Minimising surface-code failures using a color-code decoder

The development of practical, high-performance decoding algorithms reduces the resource cost of fault-tolerant quantum computing. Here we propose a decoder for the surface code that finds low-weight correction operators for errors produced by the depolarising noise model. The decoder is obtained by mapping the syndrome of the surface code onto that of the color code, thereby allowing us to adopt more sophisticated color-code decoding algorithms. Analytical arguments and exhaustive testing show that the resulting decoder can find a least-weight correction for all weight $d/2$ depolarising errors for even code distance $d$. This improves the logical error rate by an exponential factor $O(2^{d/2})$ compared with decoders that treat bit-flip and dephasing errors separately. We demonstrate this improvement with analytical arguments and supporting numerical simulations at low error rates. Of independent interest, we also demonstrate an exponential improvement in logical error rate for our decoder used to correct independent and identically distributed bit-flip errors affecting the color code compared with more conventional color-code decoding algorithms

Optimal quantum spatial search with one-dimensional long-range interactions

Continuous-time quantum walks can be used to solve the spatial search problem, which is an essential component for many quantum algorithms that run quadratically faster than their classical counterpart, in $\mathcal O(\sqrt n)$ time for $n$ entries. However the capability of models found in nature is largely unexplored - e.g., in one dimension only nearest-neighbour Hamiltonians have been considered so far, for which the quadratic speedup does not exist. Here, we prove that optimal spatial search, namely with $\mathcal O(\sqrt n)$ run time and large fidelity, is possible in one-dimensional spin chains with long-range interactions that decay as $1/r^\alpha$ with distance $r$. In particular, near unit fidelity is achieved for $\alpha\approx 1$ and, in the limit $n\to\infty$, we find a continuous transition from a region where optimal spatial search does exist ($\alpha1.5$). Numerically, we show that spatial search is robust to dephasing noise and that, for realistic conditions, $\alpha \lesssim 1.2$ should be sufficient to demonstrate optimal spatial search experimentally with near unit fidelity.Comment: 16 pages, 6 figures; accepted versio

Universality of Z3 parafermions via edge-mode interaction and quantum simulation of topological space evolution with Rydberg atoms

Parafermions are Zn generalizations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. Here we investigate the generation of quantum gates by allowing Z3 parafermions to interact in order to achieve universality. In particular, we study the form of the nontopological gate that arises through direct short-range interaction of the parafermion edge modes in a Z3 parafermion chain. We show that such an interaction gives rise to a dynamical phase gate on the encoded ground space, generating a non-Clifford gate which can be tuned to belong to even levels of the Clifford hierarchy. We illustrate how to access highly noncontextual states using this dynamical gate. Finally, we propose an experiment that simulates the braiding and dynamical evolutions of the Z3 topological states with Rydberg atom technology