We prove that the maximum eigenvalue of the (both signed and unsigned) Laplacian of level $k$ Kikuchi graph of any graph $G$ with $m$ edges is at most $m+k$. This confirms four recent conjectures of Apte, Parekh, and Sud.
As applications, we obtain that tensor products of one and two qubit product states achieve an approximation ratio of $5/8$ for Quantum Max Cut and $5/7$ for the XY Hamiltonian. Moreover, combining our bounds with the algorithms analyzed by Apte, Parekh, and Sud, yields efficient algorithms achieving an approximation ratio of $0.614$ for Quantum Max Cut and $0.674$ for the XY Hamiltonian. Finally, we also make modest progress on Brouwer's conjecture and improve Lew's bound on the sum of the top-$k$ eigenvalues of a Graph Laplacian.
We prove that the maximum eigenvalue of the (both signed and unsigned) Laplacian of level k Kikuchi graph of any graph G with m edges is at most m+k. This confirms four recent conjectures of Apte, Parekh, and Sud.
As applications, we obtain that tensor products of one and two qubit product states achieve an approximation ratio of 5/8 for Quantum Max Cut and 5/7 for the XY Hamiltonian. Moreover, combining our bounds with the algorithms analyzed by Apte, Parekh, and Sud, yields efficient algorithms achieving an approximation ratio of 0.614 for Quantum Max Cut and 0.674 for the XY Hamiltonian. Finally, we also make modest progress on Brouwer's conjecture and improve Lew's bound on the sum of the top-k eigenvalues of a Graph Laplacian.
We consider a computational problem where the goal is to approximate the maximum eigenvalue of a two-local Hamiltonian that describes Heisenberg interactions between qubits located at the vertices of a graph. Previous work has shed light on this problem's approximability by product states. For any instance of this problem the maximum energy attained by a product state is lower bounded by the Max Cut of the graph and upper bounded by the standard Goemans-Williamson semidefinite programming relaxation of it. Gharibian and Parekh described an efficient classical approximation algorithm for this problem which outputs a product state with energy at least 0.498 times the maximum eigenvalue in the worst case, and observe that there exist instances where the best product state has energy 1/2 of optimal. We investigate approximation algorithms with performance exceeding this limitation which are based on optimizing over tensor products of few-qubit states and shallow quantum circuits. We provide an efficient classical algorithm which achieves an approximation ratio of at least 0.53 in the worst case. We also show that for any instance defined by a 3- or 4-regular graph, there is an efficiently computable shallow quantum circuit that prepares a state with energy larger than the best product state (larger even than its semidefinite programming relaxation).
We study polynomial-time approximation algorithms for the Quantum Max-Cut (QMC) problem. Given an edge-weighted graph G on n vertices, the QMC problem is to determine the largest eigenvalue of a particular 2n×2n matrix that corresponds to G. We provide a sharpened analysis of the currently best-known QMC approximation algorithm for general graphs. This algorithm achieves an approximation ratio of 0.599, which our analysis improves to 0.603. Additionally, we propose two new approximation algorithms for the QMC problem on triangle-free and bipartite graphs, that achieve approximation ratios of 0.61383 and 0.8162, respectively. These are the best-known approximation ratios for their respective graph classes.
Quantum Max Cut is a problem relevant to computer science and many-body quantum physics due to its links to classical Max Cut and the anti-ferromagnetic Heisenberg Hamiltonian. We prove a lower bound to quantum Max Cut of a graph in terms of the Lovász theta function of its complement. For a graph with m edges, qmc(G)≥4m(1+3π8ϑ(Gˉ)−11), with the bound achieved by a product state. The proof can be strenghtened by the vector chromatic number and extends a result by Balla, Janzer, and Sudakov on classical Max Cut. A relaxed bound follows from ϑ(Gˉ)−1≤Δ for graphs with maximum degree Δ, making it interesting for practically relevant quantum many-body systems. We also extend results by Carlson et al. and Shearer and show that qmc(G)≥4m+3π2m3/4 for all triangle-free graphs with m edges.
Finding a high (or low) energy state of a given quantum Hamiltonian is a potential area to gain a provable and practical quantum advantage. A line of recent studies focuses on Quantum Max Cut, where one is asked to find a high energy state of a given antiferromagnetic Heisenberg Hamiltonian. In this work, we present a classical approximation algorithm for Quantum Max Cut that achieves an approximation ratio of 0.595, outperforming the previous best algorithms of Lee [Eunou Lee, 2022] (0.562, generic input graph) and King [King, 2023] (0.582, triangle-free input graph). The algorithm is based on finding the maximum weighted matching of an input graph and outputs a product of at most 2-qubit states, which is simpler than the fully entangled output states of the previous best algorithms.
Assuming the Unique Games Conjecture, we show it is NP-hard to approximate MAX-3-CUT within a multiplicative factor of α3+ε for every ε>0, where α3≈.83600811464 is the approximation ratio of Frieze-Jerrum's polynomial-time algorithm from 1995. That is, we prove sharp hardness of approximation for MAX-3-CUT. This result resolves a conjecture of Khot-Kindler-Mossel-O'Donnell from 2004 by proving the three candidate Plurality is Stablest Conjecture for correlations in [−1/2,2/5] and generalizes the Majority is Stablest Theorem of Mossel-O'Donnell-Oleszkiewicz [Annals of Math, 2010].
With a similar strategy we prove: assuming the Unique Games Conjecture, it is NP-hard to approximate the product-state value of Quantum MAX-CUT within a multiplicative factor of αBOV+ε for every ε>0, where αBOV≈0.9563372685 is the approximation ratio of the Briët-de Oliveira Filho-Vallentin algorithm. This sharp hardness result completes the conjectured hardness of Hwang-Neeman-Parekh-Thompson-Wright from 2021 by proving their Sk−1-valued Borell inequality for correlations in [−.5843,.5843] for all k≥3.
In this paper, we continue a line of research initiated by Basu, Brakensiek, and Putterman [2026] studying the sparsifiability of Hamiltonians. We focus particularly on the sparsifiability of the widely-studied Quantum Cut (QC) Hamiltonians. Our main result is that in an n-qubit system, any n-qubit QC Hamiltonian can be sparsified to O(n/ε2) many terms while preserving the energy of every state up to a factor of 1±ε. Our result can be interpreted as giving an importance sampling scheme for the edges of an arbitrary graph G such that the \emph{Kikuchi} graph at level ℓ of the sampled graph is a spectral approximation to the Kikuchi graph of G. Importantly, the \emph{same} sampling scheme works simultaneously for all ℓ.
The natural approach of leverage score sampling, analyzed via matrix concentration inequalities, yields a polynomially worse bound in our setting because the underlying matrices have dimension ∼2n. Instead, our approach relies on decomposing the action of these matrices into invariant subspaces. Then, by using an operator-valued inequality of Alon and Kozma [Ann. Henri Poincaré, 2020], itself building on an \emph{octopus inequality} of Caputo, Liggett, and Richthammer [J. AMS, 2010], we extend our sparsification technique to all expander graphs. We then invoke expander decomposition to extend our sparsifier to all graphs.