Certified algorithms for equilibrium states of local quantum Hamiltonians

Abstract

Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. In the physically relevant thermodynamic limit, certain mathematical formulations of this task have even been shown to result in undecidable problems. Using a finite-size scaling of algorithms devised for finite systems often fails due to the lack of certified convergence bounds for this limit. In this work, we design certified algorithms for computing expectation values of observables in the equilibrium states of local quantum Hamiltonians, both at zero and positive temperature. Importantly, our algorithms output rigorous lower and upper bounds on these values. This allows us to show that expectation values of local observables can be approximated in finite time, contrasting related undecidability results. When the Hamiltonian is commuting on a 2-dimensional lattice, we prove fast convergence of the hierarchy at high temperature and as a result for a desired precision $\epsilon$, local observables can be approximated by a convex optimization program of quasi-polynomial size in $1/\epsilon$.

References in corpus (9)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Approximating Gibbs states of local Hamiltonians efficiently with PEPS
• Computing Ground State Properties with Early Fault-Tolerant Quantum Computers
• Thermal State Preparation via Rounding Promises
• On the complexity of quantum partition functions
• Lower Bounding Ground-State Energies of Local Hamiltonians Through the Renormalization Group
• Certifying ground-state properties of quantum many-body systems
• Entropy Constraints for Ground Energy Optimization
• A subpolynomial-time algorithm for the free energy of one-dimensional quantum systems in the thermodynamic limit

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• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains
• Bootstrapping Nonequilibrium Stochastic Processes