Reachability Deficits in Quantum Approximate Optimization

Abstract

The quantum approximate optimization algorithm (QAOA) has rapidly become a cornerstone of contemporary quantum algorithm development. Despite a growing range of applications, only a few results have been developed towards understanding the algorithms ultimate limitations. Here we report that QAOA exhibits a strong dependence on a problem instances constraint to variable ratio-this problem density places a limiting restriction on the algorithms capacity to minimize a corresponding objective function (and hence solve optimization problems). Such reachability deficits persist even in the absence of barren plateaus [1] and are outside of the recently reported level-1 QAOA limitations [2]. Building on general numerical experiments, we compare the presence of reachability deficits with analytic solutions of the variational model of Grover's search algorithm. Comparing QAOA's performance between random 3-SAT (NP-hard) and 2-SAT (efficiently solved) instances, reachability deficits increased with problem density. *

Warm-starting quantum optimization

“…The second type of graphs are fully connected with uniformly distributed edge weights sampled from {−10, −9, ..., 0, ..., 9, 10}. We expect that finding the maximum cut for the fully connected graphs will be harder than those with p E = 1/2 [89] and that the resulting QAOA circuits will be deeper as they have more edges [90]. For each graph size and type we randomly generate 100 graphs.…”

“…RQAOA iteratively reduces the problem size and outperforms QAOA on certain forms of Ising Hamiltonians [24]. Implementing QAOA on noisy quantum hardware is challenging as the number of gates can be high for current gate fidelities [38,39]. The circuits become especially deep when large p is required or when the native hardware connectivity does not match the problem structure, thence requiring SWAP gates [40].…”

Modeling and mitigation of cross-talk effects in readout noise with applications to the Quantum Approximate Optimization Algorithm

“…6 exhibit rather poor convergence of the algorithm in a sense that adding more layers above p = 9 changed the resulting energies only slightly. This can be explained by the fact that we did not perform an exhaustive search over hyperparameters, but can also be a manifestation of the recently reported fact that QAOA might have problems with reaching global minima for relatively complicated Hamiltonians (like high-density MAX-2-SAT used in our work) [79]. Clearly, in the context of our work, only the comparison of noisy and noise-mitigated optimization to the noiseless run was of significance.…”

On the universality of the quantum approximate optimization algorithm

“…(1). Studying the sequence length at which any unitary in U(2 n ) can be approximated for certain choices of Hamiltonians or even for unitaries in a subspace A ⊆ U(2 n ) may prove useful in tasks such as state preparation [33,34], modifications of QAOA where constrains are included [35] or for understanding the limitations of this algorithm [36]. It would also be interesting to investigate universality in other variational quantum algorithms; see Ref.…”