Starting with Superconducting Qubits

I Command You To Learn!

Update: After some feedback, added some more refs, including the IBM Quantum course and some experimental milestones in QEC.

Update 11/03/2024: I learned about Hands-on Introduction to Randomized Benchmarking on the arXiv and have added it to the Gates section.

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Two years ago I wrote a post, Simulating a PhD, intended to remind our young colleagues without PhDs that, when they are hired, they have about 4-5 years until they are directly competing with new PhD grads in the job market. The post lists my view of what should be expected from people with PhD-like experience, especially in terms of autonomy, depth of expertise, and judgement. I like to think that I provided a reasonable amount of guidance with respect to how to go about closing the BS - PhD gap, but I’ve been thinking about this problem on-and-off for a while, and I believe it deserves some more attention.

A Blank Page

The hardest part of almost anything is just… how to start. At the end of 4 years of study, the BS in Physics is both armed with powerful tools for thought and also is totally ignorant. We will use the former to help remedy the latter. But how to structure a course of study to efficiently chip away at the ignorance.

Let’s take superconducting qubits as our case study, since that’s the field I know best. For those of you working on trapped ions, neutral atoms, dots, topological qubits¹, etc among the audience, just mentally substitute in the right concepts and references for a BS new hire in your field.

The field is vast. The literature could be considered 25 years deep, at least, and the amount of work being posted to the arXiv recently has really exploded. Much of it is trash, of course, but as a newly minted BS you have no idea how to even make that judgement. Eventually, as you read, a structure of the field will emerge in your mind, but right now what you need is focus.

The Basics

What I suggest, is just reading the basics. This is probably going to be 3-4ish long papers/textbooks pretty obsessively for a while. So, at the risk of boring you, let me state clearly what we are all trying to do here.

We are trying to build a fault tolerant quantum computer using superconducting qubits.

That single sentence will be our lodestone. Observe.

Circuit Quantization

We are trying to build a fault tolerant quantum computer using superconducting qubits.

To make superconducting qubits we need to know something about superconducting circuits. We need to know how to quantize them and how to put the basic elements² together to make a useful computational subspace. This means we need to know how to write down and interpret the Hamiltonian of our qubits. This, I think, is the first task.

Fig 7. from Superconducting Circuit Companion. A lumped element circuit with fluxes, currents, and voltages shown. You’ll want to know how to write down the Hamiltonian for this.

Goals:

1. Understand how to write down the Hamiltonian of an arbitrary circuit.

2. Gain intuition for how these Hamiltonians will act for common qubits³.

3. Understand how things like resonators and couplers modify the Hamiltonian. These are important, non-qubit circuits that are required for readout and multiqubit gates. This is basically circuitQED.

References:

1. I used to point people toward the Les Houches lecture notes titled Quantum Fluctuations in Electrical Circuits, but it looks like there are some more recent works that may be more accessible. Consider checking out the Superconducting Circuit Companion, or Juan Jose Garcia Ripoll’s textbook Quantum Information and Quantum Optics with Superconducting Qubits.

2. This will likely come from reading the references above and elsewhere in this post, but I suspect a numerical Hamiltonian solver that lets you actually play with ‘control knobs’ of common qubits would be indispensable here. Try SCQubits. You can see an example of what it can do in a previous post about circuit design.

3. I am not sure what the most modern source for learning cQED is. I suspect it is Prof. Ripoll’s book, linked above. I usually send out David Schuster’s dissertation, titled Circuit Quantum Electrodynamics⁴⁵. Daniel Sank’s dissertation, Fast, Accurate State Measurement in Superconducting Qubits, is also considered useful.

4. IBMQ used to have an online textbook, which I believe they’ve transformed into the IBM Quantum Learning site. I can’t speak to the quality of the courses, but since they do solid technical work and are staffed with tons of good people, I expect that this could/would be useful for someone trying to get to grips with QC for the first time.

Hardware Interlude: Microwave Engineering

If you care at all about the experimental side of superconducting quantum computing, or about the details of actually creating a superconducting qubit, you will also need to know at least a little bit of microwave/RF engineering. You will also have to cast your mind back to your undergraduate electronics course and recall the lessons you learned about putting circuits together.

Example circuit directly reproduced from the KQCircuits landing page.

Goals:

1. Understand how to create the circuit elements that define a qubit. How do you translate elements of the Hamiltonian (Ej, Ec, etc) into physical circuit elements such as inductors (linear), capacitors, and Josephson junctions?

2. Understand how to create a readout resonator with the correct properties to read out your qubit.

3. Understand how to calculate control line couplings (capacitance, mutual inductance, etc)? Understand how these coupling constrain your gate implementation.

4. Understand how intentional and parasitic couplings of the environment to your physical qubit contribute to the coherence/lifetime properties of the qubit.

5. Understand how to reduce unwanted environmental coupling (on chip, in-fridge, ex-fridge filtering).

References:

1. Many of the references above can offer insight here, but basically everyone I know who works in this field has a copy of David Pozar’s Microwave Engineering. This also serves as a nice reference for more advanced microwave topics (amplifiers, measurements, modeling, etc).

2. If you are doing self-study without access to modeling software like HFSS or Comsol, you may want to look into an open source FEA RF modeling software and start getting familiar with how physical modeling works for these devices. I don’t have a good reference here.

3. Packages like KQCircuits can help you get started with layout, and I think also plug into popular closed- and open- source FEA solves.

Gates

We are trying to build a fault tolerant quantum computer using superconducting qubits.

Unless you’re working for an annealing company⁶, your quantum computer will have to apply gates to qubits.

Two sequential Hadamard gates applied to a qubit. Reproduced from a physics stack exchange answer.

Goals:

1. Understand how are gates applied to qubits.

2. Understand the important properties of one- and two- qubit gates.

3. Understand how gates can act as a diagnostic for qubit quality.

4. Understand how we evaluate gate fidelity. What is the state of the art?

5. Understand the sources of error in gates.

6. Understand mitigation strategies (real or proposed) for one- and two- qubit gate errors.

References:

1. Matthew Reed’s dissertation, Entanglement and Quantum Error Correction with Superconducting Qubits has a nice section on single qubit gates and qubit tune up, as well as readout.

2. A quantum engineer's guide to superconducting qubits is also a nice resource for a broad swathe of relevant topics in this section and others.

3. You can also use python packages like QuTiP to visualize single qubit gates, or series of gates, on a single qubit using the Bloch sphere (see figure above).

4. A recent submission to the arXiv, Hands-on Introduction to Randomized Benchmarking, seems like a nice collection of relevant RB techniques.

Error Correction

We are trying to build a fault tolerant quantum computer using superconducting qubits.

Fault tolerance means error correction. This is a particularly rich field, but even humble experimentalists like yours truly benefit from having some understanding of how error correction might be implemented.

Figure 1 from Surface codes: Towards practical large-scale quantum computation showing small cell the surface code, as well as the operations to perform the measure-Z and measure-X operations.

Goals:

1. Understand the basic principles of error correction codes.

2. Understand the surface code.

3. Understand how error correction codes can be implemented on real qubits using real gates.

References:

1. My favorite paper on this topic is Surface codes: Towards practical large-scale quantum computation.

2. Update: I received a suggestion to add some experimental milestones in QC, so here you go.

   1. Suppressing quantum errors by scaling a surface code logical qubit from the Google Quantum AI team.

   2. Encoding a magic state with beyond break-even fidelity from IBMQ team⁷.

   3. Realizing repeated quantum error correction in a distance-three surface code from the Wallraff Group.

Advanced Topics

What I’ve listed above is just the tip of the iceberg in a rapidly expanding, vibrant field. There are tons of opportunities for additional learning- materials, novel qubits, novel architectures, exotic superconductor phenomena, RF electronics, etc. You could (and should?) spend a lifetime learning about all of these things.

You can’t learn it all at once, so try to be judicious, and make sure to learn in a way that is durable. Don’t forget to leverage the expertise of your colleagues, some of whom will have decades of experience doing this stuff.

Remember, it’s not a sprint. Consistency will be key here. You want to build a solid foundation so even if (when?) you become more senior and forget the details, the physical intuition that you’ve built up with your hard work remains.

Good luck!

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1. Lol

2. Josephson junctions, linear inductors, capacitors.

3. Transmon, flux qubit, fluxonium. Probably you can start with the Cooper Pair Box and phase qubits, but the three I listed are probably the easiest to get to grips with initially.

4. David’s dissertation is peculiar in that it happens to include homework problems, too.

5. I’m happy to include additional well-written, pedagogically focused references here.

6. lol, but with love

7. I’m actually not sure this is the best ref for IBMQ, so if you work there and have an opinion about this, please let me know.