Translation of color/toric code to a small network of solid-state spins

Question

Within Quantum Error Correction and stabilizer codes, toric codes/surface codes are very tempting, mainly for their high error threshold. For more background please check up, in our Physics sister (aunt?) site: Quantum Error Correction: Surface code vs. color code.

However, these codes require fairly specific measurements in specific bases, which I find hard to translate in practice, especially into my language of interest which is spin states in a solid-state few-spin collection. To see my motivation, here is a not-quite successful attept from a few years ago, using a more naïve QEC scheme: "Quantum Error Correction with magnetic molecules".

So, the problem:

• I am designing a realistic (molecular/supramolecular) network of 7 spins 1/2 (or something almost equivalent, see Is qubit/qudit terms, where is the experimental limit between an S=3/2 and 2·S=1/2?)
• a necessary first: what are the operations/measurements that I need to be able to do? here assume I'm able to do Pulsed electron paramagnetic resonance to address every transition coherently
• and, as a consequence, what is a good coupling scheme between my spins so that the required number of physical operations is reasonable?

Related: How does the size of a toric code torus affect its ability to protect qubits?

Answer 1

I don't know the translation into physics, but the circuit you want for the most basic demonstration is the following: Here, $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$, and the gates are controlled-not gates and controlled-phase gates. The state $|\psi\rangle$ can be any input state initially. The first time the circuit is run prepares the $|\psi\rangle$ qubits in a 4-qubit Toric code (up to some corrections depending on what measurement results you got). Repeat the measurements again, and you get a round of error correction. In effect, what the first qubit is doing is measuring the expectation value of the observable $XXXX$ on the Toric code qubits, while the second qubit measures the $ZZZZ$ observable.

I seem to remember that Jiannis Pachos (and coauthors) explicitly described the smallest Toric code implementation possible (which I guess was this version), but I can't seem to find the paper. I had assumed that JamesWooton would have jumped in by now to tell you where that paper is. It must be commented, however, that such a small size of Toric Code is completely hopeless for error correcting properties; you cannot even correct for single-qubit errors!