Stephen Piddock

The limiting constraint on simulating materials on near-term quantum hardware is the requisite circuit depths and qubit numbers, with current estimates placing them well beyond near-term capabilities. A critical subroutine of simulation algorithms is implementing a layer of unitary evolutions by each local term in the Hamiltonian. In this work we develop a new quantum algorithm which dramatically reduces the estimated cost of material simulations using this subroutine, improving circuit depths by up to 6 orders of magnitude for Strontium Vanadate, for example. We achieve this by introducing a fermionic encoding that leverages the locality of materials Hamiltonians describing an active space in the Wannier basis. This design generates quantum circuits whose depth is independent of the system’s size.

We study an elementary Markov process on graphs based on electric flow sampling (elfs). The elfs process repeatedly samples from an electric flow on a graph. While the sinks of the flow are fixed, the source is updated using the electric flow sample, and the process ends when it hits a sink vertex. We argue that this process naturally connects to many key quantities of interest. E.g., we describe a random walk coupling which implies that the elfs process has the same arrival distribution as a random walk. We also analyze the electric hitting time, which is the expected time before the process hits a sink vertex. As our main technical contribution, we show that the electric hitting time on trees is logarithmic in the graph size and weights. The initial motivation behind the elfs process is that quantum walks can sample from electric flows, and they can hence implement this process very naturally. This yields a quantum walk algorithm for sampling from the random walk arrival distribution, which has widespread applications. It complements the existing line of quantum walk search algorithms which only return an element from the sink, but yield no insight in the distribution of the returned element. By our bound on the electric hitting time on trees, the quantum walk algorithm on trees requires quadratically fewer steps than the random walk hitting time, up to polylog factors.

Estimation of the minimum eigenvalue of a quantum Hamiltonian can be formalised as the Local Hamiltonian problem. We study the natural special case of the Local Hamiltonian problem where the same 2-local interaction, with differing weights, is applied across each pair of qubits. First we consider antiferromagnetic/ferromagnetic interactions, where the weights of the terms in the Hamiltonian are restricted to all be of the same sign. We show that for symmetric 2-local interactions with no 1-local part, the problem is either QMA-complete or in StoqMA. In particular the antiferromagnetic Heisenberg and antiferromagnetic XY interactions are shown to be QMA-complete. We also prove StoqMA-completeness of the antiferromagnetic transverse field Ising model. Second, we study the Local Hamiltonian problem under the restriction that the interaction terms can only be chosen to lie on a particular graph. We prove that nearly all of the QMA-complete 2-local interactions remain QMA-complete when restricted to a 2D square lattice. Finally we consider both restrictions at the same time and discover that, with the exception of the antiferromagnetic Heisenberg interaction, all of the interactions which are QMA-complete with positive coefficients remain QMA-complete when restricted to a 2D triangular lattice.