4,787 research outputs found
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
Algorithms and complexity for approximately counting hypergraph colourings and related problems
The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of Lovász local lemma when Δ ≲ qᵏ. In prior, however, fast approximate counting algorithms exist when Δ ≲ qᵏ/³, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows.
• When q, k ≥ 4 are evens and Δ ≥ 5·qᵏ/², approximating the number of hypergraph colourings is NP-hard.
• When the input hypergraph is linear and Δ ≲ qᵏ/², a fast approximate counting algorithm does exist
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Solving graph problems with single-photons and linear optics
An important challenge for current and near-term quantum devices is finding
useful tasks that can be preformed on them. We first show how to efficiently
encode a bounded matrix into a linear optical circuit with
modes. We then apply this encoding to the case where is a matrix
containing information about a graph . We show that a photonic quantum
processor consisting of single-photon sources, a linear optical circuit
encoding , and single-photon detectors can solve a range of graph problems
including finding the number of perfect matchings of bipartite graphs,
computing permanental polynomials, determining whether two graphs are
isomorphic, and the -densest subgraph problem. We also propose
pre-processing methods to boost the probabilities of observing the relevant
detection events and thus improve performance. Finally, we present various
numerical simulations which validate our findings.Comment: 6 pages + 9 pages appendix. Comments Welcome
Quantum ergodicity on the Bruhat-Tits building for in the Benjamini-Schramm limit
We study eigenfunctions of the spherical Hecke algebra acting on
where with a
non-archimedean local field of characteristic zero, with the ring of integers of , and
is a sequence of cocompact torsionfree lattices. We prove a form of
equidistribution on average for eigenfunctions whose spectral parameters lie in
the tempered spectrum when the associated sequence of quotients of the
Bruhat-Tits building Benjamini-Schramm converges to the building itself.Comment: 111 pages, 25 figures, 2 table
Trace formulas for magnetic Schr\"odinger operators on periodic graphs and their applications
We consider Schr\"odinger operators with periodic magnetic and electric
potentials on periodic discrete graphs. The spectrum of such operators consists
of a finite number of bands. We determine trace formulas for the magnetic
Schr\"odinger operators. The traces of the fiber operators are expressed as
finite Fourier series of the quasimomentum. The coefficients of the Fourier
series are given in terms of the magnetic fluxes, electric potentials and
cycles in the quotient graph from some specific cycle sets. Using the trace
formulas we obtain new lower estimates of the total bandwidth for the magnetic
Schr\"odinger operator in terms of geometric parameters of the graph, magnetic
fluxes and electric potentials. We show that these estimates are sharp.Comment: 37 pages, 6 figures. arXiv admin note: text overlap with
arXiv:2106.04245, arXiv:2106.0866
Quantum Alternating Operator Ansatz (QAOA) beyond low depth with gradually changing unitaries
The Quantum Approximate Optimization Algorithm and its generalization to
Quantum Alternating Operator Ansatz (QAOA) is a promising approach for applying
quantum computers to challenging problems such as combinatorial optimization
and computational chemistry. In this paper, we study the underlying mechanisms
governing the behavior of QAOA circuits beyond shallow depth in the practically
relevant setting of gradually varying unitaries. We use the discrete adiabatic
theorem, which complements and generalizes the insights obtained from the
continuous-time adiabatic theorem primarily considered in prior work. Our
analysis explains some general properties that are conspicuously depicted in
the recently introduced QAOA performance diagrams. For parameter sequences
derived from continuous schedules (e.g. linear ramps), these diagrams capture
the algorithm's performance over different parameter sizes and circuit depths.
Surprisingly, they have been observed to be qualitatively similar across
different performance metrics and application domains. Our analysis explains
this behavior as well as entails some unexpected results, such as connections
between the eigenstates of the cost and mixer QAOA Hamiltonians changing based
on parameter size and the possibility of reducing circuit depth without
sacrificing performance
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