that reason that quantum computers are sometimes presented as Fan's Vision the vision to use a device operating according to the laws of quantum mechanics to simulate the dyamics um of for example a manyon um so aonian Dynamics is important for certain applications like the scattering experiments it is bqb complete so it's a very compelling application for a quantum computer uh but and is to me is one of the most fascinating aspects of quantum Computing aonian Dynamics also has deep algorithmic connections uh

with other Quantum simulation problems and I will cover this topic a bit in the last part of the presentation um so back uh with the mind to our initial diagram uh in particular what I will discuss is that hamiltonian Dynamics um can be used as a sub routine in algorithms for the preparation of aonian hien States like the ground state the lowest energy State and the low line excited States um but first um I shall give you uh let's say a disclaimer the problem of computing the miltonian igen states is fairly different from that of

simulating the miltonian Dynamics in particular it is not in bqp it is in another quantum computational complexity class called qma which is short for Quantum Marlin author again I will give you an informal discussion but I refer you for example to the book by kayv if you have interest in this topic qma is the class of problems um that whose solution can be verified though not in general produced by a quantum computer with polinomial resources for many Emil tonians the igen state problem is in the qma complexity class so we can consider it in a

worst case scenario a hard problem even for a quantum computer a very important reference in this regard is this paper by kimpe atal uh in more informal terms um a normal user someone with polinomial Quantum Resources aing to King Arthur in this metaphor can verify that some wave function is the solution of the aonian value problem but in general one cannot guarantee that such a normal user will be able to produce that wave function the production of that wave function in a general case may require up to exponential resources that only an exceptionally powerful a

agent uh presented by Merlin um um possesses hence the denomination qma and so Quantum algorithms for igen states are heuristic in nature they unavoidably give approximated results often very accurate uh but there are regimes in which they may break down and identifying and understanding these regimes is very important the first algorithm for him Onan igen States based on hamiltonian dynamics that I will present to you is a diabetic State preparation in a diabetic State preparation we are given a hiltonium which is the sum of an unperturbed term um which has known and efficiently preparable ground

state and of a perturbation V which makes the aonian complicated we can think about the electronic structure in miltonian with its one and two body part the goal is to prepare the ground state of the full aonian age not of h z the adiabetic method is based on the adiabetic theorem of quantum mechanics if a system is prepared in the ground state of a hamiltonian and then the hamiltonian changes slowly then the system remains in the instantaneous ground state meaning the ground state of the time dependent miltonian and under certain conditions so the idea is

that if we slowly transform the aonian from h0 to H we will have the ground state of h0 evolving into the ground state of H um I recommend this science paper by fari pioneering work in the field of a diabetic um State preparation in more precise terms we have a Time dependent mil onion that interpolates between h0 and H for example with a linear form and then um a quantum computer that obeys a Time dependent Shing equation and starts in the ground state of h0 starting from the ground state of h0 we converge to the

ground state of the aonian when we increase uppercase t the total simulation time when the total simulation time increases the transition from h0 to H becomes slower and slower this is true provided that the aonian remains gapped in other words the distance between the ground state of H the Target aonian and the final point of the ad diabetic State preparation of this time dependent shinger equation is bounded by one over the time and then some um combination of factors that have inside the time dependent Gap the energy difference between the ground and the lowest excited

state if the aonian remains gapped throughout the whole at diabetes State reparation then with a finite time we can converge very close to the ground state if along the ad diabetic path the hiltonium becomes gapless we have a diers urgence and this cannot be guaranteed anymore so we can think that the vanishing of the Gap is a manifestation of the qma nature of the ground state problem in the framework of a diabetic State preparation um how is a diabetic State preparation implemented in practice for example uh we can solve our time dependent shingar equation with

a product formula here for example we use a primitive approximation where the path from 0er to T is broken into an S steps the aonian is approximately considered time dep independent in each step and the exponential is approximated by the product of the exponentials of v and h0 this of course requires that both h0 and V are efficiently stimulable we also notice that the coefficient patient in front of the perturbation grows in size as the a diabetic State preparation unfolds um the a diabetic State preparation algorithm does use hamiltonian Dynamics as a sub routine to

approximate ground states there are two observations one is that the simulation time May scale unfavorably with system size if we have a gap shrinking or closure and second to achieve a certain accuracy we need a sufficiently large number of steps and on a near-term device that may mean a very deep circuit or one with many gates through which errors accumulate therefore it becomes desirable to approximate the ground state with some shallower circuit how can we do that an answer is given by the qaoa which is short for Quantum approximate optimization algorithm method qaoa again I

refer you to a paper by faral is very similar to a diabetic State preparation except the coefficients in the diabetic State preparation wave function marked here in red are replaced with free parameters the betas and gamas that you see down here so the energy the expectation value of the target miltonian over the qaoa wave function as a function of the parameters gamma and beta has favorable properties but first of all um it can be optimized variationally with respect to the parameters gamma and beta starting for example from the values corresponding to the diabetic State preparation

it is always an upper bound to the ground state energy it gives as a variational estimate and the lowest energy the minimum over gamma and beta for a given number an S of slices decreases monotonically with an S so the the number of steps now is no longer dictated by the arguments of a diabetic State preparation it's a parameter a discrete parameter and we can increase it or decrease it to capture a balance between the accur that we want and the error rates uh that we that form our computational budget qaoa is another algorithm for

approximating the aonian ground state that uses hamiltonian Dynamics as a sub routine this time in is in a very surprising way to define a variational anets and there are uh other examples of that in the literature I will end my presentation discussing another algorithm for ground state approximation and a miltonian measurement based on a miltonian Dynamics the quantum phase estimation method um in the traditional setting we assume that the register of cubits is prepared in an igen State uh lower case U of some unitary upper case U and our goal is to find the corresponding

igen value which is written as e to the i 2 pi Lambda for some number Lambda between 0 and one QP is therefore a technique to measure the unknown igen value of a unitary operation for example the time Evolution um under hamiltonian assuming the availability of an Ian State QP achieves this goal by attaching m auxiliary cubits to the main register preparing them in an equal superposition of all the 2 to the m bit strings or computational basis States then controlled powers of the unitary U are applied conditional to the state of the auxiliary cubits

since the main register is in an hen state of the unitary uh the controlled unitar is apply phases to the computational basis states of the auxiliary cubits after a f transform the auxiliary cubits are measured yielding some integer L between 0 and 2 to the m minus one the probability distribution for the measurement is picked around an integer Lambda Tilda uh who is this integer well we can write Lambda essentially the log of the an value as Lambda over 2 to the M plus Delta and in other words we can think that Lambda is an

integer approximation to 2 to the m * Lambda the higher the number of auxiliary cubits the more precisely we can therefore nail down Lambda through Lambda the probability to measure Lambda is bounded by 4 over Pi Square 0.4 for any value of M so we obtain Lambda the integer um approximation to Lambda * 2 to the m with a very high probability I can very much recommend you uh the kkit implementation of the quantum phase estimation algorithm as a very compelling learning opportunity as a way of understanding it more deeply and also experiencing experimenting with

it firsthand uh but now you could say okay but um you haven't told me how to obtain the igen state QP is fantastic if I have the igen state I can measure the igen value but in general an an igen state of a unitary transformation may not be available and that's the case for time Evolution where the the igen states are also the states of the aonian what we can have is an input an input state that can always be written as a linear combination of the igen states of U um with some coefficients if

the input state is a good approximation to the unknown Target State then the CU coefficients are picked around some particular value of U otherwise they are just very broadly distributed when Quantum phase estimation is run with this initial State the probability distribution has multiple Peaks each centered at an integer approximation Lambda till the U of A multiple of Lambda U log of the unknown I value over 2 pi and the height of those Peaks is controlled by the absolute value squared of the coefficients CU in the expansion so if with the probability proportional to the

square modules of Cu which is I if SI is close to the I State U the integer Lambda till the U is measured and the cubits collapse into an approximation of the igen state U so far we have said very little about the unitary but that can for example can be the time Evolution operator or some approximation obtained through a product formula so Quantum phase estimation shows us something very profound that aonian Dynamics can be used as as a sub routine to measure the aonian but also to probabilistically prepare a miltonian states of course for

a high probability of success we need coefficients picked around the target igen state which in a worst case scenario may be difficult to achieve due to the qma nature of the hamiltonian aan State problem uh but that can be the case thanks to other approximate juristic algorithms like a diabetic State preparation or qaoa and so in in summary um I tried to convey a sense that the miltonian Dynamics is very useful to interpret and and ideally to predict the outcomes of certain experimental observation for example scattering experiments and Quantum quenches it is also a very

compelling application for a quantum computer because it is bqp complete and it is also a surprisingly useful sub routine in the search for aonian nigen States as we have seen through the examples of a diabetic State preparation qaoa Quantum phase estimations and even other algorithms um I leave you with a few additional references um on the right um five review papers that talk about Quantum simulations of many body systems from spins to to chemistry with various algorithms including a focus on hamiltonian Dynamics and a textbook of which I happen to be a co-author which follows

the logic of this presentation and um tries to be as extensive as possible with regards to calculations explanations um examples and um and supporting material like figures with this I thank you very much for um um having attended this presentation um and hopefully um you will think about an inan Dynamics in your studies and in your research thank you