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Protected gates for superconducting qubits
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00:07
OK I want my study I think we've got the AV situational central here I'll also operated session will be satisfied in 6 minutes late so this keep that offset a schedule I on accession albeit kicked off by the John principle we talk to us about the superconducting you at the thanks Endre everybody can hear OK and talking about work done at Caltech with Peter Brooks and looks at a time and all of us here recognize
00:45
that error correction and fault tolerance you're going to be essential that were going to operate largescale quantum computers someday the standard approach to fault tolerance uses clever circuit designed to overcome the deficiencies of hardware where there's no alternative hardware topological approach in which the hardware itself has intrinsic resistance to noise because of the nature of the physical encoding of the quantum information and it may be that both of these approaches will be used together in future quantum computers but is important for the time being to seek new ways in which we can build quantum hardware with some kind of intrinsic robustness which results from the way the information is encoded 1 suggestion
01:30
has been that we can make robust encodings of quantum information using superconducting circuits in particular this 0 haiku that which was suggested by Katire and others the phase in a superconductor is a periodic variable period to apply but for a suitably designed circuit the dependence of that circuit energy on the phase difference between its 2 leads in very good approximation of periodic function of the data with period pi aside from corrections that become very small as a device gets a lot and in that case the energy as a function of the data has 2 minima which can occur at they'd equal 0 and the 80 plus pie which have very nearly the same energy and if there's a large barriers separating those minima those 2 minima can be the basis states of a robust cube is protected against bit flips by the high barrier in between 0 and 1 and is protected against facing because the degenerative the generosity of the 2 state is very stable with respect to perturbations of the Hamiltonian but of course a B.
02:45
wanted robust quantum computing we need more than just stable cubits we also need to be able to do very accurate gates and that's what I wanted described in the talk how in particular we can do highly accurate Clifford it's acting on 0 pi cubits by turning on and then turning off the coupling of the qubit to opera monarch oscillator an LC circuit and the gate will work very effectively be highly accurate if the inductance in the LC circuit in a natural units is a very large number the secret that underlies the reliability the gate is actually a quantum code of a continuous variable 1 coding which Cuba the twodimensional system is embedded in the infinite dimensional hilbert space of a harmonic oscillator and I'll be explaining how that works and
03:40
understand the physics of the 0 pi Q itself and also the protective gates which you know a little bit about Josephson junctions a dozen junction is a device in which a layer of insulating material seperates 2 pieces of superconductors such it there is a nonzero amplitude for the charge carriers in Mr. Cooper it the tunnel from 1 side of the barrier to the other side we can go that come tumbling an effective Hamiltonian if we think of a lead attention via a Josephson junction to an island that carries and charge superconducting island than the effective Hamiltonian describes processes in which the charge on the island can either increase by 1 a decrease by 1 1 a Cooperpair tunnels through A Q is the charge in units of 2 the Cooperpair charge and there's some amplitude for the tunneling which I called J. the Josephson coupling it's often convenient to describe the system using conjugate variables the superconducting phase in the charge uses the 48 conjugate of the charge so the phase and the charred debate commutation relations like the position and momentum of a particle in 1 dimension and if we express the tunneling Hamiltonian in terms of phase proportional to cosine phi with the coefficient given by the tunneling amplitude jet so 1
05:05
other thing we should know about superconductors is the flux is quantized the phase in the superconductor is actually a convention dependent concept that I wanna compare the phase at 2 different points in the superconductor I need some notion of parallel transport that tells me how they should changes and move from 1 point to another that notion of parallel transport is provided by a connection which is just the electromagnetic vector potential and those parallel transport of phases actually path dependent that means that connection has curvature the curvature is just a magnetic field so when there's a nontrivial magnetic field the face picked up on the parallel transport can depend upon hat but we have a close ring of superconductor and there's some applied flux it might be that the effect of parallel transport of the face around the loop is nontrivial and that's energetically unfavorable so what happened that some persistent current flows in the ring which adds to the applied field some additional fuel produced by the current in such a way that the effect of parallel transport around the ring really is trivial and that means that the magnetic flux enclosed by the ring is an integer multiple of a quantum of flux age over to but I think of a ring with a judge's injunction inserted inside I don't think of the phase difference between 1 side of the Josephson junction in the other as not a periodic variable with period 2 pi but a real variable because if I wines by 2 pi that just means we inserted 1 form of flux through the now I'm also going to wanna
06:38
consider LC circuits and superconductors this is really just like a particle and a harmonic well where the hot charge on the pasta corresponds to the momentum of the particle and the superconducting phase is like duration of a particle I can think about is the phase drop across the inductance which is proportional to the magnetic flux linking the circuit so it's convenient to use natural units for this problem to express charge in terms of the Cooperpair charge to e and to express the flux linking the circuit in terms of the superconducting phase drops across the inductor and then in terms of those units the square root of all over C is a dimensionless number it is the conventional compete impedance of the circuit expressed in terms of these units which is about 1 cologne so again as what's the significance of that dimensionless number being large and the answer is that in the ground state of this isolated the circuit is very large phase fluctuations the ground state is a Gaussian wave function where the distribution and 5 is wide compared to to part with that with determined by the square root of L oversee and we can imagine incessant junction into a circuit which has a large inductance in this sense and then its energy a we proportional to this so difference of phases but because of the large fluctuations in FY the cosine gets an average over many cycles of the cosine and that makes their contribution to the energy that depends on the state of very small because of the averaging exponentially small and with 5 space which is determined by the large inductance so now we know enough to understand the idea of the
08:29
protected 0 pi Q but this is the form in which it Hyatt suggested that we can think of a superconducting circuit with 2 wrongs on each wrong there's a large inductance obeying on this condition and the 2 wrongs a couple together by some very large capacity so I consider the variable which I consider the phase in the superconductor on either side of the capacitor and then the sum of the phases by 1 and phi to the plus is unaffected by the large capacity ants and so it's a like variable because of large inductance which has very large fluctuations but the difference variable phi 1 minus by 2 feels a large capacity and and that makes it a very heavy variable which wants to be well localized phase and so it lacks to a value which is determined by the phases on the external leads odd to the difference between the 2 foreign theta 1 and minus the largescreen theta 3 and they data to and now we can imagine taking the top and wrong and twisting and by 180 degrees and connecting leads together so there were identifying fated to for they 3 with that 1 and then the energy of the circuit becomes a function of the value that the heavy variable locks to which is now to think the 2 minus A 1 and then there's some correction but it's very small because of the large nations of the light variable and that's how we get a 0 by Cuban in which the potential essentially a periodic function with period pipe which has 2 minima which are very nearly degenerate in fact we can think of the protection is arising from very nonlocal encoding of information like another context where we have topological of protection if we imagine as Kotite suggested that we realize the large inductance by a long chain of Josephson junctions and then the phase difference between 1 end of the chain and the other is distributed very non locally among many devices in the chain and because of the large fluctuations in the phase of the state's locally look almost identical and the breaking of the the generous is associated with tunneling phenomena which run all the way from 1 end of the chain to the other which have an amplitude which is exponentially small and that's the reason why the corrections generously is very small when the device scales up also wanna be able to measure the 0
11:04
pi that if we measure it in the standard a computational basis we need to distinguish a phase drops across the cube it of 0 from phase drop a pipe and we could do that by coupling the cube bed to the Josephson junction inserting quarter of a flux quantum and then the current that flows will either be clockwise or counterclockwise depending on whether the phase difference between the 2 sides of the cube it is 0 or pi I also want to be able to measure in the dual basis the E R X basis dual this computational basis among where we can imagine doing that is by breaking the connection that identifies state 0 1 and theta 3 and then measuring the charge variable which is conjugate to the difference between the 2 1 that is free because if we look at how the energy depends on those if I slowly very theta 1 with data 3 fixed and the state the minimum the energy if they do 1 rotates by 2 pi fated to rotate by pie and then this state would either be invariant if X equals 1 if it's a superposition of the 0 1 1 in the computational basis or it will change sign if x is equal to minus 1 and I can distinguish those possibilities by making a charge measurement distinguishing between a charge which is an integer or while in order even integer multiple of the electron charge that is half the group for future work these measurements might be kind of noisy but we can imagine making them more robust by repeating the measurements or by using coding which I'll mention later OK so now I can start to explain how 1 does the protected gate we've seen the large inductance is important for the operation of 0 pi Cubitt but from now going to talk about the internal structure the cubic it will just assume it's a very good Cuban but a large inductance wanted the discussion a 2nd time for a rather different reasons than before when I perform the protected gate either a single Cupid out the 4 group data rotation about the z axis by power over to or to Cubitt entangling gates I will couple the 1 Cubitt or pair of qubits connected in series to an oscillator another LC circuit with a large inductive and what 1 finds if 1 simulates the process in which we have close a switch to couple the cubit to the LC circuit leaves which close for a while for some prescribed amount of time and then open the switch decoupling the oscillator from the you better pair qubits is that a superposition of the state 0 and 1 of his reply Cupid because a superposition with the modified phase and though the phase change if I keep this which close for the right amount of time will be pi over 2 so we've done a rotation about the z axis of considering the coupling single Cuba to the oscillator by by over 2 and the final state of the oscillator is essentially independent of whether the state of the qubit was 0 or one so there's no entanglement between the oscillator and is 0 by and phase eta minus pi over 2 is extremely stable with respect to deformations and false and fluctuations in the Hamiltonian so for example if this inductance parameter is chosen to be 80 and then we get the ideal gate to an accuracy in diamond norm which is about 10 to the minus even if the errors in the polls that implements the gate heard at about the 1 per cent of all so now would like to understand how this can be wise the gate so robust and the answer
14:55
really is found by thinking about upon encoder codon which cubit it is embedded in the hilbert space of a harmonic oscillator we can think of it as a stabilizer code a simultaneous eigen on state with eigen value want to poly operators where the poly operators in this case our to the 2 I ifI which translates into space by 2 and the minus 2 pie IQ which translates and phi space by 2 pi find Q don't to me with 1 another but these they space translations do can you so they can be simultaneously diagonalize if I choose both these operators to be 1 there's a twodimensional code space and I can choose work as a basis for the space the code states 0 and 1 where 0 is a superposition of phi eigenstates where the value 5 is pi times and even integer and 1 is a superposition of phi eigenstates where the value fi times and I'd integer for the bit flip operation the changes a 0 2 1 1 2 0 is a translation and phi space by the time that's logical X acting on the code and I can also Fourier transform these codewords and that's what they look like in Q space and I can't use as a basis the eigenstates of acts with eigen value plus or minus where the plus state is a superposition of Q eigenstates where the value of Q is an even integer and the minus eigenstate is a superposition of 2 eigenstates where the value q is not integer and the logical operator that flips the plus to the minus is just a translation into space by 1 now this code is protected against errors which can be described as distributions on small shifts in phi space and Q spec so if there's a shift in Vice space which is less than pi over to less than half the translation that we would do to perform the logical acts that clips of that then that's a correctable error and the distinguishability of 0 and 1 is not lost and if we have a shipped in Q which is less than 1 have less than half of the ship flips the plus and minus eigenstates that's also a correctable error which in principle can be reversed now these
17:18
codewords that I've described these ideal code words are actually non normalizable and on physical but I can consider instead 8 approximate codeword which is normalizable and physical we can think of it for example as a sequence of high equallyspaced galaxy and peaks and phi space governed by a broad gussy an where the width of the individual peaks delta is small compared to 2 pi and the width of the envelope kept inverses broad compared and apply for a transform this approximate codeword to see what it looks like into space it has a similar go grade appearance that is now there will be a broad envelope whose with these delta inverse and periodically space on narrow functions in Q space whose width is given by cap this intrinsic error in the approximate codewords because there isn't there can be a little bit of leakage of the individual peaks outside the size of correctable error but if for example each 1 of the peaks has a width which is less by a factor of 5 than the size of a ship which gives an uncorrectable error then we can make the intrinsic error the codewords less than 1 part in a million it's not really necessary for the peak functions of the envelope function to be gas and I and consider any narrow function periodically repeated governed by some slowly varying broad envelope function in that to approximate codeword OK so now let's come back to how the protected gate is done I did say so before but when I close the switch couple of the cube to the LC circuit that we done by a tunable Josephson coupling so I'll turn on Gaiety introducing in the hamiltonian of the oscillator a cosine potential where the cosine depends on the difference between the phase drop across the inductor in the LC circuit and the phase drops theta across the Cuban which is either 0 or pi depending on the state of the the 0 pi so the 0 pi but it's in the state 0 that means our data is 0 and this just a cosine potential energy is large enough the potential which was a harmonic well now becomes modulated by the cosine and has many local minima which are located at values of which I 2 pi times an integer but if they or equal to pi then the cosine would be shifted by high the minimal would occur had values of phi which applied times an auditory and now if we turn on GATE obeying suitable patisserie conditions not too slowly and not too quickly the if we started out in the ground state of the harmonic oscillator the broad Garcia and I with large look tuitions because the inductance of the LC circuit is large it would evolve to 1 of these and gonna in grid states in a box code word of the continuous variable code and it we would have you the encoded 0 or the encoded 1 according to whether the minima of the cosine occurred even or odd values of pi so 0 1 of the 0 by it would become imprinted on the code words for this oscillator code wanna turn on J. T on a time scale which is short compared to the Piercy circuit itself but which is long compared to the period of the oscillations in the local cosine wells which is determined that means we have to add j large enough
21:05
OK so now wanted prepared these codewords how we teach a fixed for a while and what we do sell the how quadratic term in the potential induces a gussy an operation on our code words an operation which if we choose the elapsed time that the switch is closed at the US leader and you better coupled appropriately can be expressed as E to the minus 5 squared over 2 pi so that means that in the cold stage 0 4 which Phys and even multiple pie this is a trivial phase acting on the stage but a 4 in the code state 1 in which a fly is an odd multiple pie there's a nontrivial phase which is E to the minus iPod over to performing a rotation by pi over to about the c axis in the code space so what happens once we prepared the code word and we keep the switch close for a while is that the state of the oscillator makes a big excursion in our face space it leaves the code space but it eventually comes back to the code space but it comes back with a twist with the nontrivial holonomy or bury phase which is just this encoding gate so this is a type of geometric phase gate yes later goes on a trip and comes back with some statedependent phase but it's especially robust type of geometric faced and if we have 2 cubits connected in parallel then whether we get the nontrivial phase or not depends on whether the on the value of the total phase drop across a pair of 0 pi Q bits and that means we've done this entangling phase now costs the pulse might not be perfect we might keep this switch close a little bit too long or open to soon for example and that means apart from this ideal gas an operation that we would like to do to perform the encoder gate there will be some additional go an error error wait a rotation by an amount that's 1 which is the fractional error and the timing of the Paul's but if epsilon a small that additional gussy in operation just causes some modest spreading Q space it's a correctable error for the continuous variable code so we do the ideal of the groups single Cuban rotation I together with some correctable error anchors since everything is calcium in this case where we started out in the ocelot oscillator ground state we can calculate everything explicitly if the timing of the pulses chosen optimally then there's some intrinsic error in the gate which comes from the nonzero with that of the individual peaks in cues that gets exponentially small when the inductance of a circuit is very large and then we can compute how the error is enhanced when there's a nonzero epsilon do you an imperfect the rotation of the code word but that is induces a multiplicative factor in the air which is close to 1 of epsilon is small on a scale which is set by the width of the peaks in Q space but we have to understand finally what happens to this correctable error when we open the switch again we d couple the oscillator from the cube it well the execution
24:30
Gators earliest half of a threestep process in the 1st step we close this which we turn on the coupling between the oscillator and Cuban that we keep it close for a while in the state goes on this excursion eventually coming back to the code space so I'll say that the state is outside began at the begin excursion inside and at the end and then finally we open the switch D coupling the oscillator and the Cuban and we get some final state of the so let's look at the steps 1 by 1 so 1st of all while the switches closing returning on the coupling and because the were considering the turning on to be fast compared to the period of the oscillator the quadratic term the potential the phi squared term the potential has a relatively small effects while this which closing so a 1st approximation went ignore that and then the Hamiltonian just has the cosine potential and the kinetic energy but the Hamiltonian has a cosine potential whose coefficient has assigned that depends on whether the cube is 0 or one if a cube is 1 that means the argument the cosine did shifted by time that changes the sine and cosine cosine fighter term Q space is a translation by the plus one or minus one what what that means is that the cosine aren't I commutes with the logical exair acting on the code space the operator which has the value plus 1 when Q. is close to an even integer and the value minus 1 when he was close to an odd integer so that means that the Hamiltonian and the 2 cases depending on the state of the qubit are related by conjugation by acts and the same is true for the evolution operators that we obtained by integrating the Schrödinger equation using that Hamiltonian and so that allows us to see that when the switch is closed and the rotation of the state begins that side began with in the case where the cube is in the state 1 is related to this I began when our the cube it is in the state 0 by well in this way that to us I 1 began is X u 0 x acting on SI initial but as I initially initial state a cube it but now remember we started out with very large flat these fluctuations in phi space were very narrow state in Q space for the harmonic oscillator and that means it's very nearly an eigenstate of acts if the state has it has its support in the interval between minus 1 half and 1 half inches space an X equals 1 state so it's a very good approximation say that X acting on the initial state is 1 and that means that the beginning stages of a cube is 1 and the beginning stated cube is 0 or just are related by the action of the logical operator X
27:41
and now we have let the state rotate goes on excursion eventually it comes back to the code space and this relationship between the state when the cubit is 1 1 and 0 will actually be protected if we don't have a logical phase in other words I did take the linear combination of CI 1 began in size 0 begin with the plus and that's an X equals 1 eigenstate and their linear combination with the minus is an X equals minus 1 I'd say and because of small error will still be a correctable phase error that relationship will still be satisfied when we come back to the code space even if we do a little over rotation or the wonder rotation and then finally we have to decouple the oscillator and the cube it by opening switch and we can argue as before or that the operator that acts during the opening of this which in the case where the cube is 0 in the case where the Cuban is 1 they're related by conjugation by acts and that allows us to say that the final state of the other 1 is finally decoupled from the cube it is is given by a logical lacks acting on the final state of the oscillator when the cube is 0 X acting on the state when Q. is 0 8 is the state of the oscillator when the cubit is 1 to finally 1 more step if we done everything nicely idiomatically the oscillator won't get highly excited it started out very narrow and Q space and a wine that being near very narrow in Q space at the end and that means x acting on the final state of the oscillator will be essentially 1 and so that allows us to conclude that the final state of the oscillator is to a very good approximation the same irrespective of whether the state of the qubit was 0 or one so there's no entanglement between the oscillator and Q and furthermore the phase that's acquired is extremely insensitive to the way we implement again and these conclusions still hold we include the 5 squared over 2 well term in the potential bring the closing and opening this wage since that is produces some modest additional spreading in Q space and because these are errors tennis like and because the states are well protected against translations in Q space the conclusions and change very much so sorry that was a lot of steps so let me
30:09
summarize them on 1 slide these were the ingredients that make a protected gave 1st of all there's some symmetry principle which relates the Hamiltonian while this which is opening or closing in the case where the 0 pi Q but it's in the state 0 to the state of a Hamiltonian 1 0 point here is in the state 1 of the best because the difference is just a share in FY space by high this state initially has large phase fluctuations that is because the large inductance spies very broad and correspondingly Q is very narrow then it's important that we prepare good code this continuous variable code and that's what protects the symmetry that relates this state the cubit is 0 and state when the Cuban is 1 how when we perform the gate even if we don't and perform it perfectly there's Nadia Batista the principle that we lined up with the final state of the oscillator which is not very excited so it's still narrowing Q space and we need some separation of time scales for things to work in other words the opening and closing of the gate has to be fast on the scale of the period the oscillator so that the effect of the harmonic term in the potential is not very big while the switch is opening and closing so the conclusion is that the oscillator really acts like the and so on which should be syndrome information gets imprinted any noise it's introduced by the imperfect implementation of the gate 1 7 mussing up the oscillator a little bit but the final state of the oscillator doesn't know anything about the state of the qubit and the phase is very stable with respect to imperfect implementation of the gay
31:56
so I try and decimate the 980 advantage of facts which contribute to the error probability is the harder do analytically we can predict how they scale using LandauZener theory but we can do very precise calculations we can simulate the process in which we close the switch leave it close for a while and then open at so here I've plotted the diamond norm deviations from the ideal phase gate home as a function of this angle epsilon which tells me the fractional error in the timing of the polls and if we didn't have any non I media batik affects the other day we get down to 0 error below 10 to the minus 9 so when the air is very small but transitions are a nonnegligible so it becomes something a little bit about 10 to the minus 8 but it stays pretty stable so that the air is about the same if we introduce errors of the order of 1 per cent in the implementation of a pulse actually that was for the case where we start
33:07
ground state of the harmonic oscillator but a similar conclusion will hold if the initial state of the oscillator is not too highly excited if some lowlying of excitation of the oscillator because I will still be narrowing Q space which is the key thing that we need initially and so the temperature is low compared to the frequency of the oscillator so that we have of thermal state to begin with the error in the gate will be much affected by temperature and we can also checked that the implementation of the gate is robust with respect to fluctuations in the Hamiltonian which introduce some anharmonicity in the US alone and for some higher harmonics in the Josephson energy now now
33:53
the remarkable accuracy the gate really hinged on this assumption that we have a very large inductance and 1 can ask whether that's reasonable I guess I forgot to say it should
34:02
have I showed you the plot this is for particular values of the parameters of the square developer see this dimensionless inductance parameter was chosen to be 80 and the square root of J psi being 8 means that it's the reciprocal of this number which determines the width of the individual peaks and phi space and we chose the optimal time scale for the switch to open and close in order to get of this performance so we need that large
34:31
inductance for the thing to work well and you can ask whether it's reasonable for this number to be this big and it may in fact be a big practical problem to get such a large inductance in a superconducting circuit the kind of once on geometrical grounds to be of order 1 and we want to be quite large 1 way in which you can get a large inductance in principle as I mentioned earlier is to chain together many junction I see that was done with a different motivation by the Yale group and they observed a value of this parameter which was about 6 per chain of 43 GeV just injunctions and in principle you can scale it up too much longer chain you get are much larger value of this impedance but there may be reasons why that's not so practical may be a more promising approach is to use materials which have a large intrinsic kinetic inductance to on build the superconducting circuits of course that what I described is just some Clifford of phase gates single Cuban into cubic gates which are not enough for universality by themselves but we can boost to a universal set of gates by using state distillation ideas all abroad in town so that if we in addition can perform measurements of a Cuban in the EC spaces that are accurate and of high were for rotation about the z axis which is not protected and has only a soso fidelity though it can new state distillation to get accurate universal gates it may be that the measurements are noisy and this type of scheme I'll come to that in a 2nd but if we do something about controlling the noise in the measurements then the 2 cubic phase gate a really is a useful tool for fault tolerance of begin do with extremely high fidelity even though it's a non universe of the 4 groups get well as far
36:29
as the noise in the measurements is concerned if we can do the measurements nondestructively that is with a small probability of flipping the value of the measured observable when we do the measurement than we can imagine repeating measurements many times to make them more reliable alternatively we can use repetition code make measurements more reliable I won't go into the details but what does is shown here is a way of teleporting us see 98 from encoded using a repetition code where the 2 cubic gage shown our 2 Cupid phase gates of the type that we know how to reliably and the arrows pointing to the right of our X eigenstate preparations and the arrows pointing to the left our X basis measurements and in this case not to repeat the measurement we can decode the results in the repetition code by doing majority voting every time you make a measurement and if for example we can do a see phase gate which has of accuracy arose the error rate for data 10 to the minus 5 and even if the measurements have a 1 per cent error rate we could do this teleported C 98 to 10 to the minus 6 accuracy
37:43
OK so memorized of course we expect to be have to use faulttolerance ideas to operate a largescale quantum computers and if we can manage to do a protected to Q booed to Cuba Clifford phase gate with very high fidelity that could be a powerful tool for fault tolerance even if measurements another dates are noisy in the case of the 0 pipe protected cubits superconducting circuits if we can turn on and off the tunable coupling to an oscillator a tunable Josephson junction that we can do single qubit and Cubitt Clipper gates with very high fidelity fidelity which gets exponentially close to 1 of you can choose the system parameters appropriately what makes a gate work is an underlying continuous variable quantum code which protects against phase errors but the 0 by Cuban itself and the protected gate requires building circuits which have a very large inductance in natural units and their minds and to achieve the game itself is robust against nonzero temperature and fluctuations in the Hamiltonian of the system so I don't know at this particular approach to doing protected gates will turn out to be practical but I do think it's important to continue to seek ways of doing reliable gates that rely on the nature of the physical encoding of quantum information and here Europe suggested 1 possible step in that direction thanks for listening questions that might in the back next there is a really nice top and it's extremely interesting in particular me because there is some similarity at least in feel to the idea of dynamic modulation which I know a little better than this I guess but I was wondering about the interview talk where he showed some of the downsides of trying to achieve this high L and C ratio is fact in trapped ions as you know we do these geometric phase gates by exciting the exciting harmonic oscillator and initially there is no intrinsic protection against that timing errors you really have to make a close the base space otherwise there's a very significant getting fidelity 1st firstorder but if you moduli the coupling to that to that emotional mood in a way that's reminiscent of dynamical the coupling you can actually build intrinsic a building robustness against these timing errors so I was curious if you think there's any possibility that you can reduce the requirement for airlines to being 80 to something closer to 0 only off by a factor of 10 so from what you said Every had achieved if you can add in some other concepts of dynamic modulation OK this so the question was how is it possible that we could combine together having a very large square develop receiver not quite as large as they seem to require with some of the modulation in time of the coupling of the OS later to the cubic a since it's of no other context a geometric phase gates can be made more robust by such modulation of that repeating the question because I'm stalling really it and also it's out it's a good question that I have an early thought about that the question the product all the state preparation for the on oscillator right well I just assumed it was a thermal stated some temperature which all compared to the oscillator frequency so the explicit calculation that I described in particular the plot was for the case where the initial state of the oscillator is the ground state of the LC circuit and or if the so that's strictly the 0 temperature if the temperature is nonzero but small compared to the frequency so it might be an excited state but with some suppressed probability of them that still works after we do the gate the oscillator doesn't return to the initial for most state or ground state in fact It's so becomes the repository for the entropy that's introduced by the noise and if I tried to use it again the gate wouldn't be quite as accurate so at some point and why no 1 a cool the oscillator back down in order to use it in highfidelity get this question when DEC yet most posession where is it when you describe of the tuning of better protection with increasing the number so with increasing the chain of the Josephson junction of possible variations in the between the individual junctions in the chain or the effect of the thing I was so I'm sorry you're reading his last part of the question the question is about getting our high inductance it from a long chain of Josephson junctions and you listen you sold small regions they wouldn't affect anything in the sky or a small variation along the chain advised parameters lowered so in a in other words I added some disorder to the changes are how would it affect the performance well I think it would probably be fairly tolerant of disorder because the the key thing is just to have the inductance of the whole composite device large compared to the I think 1 of the practical problems is if you really try to build a long thing because it's getting geometrically large it's hard to keep the capacity and sperm in a scaling up with the size of the device as well as the inductance but I don't think disorder is a fundamental problem Let's think John again and that's because the ransom
00:00
Beobachtungsstudie
Scheduling
00:43
Bit
Quantencomputer
Extrempunkt
Natürliche Zahl
Netzwerktopologie
Fehlertoleranz
Pi <Zahl>
Äußere Algebra eines Moduls
Quantisierung <Physik>
Qubit
Phasenumwandlung
Feuchteleitung
Nichtlinearer Operator
Lineares Funktional
Fehlererkennungscode
Hardware
Approximation
Gebäude <Mathematik>
Güte der Anpassung
Quantencomputer
Störungstheorie
Frequenz
Energiedichte
Würfel
Digitaltechnik
Basisvektor
HillDifferentialgleichung
Decodierung
Information
Aggregatzustand
02:43
Impuls
Bit
Prozess <Physik>
Ortsoperator
Statistische Schlussweise
Physikalismus
Zahlenbereich
Term
HamiltonOperator
Code
Variable
Harmonischer Oszillator
Einheit <Mathematik>
Quantisierung <Physik>
Phasenumwandlung
Feuchteleitung
Kanonische Vertauschungsrelation
Qubit
Freier Ladungsträger
Quantencomputer
Physikalisches System
CliffordAlgebra
Unendlichkeit
HilbertRaum
Verknüpfungsglied
Knotenpunkt
Digitaltechnik
Phasenumwandlung
Windkanal
Codierung
Ablöseblase
Partikelsystem
Pendelschwingung
Innerer Automorphismus
05:04
Impuls
Distributionstheorie
Subtraktion
Vektorpotenzial
Punkt
Statistische Schlussweise
Wellenfunktion
Abgeschlossene Menge
Zahlenbereich
Fluss <Mathematik>
EMail
Extrempunkt
Term
RaumZeit
Fluss <Mathematik>
Loop
Arithmetischer Ausdruck
Bildschirmmaske
Multiplikation
Unterring
Einheit <Mathematik>
Mittelwert
Wellenwiderstand <Strömungsmechanik>
Spektrum <Mathematik>
Quantisierung <Physik>
Wurzel <Mathematik>
Tropfen
Parallele Schnittstelle
Phasenumwandlung
Gammafunktion
Soundverarbeitung
Einfach zusammenhängender Raum
Krümmung
Fluktuation <Physik>
Geometrische Quantisierung
Datenfluss
Frequenz
Energiedichte
Knotenpunkt
Datenfeld
Quadratzahl
Dreiecksfreier Graph
Digitaltechnik
Mereologie
Elektronischer Fingerabdruck
Partikelsystem
Trigonometrische Funktion
Aggregatzustand
08:26
Prozess <Physik>
Gewichtete Summe
Statistische Schlussweise
Extrempunkt
Gruppenkeim
Kartesische Koordinaten
Drehung
Extrempunkt
Superposition <Mathematik>
HamiltonOperator
LieGruppe
Thetafunktion
Vorzeichen <Mathematik>
Kontrollstruktur
Qubit
Tropfen
Phasenumwandlung
Einflussgröße
Zentrische Streckung
Lineares Funktional
Parametersystem
Nichtlinearer Operator
Reihe
Kontextbezogenes System
Frequenz
Rhombus <Mathematik>
Verkettung <Informatik>
Knotenpunkt
Verknüpfungsglied
Ganze Zahl
Würfel
Rechter Winkel
Konditionszahl
Hochvakuum
Dynamisches RAM
Messprozess
Decodierung
Information
Ordnung <Mathematik>
Pendelschwingung
Fehlermeldung
Aggregatzustand
Standardabweichung
Telekommunikation
Subtraktion
Fluss <Mathematik>
Multiplikation
Variable
Bildschirmmaske
Pi <Zahl>
Quantisierung <Physik>
Datenstruktur
Zehn
Leistung <Physik>
Einfach zusammenhängender Raum
Qubit
Konvexe Hülle
Fluktuation <Physik>
Kanalkapazität
Einfache Genauigkeit
Energiedichte
Minimalgrad
Verschränkter Zustand
Digitaltechnik
Basisvektor
Codierung
Normalvektor
14:53
Distributionstheorie
Vektorpotenzial
Bit
Statistische Schlussweise
Extrempunkt
Extrempunkt
HamiltonOperator
Superposition <Mathematik>
RaumZeit
Gradient
Potenzielle Energie
Eigenwert
Nichtunterscheidbarkeit
Translation <Mathematik>
Umkehrung <Mathematik>
Qubit
Tropfen
Phasenumwandlung
Verschiebungsoperator
Dimension 2
Schreiben <Datenverarbeitung>
Lineares Funktional
Nichtlinearer Operator
Zentrische Streckung
Inverse
Stellenring
Cliquenweite
Ideal <Mathematik>
Ähnlichkeitsgeometrie
Frequenz
Teilbarkeit
Kugelkappe
Verknüpfungsglied
Einheit <Mathematik>
Würfel
Ganze Zahl
Konditionszahl
Phasenumwandlung
Decodierung
Pendelschwingung
Trigonometrische Funktion
Aggregatzustand
Fehlermeldung
Eigenwertproblem
Stabilitätstheorie <Logik>
Folge <Mathematik>
Subtraktion
Quader
Mathematisierung
Polygon
Code
Harmonischer Oszillator
Pi <Zahl>
Gammafunktion
Einhüllende
HilbertRaum
Schwingung
Mereologie
Basisvektor
Digitaltechnik
Wort <Informatik>
Brennen <Datenverarbeitung>
21:03
Bit
Vektorpotenzial
Prozess <Physik>
Statistische Schlussweise
BerryPhase
Gruppenkeim
Drehung
Extrempunkt
HamiltonOperator
LieGruppe
RaumZeit
Puls <Technik>
Translation <Mathematik>
Tropfen
Phasenumwandlung
Sinusfunktion
Nichtlinearer Operator
Zentrische Streckung
Addition
Parametersystem
DeanZahl
Approximation
Theoretische Physik
Cliquenweite
Ideal <Mathematik>
Kommutator <Quantentheorie>
Frequenz
Verknüpfungsglied
Würfel
Kinetische Energie
Koeffizient
Phasenumwandlung
Decodierung
Pendelschwingung
Trigonometrische Funktion
Fehlermeldung
Aggregatzustand
Eigenwertproblem
Zeitabhängigkeit
Total <Mathematik>
Fächer <Mathematik>
Gruppenoperation
Abgeschlossene Menge
Term
Mathematische Logik
Code
Harmonischer Oszillator
Datentyp
Pi <Zahl>
Wellenmechanik
Qubit
Soundverarbeitung
Fluktuation <Physik>
sincFunktion
Einfache Genauigkeit
Verschränkter Zustand
Holonomiegruppe
Digitaltechnik
Wort <Informatik>
Innerer Automorphismus
Grenzwertberechnung
27:40
Eigenwertproblem
Dualitätstheorie
Subtraktion
Bit
Vektorpotenzial
Punkt
Statistische Schlussweise
Gemeinsamer Speicher
Schaltnetz
Ablöseblase
Geräusch
Implementierung
Abgeschlossene Menge
Drehung
Term
Mathematische Logik
HamiltonOperator
RaumZeit
Code
Maßstab
Symmetrie
Translation <Mathematik>
MIDI <Musikelektronik>
Phasenumwandlung
Trennungsaxiom
Soundverarbeitung
Qubit
Zentrische Streckung
Nichtlinearer Operator
Approximation
Codierungstheorie
Fluktuation <Physik>
Frequenz
Linearisierung
Rechenschieber
Verknüpfungsglied
Verschränkter Zustand
Einheit <Mathematik>
Offene Menge
Würfel
Phasenumwandlung
Wort <Informatik>
Information
Pendelschwingung
Innerer Automorphismus
Symmetrie
Aggregatzustand
Fehlermeldung
31:55
Harmonische Analyse
Bit
Prozess <Physik>
Gruppenoperation
Implementierung
Extrempunkt
RaumZeit
Physikalische Theorie
Puls <Technik>
Harmonischer Oszillator
Phasenumwandlung
Lineares Funktional
Winkel
Fluktuation <Physik>
Ideal <Mathematik>
Rechnen
Frequenz
Rhombus <Mathematik>
Energiedichte
Verknüpfungsglied
Hypermedia
Pendelschwingung
Normalvektor
Ordnung <Mathematik>
Fehlermeldung
Standardabweichung
Grenzwertberechnung
Aggregatzustand
33:51
Zentrische Streckung
Parametersystem
Statistische Schlussweise
Minimierung
Cliquenweite
Zahlenbereich
Plot <Graphische Darstellung>
Extrempunkt
RaumZeit
Quantisierung <Physik>
Verknüpfungsglied
Quadratzahl
Wurzel <Mathematik>
Ordnung <Mathematik>
Softwareentwickler
34:29
Resultante
Eigenwertproblem
Abstimmung <Frequenz>
Statistische Schlussweise
Virtuelle Maschine
Gruppenkeim
Geräusch
Zahlenbereich
Drehung
Extrempunkt
Kubischer Graph
Code
RaumZeit
Fehlertoleranz
DedekindSchnitt
Datentyp
Wellenwiderstand <Strömungsmechanik>
Ordnungsbegriff
Zeitrichtung
Eins
Grundraum
Zehn
Einflussgröße
Phasenumwandlung
Addition
Parametersystem
Fehlermeldung
Materialisation <Physik>
Einfache Genauigkeit
Nummerung
Bitrate
Quantisierung <Physik>
Verknüpfungsglied
Knotenpunkt
Verkettung <Informatik>
Rechter Winkel
Basisvektor
Digitaltechnik
Messprozess
Ordnung <Mathematik>
Fehlermeldung
Aggregatzustand
37:42
TVDVerfahren
Punkt
Quilt <Mathematik>
Statistische Schlussweise
Natürliche Zahl
BerryPhase
HamiltonOperator
Richtung
Fehlertoleranz
Einheit <Mathematik>
TUNIS <Programm>
Einflussgröße
Phasenumwandlung
Parametersystem
Dokumentenserver
Gebäude <Mathematik>
Quantencomputer
Ähnlichkeitsgeometrie
Plot <Graphische Darstellung>
Biprodukt
Kontextbezogenes System
Rechnen
Frequenz
Dialekt
Digital Equipment Corporation
Teilbarkeit
Quantisierung <Physik>
Verknüpfungsglied
Verkettung <Informatik>
Knotenpunkt
Gruppenkeim
Phasenumwandlung
Decodierung
Information
Pendelschwingung
Ordnung <Mathematik>
Faserbündel
Aggregatzustand
Fehlermeldung
Lesen <Datenverarbeitung>
Mathematisierung
Physikalismus
Zahlenbereich
Code
Variable
Harmonischer Oszillator
Spieltheorie
Quantisierung <Physik>
Softwareentwickler
Qubit
Soundverarbeitung
Transinformation
Diskretes System
Fluktuation <Physik>
Kanalkapazität
Einfache Genauigkeit
Winkel
Physikalisches System
Quadratzahl
Mereologie
Digitaltechnik
Wort <Informatik>
Entropie
Metadaten
Formale Metadaten
Titel  Protected gates for superconducting qubits 
Serientitel  Second International Conference on Quantum Error Correction (QEC11) 
Autor 
Preskill, John

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  10.5446/35317 
Herausgeber  University of Southern California (USC) 
Erscheinungsjahr  2011 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Informatik, Mathematik, Physik 
Abstract  We explain how continuousvariable quantum errorcorrecting codes can be invoked to protect quantum gates in superconducting circuits against thermal and Hamiltonian noise. The gates are executed by turning on and off a tunable Josephson coupling between an LC oscillator and a qubit or pair of quits; assuming perfect qubits, we show that the gate errors are exponentially small when the oscillator's impedance is large in natural units. The protected gates are not computationally universal by themselves, but a scheme for universal faulttolerant quantum computation can be constructed by combining them with unprotected noisy operations. 