We design two variational algorithms to optimize specific 2-local Hamiltonians defined on graphs. Our algorithms are inspired by the Quantum Approximate Optimization Algorithm. We develop formulae to analyze the energy achieved by these algorithms with high probability over random regular graphs in the infinite-size limit, using techniques from [arXiv:2110.14206]. The complexity of evaluating these formulae scales exponentially with the number of layers of the algorithms, so our numerical evaluation is limited to a small constant number of layers. We compare these algorithms to simple classical approaches and a state-of-the-art worst-case algorithm. We find that the symmetry inherent to these specific variational algorithms presents a major \emph{obstacle} to successfully optimizing the Quantum MaxCut (QMC) Hamiltonian on general graphs. Nonetheless, the algorithms outperform known methods to optimize the EPR Hamiltonian of [arXiv:2209.02589] on random regular graphs, and the QMC Hamiltonian when the graphs are also bipartite. As a special case, we show that with just five layers of our algorithm, we can already prepare states within 1.62% error of the ground state energy for QMC on an infinite 1D ring, corresponding to the antiferromagnetic Heisenberg spin chain.
Local Hamiltonian Problems (LHPs) are important problems that are computationally QMA-complete and physically relevant for many-body quantum systems. Quantum MaxCut (QMC), which equates to finding ground states of the quantum Heisenberg model, is the canonical LHP for which various algorithms have been proposed, including semidefinite programs and variational quantum algorithms. We propose and analyze a quantum approximation algorithm which we call the Hamiltonian Quantum Approximate Optimization Algorithm (HamQAOA), which builds on the well-known scheme for combinatorial optimization and is suitable for implementations on near-term hardware. We establish rigorous performance guarantees of the HamQAOA for QMC on high-girth regular graphs, and our result provides bounds on the ground energy density for quantum Heisenberg spin glasses in the infinite size limit that improve with depth. Furthermore, we develop heuristic strategies with which to efficiently obtain good HamQAOA parameters. Through numerical simulations, we show that the HamQAOA empirically outperforms prior algorithms on a wide variety of QMC instances. In particular, our results indicate that the linear-depth HamQAOA can deterministically prepare exact ground states of 1-dimensional antiferromagnetic Heisenberg spin chains described by the Bethe ansatz, in contrast to the exponential depths required in previous protocols for preparing Bethe states.
We explain how the maximum energy of the Quantum MaxCut, XY, and EPR Hamiltonians on a graph
G are related to the spectral radii of the token graphs of
G. From numerical study, we conjecture new bounds for these spectral radii based on properties of
G. We show how these conjectures tighten the analysis of existing algorithms, implying state-of-the-art approximation ratios for all three Hamiltonians. Our conjectures also provide simple combinatorial bounds on the ground state energy of the antiferromagnetic Heisenberg model, which we prove for bipartite graphs.
Combinatorial optimization is regarded as a potentially promising application of near and long-term quantum computers. The best-known heuristic quantum algorithm for combinatorial optimization on gate-based devices, the Quantum Approximate Optimization Algorithm (QAOA), has been the subject of many theoretical and empirical studies. Unfortunately, its application to specific combinatorial optimization problems poses several difficulties: among these, few performance guarantees are known, and the variational nature of the algorithm makes it necessary to classically optimize a number of parameters. In this work, we partially address these issues for a specific combinatorial optimization problem: diluted spin models, with MAX-CUT as a notable special case. Specifically, generalizing the analysis of the Sherrington-Kirkpatrick model by Farhi et al., we establish an explicit algorithm to evaluate the performance of QAOA on MAX-CUT applied to random Erdos-Renyi graphs of expected degree
d for an arbitrary constant number of layers
p and as the problem size tends to infinity. This analysis yields an explicit mapping between QAOA parameters for MAX-CUT on Erdos-Renyi graphs of expected degree
d, in the limit
d→∞, and the Sherrington-Kirkpatrick model, and gives good QAOA variational parameters for MAX-CUT applied to Erdos-Renyi graphs. We then partially generalize the latter analysis to graphs with a degree distribution rather than a single degree
d, and finally to diluted spin-models with
D-body interactions (
D≥3). We validate our results with numerical experiments suggesting they may have a larger reach than rigorously established; among other things, our algorithms provided good initial, if not nearly optimal, variational parameters for very small problem instances where the infinite-size limit assumption is clearly violated.
Quantum Max-Cut is a well-studied local Hamiltonian problem for which the classical complexity of the bipartite case remains open. We give a polynomial-time classical randomized algorithm for the spin-
21 antiferromagnetic Heisenberg model, equivalently Quantum Max-Cut, on dense balanced bipartite expanders. This class includes
G(n,n,p) with high probability for every fixed
p>0. The algorithm estimates the ground energy and any ground state edge correlation to additive error
ε in time polynomial in
n and
1/ε. We construct a Markov chain on perfect matchings of the complete bipartite graph, each matching representing a product of singlets, that converges to the ground state, and use the expansion of the graph to show that polynomially many steps and samples of this chain suffice.
The Quantum Approximate Optimization Algorithm can be applied to search problems on graphs with a cost function that is a sum of terms corresponding to the edges. When conjugating an edge term, the QAOA unitary at depth p produces an operator that depends only on the subgraph consisting of edges that are at most p away from the edge in question. On random d-regular graphs, with d fixed and with p a small constant time log n, these neighborhoods are almost all trees and so the performance of the QAOA is determined only by how it acts on an edge in the middle of tree. Both bipartite random d-regular graphs and general random d-regular graphs locally are trees so the QAOA's performance is the same on these two ensembles. Using this we can show that the QAOA with
(d−1)2p<nA for any
A<1, can only achieve an approximation ratio of 1/2 for Max-Cut on bipartite random d-regular graphs for d large. For Maximum Independent Set, in the same setting, the best approximation ratio is a d-dependent constant that goes to 0 as d gets big.
We consider the maximum cut and maximum independent set problems on random regular graphs in the infinite-size limit, and calculate the energy densities achieved by QAOA for high degrees up to d = 100 . Such an analysis is possible because the reverse causal cones of the operators in the Hamiltonian are with high probability associated with tree subgraphs, for which efficient classical contraction schemes can be developed. We combine the QAOA analysis with state-of-the-art upper bounds on optimality for both problems. This yields novel and better bounds on the approximation ratios achieved by QAOA for large problem sizes. We show that the approximation ratios achieved by QAOA improve as the graph degree increases for the maximum cut problem. However, QAOA exhibits the opposite behavior for the maximum independent set problem, i.e. the achieved approximation ratios decrease when the degree of the problem is increased. This phenomenon is explainable by the overlap gap property for large d , which restricts local algorithms (like QAOA) from reaching near-optimal solutions with high probability. In addition, we use the QAOA parameters determined on the tree subgraphs for small graph instances, and in that way outperform classical algorithms like Goemans-Williamson for the maximum cut problem and minimal greedy for the maximum independent set problem. In this way we circumvent the parameter optimization problem and are able to compute the expected approximation ratios.