  Rapidly developing experiments across multiple platforms now aim to realise
small quantum codes, and so demonstrate a memory within which a logical qubit
can be protected from noise. There is a need to benchmark the achievements in
these diverse systems, and to compare the inherent power of the codes they rely
upon. We describe a recently-introduced performance measure called integrity,
which relates to the probability that an ideal agent will successfully 'guess'
the state of a logical qubit after a period of storage in the memory. Integrity
is straightforward to evaluate experimentally without state tomography and it
can be related to various established metrics such as the logical fidelity and
the pseudo-threshold. We offer a set of experimental milestones that are steps
towards demonstrating unconditionally superior encoded memories. Using
intensive numerical simulations we compare memories based on the five-qubit
code, the seven-qubit Steane code, and a nine-qubit code which is the smallest
instance of a surface code; we assess both the simple and fault-tolerant
implementations of each. While the 'best' code upon which to base a memory does
vary according to the nature and severity of the noise, nevertheless certain
trends emerge.

Quantum Physics