In computer science, the Aharonov–Jones–Landau algorithm is an efficient quantum algorithm for obtaining an additive approximation of the Jones polynomial of a given link at an arbitrary root of unity.
However, it is known that computing an additive approximation of the Jones polynomial is a BQP-complete problem.
[2] The algorithm was published in 2009 in a paper written by Dorit Aharonov, Vaughan Jones and Zeph Landau.
In the early 2000s, a series of papers by Michael Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang demonstrated that topological quantum computers based on topological quantum field theory had the same computational power as quantum circuits.
In particular, they showed that the braiding of Fibonacci anyons could be used to approximate the Jones polynomial evaluated at a primitive 5th root of unity.
Putting these results together, this implies that there is a polynomial length quantum circuit which approximates the Jones polynomial at 5th roots of unity.
This algorithm was completely inaccessible to ordinary quantum computer scientists, however, since the papers by Freedman-Kitaev-Larsen-Wang used heavy machinery from manifold topology.
The contribution of Aharanov-Jones-Landau was to simplify this complicated implicit algorithm in such a way that it would be palatable to a larger audience.
The first idea behind the algorithm is to find a more tractable description for the operation of evaluating the Jones polynomial.
is the number of loops attained by identifying each point in the bottom of
's Kauffman diagram with the corresponding point on top.
is a Kauffman diagram whose rightmost strand goes straight up then
A useful fact exploited by the AJL algorithm is that the Markov trace is the unique trace operator on
be the link attained by identifying the bottom of the diagram with its top like in the definition of a Markov trace, and let
be the result link's Jones polynomial.
As the writhe can be easily calculated classically, this reduces the problem of approximating the Jones polynomial to that of approximating the Markov trace.
We also wish that our representation will have a straightforward encoding into qubits.
We choose define a linear map
below and above the Kauffman diagram in the gaps between the strands then no connectivity component will touch two gaps which are labeled by different numbers.
In order to be able to act on elements of the path model representation by means of quantum circuits, we need to encode the elements of
into qubits in a way which allows us to easily describe the images of the generators
We represent each path as a sequence of moves, where
retains all the properties of the path model representation.
Specifically, it induces a unitary representation
The benefit of this construction is that it gives us a way to represent the Markov trace in a way which can be easily approximated.
be the subspace of paths we described in the previous clause, and let
be the subspace spanned by basis elements which represent walks which end on the
It turns out that this operator is a trace operator with the Markov property, so by the theorem stated above it has to be the Markov trace.
This finishes the required reductions as it establishes that to approximate the Jones polynomial it suffices to approximate
The correctness of this algorithm is established by applying the Hoeffding bound to