Cunlu Zhou

Understanding and approximating extremal energy states of local Hamiltonians is a central problem in quantum physics and complexity theory. Recent work has focused on developing approximation algorithms for local Hamiltonians, and in particular the ``Quantum Max Cut'' (QMaxCut) problem, which is closely related to the antiferromagnetic Heisenberg model. In this work, we introduce a family of semidefinite programming (SDP) relaxations based on the Navascues-Pironio-Acin (NPA) hierarchy which is tailored for QMaxCut by taking into account its SU(2) symmetry. We show that the hierarchy converges to the optimal QMaxCut value at a finite level, which is based on a characterization of the algebra of SWAP operators. We give several analytic proofs and computational results showing exactness/inexactness of our hierarchy at the lowest level on several important families of graphs. We also discuss relationships between SDP approaches for QMaxCut and frustration-freeness in condensed matter physics and numerically demonstrate that the SDP-solvability practically becomes an efficiently-computable generalization of frustration-freeness. Furthermore, by numerical demonstration we show the potential of SDP algorithms to perform as an approximate method to compute physical quantities and capture physical features of some Heisenberg-type statistical mechanics models even away from the frustration-free regions.

We present a symmetry-enabled direct quantum algorithm for computing many-body Green’s functions, a central tool for studying strongly correlated quantum systems. Our protocol relies only on native time evolution and straightforward measurements available on current hardware platforms. By exploiting parity symmetry—satisfied by a broad class of Hamiltonians in condensed matter physics and quantum chemistry, including the Fermi–Hubbard and Heisenberg models—we introduce a tailored quench spectroscopy scheme that recovers both the real and imaginary parts of two-point time correlators, from which Green’s functions can be reconstructed via efficient classical signal analysis. We further develop a tailored quantum Gibbs sampler that prepares parity-resolved (symmetric and antisymmetric) thermal states, enabling finite-temperature applications within the same framework. Finally, we show that the same symmetry-based measurement primitive extends naturally to out-of-time-ordered correlators (OTOCs), providing a practical path toward probing finite-temperature dynamics of strongly correlated quantum systems on near-term and early fault-tolerant quantum hardware.