Protected gates for superconducting qubits
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Preskill, John
We explain how continuous-variable quantum error-correcting 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 fault-tolerant quantum computation can be constructed by combining them with unprotected noisy operations.
 Okay, why don't we start here? I think we've got the AV situation mostly under control here. So for our next session, we'll be starting, I guess, five to six minutes late, so we'll just sort of keep that offset in our schedule. Our next session will be kicked off by John Preskel who will be talking to us about superconducting Q-bits. Thanks, Andrew. Everybody can hear okay? I'm talking about work done at Caltech with Peter Brooks and Alexei Kutayev. All of us here recognize that error correction and fault tolerance are going to be essential if we're going to operate large-scale quantum computers someday. The standard approach to fault tolerance uses clever circuit design to overcome the deficiencies of hardware. There's an alternative hardware or 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 it's 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. One suggestion has been that we can make robust encodings of quantum information using superconducting circuits, in particular the zero pi Q-bit which was suggested by Kutayev and others. The phase in a superconductor is a periodic variable with period two pi. But for a suitably designed circuit, the dependence of that circuit's energy on the phase difference between its two leads is a very good approximation of periodic function of theta with period pi aside from corrections that become very small as the device gets large. And in that case, the energy as a function of theta has two minima which can occur at theta equals zero and theta equals pi which have very nearly the same energy. And if there's a large barrier separating those minima, those two minima can be the basis states of a robust Q-bit. It's protected against bit flips by the high barrier in between zero and one. And it's protected against defacing because the degeneracy of the two states is very stable with respect to perturbations of the Hamiltonian. But of course if we want to do robust quantum computing, we need more than just stable Q-bits. We also need to be able to do very accurate gates. And that's what I want to describe in the talk, how in particular we can do highly accurate Clifford gates acting on zero pi Q-bits by turning on and then turning off the coupling of the Q-bit to our harmonic oscillator and LC circuit. And the gate will work very effectively, be highly accurate, if the inductance in the LC circuit in natural units is a very large number. The secret that underlies the reliability of the gate is actually a quantum code, a continuous variable quantum code in which Q-bit, a two-dimensional system, is embedded in the infinite dimensional Hilbert space of a harmonic oscillator. And I'll be explaining how that works. Now to understand the physics of the zero pi Q-bit itself and also the protected gates, we should know a little bit about Josephson junctions. A Josephson junction is a device in which a layer of insulating material separates two pieces of superconductors such that there's a non-zero amplitude for the charge carriers in the stirred Cooper pairs to tunnel from one side of the barrier to the other side. And we can encode that tunneling in an effective Hamiltonian. If we think of a lead attached via Josephson junction to an island that carries some charge, a superconducting island, then the effective Hamiltonian describes processes in which the charge on the island can either increase by one or decrease by one when a Cooper pair tunnels through. If Q is the charge in units of 2e, the Cooper pair charge, and there's some amplitude for the tunneling which I've called J, the Josephson coupling. It's often convenient to describe the system using conjugate variables, the superconducting phase rather than the charge, which is just the Fourier conjugate of the charge. So the phase and the charge obey commutation relations like the position and momentum of a particle in one dimension. And if we express the tunneling Hamiltonian in terms of the phase, proportional to cosine phi with the coefficient given by the tunneling amplitude J. So one other thing we should know about superconductors is that flux is quantized. The phase in a superconductor is actually a convention dependent concept. If I want to compare the phase at two different points in the superconductor, I need some notion of parallel transport that tells me how the phase should change as I move from one point to another. That notion of parallel transport is provided by a connection, which is just the electromagnetic vector potential. And the parallel transport of phase is actually path dependent. That means that connection has curvature. The curvature is just the magnetic field. So when there's a non-trivial magnetic field, the phase picked up under parallel transport can depend upon the path. If we have a closed ring of superconductor and there's some applied flux, it might be that the effect of parallel transport of the phase around the loop is non-trivial. And that's energetically unfavorable. So what happened Ace is that some persistent current flows in the ring, which adds to the applied field, some additional field 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, H over 2e. If I think of a ring with a Josephson junction inserted inside, I can think of the phase difference between one side of the Josephson junction and the other as not a periodic variable with period 2 pi, but a real variable. Because if I winds by 2 pi, that just means we've inserted one quantum of flux through the ring. Now I'm also going to want to consider LC circuits in superconductors. This is really just like a particle in a harmonic well where the charge on the capacitor corresponds to the momentum of the particle and the superconducting phase is like the addition of a particle. I can think of that as 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 Cooper pair charge 2e and to express the flux linking the circuit in terms of the superconducting phase drop across the inductor. And then in terms of those units, the square root of L over C is a dimensionless number. It's just the conventional impedance of the circuit expressed in terms of these units, which is about one kilo ohm. So we can ask, what's the significance of that dimensionless number being large? And the answer is that in the ground state of this oscillator, the circuit has very large phase fluctuations. The ground state is a Gaussian wave function where the distribution in phi is wide compared to 2 pi with that width determined by the square root of L over C. Now we can imagine in certain junction into a circuit which has a large inductance in this sense, and then its energy of a proportional to this difference of phases. But because of the large fluctuations in phi, the cosine gets averaged over many cycles of the cosine, and that makes the contribution to the energy that depends on the phase theta very small because of the averaging exponentially small in the width in five space, which is determined by the large inductance. So now we know enough to understand the idea of the protected zero pi qubit. This is the form in which Katayev suggested it. We can think of a superconducting circuit with two rungs. So on each rung there's a large inductance, well bang, this condition, and the two rungs are coupled together by some very large capacitance. So I can consider the variable which I can consider the phase in the superconductor on either side of the capacitor, and then the sum of the phases phi 1 and phi 2, plus, is unaffected by the large capacitance, and so it's a light variable because of the large inductance which has very large fluctuations. But the difference variable phi 1 minus phi 2 feels the large capacitance, and that makes it a very heavy variable which wants to be well localized in phase. And so it locks to a value which is determined by the phases on the external leads to the difference between theta 4 and theta 1, and minus the difference between theta 3 and theta 2. And now we can imagine taking the top rung and twisting it by 180 degrees and connecting the leads together so that we're identifying theta 2 with theta 4 and theta 3 with theta 1. And then the energy of the circuit becomes a function of the value that the heavy variable locks to which is now 2 theta 2 minus theta 1. And then there's some correction, but it's very small because of the large variations of the light variable. And that's how we can get a zero pi qubit in which the potential is essentially a periodic function with period pi, which has two minima which are very nearly degenerate. In fact, we can think of the protection as arising from a very non-local encoding of information like in other contexts where we have topological protection. If we imagine, as Katayev suggested, that we realize the large inductance by a long chain of joseph's injunctions, and then the phase difference between one 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, those states locally look almost identical. And the breaking of the degeneracy is associated with tunneling phenomena which run all the way from one end of the chain to the other, which have an amplitude which is exponentially small. And that's the reason why the corrections, degeneracy, get very small when the device scales up. We'll also want to be able to measure the zero pi qubit. If we measure it in the standard or computational basis, we need to distinguish a phase drop across the qubit of zero from a phase drop of pi. And we could do that by coupling the qubit to the joseph's injunction and inserting a 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 two sides of the qubit is zero or pi. We'll also want to be able to measure in the dual basis, the x basis, dual to this computational basis. And one way we can imagine doing that is by breaking the connection that identifies theta one and theta three, and then measuring the charge variable, which is conjugate to the difference between theta one and theta three. Because if we look at how the energy depends, if I slowly vary theta one with theta three fixed, then to stay at the minimum of the energy, if theta one rotates by two pi, theta two would rotate by pi. And then the state would either be invariant if x equals one, if it's a superposition of the zero and one in the computational basis, or it will change sign if x is equal to minus one. And I can distinguish those possibilities by making a charge measurement, distinguishing between a charge which is an integer or, well, an odd or an even integer multiple of the electron charge, that is half the Cooper pair charge. Now, 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. Okay, so now I can start to explain how one does the protected gate. We've seen that a large inductance is important for the operation of the zero pi qubit, but from now on, I'm going to talk about the internal structure of the qubit. We'll just assume it's a very good qubit. But a large inductance will enter the discussion a second time for a rather different reason than before. When I perform the protected gate, either a single qubit, a Clifford group gate, a rotation about the z axis by pi over two, or a two qubit entangling gate, I will couple either one qubit or a pair of qubits connected in series to an oscillator, another LC circuit with a large inductance. And what one finds if one simulates the process in which we close a switch to couple the qubit to the LC circuit, leave the switch closed for a while for some prescribed amount of time, and then open the switch decoupling the oscillator from the qubit or a pair of qubits, is that a superposition of the state zero and one of the zero pi qubit becomes a superposition with a modified phase. And the phase change, if I keep the switch closed for the right amount of time, will be pi over two. So we've done a rotation about the z axis if I'm considering coupling a single qubit to the oscillator by pi over two. And the final state of the oscillator is essentially independent of whether the state of the qubit was zero or one. So there's no entanglement between the oscillator and the zero pi qubit. And the phase, e to the minus i pi over two, is extremely stable with respect to deformations in the pulse and fluctuations in the Hamiltonian. So for example, if this inductance parameter is chosen to be 80, then we get the ideal gate to an accuracy in the diamond norm, which is about ten to the minus eight, even if the error is in the pulse that implements the gate are at about the one percent level. So now we'd like to understand how this can be. Why is the gate so robust? And the answer really is found by thinking about a quantum code, a code in which qubit is embedded in the Hilbert space of a harmonic oscillator. We can think of it as a stabilizer code, as a simultaneous eigenstate with eigenvalue one of two poly operators, where the poly operators in this case are e to the two i phi, which translates in q space by two, and e to the minus two pi i q, which translates in phi space by two pi. Phi and q don't commute with one another, but these phase space translations do commute, and so they can be simultaneously diagonalized. If I choose both these operators to be one, there's a two-dimensional code space, and I can choose for, as a basis for the space, the code states zero and one, where zero is a superposition of phi eigenstates, where the value of phi is pi times an even integer, and one is a superposition of phi eigenstates, where the value of phi times an odd integer. So the bit flip operation that changes a zero to a one and a one to a zero is a translation in phi space by pi, that's the logical x acting on the code. And then I can also Fourier transform these code words and ask what they look like in q space, and I can choose, as a basis, the eigenstates of x with eigenvalue plus or minus, where the plus eigenstate is a superposition of q eigenstates, where the value of q is an even integer, and the minus eigenstate is a superposition of q eigenstates, where the value of q is an odd integer, and the logical z operator that flips the plus to the minus is just a translation in q space by one. Now, this code is protected against errors which can be described as distributions on small shifts in phi space and q space. So if there's a shift in phi space which is less than pi over two, less than half the translation that we would do to perform the logical x that flips the bit, then that's a correctable error, and the distinguishability of zero and one is not lost. And if we have a shift in q which is less than one half, less than half of the shift that flips the plus and minus eigenstates, that's also a correctable error which, in principle, can be reversed. Now, these code words that I've described, these ideal code words are actually non-normalizable and unphysical, but I can consider instead a approximate code word which is normalizable and physical. We could think of it, for example, as a sequence of equally spaced Gaussian peaks in phi space governed by a broad Gaussian envelope where the width of the individual peaks, delta, is small compared to two pi, and the width of the envelope, kappa inverse, is broad compared. And then if I foray transform this approximate code word to see what it looks like in q space, it has a similar Gaussian grid appearance. That is, now there will be a broad envelope whose width is delta inverse and periodically spaced on narrow functions in q space whose width is given by kappa. So there's some intrinsic error in the approximate code words because there can be a little bit of leakage of the individual peaks outside the size of a correctable error. But if, for example, each one of the peaks has a width which is less by a factor of five than the size of a shift which gives an uncorrectable error, then we can make the intrinsic error of the code words less than one part in a million. It's not really necessary for the peak functions or the envelope function to be Gaussian. I can consider any narrow function periodically repeated governed by some slowly varying broad envelope function in that tube approximate code word. Okay, so now let's come back to how the protected gate is done. I didn't say so before, but when I close the switch and couple the qubit to the LC circuit, that will be done by a tunable Josephson coupling. So I'll turn on J of t introducing into the Hamiltonian of the oscillator a cosine potential where the cosine depends on the difference between the phase drop across the inductor and the LC circuit and the phase drop theta across the qubit which is either zero or pi depending on the state of the zero pi qubit. So if the zero pi qubit is in the state zero, that means theta is zero and this is just a cosine potential and if J 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 pi which are two pi times an integer, but if theta were equal to pi then the cosine would be shifted by pi and the minima would occur at values of pi which are pi times an odd integer. And now if we turn on J of t obeying suitable adiabaticity conditions not too slowly and not too quickly, if we started out in the ground state of the harmonic oscillator, the broad Gaussian with large five fluctuations because the inductance of the LC circuit is large, it would evolve to one of these Gaussian grid states and approximate code word of the continuous variable code and we would have either the encoded zero or the encoded one according to whether the minima of the cosine occurred at even or odd values of pi. So the zero one of the zero pi qubit would become imprinted on the code words for this oscillator code. We'll want to turn on J of t on a time scale which is short compared to the periodic 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 have J large enough. Okay, so now once we've prepared these code words we keep J fixed for a while and while we do so the quadratic term and the potential induces a Gaussian operation on our code words. An operation which if we choose the elapsed time that the switch is closed at the oscillator and qubit are coupled appropriately can be expressed as e to the minus i phi squared over two pi. So that means that if we're in the code state zero for which phi is an even multiple of pi this is a trivial phase acting on the state but if we're in the code state one in which phi is an odd multiple of pi there's a non-trivial phase which is e to the minus i pi over two performing a rotation by pi over two about the z axis in the code space. So what happens once we've prepared the code word and we keep the switch closed for a while is that the state of the oscillator makes a big excursion in a phase space. It leaves the code space but it eventually comes back to the code space but it comes back with a twist with a non-trivial halonomy or berry phase which is just this encoded gate. So this is a type of geometric phase gate. The oscillator goes on a trip and comes back with some state dependent phase but it's an especially robust type of geometric phase gate. And if we have two qubits connected in parallel then whether we get the non-trivial phase or not depends on whether the on the value of the total phase drop across a pair of zero pi qubits and that means we've done this entangling phase gate. Now of course the pulse might not be perfect we might keep the switch closed a little bit too long or open it too soon for example and that means apart from this ideal Gaussian operation that we would like to do to perform the encoded gate there will be some additional Gaussian error with a rotation by an amount epsilon which is the fractional error in the timing of the pulse. But if epsilon is small that additional Gaussian operation just causes some modest spreading in Q space it's a correctible error for the continuous variable code. So we do the ideal Clifford group single qubit rotation together with some correctible error. And since everything is Gaussian in this case where we started out in the oscillator ground state we can calculate everything explicitly. If the timing of the pulse is chosen optimally then there's some intrinsic error in the gate which comes from the nonzero width of the individual peaks and queues but that gets exponentially small when the inductance of the circuit is very large. And then we can compute how the error is enhanced when there's a nonzero epsilon due to an imperfect rotation of the code word. But that just induces some multiplicative factor in the error which is close to one if 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 correctible error when we open the switch again when we decouple the oscillator from the qubit. Well the execution of the gate is really a three step process. In the first step we close the switch we turn on the coupling between the oscillator and the qubit. Then we keep it closed for a while and the state goes on this excursion eventually coming back to the code space. So I'll say that the state is psi begin at the begin excursion and psi end at the end. And then finally we open the switch decoupling the oscillator and the qubit and we get some final state of the oscillator. So let's look at these steps one by one. So first of all while the switch is closing we're turning on the coupling and because we're considering the turning on to be fast compared to the period of the oscillator the quadratic term and the potential the phi squared term and the potential has a relatively small effects while the switch is closing. So in a first approximation let's ignore that and then the Hamiltonian just has the cosine potential and the kinetic energy term. But the Hamiltonian has a cosine potential whose coefficient has a sign that depends on whether the qubit is zero or one. If the qubit is one that means the argument of the cosine gets shifted by pi and that changes the sign of the cosine. Cosine phi term acting in Q space is a translation by either plus one or minus one. And what that means is that the cosine anti commutes with the logical x error acting on the code space. The operator which has the value plus one when Q is close to an even integer and the value minus one when Q is close to an odd integer. And so that means that the Hamiltonian in the two cases depending on the state of the qubit are related by conjugation by x and the same is true for the evolution operators that we obtained by integrating the Schrodinger 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 psi begin in the case where the qubit is in the state one is related to psi begin when the qubit is in the state zero by the, well in this way that psi one begin is x u zero x acting on psi initial but psi initial the initial state of the qubit. But now remember we started out with very large phase fluctuations in phi space or a very narrow state in Q space for the harmonic oscillator and that means it's very nearly an eigenstate of x if the state has its support in the interval between minus one half and one half in Q space it's an x equals one state. So it's a very good approximation to say that x acting on the initial state is one and that means that the beginning state if the qubit is one and the beginning state if the qubit is zero are just related by the action of the logical operator x. And now we let the state rotate goes on an excursion eventually it comes back to the code space and this relationship between the state when the qubit is one and one at zero will actually be protected if we don't have a logical phase error. In other words I can take the linear combination of psi one begin and psi zero begin with a plus sign and that's an x equals one eigenstate and the linear combination with a minus sign is an x equals minus one eigenstate. And because a 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 a little under rotation. And then finally we have to decouple the oscillator and the qubit by opening the switch and we can argue as before that the operator that acts during the opening of the switch in the case where the qubit is zero and the case where the qubit is one they're related by conjugation by x and that allows us to say that the final state of the other one is finally decoupled from the qubit is just given by the logical x acting on the final state of the oscillator when the qubit is zero. X acting on the state when the qubit is zero is the state of the oscillator when the qubit is one. So finally one more step if we've done everything nicely adiabatically the oscillator won't get highly excited it started out very narrow in Q space and it'll wind up being very narrow in Q space at the end and that means x acting on the final state of the oscillator will be essentially one. 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 zero or one. So there's no entanglement between the oscillator and the qubit and furthermore the phase that's acquired is extremely insensitive to the way we implement the gate. And these conclusions still hold if we include the phi squared over 2L term in the potential bring the closing and opening of the switch since that just produces some modest additional spreading in Q space and because the errors tend to slip and because the states are well protected against translations in Q space the conclusion isn't changed very much. So sorry that was a lot of steps so let me summarize them on one slide. These are the ingredients that make the protected gate work. First of all there's some symmetry principle which relates the Hamiltonian while the switch is opening or closing in the case where the zero pi qubit is in the state zero to the state of the Hamiltonian when the zero pi qubit is in the state one. That's because the difference is just a shift in phi space by pi. The state initially has large phase fluctuations that is because of the large inductance phi is very broad and correspondingly Q is very narrow. And it's important that we prepare good code work this continuous variable code and that's what protects the symmetry that relates the state when the qubit is zero and the state when the qubit is one. When we perform the gate even if we don't perform it perfectly there's an adiabaticity principle that we wind up with a final state of the oscillator which is not very excited so it's still narrow in 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 of the oscillator so that the effect of the harmonic term and the potential is not very big while the switch is opening and closing. So the conclusion is that the oscillator really acts like the ancilla on which the syndrome information gets imprinted. Any noise that's introduced by the imperfect implementation of the gate winds up messing 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 gate. So trying to estimate the non-adiabatic effects which contribute to the error probability is hard to do analytically. We can predict how they scale using Landau-Zener theory but we can't do very precise calculations. We can simulate the process in which we close the switch, leave it closed for a while and then open it. So here I've plotted the diamond norm deviation from the ideal phase gate as a function of this angle epsilon which tells me the fractional error in the timing of the pulse and if we didn't have any non-adiabatic effects the gate would get down to an error below 10 to the minus 9 when the error is very small the static transitions are non-negligible so it becomes something a little bit above 10 to the minus 8 but it stays pretty stable so that the error is about the same if we introduce errors of the order of 1% in the implementation of the pulse. Actually that was for the case where we start out in the 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 it's a low lying excitation of the oscillator because that will still be narrow in Q space which is the key thing that we need initially. And so if the temperature is low compared to the frequency of the oscillator so that we have a thermal state to begin with the error in the gate won't be much affected by the temperature. And we can also check that the implementation of the gate is robust with respect to fluctuations in the Hamiltonian which introduce some anharmonicity in the oscillation or some higher harmonics in the geosysm energy. Now the remarkable accuracy of the gate really hinged on this assumption that we have a very large inductance and one can ask whether that's reasonable. I guess I forgot to say, I should have when I showed you the plot, this was for particular values of the parameters. The square root of L over C, this dimensionless inductance parameter was chosen to be 80 and the square root of JC being 8 means it's the reciprocal of this number which determines the width of the individual peaks in phi space and we chose the optimal time scale for the switch to open and close in order to get this performance. So we need that large 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 the superconducting circuit. The kind of wants on geometrical grounds to be of order one and we want it to be quite large. One way in which you can get a large inductance in principle as I mentioned earlier is to chain together many jossus injunctions and actually that was done with a different motivation by the Yale group and they observed a value of this parameter which was about 6 for a chain of 43 jossus injunctions and in principle you could scale that up to a much longer chain to get a much larger value of this impedance but there may be reasons why that's not so practical. Maybe a more promising approach is to use materials which have a large intrinsic kinetic inductance to build the superconducting circuits. Of course what I've described is just some Clifford phase gates, single qubit and two qubit gates which are not enough for universality by themselves but we can boost to a universal set of gates by using state distillation ideas, a la Bravye and Katah. So that if we in addition can perform measurements of our qubit in the X basis that are accurate and pi over 4 rotation about the Z axis which is not protected and has only a so-so fidelity then we can use state distillation to get accurate universal gates. It may be that the measurements are noisy in this type of scheme, I'll come to that in a second. If we can do something about controlling the noise in the measurements then a two qubit phase gate really is a useful tool for fault tolerance if we can do it with extremely high fidelity even though it's a non-universal Clifford group gate. Well as far as the noise in the measurements is concerned if we can do the measurements non-destructively that is with the small probability of flipping the value of the measured observable when we do the measurement then we can imagine repeating the measurements many times to make them more reliable. Alternatively we can use repetition coding to make the measurements more reliable. I won't go into the details but what is shown here is a way of teleporting a C-naught gate encoded using a repetition code where the two qubit gates shown are two qubit phase gates of the type that we know how to do reliably and the arrows pointing to the right are X eigenstate preparations and the arrows pointing to the left are X basis measurements and in this case we don't have to repeat the measurements we can decode the results in the repetition code by doing majority voting every time we make a measurement and if for example we can do a C phase gate which has accuracy or error rate per gate of 10 to the minus 5 then even if the measurements have a 1% error rate we could do this teleported C-naught gate to 10 to the minus 6 accuracy. So MRIs of course we expect to have to use fault tolerance ideas to operate large scale quantum computers and if we can manage to do a protected two qubit two qubit clifford phase gate with very high fidelity that could be a powerful tool for fault tolerance even if measurements and other gates are noisy. In the case of these zero pi protected qubit superconducting circuits if we can turn on and off a tunable coupling to an oscillator a tunable josephson junction then we can do single qubit and two qubit clifford gates with very high fidelity fidelity which gets exponentially close to one if we can choose the system parameters appropriately. What makes the gate work is an underlying continuous variable quantum code which protects against phase errors but the zero pi qubit itself and the protected gate requires building circuits which have a very large inductance in natural units and that might not be easy to achieve. The gate itself is robust against non-zero temperature and fluctuations in the Hamiltonian of the system so I don't know if 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 the quantum information and here I've suggested one possible step in that direction. Thanks for listening. Questions? Well done. Mike in the back. Thanks. That was a really nice talk and it's extremely interesting in particular to me because there's some similarity at least in feel to the idea of dynamical modulation which I know a little better than this I guess but what I was wondering about at the end of your talk where you showed some of the downsides of trying to achieve this high L on C ratio is the fact that in trapped ions as you know we do these geometric phase gates by exciting and de-exciting a harmonic oscillator and initially there's no intrinsic protection against timing errors. You really have to make a closed loop in phase space otherwise there's a very significant gate infidelity to first order but if you modulate the coupling to that emotional mode in a way that's reminiscent of dynamical decoupling you can actually build in robustness against these timing errors. So I was curious if you think there's any possibility that you can reduce the requirement for L on C being 80 to something closer because you're only off by a factor of 10 or so from what you said Devereux had achieved if you can add in some other concepts of dynamic modulation. Okay so the question was is it possible that we could combine together having the very large square root of L over C but not quite as large as I seem to require with some modulation in time of the coupling of the oscillator to the qubit since it's known in other contexts that geometric phase gates can be made more robust by such a modulation. I'm repeating the question because I'm stalling I don't really have an answer and also it's a good question but I haven't really thought about that. Another question. Hold on. How do you see this with the error such as state preparation? The state preparation for the oscillator? Right well I just assumed it was a thermal state at 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 if the so that's strictly zero temperature if the temperature is nonzero but small compared to the frequency so it might be in an excited state but with some suppressed probability then it still works. After we do the gate the oscillator doesn't return to the initial thermal state or ground state in fact it 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 I'm going to want to cool the oscillator back down in order to reuse it in a high fidelity gate. There's a question way in the back. Yeah time for one more question where is it over there. When you described achieving better protection with increasing the number with increasing the chain of the Josephson junction possible variations in the between the individual junctions in the chain would they affect anything. I'm sorry you're very I didn't hear the last part of the question the question is about getting high inductance from a long chain of Josephson junctions and the rest of it. Yes so small variations they wouldn't affect anything in this game. A small variation along the chain and the device parameters. So in other words if I added some disorder to the chain how would it affect the performance. Well I think it would probably be fairly tolerant of disorder because the key thing is just to have the inductance of the whole composite device large compared to the I think one 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 capacitance from you know scaling up with the size of the device as well as the inductance but I don't think disorder is a fundamental problem. Yeah let's thank John again. Okay our next speaker is Lorenzo.
 Okay, why don't we start here? I think we've got the AV situation mostly under control  here. So for our next session, we'll be starting, I guess, five to six minutes late, so we'll  just sort of keep that offset in our schedule. Our next session will be kicked off by John  Preskel who will be talking to us about superconducting Q-bits.  Thanks, Andrew. Everybody can hear okay? I'm talking about work done at Caltech with Peter  Brooks and Alexei Kutayev. All of us here recognize that error correction and fault  tolerance are going to be essential if we're going to operate large-scale quantum computers  someday. The standard approach to fault tolerance uses clever circuit design to overcome the  deficiencies of hardware. There's an alternative hardware or 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 it's 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. One suggestion has been that we  can make robust encodings of quantum information using superconducting circuits, in particular  the zero pi Q-bit which was suggested by Kutayev and others. The phase in a superconductor  is a periodic variable with period two pi. But for a suitably designed circuit, the dependence  of that circuit's energy on the phase difference between its two leads is a very good approximation  of periodic function of theta with period pi aside from corrections that become very  small as the device gets large. And in that case, the energy as a function of theta has  two minima which can occur at theta equals zero and theta equals pi which have very nearly  the same energy. And if there's a large barrier separating those minima, those two minima  can be the basis states of a robust Q-bit. It's protected against bit flips by the high  barrier in between zero and one. And it's protected against defacing because the degeneracy  of the two states is very stable with respect to perturbations of the Hamiltonian. But of  course if we want to do robust quantum computing, we need more than just stable Q-bits. We also  need to be able to do very accurate gates. And that's what I want to describe in the  talk, how in particular we can do highly accurate Clifford gates acting on zero pi Q-bits by  turning on and then turning off the coupling of the Q-bit to our harmonic oscillator and  LC circuit. And the gate will work very effectively, be highly accurate, if the inductance in the  LC circuit in natural units is a very large number. The secret that underlies the reliability  of the gate is actually a quantum code, a continuous variable quantum code in which  Q-bit, a two-dimensional system, is embedded in the infinite dimensional Hilbert space  of a harmonic oscillator. And I'll be explaining how that works. Now to understand the physics  of the zero pi Q-bit itself and also the protected gates, we should know a little bit about Josephson  junctions. A Josephson junction is a device in which a layer of insulating material separates  two pieces of superconductors such that there's a non-zero amplitude for the charge carriers  in the stirred Cooper pairs to tunnel from one side of the barrier to the other side. And  we can encode that tunneling in an effective Hamiltonian. If we think of a lead attached  via Josephson junction to an island that carries some charge, a superconducting island, then  the effective Hamiltonian describes processes in which the charge on the island can either  increase by one or decrease by one when a Cooper pair tunnels through. If Q is the charge  in units of 2e, the Cooper pair charge, and there's some amplitude for the tunneling which  I've called J, the Josephson coupling. It's often convenient to describe the system using  conjugate variables, the superconducting phase rather than the charge, which is just the  Fourier conjugate of the charge. So the phase and the charge obey commutation relations  like the position and momentum of a particle in one dimension. And if we express the tunneling  Hamiltonian in terms of the phase, proportional to cosine phi with the coefficient given by  the tunneling amplitude J. So one other thing we should know about superconductors is that  flux is quantized. The phase in a superconductor is actually a convention dependent concept.  If I want to compare the phase at two different points in the superconductor, I need some  notion of parallel transport that tells me how the phase should change as I move from  one point to another. That notion of parallel transport is provided by a connection, which  is just the electromagnetic vector potential. And the parallel transport of phase is actually  path dependent. That means that connection has curvature. The curvature is just the  magnetic field. So when there's a non-trivial magnetic field, the phase picked up under  parallel transport can depend upon the path. If we have a closed ring of superconductor  and there's some applied flux, it might be that the effect of parallel transport of the  phase around the loop is non-trivial. And that's energetically unfavorable. So what  happened Ace is that some persistent current flows in the ring, which adds to the applied  field, some additional field 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, H over 2e. If I think  of a ring with a Josephson junction inserted inside, I can think of the phase difference  between one side of the Josephson junction and the other as not a periodic variable with  period 2 pi, but a real variable. Because if I winds by 2 pi, that just means we've inserted  one quantum of flux through the ring. Now I'm also going to want to consider LC circuits  in superconductors. This is really just like a particle in a harmonic well where the charge  on the capacitor corresponds to the momentum of the particle and the superconducting phase  is like the addition of a particle. I can think of that as 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 Cooper pair charge  2e and to express the flux linking the circuit in terms of the superconducting phase drop  across the inductor. And then in terms of those units, the square root of L over C is  a dimensionless number. It's just the conventional impedance of the circuit expressed in terms  of these units, which is about one kilo ohm. So we can ask, what's the significance of  that dimensionless number being large? And the answer is that in the ground state of  this oscillator, the circuit has very large phase fluctuations. The ground state is a  Gaussian wave function where the distribution in phi is wide compared to 2 pi with that  width determined by the square root of L over C. Now we can imagine in certain junction  into a circuit which has a large inductance in this sense, and then its energy of a proportional  to this difference of phases. But because of the large fluctuations in phi, the cosine  gets averaged over many cycles of the cosine, and that makes the contribution to the energy  that depends on the phase theta very small because of the averaging exponentially small  in the width in five space, which is determined by the large inductance. So now we know enough  to understand the idea of the protected zero pi qubit. This is the form in which Katayev  suggested it. We can think of a superconducting circuit with two rungs. So on each rung there's  a large inductance, well bang, this condition, and the two rungs are coupled together by  some very large capacitance. So I can consider the variable which I can consider the phase  in the superconductor on either side of the capacitor, and then the sum of the phases  phi 1 and phi 2, plus, is unaffected by the large capacitance, and so it's a light variable  because of the large inductance which has very large fluctuations. But the difference  variable phi 1 minus phi 2 feels the large capacitance, and that makes it a very heavy  variable which wants to be well localized in phase. And so it locks to a value which  is determined by the phases on the external leads to the difference between theta 4 and  theta 1, and minus the difference between theta 3 and theta 2. And now we can imagine  taking the top rung and twisting it by 180 degrees and connecting the leads together so  that we're identifying theta 2 with theta 4 and theta 3 with theta 1. And then the energy  of the circuit becomes a function of the value that the heavy variable locks to which is  now 2 theta 2 minus theta 1. And then there's some correction, but it's very small because  of the large variations of the light variable. And that's how we can get a zero pi qubit in  which the potential is essentially a periodic function with period pi, which has two minima  which are very nearly degenerate. In fact, we can think of the protection as arising  from a very non-local encoding of information like in other contexts where we have topological  protection. If we imagine, as Katayev suggested, that we realize the large inductance by a  long chain of joseph's injunctions, and then the phase difference between one 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, those states locally look almost identical. And  the breaking of the degeneracy is associated with tunneling phenomena which run all the  way from one end of the chain to the other, which have an amplitude which is exponentially  small. And that's the reason why the corrections, degeneracy, get very small when the device  scales up. We'll also want to be able to measure the zero pi qubit. If we measure it in the  standard or computational basis, we need to distinguish a phase drop across the qubit  of zero from a phase drop of pi. And we could do that by coupling the qubit to the joseph's  injunction and inserting a 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  two sides of the qubit is zero or pi. We'll also want to be able to measure in the dual  basis, the x basis, dual to this computational basis. And one way we can imagine doing that  is by breaking the connection that identifies theta one and theta three, and then measuring  the charge variable, which is conjugate to the difference between theta one and theta  three. Because if we look at how the energy depends, if I slowly vary theta one with theta  three fixed, then to stay at the minimum of the energy, if theta one rotates by two pi,  theta two would rotate by pi. And then the state would either be invariant if x equals  one, if it's a superposition of the zero and one in the computational basis, or it will  change sign if x is equal to minus one. And I can distinguish those possibilities by making  a charge measurement, distinguishing between a charge which is an integer or, well, an  odd or an even integer multiple of the electron charge, that is half the Cooper pair charge.  Now, 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.  Okay, so now I can start to explain how one does the protected gate. We've seen that a  large inductance is important for the operation of the zero pi qubit, but from now on, I'm  going to talk about the internal structure of the qubit. We'll just assume it's a very  good qubit. But a large inductance will enter the discussion a second time for a rather  different reason than before. When I perform the protected gate, either a single qubit,  a Clifford group gate, a rotation about the z axis by pi over two, or a two qubit entangling  gate, I will couple either one qubit or a pair of qubits connected in series to an oscillator,  another LC circuit with a large inductance. And what one finds if one simulates the process  in which we close a switch to couple the qubit to the LC circuit, leave the switch closed  for a while for some prescribed amount of time, and then open the switch decoupling  the oscillator from the qubit or a pair of qubits, is that a superposition of the state  zero and one of the zero pi qubit becomes a superposition with a modified phase. And  the phase change, if I keep the switch closed for the right amount of time, will be pi over  two. So we've done a rotation about the z axis if I'm considering coupling a single  qubit to the oscillator by pi over two. And the final state of the oscillator is essentially  independent of whether the state of the qubit was zero or one. So there's no entanglement  between the oscillator and the zero pi qubit. And the phase, e to the minus i pi over two,  is extremely stable with respect to deformations in the pulse and fluctuations in the Hamiltonian.  So for example, if this inductance parameter is chosen to be 80, then we get the ideal  gate to an accuracy in the diamond norm, which is about ten to the minus eight, even if the  error is in the pulse that implements the gate are at about the one percent level. So now  we'd like to understand how this can be. Why is the gate so robust? And the answer really  is found by thinking about a quantum code, a code in which qubit is embedded in the Hilbert  space of a harmonic oscillator. We can think of it as a stabilizer code, as a simultaneous  eigenstate with eigenvalue one of two poly operators, where the poly operators in this  case are e to the two i phi, which translates in q space by two, and e to the minus two  pi i q, which translates in phi space by two pi. Phi and q don't commute with one another,  but these phase space translations do commute, and so they can be simultaneously diagonalized.  If I choose both these operators to be one, there's a two-dimensional code space, and  I can choose for, as a basis for the space, the code states zero and one, where zero is  a superposition of phi eigenstates, where the value of phi is pi times an even integer,  and one is a superposition of phi eigenstates, where the value of phi times an odd integer.  So the bit flip operation that changes a zero to a one and a one to a zero is a translation  in phi space by pi, that's the logical x acting on the code. And then I can also Fourier transform  these code words and ask what they look like in q space, and I can choose, as a basis,  the eigenstates of x with eigenvalue plus or minus, where the plus eigenstate is a superposition  of q eigenstates, where the value of q is an even integer, and the minus eigenstate  is a superposition of q eigenstates, where the value of q is an odd integer, and the  logical z operator that flips the plus to the minus is just a translation in q space  by one. Now, this code is protected against errors which can be described as distributions  on small shifts in phi space and q space. So if there's a shift in phi space which is  less than pi over two, less than half the translation that we would do to perform the  logical x that flips the bit, then that's a correctable error, and the distinguishability  of zero and one is not lost. And if we have a shift in q which is less than one half,  less than half of the shift that flips the plus and minus eigenstates, that's also a  correctable error which, in principle, can be reversed. Now, these code words that I've  described, these ideal code words are actually non-normalizable and unphysical, but I can  consider instead a approximate code word which is normalizable and physical. We could think  of it, for example, as a sequence of equally spaced Gaussian peaks in phi space governed  by a broad Gaussian envelope where the width of the individual peaks, delta, is small compared  to two pi, and the width of the envelope, kappa inverse, is broad compared. And then  if I foray transform this approximate code word to see what it looks like in q space,  it has a similar Gaussian grid appearance. That is, now there will be a broad envelope  whose width is delta inverse and periodically spaced on narrow functions in q space whose  width is given by kappa. So there's some intrinsic error in the approximate code words because  there can be a little bit of leakage of the individual peaks outside the size of a correctable  error. But if, for example, each one of the peaks has a width which is less by a factor  of five than the size of a shift which gives an uncorrectable error, then we can make the  intrinsic error of the code words less than one part in a million. It's not really necessary  for the peak functions or the envelope function to be Gaussian. I can consider any narrow function  periodically repeated governed by some slowly varying broad envelope function in that tube  approximate code word. Okay, so now let's come back to how the protected gate is done.  I didn't say so before, but when I close the switch and couple the qubit to the LC circuit,  that will be done by a tunable Josephson coupling. So I'll turn on J of t introducing  into the Hamiltonian of the oscillator a cosine potential where the cosine depends on the  difference between the phase drop across the inductor and the LC circuit and the phase  drop theta across the qubit which is either zero or pi depending on the state of the zero  pi qubit. So if the zero pi qubit is in the state zero, that means theta is zero and this  is just a cosine potential and if J 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 pi which are two pi times an integer, but if theta were equal to pi then the cosine  would be shifted by pi and the minima would occur at values of pi which are pi times an  odd integer. And now if we turn on J of t obeying suitable adiabaticity conditions not  too slowly and not too quickly, if we started out in the ground state of the harmonic oscillator,  the broad Gaussian with large five fluctuations because the inductance of the LC circuit is  large, it would evolve to one of these Gaussian grid states and approximate code word of the  continuous variable code and we would have either the encoded zero or the encoded one  according to whether the minima of the cosine occurred at even or odd values of pi. So the  zero one of the zero pi qubit would become imprinted on the code words for this oscillator  code. We'll want to turn on J of t on a time scale which is short compared to the periodic  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 have J large enough. Okay, so now  once we've prepared these code words we keep J fixed for a while and while we do so the  quadratic term and the potential induces a Gaussian operation on our code words. An operation  which if we choose the elapsed time that the switch is closed at the oscillator and qubit  are coupled appropriately can be expressed as e to the minus i phi squared over two pi.  So that means that if we're in the code state zero for which phi is an even multiple of  pi this is a trivial phase acting on the state but if we're in the code state one in which  phi is an odd multiple of pi there's a non-trivial phase which is e to the minus i pi over two  performing a rotation by pi over two about the z axis in the code space. So what happens  once we've prepared the code word and we keep the switch closed for a while is that the  state of the oscillator makes a big excursion in a phase space. It leaves the code space  but it eventually comes back to the code space but it comes back with a twist with a non-trivial  halonomy or berry phase which is just this encoded gate. So this is a type of geometric  phase gate. The oscillator goes on a trip and comes back with some state dependent phase  but it's an especially robust type of geometric phase gate. And if we have two qubits connected  in parallel then whether we get the non-trivial phase or not depends on whether the on the  value of the total phase drop across a pair of zero pi qubits and that means we've done  this entangling phase gate. Now of course the pulse might not be perfect we might keep  the switch closed a little bit too long or open it too soon for example and that means  apart from this ideal Gaussian operation that we would like to do to perform the encoded  gate there will be some additional Gaussian error with a rotation by an amount epsilon  which is the fractional error in the timing of the pulse. But if epsilon is small that  additional Gaussian operation just causes some modest spreading in Q space it's a correctible  error for the continuous variable code. So we do the ideal Clifford group single qubit  rotation together with some correctible error. And since everything is Gaussian in this case  where we started out in the oscillator ground state we can calculate everything explicitly.  If the timing of the pulse is chosen optimally then there's some intrinsic error in the gate  which comes from the nonzero width of the individual peaks and queues but that gets exponentially  small when the inductance of the circuit is very large. And then we can compute how the  error is enhanced when there's a nonzero epsilon due to an imperfect rotation of the code word.  But that just induces some multiplicative factor in the error which is close to one if 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 correctible error when we open the switch again when we decouple  the oscillator from the qubit. Well the execution of the gate is really a three step process.  In the first step we close the switch we turn on the coupling between the oscillator and  the qubit. Then we keep it closed for a while and the state goes on this excursion eventually  coming back to the code space. So I'll say that the state is psi begin at the begin excursion  and psi end at the end. And then finally we open the switch decoupling the oscillator  and the qubit and we get some final state of the oscillator. So let's look at these  steps one by one. So first of all while the switch is closing we're turning on the coupling  and because we're considering the turning on to be fast compared to the period of the  oscillator the quadratic term and the potential the phi squared term and the potential has  a relatively small effects while the switch is closing. So in a first approximation let's  ignore that and then the Hamiltonian just has the cosine potential and the kinetic energy  term. But the Hamiltonian has a cosine potential whose coefficient has a sign that depends  on whether the qubit is zero or one. If the qubit is one that means the argument of the  cosine gets shifted by pi and that changes the sign of the cosine. Cosine phi term acting  in Q space is a translation by either plus one or minus one. And what that means is that  the cosine anti commutes with the logical x error acting on the code space. The operator  which has the value plus one when Q is close to an even integer and the value minus one  when Q is close to an odd integer. And so that means that the Hamiltonian in the two cases  depending on the state of the qubit are related by conjugation by x and the same is true for  the evolution operators that we obtained by integrating the Schrodinger 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 psi begin in the case where the qubit is in the state  one is related to psi begin when the qubit is in the state zero by the, well in this way  that psi one begin is x u zero x acting on psi initial but psi initial the initial state  of the qubit. But now remember we started out with very large phase fluctuations in phi  space or a very narrow state in Q space for the harmonic oscillator and that means it's  very nearly an eigenstate of x if the state has its support in the interval between minus  one half and one half in Q space it's an x equals one state. So it's a very good approximation  to say that x acting on the initial state is one and that means that the beginning state  if the qubit is one and the beginning state if the qubit is zero are just related by the  action of the logical operator x. And now we let the state rotate goes on an excursion  eventually it comes back to the code space and this relationship between the state when  the qubit is one and one at zero will actually be protected if we don't have a logical phase  error. In other words I can take the linear combination of psi one begin and psi zero  begin with a plus sign and that's an x equals one eigenstate and the linear combination  with a minus sign is an x equals minus one eigenstate. And because a 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 a little under rotation. And then  finally we have to decouple the oscillator and the qubit by opening the switch and we  can argue as before that the operator that acts during the opening of the switch in the  case where the qubit is zero and the case where the qubit is one they're related by conjugation  by x and that allows us to say that the final state of the other one is finally decoupled  from the qubit is just given by the logical x acting on the final state of the oscillator  when the qubit is zero. X acting on the state when the qubit is zero is the state of the  oscillator when the qubit is one. So finally one more step if we've done everything nicely  adiabatically the oscillator won't get highly excited it started out very narrow in Q space  and it'll wind up being very narrow in Q space at the end and that means x acting on the  final state of the oscillator will be essentially one. 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 zero or one. So there's no entanglement between  the oscillator and the qubit and furthermore the phase that's acquired is extremely insensitive  to the way we implement the gate. And these conclusions still hold if we include the phi  squared over 2L term in the potential bring the closing and opening of the switch since  that just produces some modest additional spreading in Q space and because the errors  tend to slip and because the states are well protected against translations in Q space the  conclusion isn't changed very much. So sorry that was a lot of steps so let me summarize  them on one slide. These are the ingredients that make the protected gate work. First of  all there's some symmetry principle which relates the Hamiltonian while the switch is  opening or closing in the case where the zero pi qubit is in the state zero to the state  of the Hamiltonian when the zero pi qubit is in the state one. That's because the difference  is just a shift in phi space by pi. The state initially has large phase fluctuations that  is because of the large inductance phi is very broad and correspondingly Q is very narrow.  And it's important that we prepare good code work this continuous variable code and that's  what protects the symmetry that relates the state when the qubit is zero and the state  when the qubit is one. When we perform the gate even if we don't perform it perfectly  there's an adiabaticity principle that we wind up with a final state of the oscillator  which is not very excited so it's still narrow in 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 of the oscillator so that the effect of the harmonic  term and the potential is not very big while the switch is opening and closing. So the  conclusion is that the oscillator really acts like the ancilla on which the syndrome information  gets imprinted. Any noise that's introduced by the imperfect implementation of the gate  winds up messing 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 gate. So trying to estimate the non-adiabatic effects  which contribute to the error probability is hard to do analytically. We can predict  how they scale using Landau-Zener theory but we can't do very precise calculations. We  can simulate the process in which we close the switch, leave it closed for a while and  then open it. So here I've plotted the diamond norm deviation from the ideal phase gate as  a function of this angle epsilon which tells me the fractional error in the timing of the  pulse and if we didn't have any non-adiabatic effects the gate would get down to an error  below 10 to the minus 9 when the error is very small the static transitions are non-negligible  so it becomes something a little bit above 10 to the minus 8 but it stays pretty stable  so that the error is about the same if we introduce errors of the order of 1% in the  implementation of the pulse. Actually that was for the case where we start out in the  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 it's a low lying excitation of  the oscillator because that will still be narrow in Q space which is the key thing that  we need initially. And so if the temperature is low compared to the frequency of the oscillator  so that we have a thermal state to begin with the error in the gate won't be much affected  by the temperature. And we can also check that the implementation of the gate is robust  with respect to fluctuations in the Hamiltonian which introduce some anharmonicity in the  oscillation or some higher harmonics in the geosysm energy. Now the remarkable accuracy  of the gate really hinged on this assumption that we have a very large inductance and one  can ask whether that's reasonable. I guess I forgot to say, I should have when I showed  you the plot, this was for particular values of the parameters. The square root of L over  C, this dimensionless inductance parameter was chosen to be 80 and the square root of  JC being 8 means it's the reciprocal of this number which determines the width of the individual  peaks in phi space and we chose the optimal time scale for the switch to open and close  in order to get this performance. So we need that large 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 the superconducting  circuit. The kind of wants on geometrical grounds to be of order one and we want it  to be quite large. One way in which you can get a large inductance in principle as I  mentioned earlier is to chain together many jossus injunctions and actually that was done  with a different motivation by the Yale group and they observed a value of this parameter  which was about 6 for a chain of 43 jossus injunctions and in principle you could scale  that up to a much longer chain to get a much larger value of this impedance but there may  be reasons why that's not so practical. Maybe a more promising approach is to use materials  which have a large intrinsic kinetic inductance to build the superconducting circuits. Of  course what I've described is just some Clifford phase gates, single qubit and two qubit gates  which are not enough for universality by themselves but we can boost to a universal set of gates  by using state distillation ideas, a la Bravye and Katah. So that if we in addition can perform  measurements of our qubit in the X basis that are accurate and pi over 4 rotation about  the Z axis which is not protected and has only a so-so fidelity then we can use state  distillation to get accurate universal gates. It may be that the measurements are noisy  in this type of scheme, I'll come to that in a second. If we can do something about  controlling the noise in the measurements then a two qubit phase gate really is a useful  tool for fault tolerance if we can do it with extremely high fidelity even though it's  a non-universal Clifford group gate. Well as far as the noise in the measurements is concerned  if we can do the measurements non-destructively that is with the small probability of flipping  the value of the measured observable when we do the measurement then we can imagine  repeating the measurements many times to make them more reliable. Alternatively we can  use repetition coding to make the measurements more reliable. I won't go into the details  but what is shown here is a way of teleporting a C-naught gate encoded using a repetition  code where the two qubit gates shown are two qubit phase gates of the type that we know  how to do reliably and the arrows pointing to the right are X eigenstate preparations  and the arrows pointing to the left are X basis measurements and in this case we don't  have to repeat the measurements we can decode the results in the repetition code by doing  majority voting every time we make a measurement and if for example we can do a C phase gate  which has accuracy or error rate per gate of 10 to the minus 5 then even if the measurements  have a 1% error rate we could do this teleported C-naught gate to 10 to the minus 6 accuracy.  So MRIs of course we expect to have to use fault tolerance ideas to operate large scale  quantum computers and if we can manage to do a protected two qubit two qubit clifford  phase gate with very high fidelity that could be a powerful tool for fault tolerance even  if measurements and other gates are noisy. In the case of these zero pi protected qubit  superconducting circuits if we can turn on and off a tunable coupling to an oscillator  a tunable josephson junction then we can do single qubit and two qubit clifford gates  with very high fidelity fidelity which gets exponentially close to one if we can choose  the system parameters appropriately. What makes the gate work is an underlying continuous  variable quantum code which protects against phase errors but the zero pi qubit itself  and the protected gate requires building circuits which have a very large inductance in natural  units and that might not be easy to achieve. The gate itself is robust against non-zero  temperature and fluctuations in the Hamiltonian of the system so I don't know if 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 the quantum information and here I've suggested one possible step in that direction.  Thanks for listening.  Questions?  Well done. Mike in the back.  Thanks. That was a really nice talk and it's extremely interesting in particular to me  because there's some similarity at least in feel to the idea of dynamical modulation which  I know a little better than this I guess but what I was wondering about at the end of your  talk where you showed some of the downsides of trying to achieve this high L on C ratio  is the fact that in trapped ions as you know we do these geometric phase gates by exciting  and de-exciting a harmonic oscillator and initially there's no intrinsic protection  against timing errors. You really have to make a closed loop in phase space otherwise  there's a very significant gate infidelity to first order but if you modulate the coupling  to that emotional mode in a way that's reminiscent of dynamical decoupling you can actually build  in robustness against these timing errors. So I was curious if you think there's any  possibility that you can reduce the requirement for L on C being 80 to something closer because  you're only off by a factor of 10 or so from what you said Devereux had achieved if you  can add in some other concepts of dynamic modulation.  Okay so the question was is it possible that we could combine together having the very large  square root of L over C but not quite as large as I seem to require with some modulation  in time of the coupling of the oscillator to the qubit since it's known in other contexts  that geometric phase gates can be made more robust by such a modulation. I'm repeating  the question because I'm stalling I don't really have an answer and also it's a good  question but I haven't really thought about that.  Another question. Hold on.  How do you see this with the error such as state preparation?  The state preparation for the oscillator? Right well I just assumed it was a thermal  state at 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 if the so that's strictly  zero temperature if the temperature is nonzero but small compared to the frequency so it  might be in an excited state but with some suppressed probability then it still works.  After we do the gate the oscillator doesn't return to the initial thermal state or ground  state in fact it 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 I'm  going to want to cool the oscillator back down in order to reuse it in a high fidelity  gate. There's a question way in the back.  Yeah time for one more question where is it over there.  When you described achieving better protection with increasing the number with increasing  the chain of the Josephson junction possible variations in the between the individual junctions  in the chain would they affect anything. I'm sorry you're very I didn't hear the last  part of the question the question is about getting high inductance from a long chain  of Josephson junctions and the rest of it. Yes so small variations they wouldn't affect  anything in this game. A small variation along the chain and the device parameters.  So in other words if I added some disorder to the chain how would it affect the performance.  Well I think it would probably be fairly tolerant of disorder because the key thing is just  to have the inductance of the whole composite device large compared to the I think one 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 capacitance from you know scaling up with  the size of the device as well as the inductance but I don't think disorder is a fundamental  problem.  Yeah let's thank John again.  Okay our next speaker is Lorenzo.
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