RFP: Request for Problems

Hunting Nerds

As part of my ongoing interest in Proof of Competence, I come to you, dear reader, with a request:

Send me your best Problems¹.

I’m especially interested in QC/QIS problems from any subfield/discipline. Intuition Builders are probably the easiest to communicate, but I’ll accept some Real Bastards if you’ve got ‘em.

What Kinds Of Problems?

A Simple Trick: These problems seem hard, but require a single simple insight/bit of knowledge that makes them trivial. Fast to ask and answer, they can filter for people who either lack insight, or lack a mastery of the basics. In more mathematical fields, they’d be straightforward proofs with elegant one line solutions that don’t require deep domain expertise.

Intuition Builders: These exercises are basically undergrad level homework problems, but the best of their class. They are usually multi-step questions that lead you through developing the correct physical intuition for a broad class of problems. They are not necessarily mathematically trivial, but too much computation here takes away from the quality of the learning, rather than adds to it. I’d ballpark the maths to be something like- simple integrals up to basic vector calculus (integrating over spheres, for example) and simple differentiation up to partial derivatives and maybe Taylor expansions. People with good physical intuition should be able to simply state the correct answer.

Real Bastards: I am not sure what to call these, but they’re very much at the graduate level. I’m thinking Jackson E&M type problems, where working through the horrible integrals and understanding the details of how to construct and execute the computation is prerequisite to delivering a useful solution. Sometimes real-world problems are hard and require serious effort².

Relevant Problems: These are problems specifically relevant to a particular application, and could fall into one of the categories above, as well³. I call them out because their relevance makes them interesting and also unlikely to be covered in the standard curricula. These problems aren’t likely to be found in any textbook, or at least not in their most useful form, and require some kind of synthesis of accumulated knowledge/experience to solve correctly. Sometimes these might also require a little bit of light reading⁴. I like these because they can often be constructed to explain why the practical implementations of, say, superconducting qubits are the way they are. For example, you could construct a fairly straightforward problem that might give the student a good feel for the practical considerations behind the operating frequency selection for superconducting qubits.

My Offerings

The first problem is, I think, relatively straight forward, but builds a decent intuition for how challenging it can be to use quantum systems with discrete energy levels as thermometers⁵. In the proud tradition of physics, the problem has a million parts, but I think any one subquestion is pretty interesting. The later parts rhyme with things I have actually asked in the workplace, the answers to which were pretty strong indicators of competence.

The second problem is more of a fun one⁶ that I got from Gavin Crooks’ twitter feed.

Q1- Quantum Thermometry

a) For a quantum system with Hamiltonian given below, write down the expression for the occupation probability of each energy eigenstate in thermal equilibrium at temperature T.

\( \hat{H} = \sum_{i=0}^n \epsilon_i |i\rangle \langle i|\)

b) Let the total number of states, n = 2 (a two level system). Plot the occupation probability of the ground and excited states as a function of eigenstate energy, temperature (in Kelvin), and physical constants.

c) How does this occupation probability change as a function of TLS energy spacing when T is fixed?

d) Systems with n > 2 can either be harmonic or anharmonic. Show that for a harmonic system, the partition function from (a) converges to an analytic expression.

e) Many quantum systems have many more than 2 energy levels, and exhibit some amount of anharmonicity. We want to know when it’s OK to use the harmonic approximation (pretend anharmonicity = 0) and when we have to revert to summing up the partition function. Perform this analysis and describe the regime where the approximation is valid in terms of energy spacing, anharmonicity, and temperature.

f) Using the harmonic approximation, write the expression for temperature as a function of ground state occupation probability.

g) In a real system, we get to choose how many times to measure our quantum thermometer. Describe a reasonable way to estimate the uncertainty on the temperature you wrote down in (f) as a function of number of measurements. What is the temperature floor of the measurement?

h) In a real system, we will undoubtedly have some measurement error, so our knowledge of occupation probability will be imperfect. That is, sometimes a measurement will read |0⟩ when it should read |1⟩ and vice versa. How does the introduction of a readout fidelity, F < 1, impact the temperature floor and uncertainty of the thermometer you described in (g)?

Q2- Reviewer 2

Dear Reviewer 2,

I hope this message finds you well. I am writing to kindly request your expertise in reviewing a recent submission, titled "Molecular thermal motion harvester for electricity conversion." Given your extensive knowledge and contributions to the field, we believe your insights would be invaluable in evaluating the rigor and significance of this work.

We understand that you have a busy schedule, but we would greatly appreciate it if you could provide us with your review at your earliest convenience. Your feedback will be instrumental in ensuring the quality and impact of our research published in our journal.

Thank you very much for considering this request. We look forward to hearing from you soon.

Best regards,
QuantumObserver
Editor in Chief
Quantum Quarterly

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1. Or perhaps it is more accurate to say ‘Exercises’.

2. I’ve mellowed out on my hatred for Jackson’s E&M and the time it stole for me in grad school. Part of this is understanding and accepting that I’m an idiot, and the other part is realizing that those problems were building the next level of physical intuition where details mattered and the process of grinding through a million weird integrals was how you got a useful answer. The real world isn’t always as nice as a Griffith’s textbook.

3. The Schuster Dissertation Problems could be examples, here.

4. What’s a few arXiv papers among friends? I once spent about 10 hours reading papers to solve a ‘homework’ question as part of a QC interview process. It was a lot of work, but they nerd-sniped the hell out of me with the question so I ended up enjoying the process.

5. Writing these questions out caused me to go back and sharpen my understanding that there were large parts of the Cramer-Rao stuff that still don’t make much sense to me.

6. I also enjoyed Smeared Out Sun and Measurement Noise from this collection of 5 Problems on Less Wrong.