Certifying ground-state properties of quantum many-body systems

Abstract

A ubiquitous problem in quantum physics is to understand the ground-state properties of many-body systems. Confronted with the fact that exact diagonalisation quickly becomes impossible when increasing the system size, variational approaches are typically employed as a scalable alternative: energy is minimised over a subset of all possible states and then different physical quantities are computed over the solution state. Despite remarkable success, rigorously speaking, all what variational methods offer are upper bounds on the ground-state energy. On the other hand, so-called relaxations of the ground-state problem based on semidefinite programming represent a complementary approach, providing lower bounds to the ground-state energy. However, in their current implementation, neither variational nor relaxation methods offer provable bound on other observables in the ground state beyond the energy. In this work, we show that the combination of the two classes of approaches can be used to derive certifiable bounds on the value of any observable in the ground state, such as correlation functions of arbitrary order, structure factors, or order parameters. We illustrate the power of this approach in paradigmatic examples of 1D and 2D spin-one-half Heisenberg models. To improve the scalability of the method, we exploit the symmetries and sparsity of the considered systems to reach sizes of hundreds of particles at much higher precision than previous works. Our analysis therefore shows how to obtain certifiable bounds on many-body ground-state properties beyond energy in a scalable way.

25 pages, 14 figures, 15 tables

References in corpus (36)

• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• Detecting topological order in a ground state wave function
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Entanglement renormalization
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Zoo of quantum-topological phases of matter
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
• Stripe order in the underdoped region of the two-dimensional Hubbard model
• Finite-Size Scaling of the Ground State Parameters of the Two-Dimensional Heisenberg Model
• Spin Liquid Ground State of the Spin-1/2 Square $J_1$-$J_2$ Heisenberg Model
• The computational complexity of PEPS
• Plaquette Ordered Phase and Quantum Phase Diagram in the Spin-1/2 J1-J2 Square Heisenberg Model
• Study of the Two-Dimensional Frustrated J1-J2 Model with Neural Network Quantum States
• N-representability is QMA-complete
• Spontaneous plaquette dimerization in the $J_1-J_2$ Heisenberg model
• Critical level crossings and gapless spin liquid in the square-lattice spin-$1/2$ $J_1$-$J_2$ Heisenberg antiferromagnet
• Convergent relaxations of polynomial optimization problems with non-commuting variables
• Constructing gapless spin liquid state for the spin-1/2 J1-J2 Heisenberg model on a square lattice
• Dirac-type nodal spin liquid revealed by refined quantum many-body solver using neural-network wave function, correlation ratio, and level spectroscopy
• Approximating strongly correlated spin and fermion wavefunctions with correlator product states
• Ground-State Properties of Quantum Many-Body Systems: Entangled-Plaquette States and Variational Monte Carlo
• Bootstrapping Matrix Quantum Mechanics
• Gapless quantum spin liquid and global phase diagram of the spin-1/2 $J_1$-$J_2$ square antiferromagnetic Heisenberg model
• Gapless spin liquid and valence-bond solid in the $J_1-J_2$ Heisenberg model on the square lattice: insights from singlet and triplet excitations
• Quantum Bounds on Bell inequalities
• Exploiting symmetries in SDP-relaxations for polynomial optimization
• Algorithms for entanglement renormalization: boundaries, impurities and interfaces
• Solving condensed-matter ground-state problems by semidefinite relaxations
• Lower Bounds for Ground States of Condensed Matter Systems
• Long-Range Entangled-Plaquette States for Critical and Frustrated Quantum Systems on a Lattice
• Lower Bounding Ground-State Energies of Local Hamiltonians Through the Renormalization Group
• Certificates of quantum many-body properties assisted by machine learning

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• Bootstrapping Flat-band Superconductors: Rigorous Lower Bounds on Superfluid Stiffness
• Nonadiabatic Self-Healing of Trotter Errors in Digitized Counterdiabatic Dynamics