11 research outputs found
Efficient unitary approximations in quantum computing: the Solovay-Kitaev theorem
Treballs Finals de Grau de MatemĂ tiques, Facultat de MatemĂ tiques, Universitat de Barcelona, Any: 2019, Director: Joana Cirici[en] Over the past few years, quantum computing has become more plausible due to the great advances in technology. While quantum computers are on their birth, the underlying mathematics have evolved to the point of proving that some quantum algorithms can solve problems that were unsolvable in classic computers. In order to implement these algorithms in a real machine, it is important to develop efficient ways to do it. The Solovay-Kitaev Theorem states that is possible. This work pretends to offer a complete review of the Solovay-
Kitaev Theorem giving all the necessary tools to prove it. Moreover, we offer a brief introduction to the standard mathematical model of quantum computing, based on unitary operations
Complexity phase diagram for interacting and long-range bosonic Hamiltonians
Recent years have witnessed a growing interest in topics at the intersection of many-body physics and complexity theory. Many-body physics aims to understand and classify emergent behavior of systems with a large number of particles, while complexity theory aims to classify computational problems based on how the time required to solve the problem scales as the problem size becomes large. In this work, we use insights from complexity theory to classify phases in interacting many-body systems. Specifically, we demonstrate a "complexity phase diagram" for the Bose-Hubbard model with long-range hopping. This shows how the complexity of simulating time evolution varies according to various parameters appearing in the problem, such as the evolution time, the particle density, and the degree of locality. We find that classification of complexity phases is closely related to upper bounds on the spread of quantum correlations, and protocols to transfer quantum information in a controlled manner. Our work motivates future studies of complexity in many-body systems and its interplay with the associated physical phenomena
Complexity phase diagram for interacting and long-range bosonic Hamiltonians
We classify phases of a bosonic lattice model based on the computational
complexity of classically simulating the system. We show that the system
transitions from being classically simulable to classically hard to simulate as
it evolves in time, extending previous results to include on-site
number-conserving interactions and long-range hopping. Specifically, we
construct a "complexity phase diagram" with "easy" and "hard" phases, and
derive analytic bounds on the location of the phase boundary with respect to
the evolution time and the degree of locality. We find that the location of the
phase transition is intimately related to upper bounds on the spread of quantum
correlations and protocols to transfer quantum information. Remarkably,
although the location of the transition point is unchanged by on-site
interactions, the nature of the transition point changes dramatically.
Specifically, we find that there are two kinds of transitions, sharp and
coarse, broadly corresponding to interacting and noninteracting bosons,
respectively. Our work motivates future studies of complexity in many-body
systems and its interplay with the associated physical phenomena.Comment: 15 pages, 5 figures. v2: 19 pages, 7 figure
Simultaneous Linearization of Diffeomorphisms of Isotropic Manifolds
Suppose that is a closed isotropic Riemannian manifold and that
generate the isometry group of . Let be smooth
perturbations of these isometries. We show that the are simultaneously
conjugate to isometries if and only if their associated uniform Bernoulli
random walk has all Lyapunov exponents zero. This extends a linearization
result of Dolgopyat and Krikorian from to real, complex, and quaternionic
projective spaces. In addition, we identify and remedy an oversight in that
earlier work.Comment: 54 pages; comments are welcom
Applications of coherent classical communication and the Schur transform to quantum information theory
Quantum mechanics has led not only to new physical theories, but also a new
understanding of information and computation. Quantum information began by
yielding new methods for achieving classical tasks such as factoring and key
distribution but also suggests a completely new set of quantum problems, such
as sending quantum information over quantum channels or efficiently performing
particular basis changes on a quantum computer. This thesis contributes two
new, purely quantum, tools to quantum information theory--coherent classical
communication in the first half and an efficient quantum circuit for the Schur
transform in the second half.Comment: 176 pages. Chapters 1 and 4 are a slightly older version of
quant-ph/0512015. Chapter 2 is quant-ph/0205057 plus unpublished extensions
(slightly outdated by quant-ph/0511219) and chapter 3 is quant-ph/0307091,
quant-ph/0412126 and change. Chapters 5-8 are based on quant-ph/0407082, but
go much furthe
Trading inverses for an irrep in the Solovay-Kitaev theorem
The Solovay-Kitaev theorem states that universal quantum gate sets can be exchanged with low overhead. More specifically, any gate on a fixed number of qudits can be simulated with error epsilon using merely polylog(1/epsilon) gates from any finite universal quantum gate set G. One drawback to the theorem is that it requires the gate set G to be closed under inversion. Here we show that this restriction can be traded for the assumption that G contains an irreducible representation of any finite group G. This extends recent work of Sardharwalla et al. [Sardharwalla et al., 2016], and applies also to gates from the special linear group. Our work can be seen as partial progress towards the long-standing open problem of proving an inverse-free Solovay-Kitaev theorem [Dawson and Nielsen, 2006; Kuperberg, 2015]
Trading inverses for an irrep in the Solovay-Kitaev theorem
The Solovay-Kitaev theorem states that universal quantum gate sets can be
exchanged with low overhead. More specifically, any gate on a fixed number of
qudits can be simulated with error using merely
gates from any finite universal quantum gate set
. One drawback to the theorem is that it requires the gate set
to be closed under inversion. Here we show that this restriction
can be traded for the assumption that contains an irreducible
representation of any finite group . This extends recent work of
Sardharwalla et al. [arXiv:1602.07963], and applies also to gates from the
special linear group. Our work can be seen as partial progress towards the
long-standing open problem of proving an inverse-free Solovay-Kitaev theorem
[arXiv:quant-ph/0505030, arXiv:0908.0512].Comment: 16 pages, TQC 2018 proceedings versio