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arXiv:2102.02991 · Nature Communications

Universal Hamiltonian simulators in one and two dimensions

Leo Zhou, Dorit Aharonov

Published Apr 21, 2026• 6 citations
arXivHTMLPDF

Abstract

encodes the physics of a target Hamiltonian H. Some families of 2D spin-lattice Hamiltonians-such as the Heisenberg or XY model on the square lattice-are known to be "universal" simulators, in the sense that they can simulate any target Hamiltonian. Unfortunately, while the known simulations of 1D or 2D target Hamiltonians are efficient, they require resources growing exponentially with system sizes for target Hamiltonians with general connectivity. This leaves many important cases such as 3D or all-to-all-connected target Hamiltonians out of reach in practice. Here, we remedy this situation and show how the known 2D universal families of Hamiltonians and a new 1D family can simulate all target local Hamiltonians, with only polynomial-scaling overhead. This exponential improvement is achieved by a nonperturbative method combining the quantum phase-estimation algorithm and the circuit-to-Hamiltonian construction. Furthermore, all Hamiltonians efficiently simulable by quantum circuits, including nonlocal ones, also have efficient analog simulators in our 2D and 1D universal families. Our work establishes that analog simulations of general Hamiltonians can be made efficient, significantly expanding the application potential of analog Hamiltonian simulations in near-term quantum technologies.

DOI
10.1038/s41467-026-71686-4

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encodes the physics of a target Hamiltonian H. Some families of 2D spin-lattice Hamiltonians-such as the Heisenberg or XY model on the square lattice-are known to be "universal" simulators, in the sense that they can simulate any target Hamiltonian. Unfortunately, while the known simulations of 1D or 2D target Hamiltonians are efficient, they require resources growing exponentially with system sizes for target Hamiltonians with general connectivity. This leaves many important cases such as 3D or all-to-all-connected target Hamiltonians out of reach in practice. Here, we remedy this situation and show how the known 2D universal families of Hamiltonians and a new 1D family can simulate all target local Hamiltonians, with only polynomial-scaling overhead. This exponential improvement is achieved by a nonperturbative method combining the quantum phase-estimation algorithm and the circuit-to-Hamiltonian construction. Furthermore, all Hamiltonians efficiently simulable by quantum circuits, including nonlocal ones, also have efficient analog simulators in our 2D and 1D universal families. Our work establishes that analog simulations of general Hamiltonians can be made efficient, significantly expanding the application potential of analog Hamiltonian simulations in near-term quantum technologies.
Digital-analog quantum computation aims to reduce the currently infeasible resource requirements needed for near-term quantum information processing by replacing sequences of one-and two-qubit gates with a unitary transformation generated by the systems' underlying Hamiltonian.Inspired by this paradigm, we consider superconducting architectures and extend the cross-resonance effect, up to first order in perturbation theory, from a two-qubit interaction to an analog Hamiltonian acting on onedimensional (1D) chains and two-dimensional (2D) square lattices, which, in an appropriate reference frame, results in a purely two-local Hamiltonian.By augmenting the analog Hamiltonian dynamics with single-qubit gates we show how one may generate a larger variety of distinct analog Hamiltonians.We then synthesize unitary sequences, in which we toggle between the various analog Hamiltonians as needed, simulating the dynamics of Ising, XY, and Heisenberg spin models.Our dynamics simulations are Trotter error-free for the Ising and XY models in 1D.We also show that the Trotter errors for 2D XY and 1D Heisenberg chains are reduced, with respect to a digital decomposition, by a constant factor.In order to realize these important near-term speedups, we discuss the practical considerations needed to accurately characterize and calibrate our analog Hamiltonians for use in quantum simulations.We conclude with a discussion of how the Hamiltonian toggling techniques could be extended to derive new analog Hamiltonians, which may be of use in more complex digital-analog quantum simulations for various models of interacting spins.
The Hamiltonians of quantum many-body systems can display an extremely diverse range of behaviours.In this thesis we show that there exist simple systems which can simulate all other quantum many-body systems, including all of their physical properties.We call these universal Hamiltonians.We first define a notion of analogue simulation which captures what it means for the Hamiltonian of one system to be reproduced in the Hamiltonian of another physical system.Then we obtain a full classification of simulation power in the case of families of Hamiltonians built up as a linear combination of 2-local qubit interactions.The generalised Heisenberg model and generalised XY model are important special cases which are universal in this classification.We also prove universality for other restricted classes of Hamiltonians such as those for which all the interaction strengths have the same sign or for Hamiltonians with spatially restricted interaction graphs.And we prove universality for large classes of qudit Hamiltonians (where the local dimension d > 2).Universality of a family of Hamiltonians implies that the problem of estimating the ground state energy, known as the Local Hamiltonian problem, is QMA-complete for Hamiltonians from that family.Our results prove that this is the case for many restricted families of Hamiltonians, for which the complexity of the Local Hamiltonian problem was not previously known.We also show that the Local Hamiltonian problem is QMA EXPcomplete for a translationally invariant Hamiltonian of 4-local interactions between spins of dimension d = 4, arranged on a face-centred cubic lattice, by directly encoding a verifier circuit into the interaction terms.Finally we consider the problem of simulating a local measurement in the ground space of a local Hamiltonian, introduced by Ambainis as APX-SIM.We simplify some of the prior work on this problem and prove some previously unknown complexity class inequalities.We show that our notion of simulation preserves hardness of this problem and obtain a complexity classification of APX-SIM for Hamiltonians of 2-local qubit interactions.i Dedication and acknowledgements First and foremost, I must thank my supervisor, Ashley Montanaro, for all of his help over the last few years.He has always made himself available to discuss work, and I am grateful for all of his advice, his problem suggestions, and his excellent proof-reading skills!I must also thank all of my other co-
We construct a family of time-independent nearest-neighbor Hamiltonians coupling eight-state systems on a 1D ring that enables universal quantum computation. Hamiltonians in this family can achieve universality either by driving a continuous-time quantum walk or by terminating an adiabatic algorithm. In either case, the universality property can be understood as arising from an efficient simulation of a programmable quantum circuit. Using gadget perturbation theory, one can demonstrate the same kind of universality for related Hamiltonian families acting on qubits in 2D. Our results demonstrate that simulating 1D chains of spin-7/2 particles is BQP-hard, and indeed BQP-complete because the outputs of decision problems can be encoded in the outputs of such simulations.
Abstract Recent work has characterized rigorously what it means for one quantum system to simulate another and demonstrated the existence of universal Hamiltonians—simple spin lattice Hamiltonians that can replicate the entire physics of any other quantum many-body system. Previous universality results have required proofs involving complicated ‘chains’ of perturbative ‘gadgets.’ In this paper, we derive a significantly simpler and more powerful method of proving universality of Hamiltonians, directly leveraging the ability to encode quantum computation into ground states. This provides new insight into the origins of universal models and suggests a deep connection between universality and complexity. We apply this new approach to show that there are universal models even in translationally invariant spin chains in 1D. This gives as a corollary a new Hamiltonian complexity result that the local Hamiltonian problem for translationally invariant spin chains in one dimension with an exponentially small promise gap is PSPACE-complete. Finally, we use these new universal models to construct the first known toy model of 2D–1D holographic duality between local Hamiltonians.
Analogue and digital Hamiltonian simulation represent two distinct approaches to making one quantum system replicate the physical properties of another, with consequences ranging from practical applications to complexity theory and quantum gravity. They are rooted in two distinct regimes of quantum dynamics: time-independent Hamiltonians for the former; fully controllable time-dependent quantum circuits in the latter. Time-periodic Hamiltonians represent an intermediate regime. In this work, we establish a theory of Floquet simulation where we use periodically driven Hamiltonians to synthesise time-independent ones. We show that a set of interactions SSS can Floquet-simulate every Hamiltonian in the Lie algebra Lie(S)\mathrm{Lie}(S)Lie(S), and consequently we give a complete and constructive characterisation of Floquet-universal Hamiltonians that are able to produce any target Hamiltonian. Notably, our construction uses only O(1)O(1)O(1) local interaction strengths and ratios thereof, avoiding the impractical multi-scale local interaction strengths often required in analogue simulation via time-independent Hamiltonians. Instead, the construction requires a range of driving frequencies which scales polynomially with system size for important classes such as kkk-local lattice Hamiltonians. This has immediate complexity-theoretic implications, including numerous BQP-completeness and QMA-hardness results for natural physical properties and quantities studied in Floquet physics.
Recent work has demonstrated the existence of universal Hamiltonians-simple spin-lattice models that can simulate any other quantum many-body system to any desired level of accuracy.Until now, proofs of universality have relied on explicit constructions, tailored to each specific family of universal Hamiltonians.In this work, we go beyond this approach and completely classify the simulation ability of quantum Hamiltonians by their complexity classes.We do this by deriving necessary and sufficient complexitytheoretic conditions characterizing universal quantum Hamiltonians.Although the result concerns the theory of analog Hamiltonian simulation-a promising application of near-term quantum technology-the proof relies on abstract complexity-theoretic concepts and the theory of quantum computation.As well as providing simplified proofs of previous Hamiltonian universality results and offering a route to new universal constructions, the results in this paper give insight into the origins of universality; for example, finally explaining the previously noted coincidences between families of universal Hamiltonian and classes of Hamiltonians appearing in complexity theory.