11,622 research outputs found
Approximate unitary -designs by short random quantum circuits using nearest-neighbor and long-range gates
We prove that -depth local random quantum circuits
with two qudit nearest-neighbor gates on a -dimensional lattice with n
qudits are approximate -designs in various measures. These include the
"monomial" measure, meaning that the monomials of a random circuit from this
family have expectation close to the value that would result from the Haar
measure. Previously, the best bound was due to
Brandao-Harrow-Horodecki (BHH) for . We also improve the "scrambling" and
"decoupling" bounds for spatially local random circuits due to Brown and Fawzi.
One consequence of our result is that assuming the polynomial hierarchy (PH)
is infinite and that certain counting problems are -hard on average,
sampling within total variation distance from these circuits is hard for
classical computers. Previously, exact sampling from the outputs of even
constant-depth quantum circuits was known to be hard for classical computers
under the assumption that PH is infinite. However, to show the hardness of
approximate sampling using this strategy requires that the quantum circuits
have a property called "anti-concentration", meaning roughly that the output
has near-maximal entropy. Unitary 2-designs have the desired anti-concentration
property. Thus our result improves the required depth for this level of
anti-concentration from linear depth to a sub-linear value, depending on the
geometry of the interactions. This is relevant to a recent proposal by the
Google Quantum AI group to perform such a sampling task with 49 qubits on a
two-dimensional lattice and confirms their conjecture that depth
suffices for anti-concentration. We also prove that anti-concentration is
possible in depth O(log(n) loglog(n)) using a different model
New strongly regular graphs from finite geometries via switching
We show that the strongly regular graph on non-isotropic points of one type of the polar spaces of type U(n, 2), O(n, 3), O(n, 5), O+ (n, 3), and O- (n, 3) are not determined by its parameters for n >= 6. We prove this by using a variation of Godsil-McKay switching recently described by Wang, Qiu, and Hu. This also results in a new, shorter proof of a previous result of the first author which showed that the collinearity graph of a polar space is not determined by its spectrum. The same switching gives a linear algebra explanation for the construction of a large number of non-isomorphic designs. (C) 2019 Elsevier Inc. All rights reserved
New Strongly Regular Graphs from Finite Geometries via Switching
We show that the strongly regular graph on non-isotropic points of one type
of the polar spaces of type , , , , and
are not determined by its parameters for . We prove this
by using a variation of Godsil-McKay switching recently described by Wang, Qiu,
and Hu. This also results in a new, shorter proof of a previous result of the
first author which showed that the collinearity graph of a polar space is not
determined by its spectrum. The same switching gives a linear algebra
explanation for the construction of a large number of non-isomorphic designs.Comment: 13 pages, accepted in Linear Algebra and Its Application
P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches
While the P vs NP problem is mainly approached form the point of view of
discrete mathematics, this paper proposes reformulations into the field of
abstract algebra, geometry, fourier analysis and of continuous global
optimization - which advanced tools might bring new perspectives and approaches
for this question. The first one is equivalence of satisfaction of 3-SAT
problem with the question of reaching zero of a nonnegative degree 4
multivariate polynomial (sum of squares), what could be tested from the
perspective of algebra by using discriminant. It could be also approached as a
continuous global optimization problem inside , for example in
physical realizations like adiabatic quantum computers. However, the number of
local minima usually grows exponentially. Reducing to degree 2 polynomial plus
constraints of being in , we get geometric formulations as the
question if plane or sphere intersects with . There will be also
presented some non-standard perspectives for the Subset-Sum, like through
convergence of a series, or zeroing of fourier-type integral for some natural . The last discussed
approach is using anti-commuting Grassmann numbers , making nonzero only if has a Hamilton cycle. Hence,
the PNP assumption implies exponential growth of matrix representation of
Grassmann numbers. There will be also discussed a looking promising
algebraic/geometric approach to the graph isomorphism problem -- tested to
successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure
Entanglement Typicality
We provide a summary of both seminal and recent results on typical
entanglement. By typical values of entanglement, we refer here to values of
entanglement quantifiers that (given a reasonable measure on the manifold of
states) appear with arbitrarily high probability for quantum systems of
sufficiently high dimensionality. We work within the Haar measure framework for
discrete quantum variables, where we report on results concerning the average
von Neumann and linear entropies as well as arguments implying the typicality
of such values in the asymptotic limit. We then proceed to discuss the
generation of typical quantum states with random circuitry. Different phases of
entanglement, and the connection between typical entanglement and
thermodynamics are discussed. We also cover approaches to measures on the
non-compact set of Gaussian states of continuous variable quantum systems.Comment: Review paper with two quotes and minimalist figure
An evolutionary advantage of cooperation
Cooperation is a persistent behavioral pattern of entities pooling and
sharing resources. Its ubiquity in nature poses a conundrum. Whenever two
entities cooperate, one must willingly relinquish something of value to the
other. Why is this apparent altruism favored in evolution? Classical solutions
assume a net fitness gain in a cooperative transaction which, through
reciprocity or relatedness, finds its way back from recipient to donor. We seek
the source of this fitness gain. Our analysis rests on the insight that
evolutionary processes are typically multiplicative and noisy. Fluctuations
have a net negative effect on the long-time growth rate of resources but no
effect on the growth rate of their expectation value. This is an example of
non-ergodicity. By reducing the amplitude of fluctuations, pooling and sharing
increases the long-time growth rate for cooperating entities, meaning that
cooperators outgrow similar non-cooperators. We identify this increase in
growth rate as the net fitness gain, consistent with the concept of geometric
mean fitness in the biological literature. This constitutes a fundamental
mechanism for the evolution of cooperation. Its minimal assumptions make it a
candidate explanation of cooperation in settings too simple for other fitness
gains, such as emergent function and specialization, to be probable. One such
example is the transition from single cells to early multicellular life.Comment: 16 pages, 2 figure
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