[Paper Review] Classical and Quantum Bounded Depth Approximation Algorithms

M. B. Hastings|arXiv (Cornell University)|May 16, 2019

Quantum Computing Algorithms and Architecture47 citations

TL;DR

The paper analyzes local classical and quantum bounded-depth algorithms for MAX-K-LIN-2 and MAX-CUT, showing that simple one-step local classical algorithms can outperform or match one-step QAOA in several cases, and that global-depth cannot be matched by fixed-depth local schemes for certain problems.

ABSTRACT

We consider some classical and quantum approximate optimization algorithms with bounded depth. First, we define a class of "local" classical optimization algorithms and show that a single step version of these algorithms can achieve the same performance as the single step QAOA on MAX-3-LIN-2. Second, we show that this class of classical algorithms generalizes a class previously considered in the literature, and also that a single step of the classical algorithm will outperform the single-step QAOA on all triangle-free MAX-CUT instances. In fact, for all but $4$ choices of degree, existing single-step classical algorithms already outperform the QAOA on these graphs, while for the remaining $4$ choices we show that the generalization here outperforms it. Finally, we consider the QAOA and provide strong evidence that, for any fixed number of steps, its performance on MAX-3-LIN-2 on bounded degree graphs cannot achieve the same scaling as can be done by a class of "global" classical algorithms. These results suggest that such local classical algorithms are likely to be at least as promising as the QAOA for approximate optimization.

Motivation & Objective

• Motivate and formalize a class of local (bounded-depth) optimization algorithms, both classical and quantum, for MAX-K-LIN-2 and MAX-CUT.
• Compare the performance of one-step local classical algorithms against one-step QAOA on selected problems.
• Generalize prior local algorithms and identify regimes where local classical methods can outperform QAOA.
• Demonstrate limitations of local algorithms and discuss when global-parameter schemes are required.

Proposed method

• Define local algorithms as iterative updates of site-associated degrees of freedom with bounded-depth circuits or classical computation.
• Introduce a tensor-algorithm framework where updates are of the form v_{a+1} = g_a(v_a + c_a F_a) or similar, depending on the objective function.
• Specialize to MAX-3-LIN-2 with a single-step update and soft-spin assignment to obtain an objective expectation of Θ(D^{1/4} N).
• Bound higher-order terms using a tensor-network diagrammatic approach and a general lemma for tensor-network contractions to control error terms.
• Relate the local tensor algorithm to simulated annealing and show equivalence in parallel updates.
• Analyze MAX-CUT on triangle-free graphs, comparing one-step QAOA to a local threshold (and soft-threshold) classical algorithm, including optimization over a threshold parameter τ.

Experimental results

Research questions

• RQ1 Can a bounded-depth local classical algorithm match or outperform the one-step QAOA on MAX-3-LIN-2 and MAX-CUT on triangle-free graphs?
• RQ2 What is the scaling of the classical local algorithm's objective value with degree D and graph size N, compared to QAOA?
• RQ3 Do generalizations of local algorithms (tensor algorithms) provide improvements over QAOA, and where are their limitations?
• RQ4 To what extent do higher-depth or nonlocal (global-parameter) choices improve performance beyond bounded-depth local schemes?
• RQ5 How do results extend to MAX-K-LIN-2 for odd K and what are the implications for even K?

Key findings

• For MAX-3-LIN-2, a single-step local tensor algorithm achieves an objective value with expectation Θ(D^{1/4} N), matching the scaling of the one-step QAOA.
• In triangle-free MAX-CUT, a single-step local classical algorithm outperforms the one-step QAOA for all degrees except D in {3,4,6,11}, where some cases still favor the classical method after parameter generalization or numerical checks.
• For almost all degree choices, existing single-step local classical algorithms already surpass the QAOA on triangle-free graphs, with four remaining degree choices where the generalized approach outperforms QAOA.
• The analysis shows that for any fixed number of QAOA steps, its scaling on MAX-3-LIN-2 on bounded-degree graphs cannot reach the performance scaling of a class of global classical algorithms, highlighting limitations of purely local approaches.
• A local classical algorithm (Hirvonen et al. 2014 style) can be viewed as a special case of the tensor-algorithm framework and, when optimized over parameters, can beat QAOA in several settings.
• The paper discusses that while local algorithms can be very competitive, there exist problems where increasing bounded-depth depth does not yield significant improvements, underscoring a fundamental limitation of local schemes.