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arXiv:quant-ph/0607094 · Physical Review A

Quantum simulations under translational symmetry

Christina V. Kraus, Michael M. Wolf, J. I. Cirac

Published Feb 6, 2007• 19 citations
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Abstract

We investigate the power of quantum systems for the simulation of Hamiltonian time evolutions on a cubic lattice under the constraint of translational invariance. Given a set of translationally invariant local Hamiltonians and short-range interactions we determine time evolutions which can and those that cannot be simulated. Whereas for general spin systems no finite universal set of generating interactions is shown to exist, universality turns out to be generic for quadratic bosonic and fermionic nearest-neighbor interactions when supplemented by all translationally invariant on-site Hamiltonians.

DOI
10.1103/physreva.75.022303

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•Jan 1, 2019•0 citations
We investigate the power of quantum systems for the simulation of Hamiltonian time evolutions on a cubic lattice under the constraint of translational invariance. Given a set of translationally invariant local Hamiltonians and short-range interactions we determine time evolutions which can and those that cannot be simulated. Whereas for general spin systems no finite universal set of generating interactions is shown to exist, universality turns out to be generic for quadratic bosonic and fermionic nearest-neighbor interactions when supplemented by all translationally invariant on-site Hamiltonians.
In this work we extend the notion of universal quantum Hamiltonians to the setting of translationally-invariant systems. We present a construction that allows a two-dimensional spin lattice with nearest-neighbour interactions, open boundaries, and translational symmetry to simulate any local target Hamiltonian---i.e. to reproduce the whole of the target system within its low-energy subspace to arbitrarily-high precision. Since this implies the capability to simulate non-translationally-invariant many-body systems with translationally-invariant couplings, any effect such as characteristics commonly associated to systems with external disorder, e.g. many-body localization, can also occur within the low-energy Hilbert space sector of translationally-invariant systems. Then we sketch a variant of the universal lattice construction optimized for simulating translationally-invariant target Hamiltonians. Finally we prove that qubit Hamiltonians consisting of Heisenberg or XY interactions of varying interaction strengths restricted to the edges of a connected translationally-invariant graph embedded in RD\mathbb{R}^DRD are universal, and can efficiently simulate any geometrically local Hamiltonian in RD\mathbb{R}^DRD.
We construct a simple translationally invariant, nearest-neighbor Hamiltonian on a chain of ten-dimensional qudits that makes it possible to realize universal quantum computing without any external control during the computational process. We only require the ability to prepare an initial computational basis state that encodes both the quantum circuit and its input. The computational process is then carried out by the autonomous Hamiltonian time evolution. After a time polynomially long in the size of the quantum circuit has passed, the result of the computation is obtained with high probability by measuring a few qudits in the computational basis. This result also implies that there cannot exist efficient classical simulation methods for generic translationally invariant nearest-neighbor Hamiltonians on qudit chains, unless quantum computers can be efficiently simulated by classical computers (or, put in complexity theoretic terms, unless BPP=BQP\mathrm{BPP}=\mathrm{BQP}BPP=BQP).
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.
We generate translationally invariant systems exhibiting many-body localization from all-bands-flat single-particle lattice Hamiltonians dressed with suitable short-range many-body interactions. This phenomenon, dubbed many-body flatband localization, is based on symmetries of both single-particle and interaction terms in the Hamiltonian, and it holds for any interaction strength. We propose a generator of corresponding Hamiltonians which covers both interacting bosons and fermions for arbitrary lattice dimensions, and we provide explicit examples of such models in one and two lattice dimensions. We also explicitly construct an extensive set of local integrals of motion for this set of models. Our results can be further generalized to long-range interactions as well as to systems lacking translational invariance.
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-