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Zeno effect for quantum control and computation

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and so it up on the talk about those you know effect Kwan competition control this is joint work with her at silver Alireza Connie and then leader so we get through the difficult situation we're seeing all all we care about and cubits and some uncontrollable that unknown that that's gonna act on this qubits and facilitation roots would help clean up into a system that can interaction terms as usual and have the a unitary propagation on on the composite to the issues of that the coherence across because of that that that and how can we deal with it and we've seen number methods this week passive methods like the Koreans for subspaces active methods close of methods like 1 error correction and an upper lip methods like an uncle coupling and we have 1 more of open-loop method which is a quantity or effect this goes back in the seventies that's well known were protection of you in states with projected measurements although other works since since the seventies has extended stated that to the more general of artist measurements as well as protecting subspaces all it's that just given states so what we wanna do is protected arbitrary subspace using continual effect and 1 trying to understand and quantify how well we can do this with finite Resources right so 0 effectively is looking at Avila many measurements whatever's we do find that number of of of not projective measurements so it is just a review of the the as you fact this this this idea where you can do so many measurements and by doing that basically slowdown freeze the kinetics of of the system a salute to the watch applied never boils so we're going to use these measurements in some finite set time interval capital T. and that 1 for them to be so I'm issuing do not reject measures we what we so what that means is a weak interaction between the system and the emission apparatus on the other problem and though the way we describe this is with the sort of an interpolation parameters this parameter epsilon hello for a 2nd that is the which it which show up Rammrez is the the strength measurement from a playful 0 which is this the gnomon all to up slightly affinity which the objective measurement
so so how do we use had we set up but as you know frightened and and do with this week measurements to to to protect arbitrary states from from this environment so the debate these ideas were going to
incorporate some ideas from from the quarter correction room stabilizer of quantum error correction go that as a good initial state when the weekly measure the stabilizer owners from that group and then the simultaneous to doing that we can think about doing some some of the degree of computation control by using the logical operators as they will commute with our mission operates within of slide in there I wanted this as a point that we're not doing error-correction were not closing the loop and correcting errors were just doing frequent detection of those areas
so again again the announced they were that and some known code for some finite time and again this this this and I'll tell you know we we talked about this this to prayer is a lot of size J J 0 and N G 1 are gonna be priors already using this is formulations were going to grab the norm of the system and and bath notorious together and and the norm of of the interaction term and then we got this issue of superoperator from the Hamiltonian and we make this assumption that the that direction is at most a most while local so that that ensures that the system of that interaction term that is a linear combination of errors that will be detected by skin so encoded as a stabilizer group G and then that the particle has a very simple thing of of of free evolution punctuated by measurements we that M times over the course of of of the time to and projection these OK years is the is the measurement so weak projection your as the measurement of weight and either all the as the stabilizer group or just the generative vectors so those 2 different particles we can look so it evaluate
how we're doing we we describe this this distance metric between the this or or that the final-state rather having applied this this protocol I compared to what the is would then if we had no system that interaction the the internal health of the system and that that no interaction terms so thrown at in this in this notation so I I mentioned we want run quantify how we can do with with finite resources or a trend that decompose assistance credit to build the bound for this distance that we can easily compute with parameters we got that had so I can't really go into details there at the cover painful shocking and and well piece to say the least but if you would allow hear it we can decompose an action that the Dyson series so we can use this the algebraic structure of stable codes is this error subspaces that decompose the terms of the of the best sciences into those 1st subspaces and apply which a triangle inequalities and so forth and what we get is a triple song in which the the the sum looks like a comet world that of of of of clotting so they're binomial coefficients and then exponentials in there and so that that that series is is what those hypergeometric series and the
that raises 2 but the mathematical none here so this is just a very swing phase in wire who this nice work back in the forties where she cheap a some paper and pencil methods that were quite simple but it really nice for our value in a set of of hypergeometric series and what what you came up with was a way of business searching for a linear recurrence relations that are satisfied by these but is serious tissues invasion special functions these series show up a lot places include a special functions issues in it all in that these methods have been developed and and generalized and and optimized and turned into a packages for mathematic enable this is girls were systems so for at least certain subclasses of these series you can just missed the point man and get an answer is at that I I series is a little bit different it has a nonstandard boundary conditions essentially and and that that that in in gee inhomogeneity in that thing but most of this comes out right from this this work every user sensor
methods we can get what what's going like pro ugly expression for the for this bound as not again pretty when I show you what what the vision actually
is about but the nice piece of this is that these are closed expressions and they involve only the the half-dozen up premise that we talked about the norms of the Hamiltonian terms of the number measurements the the final time the the size of the of the stabilizer groups and these and strengthen measures something the same right and so so we like to further analyze a missile that further understand is it a good bound for example and and what is it really telling us about the behavior of this of this method are we all react to realizing you know effect using this week measurements so we can show that active for
any of measurements strength at that McGruder 0 released this that we can expand around the end it will affinity and the minimum measurements again and we can see that we had this 1 over M data going so so it is converging to 0 as as an grows we all realizing as he offered no matter what the strength of the of the measurement is and then I'll point out that but who was pregnant shows up in this in this time yeah so if if this if the measurements are are projective of solids infinity in this this this term becomes 0 so that measures and so this term is still the shows up when the active women in the other limit when absent going 0 this thing dominates this becomes this grows to infinity so the rate of convergence of this thing you know is while ran for any arbitrary of and the rate of convergence is is isn't but it was decreasing as as as epsilon decreases as as the stress measurement decreases so I mentioned these these of variations on the protocol 1 were measuring the full stabilizer group and 1 we're measuring just at the generator that group so that the trade off between the 2 up obviously his vision the generators it's an exponential if you were measure said you have to do at each each iteration but with the caveat that the that the bigger list of this bound just
slower convergence so this but but
but Q here which is related to but the silence stable education I is replaced by 1 in the history of the generous only so so that's all this law convergence about a month to use what I what what
that what that looks like the person set of parameters here off so chosen I cannot t 3 1 and and there are generated 3 4 and this the size of the interaction of be 1 10th the size of the of the system that employs themselves and so you can see the the bottom surface is is just when we take the limit of will with that of other at that slide 1 a 0 so this is the form of strong measurements this this purple surface and then the mel services the US full stable as a group convert Carepa quickly in the when you do the generous or liquid is a little less quickly so that's alone recap
here we we get this this method of this this protocol for for protecting systems from from what from the effects of environment with his rigorous distance that we've been able to talk with that characterizes how we can do with finite resources and in this in this game about Latin really short of which you can which shows that there it does it wouldn't matter whether the Hamilton is is time dependent time-independent that's a special work either way so you get this the potential for application to Ayurvedic want computing I mentioned we get up there should also use the logical operators of the code to do the computation Wallwork taking these things and I'll just mention briefly that what I showed you was with the results were non-selective measurements so that that final state that that I talked about is this is the average the leverage over all possible of measuring outcomes but would like to know what happens if we consider all all the individual possible punishment outcomes only into the possible outcomes states what is it just the average this can that's that's converging in this nice way words it so the country of conservation measure that happen so that with high probability your your getting near the the the ideal state with real people so that protection and I would if you review the do the same week measurements than that converges to a strong measurement so it's not surprising that if you keep the measurement strength consonant with the number of measurements go to infinity that you get is you no effect but I have a question if you
measure all the elements of the stabilizer verses measuring only the generators from measuring the generators you would that because that's to have you know they give you the the full information about the whole stabilizer you you could deduce from if if there were strong measurement you could deduce the outcome of a measurement of any element of the stabilizing just for measuring the generators so is the effect of it converging more rapidly when you measure the whole stabilizer just that you're doing more measurements in other words would would just measuring the generator it the ratio between the generators in the hall thing is and if you just at the end times as many measurements of the generators would that be just the same is measuring the the whole It's a lot to tell them the just because we were looking at here is is this bound we created and it's certain in about a dozen look quite that that clear cut of but I think they're saying that that that additional measurements are contributing in a natural way that you're not getting any new for information really but but and this thing this report yes we did for the purpose of the cell and also see so many body measurements are not always easy to come by and I'm wondering if you thought about physical systems where this can be implemented for example when I was watching you talk today and the blue cheese proposal from this morning can demand about other physical systems with this might work but I have not adapted uploaded because I am not a physicist really work we consider other possibilities of of extending this beyond are the stabilizer codes formalism to think up a petition bacon short code and antibody mentions nike's the complete opposition to disprove this conference you have and you know you're here to the but again it's hard to say because because we're looking at is bound it's early possible that sort of in the in the introductory part of that of that protocol that something bad is happening but it's it's kind of hard to see in it's not in this section we amended FIL here who could talk about the
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Metadaten

Formale Metadaten

Titel Zeno effect for quantum control and computation
Serientitel Second International Conference on Quantum Error Correction (QEC11)
Autor Dominy, Jason
Paz-Silva, Gerardo
Rezakhani, Ali
Lidar, Daniel
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 10.5446/35333
Herausgeber University of Southern California (USC)
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik, Physik
Abstract It is well known that the quantum Zeno effect can protect specific quantum states from decoherence by using projective measurements. Here we combine the theory of weak measurements with stabilizer quantum error correction and detection codes. We derive rigorous performance bounds which demonstrate that the Zeno effect can be used to protect appropriately encoded arbitrary states to arbitrary accuracy, while at the same time allowing for universal quantum computation or quantum control.

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