729 research outputs found
Entanglement and its Role in Shor's Algorithm
Entanglement has been termed a critical resource for quantum information
processing and is thought to be the reason that certain quantum algorithms,
such as Shor's factoring algorithm, can achieve exponentially better
performance than their classical counterparts. The nature of this resource is
still not fully understood: here we use numerical simulation to investigate how
entanglement between register qubits varies as Shor's algorithm is run on a
quantum computer. The shifting patterns in the entanglement are found to relate
to the choice of basis for the quantum Fourier transform.Comment: 15 pages, 4 eps figures, v1-3 were for conference proceedings (not
included in the end); v4 is improved following referee comments, expanded
explanations and added reference
Sampling of operators
Sampling and reconstruction of functions is a central tool in science. A key
result is given by the sampling theorem for bandlimited functions attributed to
Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling
theory for operators which we call bandlimited if their Kohn-Nirenberg symbols
are bandlimited. We prove sampling theorems for such operators and show that
they are extensions of the classical sampling theorem
Subtropical Real Root Finding
We describe a new incomplete but terminating method for real root finding for
large multivariate polynomials. We take an abstract view of the polynomial as
the set of exponent vectors associated with sign information on the
coefficients. Then we employ linear programming to heuristically find roots.
There is a specialized variant for roots with exclusively positive coordinates,
which is of considerable interest for applications in chemistry and systems
biology. An implementation of our method combining the computer algebra system
Reduce with the linear programming solver Gurobi has been successfully applied
to input data originating from established mathematical models used in these
areas. We have solved several hundred problems with up to more than 800000
monomials in up to 10 variables with degrees up to 12. Our method has failed
due to its incompleteness in less than 8 percent of the cases
Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
We compute the leading-color (planar) three-loop four-point amplitude of N=4
supersymmetric Yang-Mills theory in 4 - 2 epsilon dimensions, as a Laurent
expansion about epsilon = 0 including the finite terms. The amplitude was
constructed previously via the unitarity method, in terms of two Feynman loop
integrals, one of which has been evaluated already. Here we use the
Mellin-Barnes integration technique to evaluate the Laurent expansion of the
second integral. Strikingly, the amplitude is expressible, through the finite
terms, in terms of the corresponding one- and two-loop amplitudes, which
provides strong evidence for a previous conjecture that higher-loop planar N =
4 amplitudes have an iterative structure. The infrared singularities of the
amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on
resummation. Based on the four-point result and the exponentiation of infrared
singularities, we give an exponentiated ansatz for the maximally
helicity-violating n-point amplitudes to all loop orders. The 1/epsilon^2 pole
in the four-point amplitude determines the soft, or cusp, anomalous dimension
at three loops in N = 4 supersymmetric Yang-Mills theory. The result confirms a
prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the
leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and
Vogt. Following similar logic, we are able to predict a term in the three-loop
quark and gluon form factors in QCD.Comment: 54 pages, 7 figures. v2: Added references, a few additional words
about large spin limit of anomalous dimensions. v3: Expanded Sect. IV.A on
multiloop ansatz; remark that form-factor prediction is now confirmed by
other work; minor typos correcte
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