Quantum many-body systems exhibit an extremely diverse range of phases and physical phenomena. However, we prove that the entire physics of any quantum many-body system can be replicated by certain simple, "universal" spin-lattice models. We first characterize precisely what it means for one quantum system to simulate the entire physics of another. We then fully classify the simulation power of all two-qubit interactions, thereby proving that certain simple models can simulate all others, and hence are universal. Our results put the practical field of analogue Hamiltonian simulation on a rigorous footing and take a step toward justifying why error correction may not be required for this application of quantum information technology.
Quantum many-body systems exhibit an extremely diverse range of phases and physical phenomena. However, we prove that the entire physics of any quantum many-body system can be replicated by certain simple, "universal" spin-lattice models. We first characterize precisely what it means for one quantum system to simulate the entire physics of another. We then fully classify the simulation power of all two-qubit interactions, thereby proving that certain simple models can simulate all others, and hence are universal. Our results put the practical field of analogue Hamiltonian simulation on a rigorous footing and take a step toward justifying why error correction may not be required for this application of quantum information technology.
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-
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.
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 S can Floquet-simulate every Hamiltonian in the Lie algebra 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) 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 k-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.
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.
Spin models are used in many studies of complex systems because they exhibit rich macroscopic behavior despite their microscopic simplicity. Here, we prove that all the physics of every classical spin model is reproduced in the low-energy sector of certain "universal models," with at most polynomial overhead. This holds for classical models with discrete or continuous degrees of freedom. We prove necessary and sufficient conditions for a spin model to be universal and show that one of the simplest and most widely studied spin models, the two-dimensional Ising model with fields, is universal. Our results may facilitate physical simulations of Hamiltonians with complex interactions.
Quantum many body physics is an exciting research area, involving novel phases of matter with fundamentally new properties, but is also notoriously hard due to complexity of interacting quantum systems. Some popular approaches involve approximation techniques and numerical simulations, which are known to fail in several important cases. In contrast, rigorous mathematical tools, such as exact solutions and operator inequalities, have a narrower range of applicability, but provide guaranteed results and insights into the underlying physical mechanisms. This research aims to develop new techniques in this direction and use them to explore novel phases of matter. My first direction is to construct toy models where exact solutions are possible. Such models are important as they prove that certain physical phenomena are theoretically possible in nature, and often lead to the discovery of new phases. Such insights are provided by three exactly solvable models I have discovered: (1) a family of 1D quantum spin models hosting free parastatistical quasiparticles (an exotic type of identical particles beyond fermions, bosons, and anyons), proving for the first time that parastatistics is theoretically possible as an emergent phenomenon; (2) a 3D classical Ising model whose phases are characterized by topological features of certain loop observables, suggesting existence of previously unknown classical phases and phase transitions with topological order parameters; and (3) a family of models with exact p-wave superconducting ground states demonstrating the existence of Majorana quasiparticles and non-Abelian statistics in particle number-conserving systems. My second direction is to derive rigorous bounds and exact constraints on physical observables, which are applicable to large families of quantum many-body systems. I present three directions of progress: (1) a method that dramatically improves the upper bounds on the speed of information propagation in locally-interacting systems, which significantly extends the scope of these bounds and enables new applications; (2) bounds on finite-size errors in numerical simulations of lattice systems, including quench dynamics and gapped ground states; and (3) a locality bound on gapped ground states of power-law interacting systems, which leads to a generalization of the aforementioned error bounds to such systems. These error bounds have important theoretical implications such as proving the existence of the thermodynamic limit and stability of phases, and are practically useful in determining the validity of finite-size numerical simulations.