We develop an improved variant of U (1)-symmetric infinite projected entangled-pair state (iPEPS) ansatz to investigate the ground state phase diagram of the spin-1/2 square J1 − J2 Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the U (1)-symmetric iPEPS ansatz (for a fixed bond dimension D) set a better upper bound, improving previous tensor-network-based results. By studying the finite-D scaling of the magnetically order parameter, we find a Néel phase for J2/J1 < 0.53. For 0.53 < J2/J1 < 0.61, a non-magnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length ξ ∼ D 1.2 and vanishing order parameters are consistent with a deconfined Néel-to-VBS transition at J c 1 2 /J1 = 0.530(5), where estimated critical anomalous exponents are ηs ∼ 0.6 and η d ∼ 1.9 for spin and dimer correlations respectively. We show that the associated VBS order parameter monotonically increases with J2/J1 and finally a first-order quantum phase transition takes place at J c 2 2 /J1 = 0.610(2) to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.

We propose an algorithm to convert a projected entangled pair state (PEPS) into a canonical form, analogous to the well-known canonical form of a matrix product state. Our approach is based on a variational gauging ansatz for the QR tensor decomposition of PEPS columns into a matrix product operator and a finite depth circuit of unitaries and isometries. We describe a practical initialization scheme that leads to rapid convergence in the QR optimization. We explore the performance and stability of the variational gauging algorithm in norm calculations for the transverse-field Ising and Heisenberg models on a square lattice. We also demonstrate energy optimization within the PEPS canonical form for the transverse-field Ising and Heisenberg models. We expect this canonical form to open up improved analytical and numerical approaches for PEPS.

We characterize the variational power of quantum circuit tensor networks in the representation of physical many-body ground states. Such tensor networks are formed by replacing the dense block unitaries and isometries in standard tensor networks by local quantum circuits. We explore both quantum circuit matrix product states and the quantum circuit multiscale entanglement renormalization Ansatz, and introduce an adaptive method to optimize the resulting circuits to high fidelity with more than 10 4 parameters. We benchmark their expressiveness against standard tensor networks, as well as other common circuit architectures, for the 1D and 2D Heisenberg and 1D Fermi-Hubbard models. We find quantum circuit tensor networks to be substantially more expressive than other quantum circuits for these problems, and that they can even be more compact than standard tensor networks. Extrapolating to circuit depths which can no longer be emulated classically, this suggests a region of advantage in quantum expressiveness in the representation of physical ground states.

We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor J1 and the next-nearest-neighbor J2 couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the Néel magnetic state to the stripe magnetic state at J2/J1 0.549, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cylinder width Ly = 10, we find that around the same critical point the Néel and the Stripe orders are strongly suppressed, which implies the absent of an intermediate phase. Both calculations identify that the stripe order comes with a first-order transition at J2/J1 0.549. Our results indicate that unlike the spin-1/2 J1 − J2 square model, quantum fluctuations in the spin-1 model may not be strong enough to stabilize an intermediate non-magnetic phase.

Simulating complex molecules and materials is an anticipated application of quantum devices. With strong quantum advantage demonstrated in artificial tasks, we examine how such advantage translates into modeling physical problems of correlated electronic structure. We simulate static and dynamical electronic structure on a superconducting quantum processor derived from Google's Sycamore architecture for two representative correlated electron problems: the nitrogenase iron-sulfur molecular clusters, and α-ruthenium trichloride, a proximate spin-liquid material. To do so, we simplify the electronic structure into low-energy spin models that fit on the device. With extensive error mitigation and assistance from classically simulated data, we achieve quantitatively meaningful results deploying about 1/5 of the gate resources used in artificial quantum advantage experiments on a similar architecture.