1,583 research outputs found
Defects in Quantum Computers
The shift of interest from general purpose quantum computers to adiabatic
quantum computing or quantum annealing calls for a broadly applicable and easy
to implement test to assess how quantum or adiabatic is a specific hardware.
Here we propose such a test based on an exactly solvable many body system --
the quantum Ising chain in transverse field -- and implement it on the D-Wave
machine. An ideal adiabatic quench of the quantum Ising chain should lead to an
ordered broken symmetry ground state with all spins aligned in the same
direction. An actual quench can be imperfect due to decoherence, noise, flaws
in the implemented Hamiltonian, or simply too fast to be adiabatic.
Imperfections result in topological defects: Spins change orientation, kinks
punctuating ordered sections of the chain. The number of such defects
quantifies the extent by which the quantum computer misses the ground state,
and is, therefore, imperfect.Comment: 8 pages, 7 figures, to appear in Scientific Reports, authors' list
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The Complexity of Translationally Invariant Spin Chains with Low Local Dimension
We prove that estimating the ground state energy of a
translationally-invariant, nearest-neighbour Hamiltonian on a 1D spin chain is
QMAEXP-complete, even for systems of low local dimension (roughly 40). This is
an improvement over the best previously-known result by several orders of
magnitude, and it shows that spin-glass-like frustration can occur in
translationally-invariant quantum systems with a local dimension comparable to
the smallest-known non-translationally-invariant systems with similar
behaviour.
While previous constructions of such systems rely on standard models of
quantum computation, we construct a new model that is particularly well-suited
for encoding quantum computation into the ground state of a
translationally-invariant system. This allows us to shift the proof burden from
optimizing the Hamiltonian encoding a standard computational model to proving
universality of a simple model.
Previous techniques for encoding quantum computation into the ground state of
a local Hamiltonian allow only a linear sequence of gates, hence only a linear
(or nearly linear) path in the graph of all computational states. We extend
these techniques by allowing significantly more general paths, including
branching and cycles, thus enabling a highly efficient encoding of our
computational model. However, this requires more sophisticated techniques for
analysing the spectrum of the resulting Hamiltonian. To address this, we
introduce a framework of graphs with unitary edge labels. After relating our
Hamiltonian to the Laplacian of such a unitary labelled graph, we analyse its
spectrum by combining matrix analysis and spectral graph theory techniques
PDE boundary conditions that eliminate quantum weirdness: a mathematical game inspired by Kurt Gödel and Alan Turing
Although boundary condition problems in quantum mathematics (QM) are well known, no one ever used boundary conditions technology to abolish quantum weirdness. We employ boundary conditions to build a mathematical game that is fun to learn, and by using it you will discover that quantum weirdness evaporates and vanishes. Our clever game is so designed that you can solve the boundary condition problems for a single point if-and-only-if you also solve the “weirdness” problem for all of quantum mathematics. Our approach differs radically from Dirichlet, Neumann, Robin, or Wolfram Alpha. We define domain Ω in one-dimension, on which a partial differential equation (PDE) is defined. Point α on ∂Ω is the location of a boundary condition game that involves an off-center bi-directional wave solution called Æ, an “elementary wave.” Study of this unusual, complex wave is called the Theory of Elementary Waves (TEW). We are inspired by Kurt Gödel and Alan Turing who built mathematical games that demonstrated that axiomatization of all mathematics was impossible. In our machine quantum weirdness vanishes if understood from the perspective of a single point α, because that pinpoint teaches us that nature is organized differently than we expect
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Quantum Stochastic Processes and Quantum Many-Body Physics
This dissertation investigates the theory of quantum stochastic processes and its applications in quantum many-body physics.
The main goal is to analyse complexity-theoretic aspects of both static and dynamic properties of physical systems modelled by quantum stochastic processes.
The thesis consists of two parts: the first one addresses the computational complexity of certain quantum and classical divisibility questions, whereas the second one addresses the topic of Hamiltonian complexity theory.
In the divisibility part, we discuss the question whether one can efficiently sub-divide a map describing the evolution of a system in a noisy environment, i.e. a CPTP- or stochastic map for quantum and classical processes, respectively, and we prove that taking the nth root of a CPTP or stochastic map is an NP-complete problem.
Furthermore, we show that answering the question whether one can divide up a random variable into a sum of iid random variables , i.e. , is poly-time computable; relaxing the iid condition renders the problem NP-hard.
In the local Hamiltonian part, we study computation embedded into the ground state of a many-body quantum system, going beyond "history state" constructions with a linear clock.
We first develop a series of mathematical techniques which allow us to study the energy spectrum of the resulting Hamiltonian, and extend classical string rewriting to the quantum setting.
This allows us to construct the most physically-realistic QMAEXP-complete instances for the LOCAL HAMILTONIAN problem (i.e. the question of estimating the ground state energy of a quantum many-body system) known to date, both in one- and three dimensions.
Furthermore, we study weighted versions of linear history state constructions, allowing us to obtain tight lower and upper bounds on the promise gap of the LOCAL HAMILTONIAN problem in various cases.
We finally study a classical embedding of a Busy Beaver Turing Machine into a low-dimensional lattice spin model, which allows us to dictate a transition from a purely classical phase to a Toric Code phase at arbitrarily large and potentially even uncomputable system sizes
The ghost in the machine: governing ‘data’ in intellectual monopolies
The lack of data regulation enables different ways of value extraction and benefits only a few platform companies globally, writes Farwa Sia
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