I was wandering around Quantum Computing Stack Exchange when I came across this question from a few years ago:
Superconducting qubits generally have frequencies within the range of 4 - 8 GHz. What design considerations give the upper and lower bounds for what is a feasible design. I.e, why can't they be higher or lower in frequency?
It’s an interesting question, and I actually really like the top rated answer by Ofer Naaman:
A practical reason is that microwave components in this frequency range are readily available and reasonably priced.
It’s a deep answer which undergirds many decisions about what gets designed/studied when the majority of people doing the basic science have serious constraints (money, time, experience, etc). But it also relies on the reader already having a great deal of implicit knowledge regarding superconducting qubits. Not unreasonable on Quantum Computing Stack Exchange! However, I began to wonder how I would explain this to someone not within the fold, so to speak. What follows is my attempt to at an explanation. Hopefully written at a level comprehensible to a physics undergrad without any special knowledge of superconducting circuits.
Let’s start with a simple requirement and see how far it gets us. We want to use some common superconducting metals to somehow create a qubit. At least two energy levels of this device should have some sort of convenient energy spacing in order to create the qubit. In the biz we refer to energy spacings in units of h*GHz, where h is Planck’s constant. Most people simply omit the h and it is understood that GHz is a stand in for energy and a convenient reminder of the frequency of photon that will excite the qubit.
Environmental Energetics
Superconducting qubits require very low operating temperatures. Most dilution refrigerators can reach base temperatures of 10-20 mK. This is much lower than the superconducting transition temperature, Tc, of many common superconducting metals used in qubit construction. For example, aluminum superconducts at ~1.1 K, tantalum superconducts at 4.8 K, and niobium superconducts at a sweltering 9.2 K.
With Tc more than 100x greater than the base temperature of a dilution refrigerator, why bother with the expense and fussiness of the dil fridge? In principle, superconducting qubits can reach their superconducting state in a simple bath-pumped helium cryostat. These cryostats function entirely by lowering the vapor pressure of liquid helium until the desired temperature has been reached, and they can bottom out at or below 1 K.
So, aside for some practical difficulties in getting a helium bath below 1.1 K, why shouldn’t we opt for this solution? One very good reason has to do with thermal excitations of the qubit. That is to say that the temperature of our refrigeration apparatus sets the temperature of the qubit environment. Since the qubit is immersed in this environment, this also sets the minimum temperature of the qubit.
What is the energy scale of this temperature? We can quickly get a feel for the scale by calculating kT, where k is Boltzmann constant, and T is the temperature of the environment. Remember that our energies are in h*GHz, so to get frequency out, we actually want to calculate kT/h. For a temperature of ~1 K, kT/h is about equal to 21 GHz.
Ok, so we’re going to design our qubits to have a frequency of 21 GHz, right?
Not so fast.
Another important requirement for a good qubit is high fidelity operation. We need to ensure that when we prepare a qubit in the |0> state, it is in that state 99.99% of the time. Is that achievable with 21 GHz qubits in an ambient environment of 1 K? Our knowledge of statistical mechanics will provide the answer. We will use the Boltzmann distribution to calculate the expected equilibrium population of the qubit if we just leave it idle (which is the same as preparing |0>).
Disturbingly, we find that the expected population of the |0> state at 1 K is more like 73%, rather than the desired 99.99%. The hot qubit environment is causing unwanted excitations, so our qubit is spending over a quarter of its time in the |1> state! Ok, so how DO we get to 99.99%? Let’s try turning down the temperature.
We find that we have to turn the temperature waaaaaay down to about 100 mK to get thermal excitations to an acceptable level1. Well, shit, I guess we’ll have to buy that fancy Bluefors dil fridge anyway. Hope you’ve been polishing up your grant writing skills!
Qubit Fabrication
Our dilution refrigerator has been ordered and it is scheduled to be delivered in a few months. Now it is time to turn our attention to actual qubit fabrication. Remember that we’ve decided on a 21 GHz qubit, so we have to figure out how to make it. For our qubit architecture we’ll choose the workhorse of the superconducting quantum world, the trusty transmon.
Our transmon is really just a fancy LC oscillator, where the role of inductance, L, is played by a Josephson junction whose figure of merit is the critical current, Ic. There are many other details, but luckily the original transmon paper supplies us with a way to translate C and Ic into frequency.
The eigenenergies (or eigenfrequencies, if you divide by h) of the transmon are indexed by m, which are just the positive integers and 0. Ec and Ej are called the charging and Josephson energies, respectively. A little bit of mathematics and we find that our 21 GHz qubit needs a capacitor with C = 120 fF and a Josephson junction with Ic = 2000 nA.
These numbers aren’t entirely unreasonable for any competent fabrication facility to manage. In fact, you could reach our target frequency of 21 GHz with smaller junctions in a different configuration. Many flux qubits could easily be in this regime with the right parameter selection.
My next consideration might be resonator fabrication for readout, but resonators in the 15-25 GHz frequency range would not be very difficult to fabricate.
Instrumentation and Filtering
The day has come, our dilution refrigerator has been delivered, built, and installed. The transmon we’ve designed has been produced and packaged by the fab and its ready to go into the fridge. What’s left? Well, we need to decide on filtering and instrumentation!2 We need a way to address and read the qubit, and we need to add pretty aggressive filtering and amplification in the fridge to prevent hot environmental noise from traveling down the signal cables into the transmon.
To properly filter the readout line, we’ll need SMA connectorized filters in the 15 - 25 GHz range, depending on how we designed the resonators used to readout the qubit. One of the most common types of filter, the VLF(X) series from Mini-Circuits only appears to go up to 8 GHz. It’s non-trivial to find components in the 10s of GHz, and their cost grows substantially.
Addressing our high frequency transmon will require a microwave signal generator. These are instruments that require the purchaser to ‘get a quote’ which is business-ese for “you will pay us a lot of money for this”. The lowest end MXG signal generator will set you back about $20,000 (USD). Models that are capable of generating 20+ GHz signals will be substantially more expensive.
You might rightly note that all of these expenses are much, much smaller than the cost of the dilution refrigerator itself. That is a point with real merit. I think the true difficulty with respect to these RF components is actually the switch from the trusty and very reliable SMA connector to.. something else. You’ll note that SMA connectors were originally designed for DC - 12 GHz. Some have been modified to go up to 26.5 GHz. As with the filters, you will have to be very careful to make sure you buy these special SMAs, and don’t accidentally use lower-rated ones on signal paths carrying 20 GHz tones. If for some reason your qubit frequency exceeds this limit, you will have to settle for whatever specialty connector will do the job.
In terms of the higher frequencies, it may be sufficient to stop here. Since academic labs are the places where these things were invented and proven out, it’s not surprising that cost and availability of components played (and plays!) a role in what design spaces are explored.
Physical Limits- Low frequencies
We want to keep our qubit high end frequency below, say, 12 GHz because of the availability of microwave components, as the first answer to the linked question states. What about the low end? There are still many widely available microwave components down into the 100s of MHz.
Well, these days, the answer is that there are qubits that operate in the 100s (and 10s?) of MHz! I’m thinking of some fluxonium qubit implementations.3 The caveat is that qubits operating at these low frequencies need to be protected from thermal noise!
Let’s take a look at our |1> state occupation probability as a function of temperature. This time, we’ll put our theorist hats on and plot this is a function of thermal energy, kT, divided by qubit eigenenergy h*f01.
If we want to maintain thermal noise populations at or below 0.01%, then kT/hf has to be below 0.11. In other words, for a 4 GHz qubit, T ~ 20 mK to satisfy this constraint. Only dilution refrigerators are both (relatively) widely available and able to hit these temperatures. For frequencies below 4 GHz, the required temperature will drop rapidly (exponentially). Thus, at very low frequencies, we must use qubits that can forbid4 transitions between the |0> and |1> states.
Recap
Qubit maximum frequencies need to be lower than the energy of the superconducting gap of the superconductor they’re made from. The upper bound for aluminum is about 80 GHz.
The availability and expense of microwave instrumentation and components at 15 - 80 GHz makes components expensive and harder to come by, which incentivizes us to operate in more agreeable frequency regimes (< 12 GHz).
Qubits are operated at temperatures much lower than Tc of the superconductor because this reduces unwanted thermal populations in the qubits themselves. The energy of a 4 GHz qubit is about 10x that of an ambient qubit environment sitting at 20 mK. 20 mK happens to be a common base temperature for dilution refrigerators.
Qubits with frequencies much lower than 4 GHz must be noise protected in a way that makes them much less susceptible to thermal noise. This means our standard, easy to fabricate transmons are not going to be usable < 4 GHz. We’d need to switch to something like fluxonium.
You could also turn up the qubit frequency, but the temperature improvement isn’t great. A qubit at the pair-breaking frequency of aluminum, 81 GHz, would still require temperatures <= 400 mK to stay below our target thermal excitation.
To be clear: you should not plan experiments in this way, nor should you plan substantial capital expenditures in this way. The right thing to do is to have a pretty good idea of how you’ll do the experiment and what infrastructure will be required. Only then should you design your qubits and purchase your very expensive lab equipment.
It’s important to note that even protected qubits must thermally equilibrate eventually. In the reference above, the Schuster group has to use a special reset sequence to cool their fluxonium qubit (down to 190 uK!) in order to do good state preparation. At that point, the protected nature of the fluxonium prevents thermalization long enough to do something interesting with the qubit (gates, etc).