Computing with noisy fermions

Now on to some of the talks of the last day of the Workshop. The main question addressed by Fernando’s talk is: how noisy can a quantum computer be before efficient classical simulation is possible? The setting in which he addresses this question is topological quantum computation by braiding of Majorana fermions.

Majorana fermions are examples of anyons: particles with exotic exchange symmetry, which gain a complex phase when their position is exchanged, i.e., when they are braided. The estimated error-rate for a single X gate is estimated to be 10^-30, which is great news. The bad news is that the braiding operators are not universal. They can be neatly complemented by ancilla states involving two or four qubits, encoded respectively in four or eight majorana fermions. These are known as magic states, after work by Kitaev and colleagues.

Magic states can be distilled using braiding operators only, so if they have sufficient initial quality (purity), universal computation using only braiding operation becomes possible.

Gaussian states of majorana fermions are:
– quadratic in creation operators;
– fully described by its correlation matrix (that’s why it’s called Gaussian);
– their higher order correlations computable from the correlation matrix.

Fermionic linear optical operations map Gaussian states to Gaussian states; braiding operations are a subgroup of those. Without the use of non-Gaussian ancillas, the whole braiding computation can be simulated efficiently.

Gaussian states have zero volume in the set of all states, which is unhelpful for Fernando’s aproach. His solution was to take the convex hull of Gaussian states, defining the set of Convex-Gaussian states. Fernando and co-workers proved that the volume of this new set is non-zero, and that the simulation scheme for Gaussian states can be easily adapted to simulate these states efficiently.

The question now is: how noisy can the magic state ancillas be before they become convex-Gaussian? They adapted tools from entanglement theory to find bounds on the amount of noise that enables classical simulation. A lower bound for the noise could then be found by inscribing a polytope inside the set of convex-Gaussian states, which facilitates the required optimization.

In the last few minutes of his talk, Fernando presented a bonus result: a hierarchy of tests to determine whether a state is convex-Gaussian or not. Two of the remaining open questions Fernando mentioned were: is there an intermediate regime of noise that results in an intermediate quantum computational power? What’s the value of the exact threshold for simulability?

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