37 research outputs found
Area laws for the entanglement entropy - a review
Physical interactions in quantum many-body systems are typically local:
Individual constituents interact mainly with their few nearest neighbors. This
locality of interactions is inherited by a decay of correlation functions, but
also reflected by scaling laws of a quite profound quantity: The entanglement
entropy of ground states. This entropy of the reduced state of a subregion
often merely grows like the boundary area of the subregion, and not like its
volume, in sharp contrast with an expected extensive behavior. Such "area laws"
for the entanglement entropy and related quantities have received considerable
attention in recent years. They emerge in several seemingly unrelated fields,
in the context of black hole physics, quantum information science, and quantum
many-body physics where they have important implications on the numerical
simulation of lattice models. In this Colloquium we review the current status
of area laws in these fields. Center stage is taken by rigorous results on
lattice models in one and higher spatial dimensions. The differences and
similarities between bosonic and fermionic models are stressed, area laws are
related to the velocity of information propagation, and disordered systems,
non-equilibrium situations, classical correlation concepts, and topological
entanglement entropies are discussed. A significant proportion of the article
is devoted to the quantitative connection between the entanglement content of
states and the possibility of their efficient numerical simulation. We discuss
matrix-product states, higher-dimensional analogues, and states from
entanglement renormalization and conclude by highlighting the implications of
area laws on quantifying the effective degrees of freedom that need to be
considered in simulations.Comment: 28 pages, 2 figures, final versio
Adiabatic graph-state quantum computation
Measurement-based quantum computation (MBQC) and holonomic quantum
computation (HQC) are two very different computational methods. The computation
in MBQC is driven by adaptive measurements executed in a particular order on a
large entangled state. In contrast in HQC the system starts in the ground
subspace of a Hamiltonian which is slowly changed such that a transformation
occurs within the subspace. Following the approach of Bacon and Flammia, we
show that any measurement-based quantum computation on a graph state with
\emph{gflow} can be converted into an adiabatically driven holonomic
computation, which we call \emph{adiabatic graph-state quantum computation}
(AGQC). We then investigate how properties of AGQC relate to the properties of
MBQC, such as computational depth. We identify a trade-off that can be made
between the number of adiabatic steps in AGQC and the norm of as well
as the degree of , in analogy to the trade-off between the number of
measurements and classical post-processing seen in MBQC. Finally the effects of
performing AGQC with orderings that differ from standard MBQC are investigated.Comment: 25 pages, 3 figure
Flow Ambiguity: A Path Towards Classically Driven Blind Quantum Computation
Blind quantum computation protocols allow a user to delegate a computation to
a remote quantum computer in such a way that the privacy of their computation
is preserved, even from the device implementing the computation. To date, such
protocols are only known for settings involving at least two quantum devices:
either a user with some quantum capabilities and a remote quantum server or two
or more entangled but noncommunicating servers. In this work, we take the first
step towards the construction of a blind quantum computing protocol with a
completely classical client and single quantum server. Specifically, we show
how a classical client can exploit the ambiguity in the flow of information in
measurement-based quantum computing to construct a protocol for hiding critical
aspects of a computation delegated to a remote quantum computer. This ambiguity
arises due to the fact that, for a fixed graph, there exist multiple choices of
the input and output vertex sets that result in deterministic measurement
patterns consistent with the same fixed total ordering of vertices. This allows
a classical user, computing only measurement angles, to drive a
measurement-based computation performed on a remote device while hiding
critical aspects of the computation.Comment: (v3) 14 pages, 6 figures. expands introduction and definition of
flow, corrects typos to increase readability; contains a new figure to
illustrate example run of CDBQC protocol; minor changes to match the
published version.(v2) 12 pages, 5 figures. Corrects motivation for
quantities used in blindness analysi
Localization and Scrambling of Quantum Information with Applications to Quantum Computation and Thermodynamics
As our demand for computational power grows, we encounter the question: What are the physical limits to computation? An answer is necessarily incomplete unless it can incorporate physics at the smallest scales, where we expect our near-term high-performance computing to occur. Microscopic physics -- namely, quantum mechanics -- behaves counterintuitively to our everyday experience, however. Quantum matter can occupy superpositions of states and build stronger correlations than are possible classically. This affects how quantum computers and quantum thermodynamic engines will behave.
Though these properties may seem to overwhelmingly defeat our attempts to build a quantum computer at-first-glance, what is remarkable is that they can also be immensely helpful to computation. Quantum mechanics hinders and helps computation, and the nuanced details of how we perform computations are important. In this dissertation, we examine the transition between these two behaviors and connect it to a well-studied behavior in condensed matter physics, known as the many-body-localization transition.
Our idea utilizes the fact that quantum many-body systems have an intrinsic fastest speed at which signals can travel. When this speed is maximal, we expect arbitrary universal quantum computation to be achievable, since strong quantum correlations, or entanglement, can be built quickly. When it is limited, however, the difficulty of the computation is classically simulatable. We demonstrate a similar transition in the amount of thermodynamic work that can be performed by a quantum system when entanglement is present.
We first consider computations consisting of the evolution of a single particle or many noninteracting particles. When the number of such noninteracting particles is comparable to the total size of the system, we do not know of any way to simulate such computations classically. However, we find that we can still determine the fastest signal speed in such systems. We extend our result to interacting particles, which are universal for quantum computation, and observe a many-body-localization transition in a simple computational model using our algorithm. Finally, we apply ideas from quantum information to simulate the thermodynamic performance of a simple quantum system, showing that quantum effects can enable it to outperform its classical counterpart
Topological Color Codes and Two-Body Quantum Lattice Hamiltonians
Topological color codes are among the stabilizer codes with remarkable
properties from quantum information perspective. In this paper we construct a
four-valent lattice, the so called ruby lattice, governed by a 2-body
Hamiltonian. In a particular regime of coupling constants, degenerate
perturbation theory implies that the low energy spectrum of the model can be
described by a many-body effective Hamiltonian, which encodes the color code as
its ground state subspace. The gauge symmetry
of color code could already be realized by
identifying three distinct plaquette operators on the lattice. Plaquettes are
extended to closed strings or string-net structures. Non-contractible closed
strings winding the space commute with Hamiltonian but not always with each
other giving rise to exact topological degeneracy of the model. Connection to
2-colexes can be established at the non-perturbative level. The particular
structure of the 2-body Hamiltonian provides a fruitful interpretation in terms
of mapping to bosons coupled to effective spins. We show that high energy
excitations of the model have fermionic statistics. They form three families of
high energy excitations each of one color. Furthermore, we show that they
belong to a particular family of topological charges. Also, we use
Jordan-Wigner transformation in order to test the integrability of the model
via introducing of Majorana fermions. The four-valent structure of the lattice
prevents to reduce the fermionized Hamiltonian into a quadratic form due to
interacting gauge fields. We also propose another construction for 2-body
Hamiltonian based on the connection between color codes and cluster states. We
discuss this latter approach along the construction based on the ruby lattice.Comment: 56 pages, 16 figures, published version