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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• Quantum Computing for Molecular Biology
• On the representation of Boolean and real functions as Hamiltonians for quantum computing
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• Quantum linear systems algorithms: a primer
• Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames
• Toy Models of Holographic Duality between local Hamiltonians
• On the complexity of quantum partition functions
• The Feynman-Kitaev computer's clock: bias, gaps, idling and pulse tuning
• Supervised learning of time-independent Hamiltonians for gate design
• Error Bounds for Variational Quantum Time Evolution
• Reconstructing complex states of a 20-qubit quantum simulator
• Two-local qubit Hamiltonians: when are they stoquastic?
• Effective Hamiltonians for interacting superconducting qubits -- local basis reduction and the Schrieffer-Wolff transformation
• Importance of the spectral gap in estimating ground-state energies
• Uncomputability of Phase Diagrams
• A comparative study of universal quantum computing models: towards a physical unification
• Local minima in quantum systems
• State Preparation in the Heisenberg Model through Adiabatic Spiraling
• Analogue Quantum Simulation: A New Instrument for Scientific Understanding
• Digital-Analog Quantum Computation with Arbitrary Two-Body Hamiltonians
• Josephson Ladders as a Model System for 1D Quantum Phase Transitions
• General conditions for universality of Quantum Hamiltonians
• Hamiltonian sparsification and gap-simulations
• Holographic duality between local Hamiltonians from random tensor networks
• The complexity of simulating local measurements on quantum systems
• Quantum Algorithms for Testing Hamiltonian Symmetry
• Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians
• Early Fault-Tolerant Quantum Algorithms in Practice: Application to Ground-State Energy Estimation
• Stoquasticity in circuit QED
• Phase transition in a noisy Kitaev toric code model
• Symmetry Protected Quantum Computation
• Universal resources for quantum computing
• Translationally invariant universal classical Hamiltonians
• The Complexity of Translationally Invariant Problems beyond Ground State Energies
• Quantum simulation from the bottom up: the case of rebits
• Classical spin Hamiltonians are context-sensitive languages
• Quantum advantage from energy measurements of many-body quantum systems
• Simulation Methodology for Electron Transfer in CMOS Quantum Dots
• Translationally-Invariant Universal Quantum Hamiltonians in 1D
• Individually tunable tunnelling coefficients in optical lattices using local periodic driving
• Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators
• Accreditation of Analogue Quantum Simulators
• Universal Translationally-Invariant Hamiltonians
• Going Beyond Gadgets: The Importance of Scalability for Analogue Quantum Simulators
• The 7 faces of quantum NP
• Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction
• Noisy Toric code and random bond Ising model: The error threshold in a dual picture
• Analogue Quantum Simulation: A Philosophical Prospectus
• Quantum spin compass models in 2D electronic topological metasurfaces
• A Framework for Universality in Physics, Computer Science, and Beyond
• Efficiently verifiable quantum advantage on near-term analog quantum simulators
• Perturbative gadgets for gate-based quantum computing: Non-recursive constructions without subspace restrictions
• Simplifying the simulation of local Hamiltonian dynamics
• Oracle complexity classes and local measurements on physical Hamiltonians
• Expanding Hardware-Efficiently Manipulable Hilbert Space via Hamiltonian Embedding
• Security of quantum position-verification limits Hamiltonian simulation via holography
• Universal Hamiltonian simulators in one and two dimensions
• Pinned QMA: The power of fixing a few qubits in proofs
• Stoquastic ground states are classical thermal distributions
• Classical Simulation of High Temperature Quantum Ising Models
• Exponential Separation Criteria for Quantum Iterative Power Algorithms
• Waveguide QED toolboxes for universal quantum matter
• Finite-group gauge theories on lattices as Hamiltonian systems with constraints