Citation: Bravyi, Sergey B., and Alexei Yu Kitaev. "Fermionic quantum computation." Annals of Physics 298.1 (2002): 210-226.
Web: https://arxiv.org/abs/quant-ph/0003137
Tags: Computer-scientific, Fermionic-order
In this paper, Bravyi and Kitaev say all sorts of wonderful things about quantum computation using fermions and Majorana zero modes.
The first thing this paper does it formally introduce fermionic quantum computation and Majorana quantum computation as distinct modes of computation than standard quantum computation. This has to be done because the creation operators for fermions, when acting on the Fock space in a standard basis, are not local in the bosonic sense! Hence, they have different notions of locality so they are different models.
This paper introduces a generalized notion of locality in terms of C* algebras. It proves that every standard gate on M qubits can be simulated by O(1) gates on M fermions, and that every fermionic gate on M qubits can be simulated on O(log m) gates on M fermions. It defines a universal gate set in the world of Majorana and fermions. It interprets the Shor code in terms of Majorana operators.
An point to note is that there is an asymmetry between standard quantum circuits - O(1) simulations one direction and O(log m) simulation the other. This seems to imply that fermions are fundamentally harder that spins by a linear factor. This is not the case when the notion of locality is changed from "finite domain" to "geometrically local". Once we pass to geometric locality, a lattice stabilizer code is introduced which allows for O(1) simulation time. The way this is discussed in Kitaev's KITP lecture (discussed below) is much nicer - in the KITP lecture he frames it in a way specific to the honeycomb code which makes it look very natural, which I lile.
One funny thing to note about this paper is that it was discussed in Kitaev's KITP lecture "Toward Topological Classification of Phases with Short-range Entanglement", which is linked here: https://online.kitp.ucsb.edu/online/topomat11/kitaev/.
Here is what he said about this paper: "Fermions can be embedded into spins. There is a great general procedure to do this from a paper with Sergey Bravyi some 10 years ago. Unfortunately, I found that nobody read this paper, probably because it was written in computer science language". Interestingly enough, this changed. This paper went from getting five citations per year to getting hundreds. I'm not sure what happened, but the community seems to have rediscovered and fallen in love with this paper.
There's another cute quote from Kitaev's talk: "In the computer science world, the universe is a network of little quantum computers sitting everywhere. Each plank-scale volume is a little computer, and they communicate".