296 research outputs found
The Complexity of Relating Quantum Channels to Master Equations
Completely positive, trace preserving (CPT) maps and Lindblad master
equations are both widely used to describe the dynamics of open quantum
systems. The connection between these two descriptions is a classic topic in
mathematical physics. One direction was solved by the now famous result due to
Lindblad, Kossakowski Gorini and Sudarshan, who gave a complete
characterisation of the master equations that generate completely positive
semi-groups. However, the other direction has remained open: given a CPT map,
is there a Lindblad master equation that generates it (and if so, can we find
it's form)? This is sometimes known as the Markovianity problem. Physically, it
is asking how one can deduce underlying physical processes from experimental
observations.
We give a complexity theoretic answer to this problem: it is NP-hard. We also
give an explicit algorithm that reduces the problem to integer semi-definite
programming, a well-known NP problem. Together, these results imply that
resolving the question of which CPT maps can be generated by master equations
is tantamount to solving P=NP: any efficiently computable criterion for
Markovianity would imply P=NP; whereas a proof that P=NP would imply that our
algorithm already gives an efficiently computable criterion. Thus, unless P
does equal NP, there cannot exist any simple criterion for determining when a
CPT map has a master equation description.
However, we also show that if the system dimension is fixed (relevant for
current quantum process tomography experiments), then our algorithm scales
efficiently in the required precision, allowing an underlying Lindblad master
equation to be determined efficiently from even a single snapshot in this case.
Our work also leads to similar complexity-theoretic answers to a related
long-standing open problem in probability theory.Comment: V1: 43 pages, single column, 8 figures. V2: titled changed; added
proof-overview and accompanying figure; 50 pages, single column, 9 figure
Continuum limits of Matrix Product States
We determine which translationally invariant matrix product states have a
continuum limit, that is, which can be considered as discretized versions of
states defined in the continuum. To do this, we analyse a fine-graining
renormalization procedure in real space, characterise the set of limiting
states of its flow, and find that it strictly contains the set of continuous
matrix product states. We also analyse which states have a continuum limit
after a finite number of a coarse-graining renormalization steps. We give
several examples of states with and without the different kinds of continuum
limits.Comment: 7 pages, 2 figures. New version: somewhat expanded, some explanations
added. Close to published versio
The semigroup structure of Gaussian channels
We investigate the semigroup structure of bosonic Gaussian quantum channels.
Particular focus lies on the sets of channels which are divisible, idempotent
or Markovian (in the sense of either belonging to one-parameter semigroups or
being infinitesimal divisible). We show that the non-compactness of the set of
Gaussian channels allows for remarkable differences when comparing the
semigroup structure with that of finite dimensional quantum channels. For
instance, every irreversible Gaussian channel is shown to be divisible in spite
of the existence of Gaussian channels which are not infinitesimal divisible. A
simpler and known consequence of non-compactness is the lack of generators for
certain reversible channels. Along the way we provide new representations for
classes of Gaussian channels: as matrix semigroup, complex valued positive
matrices or in terms of a simple form describing almost all one-parameter
semigroups.Comment: 20 page
The problem of equilibration and the computation of correlation functions on a quantum computer
We address the question of how a quantum computer can be used to simulate
experiments on quantum systems in thermal equilibrium. We present two
approaches for the preparation of the equilibrium state on a quantum computer.
For both approaches, we show that the output state of the algorithm, after long
enough time, is the desired equilibrium. We present a numerical analysis of one
of these approaches for small systems. We show how equilibrium
(time)-correlation functions can be efficiently estimated on a quantum
computer, given a preparation of the equilibrium state. The quantum algorithms
that we present are hard to simulate on a classical computer. This indicates
that they could provide an exponential speedup over what can be achieved with a
classical device.Comment: 25 pages LaTex + 8 figures; various additional comments, results and
correction
Universal Hamiltonians for quantum simulation and their applications to holography
Recent work has demonstrated the existence of universal Hamiltonians – simple spin lattice models that can simulate any other quantum many body system. These universal Hamiltonians have applications for developing quantum simulators, as well as for Hamiltonian complexity, quantum computation, and fundamental physics. In this thesis we extend the theory of universal Hamiltonians. We begin by developing a new method for proving that a given family of Hamiltonians is indeed universal. We then use this method to construct two new universal models – both of which consist of translationally invariant interactions acting on a 1D spin chain.
But the benefit of our method doesn’t just lie in the simple universal models it allows us to construct. It also gives deeper insight into the origins of universality – and demonstrates a link between the universality and complexity. We make this insight rigorous, and derive a complexity theoretic classification of universal Hamiltonians which encompasses all known universal models. This classification provides a new, simplified route to checking whether a particular family of Hamiltonians meets the conditions to be a universal simulator.
We also consider the practical use of analogue Hamiltonian simulation. Under- standing the effect of noise on Hamiltonian simulation is a key issue in practical implementations. The first step to tackling this issue is characterising the noise processes affecting near term quantum devices. Motivated by this, we develop and numerically benchmark an algorithm which fits noise models to tomographic data from quantum devices to enable this process. This algorithm has applicability beyond analogue simulators, and could be used to investigate the physical noise processes in any quantum computing device.
Finally, we apply the theory of universal Hamiltonians to high energy physics by using them to construct toy models of holographic duality which capture more of the expected features of the AdS/CFT correspondence
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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
Using Quantum Computers for Quantum Simulation
Numerical simulation of quantum systems is crucial to further our
understanding of natural phenomena. Many systems of key interest and
importance, in areas such as superconducting materials and quantum chemistry,
are thought to be described by models which we cannot solve with sufficient
accuracy, neither analytically nor numerically with classical computers. Using
a quantum computer to simulate such quantum systems has been viewed as a key
application of quantum computation from the very beginning of the field in the
1980s. Moreover, useful results beyond the reach of classical computation are
expected to be accessible with fewer than a hundred qubits, making quantum
simulation potentially one of the earliest practical applications of quantum
computers. In this paper we survey the theoretical and experimental development
of quantum simulation using quantum computers, from the first ideas to the
intense research efforts currently underway.Comment: 43 pages, 136 references, review article, v2 major revisions in
response to referee comments, v3 significant revisions, identical to
published version apart from format, ArXiv version has table of contents and
references in alphabetical orde
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