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arXiv:2309.10957

Approximation Algorithms for Quantum Max-ddd-Cut

Charlie Carlson, Zackary Jorquera, Alexandra Kolla, Steven Kordonowy, Stuart Wayland

Published Sep 19, 2023• 0 citations
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Abstract

We initiate the algorithmic study of the Quantum Max-$d$-Cut problem, a quantum generalization of the well-known Max-$d$-Cut problem. The Quantum Max-$d$-Cut problem involves finding a quantum state that maximizes the expected energy associated with the projector onto the antisymmetric subspace of two, $d$-dimensional qudits over all local interactions. Equivalently, this problem is physically motivated by the $SU(d)$-Heisenberg model, a spin glass model that generalized the well-known Heisenberg model over qudits. We develop a polynomial-time randomized approximation algorithm that finds product-state solutions of mixed states with bounded purity that achieve non-trivial performance guarantees. Moreover, we prove the tightness of our analysis by presenting an algorithmic gap instance for Quantum Max-d-Cut problem with $d \geq 3$.

DOI
10.48550/arxiv.2309.10957

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We initiate the algorithmic study of the Quantum Max-ddd-Cut problem, a quantum generalization of the well-known Max-ddd-Cut problem. The Quantum Max-ddd-Cut problem involves finding a quantum state that maximizes the expected energy associated with the projector onto the antisymmetric subspace of two, ddd-dimensional qudits over all local interactions. Equivalently, this problem is physically motivated by the SU(d)SU(d)SU(d)-Heisenberg model, a spin glass model that generalized the well-known Heisenberg model over qudits. We develop a polynomial-time randomized approximation algorithm that finds product-state solutions of mixed states with bounded purity that achieve non-trivial performance guarantees. Moreover, we prove the tightness of our analysis by presenting an algorithmic gap instance for Quantum Max-d-Cut problem with d≥3d \geq 3d≥3.
Approximation algorithms for constraint satisfaction problems (CSPs) are a central direction of study in theoretical computer science. In this work, we study classical product state approximation algorithms for a physically motivated quantum generalization of Max-Cut, known as the quantum Heisenberg model. This model is notoriously difficult to solve exactly, even on bipartite graphs, in stark contrast to the classical setting of Max-Cut. Here we show, for any interaction graph, how to classically and efficiently obtain approximation ratios 0.649 (anti-feromagnetic XY model) and 0.498 (anti-ferromagnetic Heisenberg XYZ model). These are almost optimal; we show that the best possible ratios achievable by a product state for these models is 2/3 and 1/2, respectively.
It is well-known in physics that the limit of large quantum spin SSS should be understood as a semiclassical limit. This raises the question of whether such emergent classicality facilitates the approximation of computationally hard quantum optimization problems, such as the local Hamiltonian problem. We demonstrate this explicitly for spin-SSS generalizations of Quantum Max-Cut (QMaxCutS\mathrm{QMaxCut}_SQMaxCutS​), equivalent to the problem of finding the ground state energy of an arbitrary spin-SSS quantum Heisenberg antiferromagnet (QHAS\mathrm{QHA}_SQHAS​). We prove that approximating the value of QHAS\mathrm{QHA}_SQHAS​ to inverse polynomial accuracy is QMA-complete for all SSS, extending previous results for S=1/2S=1/2S=1/2. We also present two distinct families of classical approximation algorithms for QMaxCutS\mathrm{QMaxCut}_SQMaxCutS​ based on rounding the output of a semidefinite program to a product of Bloch coherent states. The approximation ratios for both our proposed algorithms strictly increase with SSS and converge to the Briët-Oliveira-Vallentin approximation ratio αBOV≈0.956α_{\mathrm{BOV}} \approx 0.956αBOV​≈0.956 from below as S→∞S \to \inftyS→∞.
Quantum Max-Cut is a well-studied local Hamiltonian problem for which the classical complexity of the bipartite case remains open. We give a polynomial-time classical randomized algorithm for the spin-12\tfrac1221​ antiferromagnetic Heisenberg model, equivalently Quantum Max-Cut, on dense balanced bipartite expanders. This class includes G(n,n,p)G(n,n,p)G(n,n,p) with high probability for every fixed p>0p>0p>0. The algorithm estimates the ground energy and any ground state edge correlation to additive error ε\varepsilonε in time polynomial in nnn and 1/ε1/\varepsilon1/ε. We construct a Markov chain on perfect matchings of the complete bipartite graph, each matching representing a product of singlets, that converges to the ground state, and use the expansion of the graph to show that polynomially many steps and samples of this chain suffice.
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).
Quantum Max Cut, also known as the anti-ferromagnetic Heisenberg Hamiltonian, is a QMA-complete problem which serves as a benchmark for approximation algorithms in quantum physics. Here we develop a hybrid approximation algorithm to quantum Max Cut, which uses the natural quantum dynamics of Rydberg atom systems in combination with semidefinite programming and randomized rounding. It achieves a conditional approximation ratio of 0.6510.6510.651, compared to the best-known ratio of 0.6140.6140.614 that relies on semidefinite programming alone. The algorithm is robust in the sense that the advantage persists even if the annealing procedure of the Rydberg atom system obtains a state whose energy is only 89%89\%89% of its true ground state energy. Our approach opens a new route for hybrid quantum-classical algorithms that combine quantum with classical optimization methods.
We resolve the approximability of the maximum energy of the Quantum Max Cut (QMC) problem using product states. A classical 0.498-approximation, using a basic semidefinite programming relaxation, is known for QMC, paralleling the celebrated 0.878-approximation for classical Max Cut. For Max Cut, improving the 0.878-approximation is Unique-Games-hard (UG-hard), and one might expect that improving the 0.498-approximation is UG-hard for QMC. In contrast, we give a classical 1/2-approximation for QMC that is unconditionally optimal, since simple examples exhibit a gap of 1/2 between the energies of an optimal product state and general quantum state. Our result relies on a new nonlinear monogamy of entanglement inequality on a triangle that is derived from the second level of the quantum Lasserre hierarchy. This inequality also applies to the quantum Heisenberg model, and our results generalize to instances of Max 2-Local Hamiltonian where each term is positive and has no 1-local parts. Finally, we give further evidence that product states are essential for approximations of 2-Local Hamiltonian.