147 research outputs found
Universal Quantum Hamiltonians
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.Comment: 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
The complexity of antiferromagnetic interactions and 2D lattices
Estimation of the minimum eigenvalue of a quantum Hamiltonian can be
formalised as the Local Hamiltonian problem. We study the natural special case
of the Local Hamiltonian problem where the same 2-local interaction, with
differing weights, is applied across each pair of qubits. First we consider
antiferromagnetic/ferromagnetic interactions, where the weights of the terms in
the Hamiltonian are restricted to all be of the same sign. We show that for
symmetric 2-local interactions with no 1-local part, the problem is either
QMA-complete or in StoqMA. In particular the antiferromagnetic Heisenberg and
antiferromagnetic XY interactions are shown to be QMA-complete. We also prove
StoqMA-completeness of the antiferromagnetic transverse field Ising model.
Second, we study the Local Hamiltonian problem under the restriction that the
interaction terms can only be chosen to lie on a particular graph. We prove
that nearly all of the QMA-complete 2-local interactions remain QMA-complete
when restricted to a 2D square lattice. Finally we consider both restrictions
at the same time and discover that, with the exception of the antiferromagnetic
Heisenberg interaction, all of the interactions which are QMA-complete with
positive coefficients remain QMA-complete when restricted to a 2D triangular
lattice.Comment: 35 pages, 11 figures; v2 added reference
Toric codes and quantum doubles from two-body Hamiltonians
We present here a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the low-energy limits of entirely two-body Hamiltonians. Our construction makes use of a new type of perturbation gadget based on error-detecting subsystem codes. The procedure is motivated by a projected entangled pair states (PEPS) description of the target models, and reproduces the target models' behavior using only couplings that are natural in terms of the original Hamiltonians. This allows our construction to capture the symmetries of the target models
The computational complexity of density functional theory
Density functional theory is a successful branch of numerical simulations of
quantum systems. While the foundations are rigorously defined, the universal
functional must be approximated resulting in a `semi'-ab initio approach. The
search for improved functionals has resulted in hundreds of functionals and
remains an active research area. This chapter is concerned with understanding
fundamental limitations of any algorithmic approach to approximating the
universal functional. The results based on Hamiltonian complexity presented
here are largely based on \cite{Schuch09}. In this chapter, we explain the
computational complexity of DFT and any other approach to solving electronic
structure Hamiltonians. The proof relies on perturbative gadgets widely used in
Hamiltonian complexity and we provide an introduction to these techniques using
the Schrieffer-Wolff method. Since the difficulty of this problem has been well
appreciated before this formalization, practitioners have turned to a host
approximate Hamiltonians. By extending the results of \cite{Schuch09}, we show
in DFT, although the introduction of an approximate potential leads to a
non-interacting Hamiltonian, it remains, in the worst case, an NP-complete
problem.Comment: Contributed chapter to "Many-Electron Approaches in Physics,
Chemistry and Mathematics: A Multidisciplinary View
Adiabatic and Hamiltonian computing on a 2D lattice with simple 2-qubit interactions
We show how to perform universal Hamiltonian and adiabatic computing using a
time-independent Hamiltonian on a 2D grid describing a system of hopping
particles which string together and interact to perform the computation. In
this construction, the movement of one particle is controlled by the presence
or absence of other particles, an effective quantum field effect transistor
that allows the construction of controlled-NOT and controlled-rotation gates.
The construction translates into a model for universal quantum computation with
time-independent 2-qubit ZZ and XX+YY interactions on an (almost) planar grid.
The effective Hamiltonian is arrived at by a single use of first-order
perturbation theory avoiding the use of perturbation gadgets. The dynamics and
spectral properties of the effective Hamiltonian can be fully determined as it
corresponds to a particular realization of a mapping between a quantum circuit
and a Hamiltonian called the space-time circuit-to-Hamiltonian construction.
Because of the simple interactions required, and because no higher-order
perturbation gadgets are employed, our construction is potentially realizable
using superconducting or other solid-state qubits.Comment: 33 pages, 5 figure
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