Of course, what is amazing about Kitaev's scheme is the claim that Z and controlled Z gates can be protected!

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1 Kitaev's scheme for a protected qubit in a circuit. Superconducting qubits can be improved by using better materials -- e.g., by replacing the amorphous dielectrics in capacitors with crystalline materials to eliminate the two-level fluctuators. That's important, and it will improve performance. But we should also be thinking about designing more complex circuits with intrinsic robustness. Kitaev (Ioffe has related ideas) proposes to build a two-rung ladder of Josephson junctions shunted by capacitors with much larger capacitance than the inherent capacitance of the JJs. Then excitons propagate along the ladder, and it becomes a "current mirror". By connecting the leads we can construct a device for which the energy is a periodic phase of a superconducting phase difference with period Pi instead of 2*Pi. Thus, there are two degenerate minima of the potential energy, which we may identify with the computational basis states 0 and 1 or a qubit. This is true up to corrections that occur in an order of perturbation theory that is linear in the length of the ladder, and therefore exponentially small (analogous to topological protection). The barrier is high enough to prevent bit flips and the stable degeneracy of the two states controls the phase errors. As usual, protection arises because the encoding of the quantum information is highly nonlocal. Note that in other schemes, like the Saclay charge qubits, one works at a "sweet spot" such that the energy splitting between 0 and 1 is a very flat function of the perturbations that can drive dephasing. In Kitaev's scheme, the goal is to make the energy splitting a really, really, really flat function. To measure in the computational basis (Z measurement), one needs to distinguish a phase difference 0 from a phase difference 1. That should be easy... we connect the leads through a Josephson junction and measure the current. To measure in the conjugate (X) basis, we measure charge instead of phase, distinguishing integer charge (X=+1) from integer + 1/2 (X=-1). What about gates? If the qubit is so highly protected (a "neutrino") one wonders how to couple to it from outside. Yet, Kitaev says that protected gates are possible (Alas, not a universal set.) If we were not concerned about protection, we could simply connect the two leads thorough a Josephson junction (as for Z measurement) for a specified time. Then the Josephson energy distinguishes the state 0 (no phase difference) from the state 1 (phase difference Pi and hence energy J. The accumulated relative phase of 0> and 1> in time t is Jt. Of course, what is amazing about Kitaev's scheme is the claim that Z and controlled Z gates can be protected! For Kitaev's protected gates, he needs not just the current mirror that encodes the qubit, but also an additional superconducting circuit that can be coupled to it. This additional circuit is an LC resonator with a "superinductor" --- that is its inductance is large enough that the quantum fluctuations of the trapped flux are large compared to the flux quantum. Naively such a large inductance would require unreasonably large components, but Kitaev proposes that it might be achieved instead using amorphous superconducting wires. (Actually, he also proposes the use of a large inductor as an alternative to the long JJ chain to achieve a protected qubit --- two large inductors coupled by a capacitor can propagate excitonic current but not any net current.)

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4 Now we are to take into account the corrections that arise due to the width of the peaks and the uncertainty in the time tau. But Kitaev asserts that these imperfections "mainly result in oscillations of the LC circuit after the cycle is complete, leaving the qubit unaffected." This assertion seems remarkable when we consider what is happening in the phi basis. However, it is unremarkable when considered in the dual charge basis, for in that case the states +> and -> are highly distinguishable grid states, just as are 0> and 1> in the phi basis. And a phase error in the phi basis becomes a bit flip in the dual basis. Such errors are highly suppressed.

5 8 June 2008 The scheme is extremely remarkable, because, naively, a very well protected qubit should be hard to couple to --- it should be what Martinis derisively calls a "neutrino." Yet we can couple to it, and even more amazingly we can do gates that are *physically* protected. We should separate the mechanism by which the protected qubit is achieved from the mechanism for the protected gate. That is, suppose we just take it for granted that the qubit is described by a continuous variable system in a potential with a high barrier and two very nearly degenerate minima (e.g. an exponentially good degeneracy that is well protected from local noise sources). And suppose we can switch on a coupling to an oscillator with a broad Gaussian ground state. By a combination of analytic and numerical methods, we could study the performance of the protected gate. What are the effects of the nonadiabaticity and of noise? And we could hope to explain more clearly and from a broader perspective how we manage to have so well controlled a coupling to this "neutrino." Roughly analogous is the Yale group's "transmon" qubit, which has a highly stable (but nonzero) energy splitting. For the transmon, there is still dephasing, but the dephasing is limited by relaxation. How robust are the measurements and preparations? If the ground state in the phase basis has two sharp peaks, it should be easy to distinguish phi=0 from phi=pi. In the charge basis, though, the state would be broad, and we would want to distinguish even from odd charge, in units of e (half the Cooper pair charge).

6 27 June 2008 In Kitaev's protected gate, the qubit is encoded in a flux inserted in the LC circuit, where the flux can be either 0 or Pi. The switch that connects the qubit to the oscillator must be adiabatic. The switch is a tunable JJ. The Hamiltonian has three terms (Coulomb energy of the qubit is neglected). The potential energy of the LC oscillator (depending on the phase phi) is quadratic. The tunable JJ energy depends on the relative phase phi-theta, where the two states of the qubit are theta=0,pi. When the switch is closed (the JJ is turned on), the potential energy for phi has wiggles whose positions depend on the value of theta.to "lock" the phase, the tunable J_switch must be large compared to the capacitance C of the LC circuit. This means that phi will have a peak localized in each "well", where the wells have spacing delta phi =2*Pi. Adiabatic condition: The time to turn on the switch must be long compared to period of the harmonic oscillations in the local potential wells. That way the wave function will be in the ground state of each local potential well. But it must be *short* compared to the oscillation periond of the LC circuit. This is necessary to prevent the the ground state from relaxing to the lowest local minimum. The zero-point oscillations of the LC circuit are "frozen in" so that we have a superposition of many localized peaks. Kitaev says (KITP 2006): "If the time is not exact, then when the switch opens the error will be transmitted to the oscillator, not the qubit. We end up in a state where the oscillator is not in its ground state, but the qubit is intact. This is a miracle of quantum error correction." Let's review some of the improvements that Kitaev made to his scheme since the apprearance of the paper arxiv:cond-mat/ These further ideas were presented at the IPAM and QEC07 meetings in The main change is that while the paper proposed to use a two-rung ladder of Josephson junctions (with the rungs connected by large capacitors to achieve excition condensation), the new proposal is to replace each rung of the ladder by a "superinductor" in series with a tunable Josephson junction, where a single capacitor connects the two rungs.

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9 On the other hand, in the regime where JC >> 1, the energy function E(q) is extremely flat. The reason is that phi is very well localized, and q only determines the relative phases of the peaks in phi that are separated by multiples of 2 *Pi. Therefore the dependence of the energy on q arises only from the phase slips -- i.e. the tunneling from one minimum of the potential to a neighboring minimum, which is exponentially rare. We want this dependence of energy on the charge to be negligible compared to the inductive energy, or

10 By the way... how do we makes the Josephson coupling J tunable? It's done with a SQUID: the tuning is achieved by adjusting the magnetic flux through a loop with two junctions. Now consider the "current mirror" circuit.

11 Now the idea is that phi_plus is a "fast" variable with a small capacitance (small mass) and phi_minus is a "slow" variable with a large capacitance (large mass). The slow phi_minus is in the charge regime where the dependence of energy (and hence the associated current) is exponentially small, while the fast phi_plus is in the flux regime so that phi_minus locks to the phase difference theta_1 - theta_2; hence the energy depends only on the difference theta_1 - theta_2, up to exponentially small corrections. But... why doesn't averaging over phi_plus make the cosine average nearly to zero, so that the effective Josephson coupling for phi_minus is exponentially small? Now, the phase phi_1-phi_ 2 locks to (theta_4 - theta_ 1) - (theta_3 - theta_2) = (theta_2 - theta_1) + (theta_4 - theta_3) Because of phase locking, phi_minus = phi_1 - phi_2 = 2*(theta_1 - theta_2). And the energy is a function of this quantity, apart from exponentially small corrections.

12 Readout: In principle we can read out the phase difference theta_1 - theta_2 by connecting the two leads through a Josephson junction with flux inserted through the loop of wire. The current through the junction will be How to make a super-inductor? (1) A coil with many turns. We want L to be large compared to the quantum impedance, which is enhanced relative to the vacuum impedeance by a factor proportion to one over fine structure constant. Probably impracticle. (2) A long chain of Josephson junctions.

13 (3) A long wire with large normal state resistance (a "dirty" or "amorphous" superconductor). The width d of the wire must be large enough compared to coherence length (e.g. about 30 Angstroms in MoGe), to suppress phase slips, and the length l of the wire is large (e.g. about 1000 Angstroms), so that resistance R of the wire is large. The inductance is L is R/(Pi*Delta) where Delta is the SC gap (but why?). Side comments on JJ chains. For a chain of JJs where each JJ has Josephson coupling J and capacitance C, phase slips are suppressed for JC >> 1. But for a long enough chain the phase slips will disorder the chain and make it insulating. But if the chain is grounded (connected to ground by large capacitors with C_1 >> C), then there is a KT transition from high J SC phase (vortices are suppressed and phase has power law correlations) and a low J insulator phase (vortices proliferate and the phase has exponentially falling correlations). In the case of a two rung ladder, it's possible for the symmetric combination of the phases of the two rungs to be in the insulating regime while the antisymmetric combination is SC. This means that currents flow in opposite directions in the two rungs (there is a current of *excitons*).

14 26 June 2008 As a warmup example, we could consider coupling a qubit to a qudit, where the qudit can be prepared in a shift resistant code. The simplest code that resists unit shifts Z and X is the d=18 code:

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16 A quadratic Hamiltonian realizes a Clifford phase gate, and a cubic Hamiltonian realizes a non-clifford C3 phase gate. We might prefer to allow the integer k to run over the symmetric range [-9,9]. Then, in the case of the sympectic phase gate, the Hamiltonian has the appropriate periodicity property, h(9)=h(-9)

17 Let's put aside for now the idea of performing a gate on a qubit by coupling it to this d=18 qudit. Instead, let's just suppose that we want to perform a logical Clifford phase gate by turning off and on the diagonal Hamiltonian. Will this gate be protected against imperfections in the timing of the pulsed Hamiltonian? But what is more important for our purpose is the *phase* of the postmeasurement state, as that determines whether there is a logical error.

18 When we project onto the code space, the phase of the logical 0 becomes So perhaps it is essential to use the idea that the same interaction that couples the qubit to the oscillator (or qudit in this case) is also responsible for preparing the "Gaussian grid state" (?). It is essential that the coupling turns on and off adiabatically. We don't need to know the microscopic details that make the qubit work --- we only need to know that its energy is a periodic function with period *Pi* of the phase difference theta_1 - theta_2 of the qubit's two leads, that the logical qubit is encoded according to whether the multiple of Pi is even or odd, and that the amplitude for tunneling from one multiple of Pi to an adjacent one is heavily suppressed. The switch coupling the qubit to the LC circuit turns on and off adiabatically. When the switch is on, the oscillators phase phi is locked to values that are even multiples of phi if the qubit's logical state is 0>, or to odd multiples of phi if the qubit's logical state is 1>. To investigate whether there is a logical X error, we investigate the circuit's behavior using the phase basis. The X errors are rare, because the qubit needs to tunnel through the barrier to make the transition from an even to an odd multiple of Pi. To investigate whether there is a logical X error, perhaps we should use the charge basis instead. While in the phase basis the wave function of the LC circuit is broad and the qubit's wave function is narrow, in the charge basis the opposite is true.

19 We should regard the phase phi as a variable that is not necessarily periodic, and the charge q is the conjugate momentum of the phase variable. When not coupled to the LC oscillator, the qubit sees a periodic energy which is a periodic function of phi with period Pi. We can simultaneously diagonalize the Hamiltonian with a Bloch momentum that takes values from -1 to 1. The Bloch momentum determines how the wave function's phase changes under a translation by Pi, the period of the potential. The qubit's logical +> state has even charge (Bloch momentum 0) because it is periodic with period Pi, and the qubit's logical -> state has odd charge (Bloch momentum 1) because it is antiperiodic with period Pi. Thus, in order for a logical Z error to occur, unit momentum (i.e. charge) must be transferred to the qubit. But such a transfer is unlikely because the ground state of LC circuit (before it is coupled to the qubit) with L >> C is very well localized in momentum. Its variance in momentum is C/4L << 1. Equivalently, when the qubit and LC circuit are coupled, then in the phase basis the wavefunction is a Gaussian grid state --- the phase phi modulo 2*Pi of the qubit becomes locked to the value 0 or Pi, depending on qubit's logical state, and is governed by a broad Gaussian envelope. The width of each narrow peak is determined by the qubit's dynamics, and the width of the broad envelope is determined by the LC circuit's dynamics. When Fourier transformed, the wave function in momentum space is also a Gaussian grid state, The value of q is locked to 0 or 1 modulo 2, depending on the qubit's logical state, and is governed by a broad envelope. In this basis, the width of each narrow peak is determined by the LC circuit's dynamics, and the width of the broad envelope is determined by the qubits dynamics. But... something seems fishy. Because the logical gate performed by the protocol *does* transfer momentum to the qubit. How does that happen? And if it happens, what ensures that the momentum transferred is exactly what it is supposed to be? When the states 0> and 1> are superpositions of narrow peaks in the phase representation, then we can think of the states in the lowest energy band as superpositions of phi=n*pi> weighted only by phases.

20 [This is actually eq. (104) in the GKP paper.] This is a Gaussian operation --- i.e. a linear transformation in phase space, induced by a quadratic Hamiltonian.

21 So clearly the term in the Hamiltonian (phi)^2 / 2L that drives the gate does not commute with the momentum and it pumps momentum into the qubit -- this is necessary to do a logical S gate, which takes (say) a state +> with even momentum to a state a state +i> which is a superposition of even and odd momentum. If the gate is really protected something truly magical has to occur as we adiabatically turn off the interaction... *During* the execution of the gate S, the state leaves the codespace. But my worry is that if the execution time is not exactly right, there is an error with a nontrivial projection along the codespace, and that could be a problem. Well... in the GKP preskill, we claimed that the Gaussian operations are fault tolerant, and here we are applying a Gaussian operation to a continuous variable codeword. So shouldn't it be all right here (or were GKP wrong)? GKP emphasized that the symplectic operations, since they act linearly on the positions and momenta, will not take a small shift error and turn it into a large shift error. But they don't seem to have specifically addressed the question: if the symplectic operations are slightly wrong, would there be a small *logical* error, aside from errors that take a codeword out of the codespace? E.g. using standard normalization for the momentum p and position q (the quadrature amplitudes of an oscillator), the phase gate is S: q -> q; p -> p - q +constant. What if these operations are slightly wrong... is there a logical error?

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26 Something looks wrong. For epsilon = 1, this should be the phase (i). I suppose that's true, but it is not so obvious from the formula, because the formula is valid if 1/[alpha - (i)epsilon] has a large real part, which may not be satisfied for epsilon = 1 and alpha small. On the other hand, when epsilon is small compared to alpha, our approximation (replacing the sum over k by a sum over 0, +1, and -1) is justified, if alpha is also small. In that case: But for the error to be small (in this approximation), we require not only that epsilon is small compared to 1, but that it is small compared to alpha. This is somewhat unexpected: at alpha gets smaller, the codewords more closely approach the ideal codewords, yet the gates become less accurate. On the other hand, it is not really so surprising.

27 So.. if we fix a small value of epsilon, and consider alpha getting smaller... for alpha larger than epsilon the logical error is suppressed by exp(-pi / 2*alpha). Then as alpha becomes smaller than epsilon the suppression crosses over to exp[ - Pi * alpha / 2 * (epsilon)^2 ]. By the way, what if the syndrome is nontrivial, i.e. if the measurement of X^2 indicates a nonzero value of p modulo Sqrt[Pi]? Then we have

28 We still have work to do in order to understand what happens when the coupling of a qubit to the oscillator is adiabatically turned on and off. But the idea is becoming clearer. At least if epsilon is small compared to alpha (and a weaker condition may suffice), then the momentum shift of the Gaussian grid state is dominated by alpha rather than the error epsilon. And conditioned on the shift, the logical error is small. So epsilon, a measure of the gate's deviation from ideal, really doesn't matter very much --- it has a highly suppressed effect on both the logical state and on the oscillator.

29 In the leading approximation (ignoring terms that are exponentially smaller), the probability of a momentum shift by p is independent of the codeword: To get the normalization right, I suppose I should replace the q eigenstates by narrow Gaussians, as in Sec. V of GKP. But the exponential will still be as computed above. The intuition we started with is essentially correct. The error exp(- i epsilon * q^2 /2) does not shift the momentum of the Gaussian grid state much when epsilon is small (even though when epsilon is large enough the momentum shifts substantially). But we have also seen that the value of epsilon that causes a sizable shift, and the value of epsilon that causes a sizable encoded phase error conditioned on the shift, depends on alpha --- resistance to a small epsilon requires that alpha not be too small. How to do a numerical analysis? We could consider adiabatically turning the oscillator's coupling to the qubit on and off and compute the accumulated phase when the qubit is in the state 0> and in the state 1> (i.e. the phase drop across the qubit prefers to be either 0 or Pi (mod 2*Pi)? Actually, it would be preferable to simulate the gate for a coherent superposition of 0> and 1> (like +>), because there might be some residual entanglement of the qubit with the oscillator that contributes to the degradation of the qubit.

30 Phase drop across inductor is phi, which gets locked to multiple of Pi when the switch is closed (an even multiple if logical state is 0>, and odd multiple if logical state is 1>). While ordinarily we think that "many-body systems" are needed for physical protection (or circuits with many components, with protection improving exponentially with the number of components), since the quantum system is to behave like a quantum code, the GKP construction illustrates an alternative: coding based on a single oscillator. In fact, Kitaev's (new) scheme uses super-inductors in three logically independent ways: (1) to realize the protected qubit, (2) to realize the protected gate by coupling the qubit to the LC oscillator, and (3) to devise the adiabatic switch that turns the coupling on and off.