[Paper Review] The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: Worst Case Examples

Edward Farhi, David Gamarnik+1 more|arXiv (Cornell University)|May 18, 2020

Complexity and Algorithms in Graphs9 references55 citations

TL;DR

The paper shows that at shallow depths, QAOA performance on random d-regular graphs is limited because local neighborhoods dominate; in particular, bipartite random d-regular graphs yield at most a 1/2 approximation for Max-Cut and a d-dependent MIS approximation that tends to 0 as d grows.

ABSTRACT

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 $(d-1)^{2p} < n^A$ 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.

Motivation & Objective

• Motivate and analyze the limitations of the Quantum Approximate Optimization Algorithm (QAOA) at shallow depths on graph-based problems (Max-Cut and Maximum Independent Set).
• Show that for random d-regular graphs, edge-level neighborhoods dominate QAOA outputs when p is small, leading to suboptimal performance.
• Contrast bipartite and general random d-regular graphs to establish worst-case performance results.
• Highlight that when (d-1)^{2p} is o(n), QAOA cannot approximate beyond known bounds for MC and MIS on these graphs.

Proposed method

• Express the cost functions for MC and MIS on d-regular graphs and their edge-local contributions under QAOA.
• Use the locality of U(C,γ) to argue that the objective depends only on p-neighborhoods of edges.
• Show that for random d-regular graphs, almost all edge p-neighborhoods are trees when (d-1)^{2p}<n^{A} for some A<1, enabling tree-based performance bounds.
• Relate QAOA expectations to known optimums for MC (ρ_d) and MIS (σ_d) on large random d-regular graphs.
• Derive upper bounds on the QAOA cost expectation for bipartite random d-regular graphs and connect to worst-case approximation ratios.

Experimental results

Research questions

• RQ1 What is the performance limit of QAOA at shallow depth p on random d-regular graphs for Max-Cut and Maximum Independent Set?
• RQ2 How does restricting to bipartite random d-regular graphs affect QAOA outcomes compared to general random d-regular graphs?
• RQ3 To what extent do tree-like p-neighborhoods determine QAOA's ability to approach optimal solutions?
• RQ4 How do known asymptotic optima for MC and MIS bound the QAOA’s edge-level contributions and overall approximation ratios?

Key findings

• For random d-regular graphs with p = o(log n), almost all edge p-neighborhoods are trees, so QAOA performance is governed by a tree-edge analysis.
• In bipartite random d-regular graphs, despite tree-like neighborhoods, the Max-Cut value can be perfectly satisfiable, yet QAOA cannot exceed a bound implying an approximation ratio no better than 1/2 for large d.
• For MIS on the same graphs at shallow depth, the best approximation ratio is a d-dependent constant that tends to 0 as d grows.
• The expected QAOA objective for MC and MIS is bounded by nd/2 times the tree-edge contributions, yielding Ex[⟨s|U†CU|s⟩] ≤ ρ_d n + O(n^{A'}) and ≤ σ_d n + O(n^{A'}) respectively, where ρ_d and σ_d relate to graph optima.
• Consequently, on bipartite random d-regular graphs, the QAOA cannot achieve an approximation ratio better than 2ρ_d/d + O(n^{A'-1}) for MC and 2σ_d + O(n^{A'-1}) for MIS, with ρ_d ≤ d/4 + O(√d) implying a ratio near 1/2 for large d, and σ_d decaying as d grows.
• The results are proven for shallow depth p growing no faster than a constant times log n; beyond that depth, the current arguments do not apply and see-the-whole-graph effects may alter performance.