60 research outputs found
Optimal Power-Down Strategies
We consider the problem of selecting threshold times to transition a device to low-power sleep states during an idle period. The two-state case, in which there is a single active and a single sleep state, is a continuous version of the ski-rental problem. We consider a generalized version in which there is more than one sleep state, each with its own power-consumption rate and transition costs. We give an algorithm that, given a system, produces a deterministic strategy whose competitive ratio is arbitrarily close to optimal. We also give an algorithm to produce the optimal online strategy given a system and a probability distribution that generates the length of the idle period. We also give a simple algorithm that achieves a competitive ratio of for any system
Ground State Entanglement in One Dimensional Translationally Invariant Quantum Systems
We examine whether it is possible for one-dimensional
translationally-invariant Hamiltonians to have ground states with a high degree
of entanglement. We present a family of translationally invariant Hamiltonians
{H_n} for the infinite chain. The spectral gap of H_n is Omega(1/poly(n)).
Moreover, for any state in the ground space of H_n and any m, there are regions
of size m with entanglement entropy Omega(min{m,n}). A similar construction
yields translationally-invariant Hamiltonians for finite chains that have
unique ground states exhibiting high entanglement. The area law proven by
Hastings gives a constant upper bound on the entanglement entropy for 1D ground
states that is independent of the size of the region but exponentially
dependent on 1/Delta, where Delta is the spectral gap. This paper provides a
lower bound, showing a family of Hamiltonians for which the entanglement
entropy scales polynomially with 1/Delta. Previously, the best known such bound
was logarithmic in 1/Delta.Comment: 22 pages. v2 is the published version, with additional
clarifications, publisher's version available at
http://jmp.aip.org/resource/1/jmapaq/v51/i
Commuting Local Hamiltonian Problem on 2D beyond qubits
We study the complexity of local Hamiltonians in which the terms pairwise
commute. Commuting local Hamiltonians (CLHs) provide a way to study the role of
non-commutativity in the complexity of quantum systems and touch on many
fundamental aspects of quantum computing and many-body systems, such as the
quantum PCP conjecture and the area law. Despite intense research activity
since Bravyi and Vyalyi introduced the CLH problem two decades ago [BV03], its
complexity remains largely unresolved; it is only known to lie in NP for a few
special cases. Much of the recent research has focused on the physically
motivated 2D case, where particles are located on vertices of a 2D grid and
each term acts non-trivially only on the particles on a single square (or
plaquette) in the lattice. In particular, Schuch [Sch11] showed that the CLH
problem on 2D with qubits is in NP. Aharonov, Kenneth and Vigdorovich~[AKV18]
then gave a constructive version of this result, showing an explicit algorithm
to construct a ground state. Resolving the complexity of the 2D CLH problem
with higher dimensional particles has been elusive. We prove two results for
the CLH problem in 2D:
(1) We give a non-constructive proof that the CLH problem in 2D with qutrits
is in NP. As far as we know, this is the first result for the commuting local
Hamiltonian problem on 2D beyond qubits. Our key lemma works for general qudits
and might give new insights for tackling the general case.
(2) We consider the factorized case, also studied in [BV03], where each term
is a tensor product of single-particle Hermitian operators. We show that a
factorized CLH in 2D, even on particles of arbitrary finite dimension, is
equivalent to a direct sum of qubit stabilizer Hamiltonians. This implies that
the factorized 2D CLH problem is in NP. This class of CLHs contains the Toric
code as an example.Comment: 28 pages, 6 figure
Translationally Invariant Constraint Optimization Problems
We study the complexity of classical constraint satisfaction problems on a 2D
grid. Specifically, we consider the complexity of function versions of such
problems, with the additional restriction that the constraints are
translationally invariant, namely, the variables are located at the vertices of
a 2D grid and the constraint between every pair of adjacent variables is the
same in each dimension. The only input to the problem is thus the size of the
grid. This problem is equivalent to one of the most interesting problems in
classical physics, namely, computing the lowest energy of a classical system of
particles on the grid. We provide a tight characterization of the complexity of
this problem, and show that it is complete for the class . Gottesman
and Irani (FOCS 2009) also studied classical translationally-invariant
constraint satisfaction problems; they show that the problem of deciding
whether the cost of the optimal solution is below a given threshold is
NEXP-complete. Our result is thus a strengthening of their result from the
decision version to the function version of the problem. Our result can also be
viewed as a generalization to the translationally invariant setting, of
Krentel's famous result from 1988, showing that the function version of SAT is
complete for the class . An essential ingredient in the proof is a
study of the complexity of a gapped variant of the problem. We show that it is
NEXP-hard to approximate the cost of the optimal assignment to within an
additive error of , for an grid. To the best of
our knowledge, no gapped result is known for CSPs on the grid, even in the
non-translationally invariant case. As a byproduct of our results, we also show
that a decision version of the optimization problem which asks whether the cost
of the optimal assignment is odd or even is also complete for .Comment: 75 pages, 13 figure
The Subset Assignment Problem for Data Placement in Caches
We introduce the subset assignment problem in which items of varying sizes are placed in a set of bins with limited capacity. Items can be replicated and placed in any subset of the bins. Each (item, subset) pair has an associated cost. Not assigning an item to any of the bins is not free in general and can potentially be the most expensive option. The goal is to minimize the total cost of assigning items to subsets without exceeding the bin capacities. This problem is motivated by the design of caching systems composed of banks of memory with varying cost/performance specifications. The ability to replicate a data item in more than one memory bank can benefit the overall performance of the system with a faster recovery time in the event of a memory failure. For this setting, the number n of data objects (items) is very large and the number d of memory banks (bins) is a small constant (on the order of 3 or 4). Therefore, the goal is to determine an optimal assignment in time that minimizes dependence on n. The integral version of this problem is NP-hard since it is a generalization of the knapsack problem. We focus on an efficient solution to the LP relaxation as the number of fractionally assigned items will be at most d. If the data objects are small with respect to the size of the memory banks, the effect of excluding the fractionally assigned data items from the cache will be small. We give an algorithm that solves the LP relaxation and runs in time O(binom{3^d}{d+1} poly(d) n log(n) log(nC) log(Z)), where Z is the maximum item size and C the maximum storage cost
Quantum Tutte Embeddings
Using the framework of Tutte embeddings, we begin an exploration of
\emph{quantum graph drawing}, which uses quantum computers to visualize graphs.
The main contributions of this paper include formulating a model for quantum
graph drawing, describing how to create a graph-drawing quantum circuit from a
given graph, and showing how a Tutte embedding can be calculated as a quantum
state in this circuit that can then be sampled to extract the embedding. To
evaluate the complexity of our quantum Tutte embedding circuits, we compare
them to theoretical bounds established in the classical computing setting
derived from a well-known classical algorithm for solving the types of linear
systems that arise from Tutte embeddings. We also present empirical results
obtained from experimental quantum simulations.Comment: 19 pages, 6 figure
- …