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Protecting quantum gates from control noise

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OK so Constantine will give a brief talk about protecting quantum gates from control the ones as my name is constant and briefly work in new national labs and this work was done in collaboration with these Mr. Grayson came in the UK and India and to this a group of Virtual Robson Princeton University so at the
center some basic notation a mall so they consider just they control to quantum gates which is a basic element of a quantum circuit and small and the number of qubits instigates capital and the Hilbert's is the color space dimension W is the target unit transformations it to want to accomplish and U. of T. is the actual evolution operator at the final time so we want to produce the same evolution are upon any initial state in the end we want this evolution to be as close as possible to the target
suu external control as as we're here in in many experimental works and yesterday and today if you want to enact some are useful of operation onto a quantum system summer control is necessary to do so so see if a external control and that was in tone and an evolution breeders of functionals was a control functions and of the measured disability and it's that shows us call Holwell's the gate was performed so and if it's 1 minus some normalized distance between the actual evolution and the target and there are different definitions based on how you measure of how you compute this norm are 1 2 conventional definitions using Hilbert Schmidt not only gets disease or in some cases the it'll be obsolete but instead of real part out there but in so in order to doing I use this a definition how were were all of actually will be the same as this 1 is used because work very close to the optimal solution which means very close to fidelity equal to 1 a so fidelity itself is also yields a function of control and
what they're going to beat about so-called of quantum natural landscape upon eventual landscape to basically this as a functional dependence all fuel objective in this case is quantum use fidelity on their control variables so it can be continuous control field some set of control parameters a discretized version in an experiment a numerical study so a formally as a aerobic critical point this landscape each satisfies this condition and of course we're looking for an optimum so half of you will look at youth as a maximum so we want to 2 point to be defined as a point of point and their landscape we should was critical and also have that the sets the 2nd which is the Hessian of fidelity with respect to the control field will be negative semi-definite so if you want some of isn't should you of access control theory this is a reference and actually by
doing analysis of this quantum control landscape for this particular problem I mean is a given fidelity definition arabic can becomes a full and results so far so system is controllable and if you look only so-called travel a critical points I was a little there 1 marks a manifold where all optimal control from this manifold to those on this manifold optimal they produce unity fidelity as a rule we won the new manifold mu 0 fidelity and to all other critical points correspond to so Sandel manifolds and if you do optimal control of storage those can be easily avoided by the dollars so it's quite easy to get on the top of the landscape and find an optimal solution so in duality there will be an infinite number of optimal solutions you start is dishes different initial field you get 2 different optimal solutions but in ideal conditions renders there's no noise Sonoco coupling to environment every of those structural controls of the solution produces unit fidelity it meant I actually in many cases control cup coupling to controls we our moon here is just dipole operator and in the in the case when Andrew was the Hessian at the mark soon can be evaluated in given me this night expression so MILF T is just out of the dipole per a time-dependent dipole operator in the Heisenberg picture and and choose the structure this Hessian shows us how fled is is a top was a control landscape and the flatness of 2 but the top was a control landscape it determines how a biased is our gate operation to control errors so a noun
let me introduce noise errors so in immorality event that have control field It's all this will be nice thing and that just assume that as a to work very close to the optimal solution so what you want to produce fields see not but in reality to be seen not some error and here z of T will be a random variable and you have to build this deterministic function and I use this notation so I can account for 2 important cases 1 is additive noise so she's just 1 the noise just added to the control field and 2nd to important cases multiplicative noise whereas this G function is just proportional to the control field itself so for example in control of semiconductor bits by voltages to be usually additive noise some time and the Johnson Norris in other cases and it's microwave field to laser field we choose to control some part as and noise can be the effective so our approach is based on just taken the fidelity is a function of control field and expanding it into the teller serious I'm assuming the noise is small so because we are out of work close their told it's the optimum so 0 order to to is just 1 and 1st order from 0 actually for 2 reasons 1 reason is that 1st do to 4 on the top a 0 and also if you consider a semantic noise actually all but it is the expectation values of all I am better or and order contributions to 0 as well so as the leading the correction of the be because of the would be us of 2nd in order to so it's 2nd order in the control variable and this is the Hessian compute Adzic optimal control field so every I consider some random noise forces so this was the only thing which we know about basically about this control where variable assume a symmetric and of the human has some given order correlation function so because it's random variable in order to estimate the actual stability the need to ever reach over all possible liberalization of the noise so this gives us statistical expectation value of the quantum bits fidelity which the norm with this notation so then we ever reach over all the noise process is this just gives us but a correlation function so in many cases when you want to assimilate system evolution in the presence of noise we need to do for Monte-Carlo simulations which is numerically but because we just do this approximate expansion of the new closely with the always think each contributes from noise is the spot a coalition function it then this means that know some of you don't need to actually simulate noise through Monte-Carlo because in order correlation function is something conditional and so on so from this expression that our goal is to to have this additional term until always be negative because fashion is negative 7 different and you want to this absolute value of this tour be as small as possible that is as close as possible to 1 in order to do so in order to to protect a quantum from of control noise we've 1 to minimize this over a and contribute to so the ship because there isn't in 4 of control field of each of optimal and different optimal fields will have different passions and also for moved in the case of multiplicative noise those G functions will also depend on control so we can sewage ones it doubles the landscapes through all possible optimal solutions to to find those ones beach minimize absolute theory of this overlap integral and zealous optimal solution give will be more busses and others so before I go to this to a general case with me consider some important special case which is white
noise so white nicest is basic as the worst possible situation because like for example now dynamical the coupling for example cannot work against noise a 0 correlation time if like as most infinitely fast so you can do you can't really control it too but let's see what happens so because but collision function for white noise is delta function are instead of to Intergraph's lesser just fun to go for some additional useful property misuse of this fresh this Hessian it's only diagonal part of all free and diagonal elements of the Hessian actually time independent just cause trace of the square friend operators time time-dependent there and so this trade goes out the integral and just left his this expression for all expert expected value of stability in the presence of white noise controls and for any given white noise is this we'll digit of all this into go will be just a duration of control who do we just capital T. and from multiplicative white-noise this integral with just control energy always control serious like to call it fluent so they can it should compare how do
without approximation is by looking at against Monte-Carlo simulations so Zoe's once again for additive of white noise the in this game is just proportion to control time and all this lines are just from the use of approximate simple approximate form for different values of the North strength and each side this circle here is obtained by sampling Comónta call sampling over 2 to source nor Saros ations and you see that as that of our approximation is extremely dude but basically up to fidelity all flows the 0 comma decimal 9 albeit error of 0 comma decimal 1 which means in this entire area which is of the quantum information processing and similar sinking
done for the multiplicative white-noise the same thing just replace the control time by our control energy and once again agreement between our approximate result and to color simulations very good up to because of the things that area of high relatively high villages
still on the Fed when we have energy white noise about continued gate errors proportion to control time and only signature can do to increase their boss this these to minimize control time and minimizing control time that can be done without sacrificing the buttons but only up to a point there is a critical time below each have our data will be his and no longer reachable and to reduce some exploration of Z spot operator for foot 2 objectives to compete in the objectives wise maximizing control of maximizing its ability and the 2nd is to minimize the controlled I so this gave us this breed front is flat up to critical time car were were these increases very fast he chose to use it it's not very good to operate below the critical time I was there remain different
studies of time optimal control we did some additional work here this preprints with just a few on archive yesterday so you can see is the weight of critical time depends on of what a what is a target date it also depends actual a global faces a target date and to be also studied there for 2 cubits systems hollow as the rate of critical time depends upon some into cubit coupling and it's 1 over the of of sort of what 1 over J J and can be coupling for how low couplings how words in the K. were is different a fool of hot coupling strengths but the coming
so going back to the case of duplicative or white noise as a gate errors proportional to control energy so if you want to minimize control energies in different stories and minimizing control time so this this quote shows you are all is there energy of fluency of the optimal control field versus the same property of initial field from which to start optimization and you see that if you want to achieve the of small energy in the optimal field you need to start is small energy in initial field and then of the optimization optimization show which will take you to optimal field is relatively all all of this is a small through into the needed to make sure I choose it achieves a target operation is those different lines obtained for different our final times and he sees it generally is about time increases and choose the optimal fidelity goes down the hall where is this change is not 1 in 20 so this is it the final time 1 and actually father this is final time equals 3 it's blows and foreign seeks so as this changes non-monotonic
and if you look at the dependence of for optimal fluid some control time it also it's some so it has envelope which decreases as 1 over T and this is generic for all going to target dates whole were results oscillations have under 2 associations with constant for high the Margate and the gate and the decay on political for solutions the cakes financially for and politics I still with me just a few words the food in the general case when It's a caller nor it's not white noise we want to minimize of these Naur these there was this metric and I would say is always and that we can also prevent this robustness metrics in in the frequency domain using Fourier transform so as of z ies about a Croatian function which is free Fourier-transformed us so its power spectral density of noise which is Fourier-transformed of order correlation functions and 2 the the season the made the youth if it's white noise this will be constant so as there is no might you can do about changing is the use of free atoms form the Hessian whole were right you've 0 is the peak in as a power spectrum of a white noise you won the peak of the Hessian to be as far separated as possible forms of people's notes so we want them to be spectrally separated from each positive cues from Boston's and
finally I can say to use some of the genetic algorithm to but that minimizes the and robustness Medicaid so this line is obtained to and we just you spoke parameterize the control field is 1 frequency and improvement is with factor of 2 whole where when we make a control field more flexible and parameterized by force
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Metadaten

Formale Metadaten

Titel Protecting quantum gates from control noise
Serientitel Second International Conference on Quantum Error Correction (QEC11)
Autor Brif, Constantin
Lizenz CC-Namensnennung - keine kommerzielle Nutzung - keine Bearbeitung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nicht-kommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/35318
Herausgeber University of Southern California (USC)
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik, Physik
Abstract External controls are necessary to enact quantum logic operations, and the inevitable control noise will result in gate errors in a realistic quantum circuit. We investigate the robustness of quantum gates to the random noise in an optimal control field, by utilizing properties of the quantum control landscape that relates the physical objective (in the present case, the quantum gate fidelity) to the applied controls. An approximate result obtained for the statistical expectation value of the gate fidelity in the weak noise regime is shown to be in excellent agreement with direct Monte Carlo sampling over noise process realizations for fidelity values relevant for practical quantum information processing. Using this approximate result, we demonstrate that maximizing the robustness to additive/multiplicative white noise is equivalent to minimizing the total control time/fluence. Also, a genetic optimization algorithm is used to identify controls with improved robustness to a colored noise characterized by its autocorrelation function.

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