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Performance requirements of a quantum computer using surface code error correction
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new hearing to cite thank but the it this uncanny of but it but but the better about that's not the rights and OK so I'd like to thank the organizers for give me a chance to talk to following I'd also like to acknowledge all the people who work on this project I think you get a sense when I go into it that I could have done all of us alone many people contributed to this what we'd like to do is for all
01:00
the resources that go into faulttolerant quantum computing in particular we like to examine the overhead costs but you incur when you incorporate fault tolerance and I'm going to show that that's a pretty substantial and from that we'd like to you can determine the implications of this has 4 hardware so what types of hard which would be taken what types of design decisions should be making when we're and I sort of choosing directions for future research so I'll give you a preview for
01:33
my talk this is a diagram that shows operations inside of a quarter of a year or more specifically a fault Oracle in the lower left will begin with a spin state protection and although we have here we have an arbitrary logical quantum and in between these the steps then make fault tolerance errorcorrection etc. and of obscure the beautiful but on the X. axis you can see the timescales for these operations and this time scales I actually moved on a logarithmic axis and so you can see that we go from peak of seconds of 2 ms that's what the stock is gonna be about
02:18
so to organize fault tolerant quantum computer we developed a layered architecture so the lowest layer of the architecture is the physical and physical layer you have just the physical quantum processes Hubert's gates and measures are going to these more detail so this was the abstract above the physical layer is the Virtual virtual areas where you start to actually shape physical processes into information from this this is where you have dynamical the coupling that creates a better more coherent memory you also have compensation sequences the counter at gate areas might have become the courage we subspaces this is always might be called passive error correction and this is not a measurement so in other words openloop I think is what Robert Crossett or and above that we actually have active 1 the correction in this example were considering surface codes which you could also insert your favorite errorcorrection making sure that so in this layer you have circuits for the code and also the central processing which but as you might see in Alsace stock is not true most Intel the but finally the outputs of quantum error correction or not completely only have a fault tolerance set of resources bowl need to that turn that into is a logical substrate for quantum computing and this some more step actions important steps to go through we need to create a universal set of gates so in in the case of the surface code we need to distill special and so states that allows to create these university we also have to in some cases create of arbitrary quantum gates the several isn't it but finally above all that we have the application and this is where we execute quantum algorithms on 2 logarithms are analyzed today are factory something more complicated than this and the quantum simulation and particularly delicate Simulink which so the physical areas
04:25
populated with of the usual suspects that you have a physical cubic in our case we're looking at SelfAssembled quantum dots in Emerson they're controlled with laser pulses of stimulated Roman transitions and we also are proposing to use optical measurement but in addition to that what we also care about our where the engineering parameters of these processes coherence time gain execution time and what types of errors are we saying of the systematic a random because the next layer up
04:59
is going to use the outputs of the physical and the virtual there needs to shape physical processes into information primitives so dynamical coupling actually operates on physical layer to create a coherent cubits this case edition of simple the coupling sequence of that will extend our lead Jedi time but then we might instead this decoupling sequence in MPEG1 compensation SingPost also get good virtual quantum gates the reason we might do this is to cancel out laser intensity for actually so what we can do is we start going up that diagram I showed at the beginning of the talk we now have the 1st 2 layers the architecture we have and I filled in some of what those those operations are as well the spin state rotation and measurement entangling operation from this architecture art the lowest level and the relatively fast but building on we now have some of the virtual operations and they take a little bit longer because they're built of sequences and so necessarily the smaller we also have the coherence here which is owned out in the microsecond range on based on experiments and on that so moving beyond that we have a quantum error
06:16
correction there so this is a slightly tilted of set of tiles from the surface could from that you can create more you can just and extend what is the circuit for doing surface quantum correct that shows you how to extract syndrome the syndrome you can use and the matching the society the error correction process itself but it also shows you how to account for the resources were the gates that make up the surface as important to know when you're developing an architecture and you can count the gates to see how many gates required to do the correction steps or also interested in estimating how big is the code how large because do we need for a particular problem for the service because there's been some of the simulation work so that we can model this with so well with these powerlaw fits and so what you can do this you have to get our in mind with a particular logical there it's all going defiance these terms the threshold is a threshold of the code in this case it's been shown to be about point comma decimal 9 per at science and the is the error rate coming out of the virtual air member virtual air just below correction there so all of the virtual operations feeding in to the quantum errorcorrection is just a constant basically just for fitting and epsilon sub L is a logical error this is the area we need for the average so I've chosen here 10 miles 15 was something typical for Shaw's and probably also representative for simulation algorithms depending on your parameters but when you look at this and you say OK for these sets of parameters needed to 29 that allows us to start making estimates about how large of a surface could we need for this particular problem so from that we can start to fill in the next layer of the architecture what of the time scales that we need to do error correction just 1 step of error correction for example so moving out from the virtual their operations how long does it take to do the lattice refreshable that's built on several layers of virtual gates we we solve a circuit diagram on just for this and then if we wanted to defect to do logical gates of sea not that's built of several lattice refresh and also this little red arrow here shows that importantly the last refreshed which detects there's this week at a single them these to be somewhat rather probably 2 or 3 orders of magnitude faster than decoherence you have to detect errors faster than the purchased awful so the corner there will provide you with error Correcting resources that's not everything that you need for quantum computing what comes out of that is just so high fidelity and actually not quite Clifford but almost Clifford Gates is missing 1 the asking but you need to actually make a full set of the universe and so to do that you have to inject states the code the states on noisy so they have to be distilled that's process is shown here inject faulty answer lies in the stomach but even after that you just have a finite discrete set of gates and from that you can approximately arbitrary gates and is at least 2 good ways of doing this that I know about on methods that do not involve additional and so is not talking about the the State distillation cells not down here there's also a method that I sometimes noticed faced that involved special Walter Cuban and so state used for approximating arbitrary quantities and I'll just touch on this briefly because it impacts a resource analysis so as I said we needed a state installation because we need a universal universal set of K for so this is the gate here I need to make a full set of K and and so we need to and create the state reason this measurements can be promise in this many times before but what I wanna emphasize is we need very many of these gates you get a sense of how many in a few slides and I start looking at these are but it's on the order of billions trillions her quadrillions you need a lot the you also need them at very high fidelity and so what that means is that quantum computers burden require factories to bruises and words not gonna make them 1 at a time you're going have large sections of the machine devoted to pump and out it's kind of like a catch on currency fuse they take up a large section of future if you're familiar with how circuits were held processes so this for examples heuristically what factory might look like this is concatenated distillation circuits you actually see distillation circuit and this is you know if I could fit more you put more here outputs of 1 distillation circuit feeds into another 1 we might do this because you need to get very high fidelity and
11:35
so how do the resources for dates state distillation scale while the axis skill kind of well but you should still be mindful of that the error suppression is doubly exponential was pretty good and that means that the the leading order error goes like something so read to the the level of distillate p to the three of distillation so goes 1st level Q thinking into 27 so as suppressing at a very high rate but the researchers do climb up so exponentially as well so I'm looking here at circuit volume circuit volume is if I stronger a rectangle 1 side would be the number of cubism using in the other side would be the number of gates so this is due that's times gates and the reason I look at it this way is because because of there's a lot of opportunities to stretch circuits for squeeze in different ways just by parallelism sometimes you can do things in parallel to make the circuit shorter sometimes you can stretch it out because we're doing his factorystyle you're looking at doing very many in parallel and their limits to that obviously there is a minimum circuit minimum circuit that you have to be mindful of but anyway this is just to feed into our resources analysis plots the make this a little more visual arbitrary quantum gates
13:03
also really important they don't play such a big role ensures utterance of us would you care about that's a big deal but they're very important simulation and I'll show that the at the slides at the end of my talk 2 ways that I know about to do arbitrary common it's the 1st way is probably the better known of the 2 which is gait sequences that you just take a sequence of gates so H. clean and we as whatever these are just gates that are available to us from but beta the error correction or from the injected and still is that we still and we use these to approximate an arbitrary unitary the most probably the oldest and most famous example is a solid ties our them but in show you that's not the best way to do but in addition there's also a method known as phase back and is clearly good although that's not necessarily the best in context in this case there's a special on so state in what you do if you have controlled addition circuit weird idea but when you do that this controlled addition circuit actually doesn't change the and so it puts the stains on circular from could it and who was the control as lexical kickback kicks back face and have time to go through all the details in the to talk about it up to the top of that it's a neat idea on and because it has added circuits there's different opportunities by some the operation so sequence methods you
14:37
start with ready to a sequence of gates so it's only a single qubit go on a line like this so for example 1 of drawn here just approximates this phase gate and the idea that when you're setting at a sequence methods at longer sequences will produce better approximations so there in a sense when you designing are and you'll say I need a certain level of precision from my gates and if I only higher precision that's gonna come at a cost of higher circuit death higher resources in particular it may also will certainly come at a cost of hierarchy gates we've seen that these T gates are expensive so here are the 3 methods that I know about for doing an arbitrary quantum gates Sonic entitled gate sequence method in the circuit that scales poly logarithmically the power here is something like between 3 and 4 the Gowers method of there was a reference on the previous slide is also state sequence method and so the sequences looked just like the sequences from solid type the just more compact and the way they're determined is dif it's the classical algorithm for determining the and and in face bag is completely different from the 2 uses a multikey Peninsula and is added circuits in because it is not a circuits the circuit depth and CA resources depend on as as so ripple carry circuits are simpler and have higher death carry look ahead of this terminology from arithmetic circuits but the more complicated but they can be fast so I guess I'll have time now but the the classical computing time varies also with him so
16:27
what I wanna impress upon you is so they could type of even though it's probably the way the best known method for making arbitrary gates and again arbitrary gates are a necessary and important to simulation algorithms so if you need simulation algorithms need arbitrary gates so they could type skills really bad and I don't see a good reason to use instead we should use 1 of these other 2 either Falaise for face kickback back they're different scenarios were 1 might better than the other I don't see were solid K times but this plot shows up there's as a distance metric here which is basically the error in the gate on the left axis is circuit depth and some lot axes so you can just see that these 2 of scaling the other of gates hundreds of gates and selected peoples of so what is the what we will consider that Here's looking at just face kicked back into dollars method in more detail it looks like Gowers method is the better of the 2 in both circuit death nonlinear scale and gates it's lower and it turns out that that's heat that is deftly troop you're only making simple of single cue unitary however there are some more complicated SNAREs refrains can that can be advantageous I don't have time to go into experience so now we
17:57
can set up a diagram this is the diagram the way up to layer 4 which is what we saw at the beginning of the talk so if you go through all the steps from physical their ontological layer what is it take to create fault tolerance up to this arbitrary gate that we need and so we can see that all the steps in between especially going to error correction and going to the logical state prior logical OK preparation there's some big jump to in particular going from the bottom to the top is about 6 orders of magnitude so I'll quickly go through
18:35
some resulting shores algorithm but the some assumptions that led to these results a few curious but how fast you do it on this model so the bottom trace is well if you could run the circuit as fast as possible that's which would a so the that's the factor that somebody is a thousand 24 2 thousand 48 is a typical problem sizes we consider interesting they're beyond what we think is possible classically the blue trace corresponds directly to circuit that and if you can render service as possible this is also execution time today but to make this problem more interesting we also looked at 1 of the size of the quantum computers fixed so it's only 100 thousand logical cubits that's what leads to the green trace the green trace shows you that hat we cannot of you we cannot distilled those states we need the T. gates as fast as possible the size of machine is not large enough this is what I was talking about this factories going to take up the low of all of our common computer when I say the ball I mean 90 % 90 % of a complicated needs to be devoted to distillation or the algorithm will stop shores of so what's happening here in this case only 75 per cent was distillation doesn't so much that the quantity and
19:55
in this case this is 1st quantized simulation the xaxis here is number of particles so this could be electrons and nuclei so again circuit that on the left the execution time days on the right and here's an example problems in preparing elements and amino acids and and so in this case reaction are using face kick details come talk to me or particularly the simplest of games with them and and you can actually run an interesting simulation problem on the order of a few days and so the last by results slide I'll show you is of course we also want to look at 2nd quanta simulations so what if we're doing all lithium hydride we had a focus on a specific markers secondquantized doesn't you can't look at you know why printer prominence easily but in a single lithium hydride energy eigen value simulation using the S U 3 G now I'm showing the impact of the choice of phase gates so these kickback acknowledgment method are down here they take I guess this is on a scale of about around phys 8 days and solve I can't I just it's unreasonable so every lesson about the use of all my time and I should include the layered
21:23
architecture we've introduced is a promising approach for the design fault or computers but in particular the overhead costs associated with fault tolerance separately operation ties between the logical and and the physical layer by 46 orders of light and if you're curious we have a paper on the archive layer architecture for quantities in the simulation results are in preparation thanks you want to and all the reasons to what I have a question about 1 of the very early slice of that you give the formula for the residual error after error correction in a surface code yes and it goes light basic error rate epsilon to the power of D over 1 the plus 1 over 2 where d is the size of the system I believe and the is the distance with these this is the code yes exactly so so on Monday I think that we had to talk about Us surface code of thin and buy into across and it was mentioned that it looks like from his simulations that it's not wait arrows that dominant and the residual error a often error correction so there longer length error chains which are individually suppressed by the powers of epsilon but they have a combinatorial factor that's that's more important so who are based on on on that comment on that talk I would think and I would say I would expect a different power up there on a acquire correction factor because would you seem to be saying it is that the the errors that dominated the dress shows all of the shortest possible ones so there are many commons and yet do on 1st of all this is based on simulation results and 2nd of all where Sunni I don't have undergoing this but we're assuming that you are a substantially below the threshold at least an order of magnitude and I believe if I understood correctly that enters work was close to the threshold on and so that's where you would probably see higher rate errors having higher and so yeah I think that's it that as you go to the limit and epsilon simply going to 0 the dominant errors are going to be the shortest thanks to a solution market so so I'm very happy see accounting of all resources here but I can go back to your sort of your your your your later hierarchy there's a you know you you talk to the very beginning about the the control class if you like where are the physical lives with his virtual layer you talk about having dynamic coupling policy the guy and so forth right and our work is the was how you get infected the Vols pass into the interior overhead as well the cost of some time with a couple this of well they they are incorporated here for example as you can see the 1 cubic gate at the virtual air is substantially longer than the 1 cubic gate if you will at the physical layer I don't have a lot of time to go into the details scanning of I'm sorry I don't I guess I don't fully understand what you're asking then I mean this is about a factory hundred longer than it has in part to do with the fact that there's and 20 operation there but also the fact that they're separated time deliberately the BIO present as it was the good of a thing of the Anderson family rather like to thank all the speakers of decision Pentagon theft however the recovery and then and of the block
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Titel  Performance requirements of a quantum computer using surface code error correction 
Serientitel  Second International Conference on Quantum Error Correction (QEC11) 
Autor 
Jones, Cody

Mitwirkende 
Meter, Rodney Van

Lizenz 
CCNamensnennung  keine kommerzielle Nutzung  keine Bearbeitung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nichtkommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/35315 
Herausgeber  University of Southern California (USC) 
Erscheinungsjahr  2011 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Informatik, Mathematik, Physik 
Abstract  We study the various overhead costs associated with translating an abstract quantum algorithm to a practical implementation in faulttolerant, errorcorrected quantum hardware. The processes required for quantum error correction can be expensive in terms of quantum resources, and we consider the collective demands of the error correction circuits, distillation of ancilla states, and composition of arbitrary logical gates. To provide a concrete demonstration, we study a quantum computer architecture using surface code error correction, and we examine Shor's algorithm and simulation of quantum chemistry in firstquantized form as typical quantum algorithms for a largescale quantum computer. As a consequence of this investigation, we can show that practical quantum computers executing these algorithms will require quantum hardware with physical gate operation times of less than 1 microsecond, if the calculation is to complete within 30 days for problems too difficult for existing classical processors. 