Limitations of Local Quantum Algorithms on Random Max-k-XOR and Beyond

Abstract

In this work, we study the limitations of the Quantum Approximate Optimization Algorithm (QAOA) through the lens of statistical physics and show that there exists > 0, such that log(n) depth QAOA cannot arbitrarily-well approximate the ground state energy of random diluted k-spin glasses when k ≥ 4 is even. This is equivalent to the weak approximation resistance of logarithmic depth QAOA to the Max-k-XOR problem. We further extend the limitation to other boolean constraint satisfaction problems as long as the problem satisfies a combinatorial property called the coupled overlap-gap property (OGP) [Chen et al., Annals of Probability, 47(3), 2019]. As a consequence of our techniques, we confirm a conjecture of Brandao et al. [arXiv:1812.04170, 2018 asserting that the landscape independence of QAOA extends to logarithmic depth-in other words, for every fixed choice of QAOA angle parameters, the algorithm at logarithmic depth performs almost equally well on almost all instances. Our results provide a new way to study the power and limit of QAOA through statistical physics methods and combinatorial properties.

Performance and limitations of the QAOA at constant levels on large sparse hypergraphs and spin glass models

“…Lastly, we show that the QAOA at any constant p cannot approximate arbitrarily well the ground state values of q-spin models in the average-case when q ≥ 4 and is even (Theorem 4). Previously, this limitation was only shown for some COPs on sparse hypergraphs, via arguments exploiting the OGP and the locality of the QAOA that prevents it from seeing the whole graph at sufficiently low depth [FGG20a,CLSS21]. Importantly, for the fully connected q-spin models we consider, the locality-based arguments do not apply, and no limitation of this kind was known.…”

“…Moreover, recent theoretical results show that the QAOA's level p needs to grow at least logarithmically with problem size n for some COPs on graphs exhibiting locally tree-like structures [BKKT20,FGG20a,FGG20b,CLSS21]. The practical relevance of this limitation on the QAOA is yet to be understood, and furthermore these results do not apply to models of graphs exhibiting full connectivity.…”

“…• G ER d,q (n) -Max-q-XORSAT on a random Erdős-Rényi directed multi-hypergraph [CLSS21]. Here, a random directed multi-hypergraph on n vertices is obtained by first choosing the number of edges m ∼ Poisson(dn), and then choosing hyperedges e 1 , e 2 , .…”

“…This is the overlap gap property (OGP), which roughly says that for certain choices of the disorder J, specifically in the case of the pure q-spin model with q ≥ 4 even, there is a gap in the set of possible pairwise overlaps of near-optimal solutions. The use of OGP to show obstruction for quantum algorithms, specifically the QAOA, was initiated in [FGG20a] and subsequently extended in [CLSS21]. Both work prove limitation of local quantum algorithms when the COPs can be embedded on a sparse hypergraph.…”

Quantum Optimization of Maximum Independent Set using Rydberg Atom Arrays

“…The computational performance of such algorithms has been investigated theoretically [4,[8][9][10][11][12][13] and experimentally [14][15][16] in small quantum systems with shallow quantum circuits, or in systems lacking the many-body coherence believed to be central for quantum advantage [17,18]. However, these studies offer only limited insights into algorithms' performances in the most interesting regime involving large system sizes and high circuit depths [19,20].…”