Verifying the Smallest Interesting Colour Code with Quantomatic

Error correction will form a crucial layer in the software stack of any realistic quantum computer for the foreseeable future. Since the implementation of any error correction scheme will depend on the details of the actual hardware, it is generally expected that error correction will be added to a quantum program late in the compilation process (see, for example [18]). In other words, the fault-tolerant executable program will be automatically generated from a higher-level description which is unaware of the error-correction scheme to be employed. At a minimum, any such translation process from “logical” quantum circuits to their fault-tolerant versions should be proven sound—i.e. that translation does not change the meaning of the program. However we might demand more. For example, with knowledge of the hardware operations and the fault-tolerant program, it may be possible to optimise beneath the error correction scheme. Such optimisations again require correctness proofs, not just of soundness, but to guarantee that the optimisation preserves fault-tolerance. To do any of this, a language combining circuits, error-correcting schemes, and translations between the two is required, and this language should support robust and powerful automated reasoning, capable of proving that relevant properties of error-correcting schemes hold. In this paper we present a case study along these lines. Precisely, we use the zx-calculus [12] as a language, in concert with the interactive theorem prover Quantomatic [19, 24], to study the Smallest Interesting Colour Code [7]. We provide formal proofs of the basic properties of the code itself and its fault-tolerant operations. A similar study was conducted in 2013 for the 7-qubit Steane code [15] (see also the recent [10]), however the Quantomatic system has undergone significant development in the intervening four years, and is vastly more powerful. Many of our proofs can be produced automatically by Quantomatic; others require the formalisation of human insight into reusable tactics; still others resist full automation with the current technology. In the final section we discuss the obstacles encountered, and desiderata for future development of the zx-calculus/ Quantomatic system.