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.