Perfect Coding for Dephased Quantum State Transfer

Near-future realisations of quantum technologies, from simple state-generation tasks [1–3] to analogue quantum simulation [4], will be small scale and will rely on intrinsic properties of the system to facilitate as much of their functionality as possible, rather than trying to impose error-prone operations to strong-arm the system into performing tasks. For example, existing quantum simulators such as [5], while scaling to larger numbers of qubits, are not universal computational devices; they are Hamiltonian-based systems with some ability to tune their parameters. Other systems similarly sacrifice computational universality to achieve their ends, from quantum key distribution systems [6] to the DWave quantum computer [7], being tuned to do precisely what they need to. with minimal controls. The difficulty is that Hamiltonian evolution is not obviously compatible with future scaling ambitions. Error correction is a particular challenge given that noise, whose initial action may be well localised, rapidly evolves into potentially destructive correlated errors across an entire device. Can we guard against these effects? The question is especially pertinent when we recognise that interest quantum computers was largely spurred on by the development of a theory of error correction [8, 9]. It had previously been suggested that the continuum of possible errors would make quantum error correction vastly more challenging, if not impossible, compared to the discrete errors of classical theory. Implementation of Hamiltonian-based technologies might be similarly inhibited. In this paper, we study the error correction of one particular protocol, perfect state transfer [10–12], which is well suited to this scenario since the relevant observables will be well localised at particular times. We show that error correction is possible with remarkable efficiency by specifying perfect quantum codes, those which are maximally efficient in that every element in the state space is involved in detecting errors, massively improving the operating regime compared to the preliminary results in [13]. Perfect quantum state transfer is the process whereby an unknown quantum state is transported perfectly from one node of a network to another simply via the evolution of a time-invariant Hamiltonian, often across a one-dimensional spin chain in order to maximise the transfer distance. By suitably engineering the