The paper introduces a novel method to efficiently construct a quite general class of quantum states on a quantum computer. This class of quantum states is known as Projected Entangled Pair States (PEPS) and is conjectured to represent ground states of gapped Hamiltonians faithfully while having a polynomial-sized classical description. Our algorithm can reconstruct such states from their description whenever they satisfy certain additional properties (injectivity, gapped parent Hamiltonian) which are considered quite general and not too restrictive. It is clear that some restrictions must be placed on the subset of efficiently constructible PEPS as otherwise this would imply the computational power of solving PP-complete problems, which is well beyond the conjectured capabilities of a quantum computer. Our result essentially says, that all reasonably defined PEPS are also physically realizable.
Another interesting point about our result is the fact that the PEPS are classically described as the (partial) contraction of a tensor network. A tensor is simply a multi-dimensional array of complex numbers. A tensor with just one index is a vector, a tensor with two indices is a matrix, but in general tensor can have an arbitrary number of indices. A tensor network is simply a graph, where a tensor is attached to each vertex with as many indices as there are edges coming into the vertex, and where each edge can be labeled with a number from 1 to up to the dimension of the associated tensor index (of course tensor index dimensions at the two ends of the edge must match up).
A PEPS is a special class of tensor networks, where each vertex has an open index, called the physical index, representing the state of the local quantum system represented by this vertex. The quantum state represented by the PEPS is simply the contraction of (sum over) all closed edges of the tensor network.
Contracting a tensor network as described above is a quite generic primitive that can be used to model many kinds of exponential sums over products of variables. Such sums-of-products are seen for example in the partition function of systems in statistical physica, quantum spin systems of course, the classical simulation of quantum computers, and all kinds of counting problems. In general, such problems have been shown to be #P-complete and are thus considered intractable for classical computers. Interestingly, our paper shows that a specific subset of these partition functions can be represented as a quantum state and can be constructed efficiently on a quantum computer. It would be interesting to understand, if any new counting problem or other classical problem not known to be tractable classically (but also not shown #P-complete) could be solved by constructing the respective quantum state and measuring some non-trivial observable. This would exhibit a new type of quantum algorithm yielding a speedup over classical computers. First steps in this direction have been taken by Arad and Landau, and van den Nest, et al., who give efficient contractions of certain classical partition functions on quantum computers. These results, though, do not rely on explicitly constructing a quantum state implied by the partition function but use other methods to approximate this contraction value using a quantum computer. In this sense, our result provides a new, generic tool to these approaches, eventhough we have not really shown its efficiveness in this context yet.
I’ve had a lot of fun working out this result jointly with Frank and Kristan. It uses quite a few nice linear algebra tricks, such as Jordan’s Lemma on the interaction of two subspaces and the fact that the injectivity property of PEPS implies it is the unique ground state of an easily constructed parent Hamiltonian.