80 research outputs found
The Bose-Hubbard model is QMA-complete
The Bose-Hubbard model is a system of interacting bosons that live on the
vertices of a graph. The particles can move between adjacent vertices and
experience a repulsive on-site interaction. The Hamiltonian is determined by a
choice of graph that specifies the geometry in which the particles move and
interact. We prove that approximating the ground energy of the Bose-Hubbard
model on a graph at fixed particle number is QMA-complete. In our QMA-hardness
proof, we encode the history of an n-qubit computation in the subspace with at
most one particle per site (i.e., hard-core bosons). This feature, along with
the well-known mapping between hard-core bosons and spin systems, lets us prove
a related result for a class of 2-local Hamiltonians defined by graphs that
generalizes the XY model. By avoiding the use of perturbation theory in our
analysis, we circumvent the need to multiply terms in the Hamiltonian by large
coefficients
Complexity of the XY antiferromagnet at fixed magnetization
We prove that approximating the ground energy of the antiferromagnetic XY
model on a simple graph at fixed magnetization (given as part of the instance
specification) is QMA-complete. To show this, we strengthen a previous result
by establishing QMA-completeness for approximating the ground energy of the
Bose-Hubbard model on simple graphs. Using a connection between the XY and
Bose-Hubbard models that we exploited in previous work, this establishes
QMA-completeness of the XY model
Adiabatic quantum algorithm for the Bose-Hubbard model
MĂ ster Oficial de Ciència i Tecnologia QuĂ ntiques / Quantum Science and Technology, Facultat de FĂsica, Universitat de Barcelona. Curs: 2021-2022. Tutor: Axel PĂ©rez-ObiolIn recent years research has been carried out on algorithms to simulate quantum many body systems in current NISQ devices. In particular, for the ground state finding problem, known to be QMA-complete, a quantum adiabatic algorithm can be used. On the other hand, the Bose-Hubbard model has gained impact lately because of the prediction of exotic phases of matter and because its experimental realisation in a set up with cold atoms in optical lattices. In this work, an adiabatic quantum algorithm is designed to obtain
the ground state of a one-dimensional Bose-Hubbard model. The three parts of the algorithm are presented: initial state preparation, adiabatic evolution and measurement. The results presented correspond to a system of 2 sites and NP particles, although the algorithm has been tested for systems with more sites.
The algorithm has been tested by performing simulation
Quantum Hamiltonian Complexity
Constraint satisfaction problems are a central pillar of modern computational
complexity theory. This survey provides an introduction to the rapidly growing
field of Quantum Hamiltonian Complexity, which includes the study of quantum
constraint satisfaction problems. Over the past decade and a half, this field
has witnessed fundamental breakthroughs, ranging from the establishment of a
"Quantum Cook-Levin Theorem" to deep insights into the structure of 1D
low-temperature quantum systems via so-called area laws. Our aim here is to
provide a computer science-oriented introduction to the subject in order to
help bridge the language barrier between computer scientists and physicists in
the field. As such, we include the following in this survey: (1) The
motivations and history of the field, (2) a glossary of condensed matter
physics terms explained in computer-science friendly language, (3) overviews of
central ideas from condensed matter physics, such as indistinguishable
particles, mean field theory, tensor networks, and area laws, and (4) brief
expositions of selected computer science-based results in the area. For
example, as part of the latter, we provide a novel information theoretic
presentation of Bravyi's polynomial time algorithm for Quantum 2-SAT.Comment: v4: published version, 127 pages, introduction expanded to include
brief introduction to quantum information, brief list of some recent
developments added, minor changes throughou
Entanglement Theory and the Quantum Simulation of Many-Body Physics
In this thesis we present new results relevant to two important problems in
quantum information science: the development of a theory of entanglement and
the exploration of the use of controlled quantum systems to the simulation of
quantum many-body phenomena.
In the first part we introduce a new approach to the study of entanglement by
considering its manipulation under operations not capable of generating
entanglement and show there is a total order for multipartite quantum states in
this framework. We also present new results on hypothesis testing of correlated
sources and give further evidence on the existence of NPPT bound entanglement.
In the second part, we study the potential as well as the limitations of a
quantum computer for calculating properties of many-body systems. First we
analyse the usefulness of quantum computation to calculate additive
approximations to partition functions and spectral densities of local
Hamiltonians. We then show that the determination of ground state energies of
local Hamiltonians with an inverse polynomial spectral gap is QCMA-hard.
In the third and last part, we approach the problem of quantum simulating
many-body systems from a more pragmatic point of view. We analyze the
realization of paradigmatic condensed matter Hamiltonians in arrays of coupled
microcavities, such as the Bose-Hubbard and the anisotropic Heisenberg models,
and discuss the feasibility of an experimental realization with
state-of-the-art current technology.Comment: 230 pages. PhD thesis, Imperial College London. Chapters 6, 7 and 8
contain unpublished materia
Exact Gate Decompositions for Photonic Quantum Computers
The purpose of this work is to examine the use of decompositions on a continuous-variable quantum computer by both implementing and examining known methods, as well as to expand on them by developing my own. I detail the usage of known and new techniques for gate decompositions in some useful quantum algorithms such as simulating bosonic particles in a optical lattice, and solving differential equations with broad applications in other scientific fields. The new methods detailed in this work provide decompositions for continuous variable quantum computers which no longer require approximations. These methods rely on strategically using unitary conjugation and a lemma to the Baker-Campbell-Hausdorff formula to derive new exact decompositions from previously known ones, leading to exact decompositions for a large class of gates. I also demonstrate how exact decompositions can be employed in a wide range of algorithms, while requiring much fewer gates (sometimes as many as order-of-magnitude less) than equivalent decompositions with other methods. This work can potentially further bridge the gap between what is required to perform algorithms on a quantum computer and what can be done experimentally
Introduction to Quantum Algorithms for Physics and Chemistry
In this introductory review, we focus on applications of quantum computation
to problems of interest in physics and chemistry. We describe quantum
simulation algorithms that have been developed for electronic-structure
problems, thermal-state preparation, simulation of time dynamics, adiabatic
quantum simulation, and density functional theory.Comment: 44 pages, 5 figures; comments or suggestions for improvement are
welcom
Complexity Classification of Local Hamiltonian Problems
The calculation of ground-state energies of physical systems can be formalised as the k-local Hamiltonian problem, which is the natural quantum analogue of classical constraint satisfaction problems. One way of making the problem more physically meaningful is to restrict the Hamiltonian in question by picking its terms from a fixed set S. Examples of such special cases are the Heisenberg and Ising models from condensed-matter physics. In this work we characterise the complexity of this problem for all 2-local qubit Hamiltonians. Depending on the subset S, the problem falls into one of the following categories: in P, NP-complete, polynomial-time equivalent to the Ising model with transverse magnetic fields, or QMA-complete. The third of these classes contains NP and is contained within StoqMA. The characterisation holds even if S does not contain any 1-local terms, for example, we prove for the first time QMA-completeness of the Heisenberg and XY interactions in this setting. If S is assumed to contain all 1-local terms, which is the setting considered by previous work, we have a characterisation that goes beyond 2-local interactions: for any constant k, all k-local qubit Hamiltonians whose terms are picked from a fixed set S correspond to problems either in P, polynomial-time equivalent to the Ising model with transverse magnetic fields, or QMA-complete. These results are a quantum analogue of Schaefer's dichotomy theorem for boolean constraint satisfaction problems.Some of this work was completed while AM was at the
University of Cambridge. TC is supported by the Royal
Society.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/FOCS.2014.2
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