Complexity of the Guided Local Hamiltonian Problem: Improved Parameters and Extension to Excited States

Recently it was shown that the so-called guided local Hamiltonian problem -- estimating the smallest eigenvalue of a k-local Hamiltonian when provided with a description of a quantum state ('guiding state') that is guaranteed to have substantial overlap with the true groundstate -- is BQP-complete for k≥6 when the required precision is inverse polynomial in the system size n, and remains hard even when the overlap of the guiding state with the groundstate is close to a constant (12−Ω(1poly(n))). We improve upon this result in three ways: by showing that it remains BQP-complete when i) the Hamiltonian is 2-local, ii) the overlap between the guiding state and target eigenstate is as large as 1−Ω(1poly(n)), and iii) when one is interested in estimating energies of excited states, rather than just the groundstate. Interestingly, iii) is only made possible by first showing that ii) holds

Quantum algorithms for community detection and their empirical run-times

We apply our recent work on empirical estimates of quantum speedups to the practical task of community detection in complex networks. We design several quantum variants of a popular classical algorithm -- the Louvain algorithm for community detection -- and first study their complexities in the usual way, before analysing their complexities empirically across a variety of artificial and real inputs. We find that this analysis yields insights not available to us via the asymptotic analysis, further emphasising the utility in such an empirical approach. In particular, we observe that a complicated quantum algorithm with a large asymptotic speedup might not be the fastest algorithm in practice, and that a simple quantum algorithm with a modest speedup might in fact be the one that performs best. Moreover, we repeatedly find that overheads such as those arising from the need to amplify the success probabilities of quantum sub-routines such as Grover search can nullify any speedup that might have been suggested by a theoretical worst- or expected-case analysis

Quantifying Grover speed-ups beyond asymptotic analysis

The usual method for studying run-times of quantum algorithms is via an asymptotic, worst-case analysis. Whilst useful, such a comparison can often fall short: it is not uncommon for algorithms with a large worst-case run-time to end up performing well on instances of practical interest. To remedy this it is necessary to resort to run-time analyses of a more empirical nature, which for sufficiently small input sizes can be performed on a quantum device or a simulation thereof. For larger input sizes, alternative approaches are required. In this paper we consider an approach that combines classical emulation with rigorous complexity bounds: simulating quantum algorithms by running classical versions of the sub-routines, whilst simultaneously collecting information about what the run-time of the quantum routine would have been if it were run instead. To do this accurately and efficiently for very large input sizes, we describe an estimation procedure that provides provable guarantees on the estimates that it obtains. A nice feature of this approach is that it allows one to compare the performance of quantum and classical algorithms on particular inputs of interest, rather than only on those that allow for an easier mathematical analysis. We apply our method to some simple quantum speedups of classical heuristic algorithms for solving the well-studied MAX-k-SAT optimization problem. To do this we first obtain some rigorous bounds (including all constants) on the expected- and worst-case complexities of two important quantum sub-routines, which improve upon existing results and might be of broader interest: Grover search with an unknown number of marked items, and quantum maximum-finding. Our results suggest that such an approach can provide insightful and meaningful information, in particular when the speedup is of a small polynomial nature

Improved hardness results for the guided local Hamiltonian problem

Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with 6-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to 1/2 with a ground state. In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry