Universal Quantum Hamiltonians

Abstract

Quantum many-body systems exhibit an extremely diverse range of phases and physical phenomena. Here, we prove that the entire physics of any other quantum many-body system is replicated in certain simple, "universal" spin-lattice models. We first characterise precisely what it means for one quantum many-body system to replicate the entire physics of another. We then show that certain very simple spin-lattice models are universal in this very strong sense. Examples include the Heisenberg and XY models on a 2D square lattice (with non-uniform coupling strengths). We go on to fully classify all two-qubit interactions, determining which are universal and which can only simulate more restricted classes of models. Our results put the practical field of analogue Hamiltonian simulation on a rigorous footing and take a significant step towards justifying why error correction may not be required for this application of quantum information technology.

78 pages, 9 figures, 44 theorems etc. v2: Trivial fixes. v3: updated and simplified proof of Thm. 9; 82 pages, 47 theorems etc. v3: Small fix in proof of time-evolution lemma (this fix not in published version)

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• Single-Atom Resolved Fluorescence Imaging of an Atomic Mott Insulator
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• On the complexity of quantum partition functions
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• Supervised learning of time-independent Hamiltonians for gate design
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• Two-local qubit Hamiltonians: when are they stoquastic?
• Effective Hamiltonians for interacting superconducting qubits -- local basis reduction and the Schrieffer-Wolff transformation
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• Uncomputability of Phase Diagrams
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• Analogue Quantum Simulation: A New Instrument for Scientific Understanding
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• General conditions for universality of Quantum Hamiltonians
• Hamiltonian sparsification and gap-simulations
• Holographic duality between local Hamiltonians from random tensor networks
• Quantum Algorithms for Testing Hamiltonian Symmetry
• Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians
• Stoquasticity in circuit QED
• Early Fault-Tolerant Quantum Algorithms in Practice: Application to Ground-State Energy Estimation
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• Symmetry Protected Quantum Computation
• Universal resources for quantum computing
• Translationally invariant universal classical Hamiltonians
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• The Complexity of Translationally Invariant Problems beyond Ground State Energies
• Quantum advantage from energy measurements of many-body quantum systems
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• Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators
• Going Beyond Gadgets: The Importance of Scalability for Analogue Quantum Simulators
• The 7 faces of quantum NP
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• Noisy Toric code and random bond Ising model: The error threshold in a dual picture
• Simplifying the simulation of local Hamiltonian dynamics
• Analogue Quantum Simulation: A Philosophical Prospectus
• Efficiently verifiable quantum advantage on near-term analog quantum simulators
• Oracle complexity classes and local measurements on physical Hamiltonians
• A Framework for Universality in Physics, Computer Science, and Beyond
• Quantum spin compass models in 2D electronic topological metasurfaces
• Perturbative gadgets for gate-based quantum computing: Non-recursive constructions without subspace restrictions
• Security of quantum position-verification limits Hamiltonian simulation via holography
• Universal Hamiltonian simulators in one and two dimensions
• Expanding Hardware-Efficiently Manipulable Hilbert Space via Hamiltonian Embedding
• Pinned QMA: The power of fixing a few qubits in proofs
• Exponential Separation Criteria for Quantum Iterative Power Algorithms
• Classical Simulation of High Temperature Quantum Ising Models
• Stoquastic ground states are classical thermal distributions
• Finite-group gauge theories on lattices as Hamiltonian systems with constraints
• Waveguide QED toolboxes for universal quantum matter