Ashley Montanaro

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 the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model (``OR queries''), the oracle returns whether a given subset of the vertices contains any edges. In the second (``parity queries''), the oracle returns the parity of the number of edges in a subset. In the third model, we are given copies of the graph state corresponding to the graph. We give quantum algorithms that achieve speedups over the best possible classical algorithms in the OR and parity query models, for some families of graphs, and give quantum algorithms in the graph state model whose complexity is similar to the parity query model. For some parameter regimes, the speedups can be exponential in the parity query model. On the other hand, without any promise on the graph, no speedup is possible in the OR query model. A main technique we use is the quantum algorithm for solving the combinatorial group testing problem, for which a query-efficient quantum algorithm was given by Belovs. Here we additionally give a time-efficient quantum algorithm for this problem, based on the algorithm of Ambainis et al.\ for a ``gapped" version of the group testing problem. We also give simple time-efficient quantum algorithms based on Fourier sampling and amplitude amplification for learning the exact-half and majority functions, which almost match the optimal complexity of Belovs' algorithms.

Abstract Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. It can, for example, be used to amplify the correctness of a quantum device whose output is classical. However, when the output of a device is a quantum state, it is not apriori clear how to implement an analogous \emph{quantum} majority vote. To this end, we consider an extension of majority vote to quantum inputs and outputs: given a product state of the form $\ket{\phi_1, \phi_2, \dotsc ,\phi_n}$ where each qubit $\ket{\phi_i}$ is in one of two orthogonal states $\ket{\psi_0}$ or $\ket{\psi_1}$, output the majority state $\ket{\psi_0}$ or $\ket{\psi_1}$. We provide an optimal algorithm for this problem that achieves worst-case fidelity of $1/2 + \Theta(1/\sqrt{n})$. Under the promise that at least $2/3$ of the qubits are in the majority state, the fidelity increases to $1 - \Theta(1/n)$ and approaches one in the limit. More generally, we initiate the study of covariant and symmetric Boolean functions $f: \set{0,1^n} \to \set{0,1}$ with quantum inputs and outputs. We provide a simple linear program of size roughly $n/2$ for computing the optimal worst-case fidelity and show that a generalization of our algorithm is optimal for computing $f$. Our algorithm has complexity $O(n^4 \log n)$ where $n$ is the number of qubits.

In this work we revisit the Boolean Hidden Matching communication problem, which was the first communication problem in the one-way model to demonstrate an exponential classical-quantum communication separation. In this problem, Alice’s bits are matched into pairs according to a partition that Bob holds. These pairs are compressed using a Parity function and it is promised that the final bit-string is equal either to another bit-string Bob holds, or its complement. The problem is to decide which case is the correct one. Here we generalize the Boolean Hidden Matching problem by replacing the parity function with an arbitrary function $f$. Efficient communication protocols are presented depending on the sign-degree of $f$. If its sign-degree is less than or equal to 1, we show an efficient classical protocol. If its sign-degree is less than or equal to $2$, we show an efficient quantum protocol. We then completely characterize the classical hardness of all symmetric functions $f$ of sign-degree greater than or equal to $2$, except for one family of specific cases. We also prove, via Fourier analysis, a classical lower bound for any function $f$ whose pure high degree is greater than or equal to $2$. Similarly, we prove, also via Fourier analysis, a quantum lower bound for any function $f$ whose pure high degree is greater than or equal to $3$. These results give a large family of new exponential classical-quantum communication separations.

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.