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  • [MUSIC PLAYING]

  • JOE BARDIN: Welcome to virtual March meeting.

  • Today, my talk is about the control of transmon qubits

  • using a CMOS integrated chip operating

  • at cryogenic temperatures.

  • Outline of my talk is as follows.

  • I'm going to start by explaining how transmons are controlled.

  • I'm really just going to review this.

  • I'll review what transmons are, how we control them,

  • what the considerations in terms of control are.

  • Once we have the basics down, I'll

  • go over how people do this today, what kind of hardware

  • is used, and really importantly, why

  • it's not scalable to future fault tolerant systems.

  • Once I've motivated the work, I'll

  • present the design and implementation

  • of a cryo-CMOS controller followed

  • by experimental characterization.

  • So this is a basic diagram of a transmon qubit.

  • The qubit itself is made up of a nonlinear LC resonator.

  • Here we have, instead of just one Josephson junction

  • replacing the inductor in a parallel LC resonator,

  • we have two.

  • This is a squid, which acts as a flux tunable nonlinear

  • inductor.

  • You can tune the effective value of this inductance

  • through an external bias that threads

  • a flux through the loop.

  • We also have an XY drive port on here,

  • which we can use to couple energy

  • at the resonant frequency of the qubit into the circuit.

  • It turns out if you turn--

  • you cool this down to temperatures low enough, such

  • that thermal energy in the environment

  • is lower than the effective photon of a--

  • or effective temperature of a photon in the resonator,

  • then it behaves quantum mechanically.

  • And its nonlinearity gives us a really nice feature

  • called anharmonicity, which means

  • that the spacing between each of these levels is different.

  • What that means is that, if we want to drive from the 0 to 1

  • state, we want to drive at omega 01.

  • If we want to go from 1 to 2 or back, we drive at omega 1, 2,

  • and so forth.

  • And all these are different frequencies.

  • The more nonlinear the qubit is, the more--

  • the bigger the differences between each

  • of these spacings, and the easier it

  • is to address an individual spacing.

  • So if we can constrain our drive signals to just omega 01,

  • then we can just address the 01 subsystem,

  • and it behaves as an ideal 2-level qubit.

  • So the transmon qubit is this circuit,

  • and when we operate it, we typically want to be in just

  • a 01 subspace, which means our drive signal at the RF port

  • needs to be bandlimited, such that we don't hit omega 12.

  • To put some numbers on this, typical frequencies for omega

  • 01 are 4 to 8 gigahertz, which means that we really need

  • to cool to the 10 millikelvin range so that we have

  • sufficiently suppressed thermal noise in the resonator.

  • And our anharmonicity, or difference between omega 01

  • and omega 12 expressed in hertz is about 150 to 350 megahertz.

  • As I'll mention shortly, this is an engineerable parameter,

  • as you see here based on the C in the resonator.

  • If we write the Hamiltonian system in the lab frame,

  • we find we get this form.

  • You have two terms.

  • First you have, this sigma Z term,

  • which is describing a natural rotation

  • about the negative z-axis at the qubit frequency.

  • And then you have a drive term due to this--

  • this drive source here.

  • And that causes a rotation around sigma y.

  • If you look at what's going on, you

  • see that this term is causing it to rotate

  • around the negative z-axis.

  • Whereas this term is causing a y rotation.

  • We usually like to think about it not in the lab frame,

  • but in the rotating frame.

  • So if we write the rotating wave approximation,

  • we get a slightly different expression,

  • but we no longer have the sigma z rotation,

  • and things are easier to think about.

  • So if we drive at omega 01 with sine omega 01t and a phase pi

  • minus phi d, where this as a controlled variable and some

  • envelope of function, we find that the Hamiltonian looks like

  • this.

  • And we have a sigma x and sigma y term,

  • and their weights are dependent on the phase of this carrier

  • signal.

  • So if we control the carrier signal,

  • we control the rotation about sigma x and sigma y,

  • or the rotation about some axis, which

  • axis the vector we're rotating about is.

  • It's going to be in the x, y plane,

  • but we can control where it lies.

  • And the amount of rotation is determined by the envelope.

  • So we can set this envelope to determine how far we rotate.

  • And we can set the phase, as I mentioned before,

  • to set which axis we rotate about.

  • So basically, when we design our control pulses,

  • we want to engineer these things to get what we want.

  • So how do we pick the envelope?

  • Well, there are a couple of considerations.

  • We have a finite coherence time.

  • For transmon qubits, this might be 50 microseconds.

  • It might be shorter, depending on what the exact circuit is,

  • but we can engineer this anharmonicity by choosing c.

  • Larger anharmonicity means we can have more bandwidth

  • in our pulses, which means we can make them quicker.

  • So we'd want smaller c to do fast gates.

  • On the other hand, the dephasing time, T2,

  • is determined by the frequency fluctuations of the qubit,

  • and we want to be insensitive to charge noise

  • so that, if we move around--

  • if our charge moves, then the frequency doesn't jitter and--

  • causing dephasing.

  • So it turns out what the beauty of the transmon

  • is, if you put a big capacitance,

  • you really desensitize yourself to the charge.

  • You really flatten these curves out,

  • and so charge noise doesn't hurt you as bad.

  • So there's these conflicting design criteria.

  • You want a small capacitor for nonlinearity.

  • You want a big capacitor for large T2.

  • Because of this, we typically end up

  • with 150 to 350 megahertz range for our anharmonicity,

  • which gives us single qubit gate times in 10 to 30 nanoseconds

  • range.

  • And we have coherence times typically

  • 30 to 100 microseconds.

  • So how do we actually shape the pulses?

  • Well, we want to avoid the omega 12 transition.

  • We want the pulse to be as quick as possible.

  • If were to use rectangular pulse,

  • we'd get this sink side lobes.

  • And we-- it doesn't roll off very fast.

  • We want to do something more clever than that.

  • Gaussian pulses have been quite popular for a long time.

  • You usually have to truncate them or have some limiting

  • function you multiply them with so

  • that they don't go on for all of infinity,

  • because a Gaussian never really ends.

  • Another kind of convenient pulse that has well-defined start

  • and stop sign--

  • start and stop time is the raised cosine you can see here.

  • And you can see the spectrum of all these different pulses.

  • So the Gaussian is without a limiting function,

  • so it would pick up some side lobes also,

  • but you can see both the Gaussian

  • and raised cosine roll off much quicker and allow us to--

  • because of that, we can do much quicker pulses

  • than if we were just using rectangular shaping,

  • as you might do if you were driving, say, a spin qubit.

  • So in this work, we'll use raised cosine,

  • but I do want to acknowledge Gaussian, as well.

  • Is this all we want?

  • Even with this raised cosine, we'd

  • still like to make the pulses as short as possible.

  • So people like to do things like second order shaping.

  • In addition, as you drive the qubit, it's a nonlinear thing.

  • Its frequency changes.

  • So depending on, if you want to drive a pi pulse, 180 degree

  • rotation, or a pi over 2 pulse, which is typically

  • half the amplitude, you might need

  • to drive at a different frequency for the two.

  • So there's a stark shift that you might need to compensate.

  • So to get rid of--

  • so one thing that's used is drag.

  • Drag-- we put in a derivative term that's weighted.

  • This is in quadrature if you want to think in the microwave

  • terms.

  • So you have a sine and a cosine carrier term.

  • And if you do it right, with drag of 1

  • you get a notch at some frequency that you can set.

  • If you set that at omega 12, you can avoid the omega 12

  • transition by adding this extra modulation tone.

  • And typically there is-- you also

  • want to do an amplitude dependent offset

  • to compensate the stark shift.

  • When you see the work we've done,

  • we haven't done this more advanced shaping, but we'll--

  • it's something for future work.

  • So I wanted to introduce it anyways.

  • So a couple practical issues--

  • how strong should these pulses be?

  • I could go through all this math,

  • but essentially you're going to end up integrating the envelope

  • amplitude, and you're going to find, when you get out, some--

  • for a raised cosine pulse, some level.

  • And it's typically, depending on if you're

  • raised cosine, Gaussian, other types here,

  • or rectangular, you're somewhere in the 50 to 500

  • microvolt pulse amplitude at the port of a qubit.

  • Now, we need to really lightly couple

  • the qubit, because we don't want it

  • to cohere the qubit through coupling to our 50 ohm source.

  • So you're typically using [INAUDIBLE] farrad

  • type capacitor to couple.

  • So you're very weakly coupled, and only a small fraction

  • of the energy that's available at that port makes it.

  • Just to put some numbers, this is maybe

  • negative 65 DBM, negative 70 DBM of power

  • available at that port, but it's attenuated by about another 30

  • dB before you get to the qubit.

  • Sorry, another 60 dB, not 30.

  • Another important thing is how much noise

  • can there be on the drive line?

  • Noise here will cause up down transitions,

  • which will hurt your coherence.

  • We really don't want this.

  • So you can write an expression.

  • I've written it in a way that looks

  • nice to microwave engineers, where the rate is

  • the bandwidth over which you couple times

  • the ratio of the effective temperature of this resistor,

  • assuming it's going to put out white noise, thermal noise,

  • to the temperature of a photon at the qubit frequency.

  • That's-- at 1 gigahertz, a photon will be 50 millikelvin.

  • At 5 gigahertz, it would be 250 millikelvin.

  • I already said these.

  • We can plug in for this, the delta omega.

  • It's the ratio of the qubit frequency,

  • the omega 01 to the effective q due to the drive circuit.

  • So if the only thing dequeueing the qubit

  • were the drive circuit, that would be this value.

  • And this is-- inversely, this is 1 over the t1 due to the drive

  • circuit.

  • We typically set that to be about a millisecond, which

  • corresponds to a q of about 40 million

  • for a 5 gigahertz qubit.

  • You can do some simplification, and if you

  • were to set this 1 over the rate of transition,

  • so effectively how much your qubit de-coheres

  • due to the noise on this line, and make that equivalent to how

  • much the qubit relaxes due to damping by this resistor,

  • if you make those two things equal,

  • you find the amount of noise you can take on the driveline

  • is one photon worth of noise.

  • That's independent of the coupling,

  • so it's an easy thing to remember--

  • is that we really want to have thermal noise on here

  • equal or less than the equivalent photon energy.

  • We cool these things to 10 millikelvin

  • so we can take a little bit more noise than just the noise--

  • the thermal noise of the attenuator-- actually,

  • quite a bit more, because it's exponentially

  • suppressed with cooling, but not that much more.

  • So that's the basics.

  • How is it done today?

  • What kind of hardware do we use, and why do we

  • want a different approach?

  • This is a block diagram of the basic configuration

  • of