25,446 research outputs found
Obtaining the Quantum Fourier Transform from the Classical FFT with QR Decomposition
We present the detailed process of converting the classical Fourier Transform
algorithm into the quantum one by using QR decomposition. This provides an
example of a technique for building quantum algorithms using classical ones.
The Quantum Fourier Transform is one of the most important quantum subroutines
known at present, used in most algorithms that have exponential speed up
compared to the classical ones. We briefly review Fast Fourier Transform and
then make explicit all the steps that led to the quantum formulation of the
algorithm, generalizing Coppersmith's work.Comment: 12 pages, 1 figure (generated within LaTeX). To appear in Journal of
Computational and Applied Mathematic
Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms
The Schur basis on n d-dimensional quantum systems is a generalization of the
total angular momentum basis that is useful for exploiting symmetry under
permutations or collective unitary rotations. We present efficient (size
poly(n,d,log(1/\epsilon)) for accuracy \epsilon) quantum circuits for the Schur
transform, which is the change of basis between the computational and the Schur
bases. These circuits are based on efficient circuits for the Clebsch-Gordan
transformation. We also present an efficient circuit for a limited version of
the Schur transform in which one needs only to project onto different Schur
subspaces. This second circuit is based on a generalization of phase estimation
to any nonabelian finite group for which there exists a fast quantum Fourier
transform.Comment: 4 pages, 3 figure
Theoretical derivation of 1/f noise in quantum chaos
It was recently conjectured that 1/f noise is a fundamental characteristic of
spectral fluctuations in chaotic quantum systems. This conjecture is based on
the behavior of the power spectrum of the excitation energy fluctuations, which
is different for chaotic and integrable systems. Using random matrix theory we
derive theoretical expressions that explain the power spectrum behavior at all
frequencies. These expressions reproduce to a good approximation the power laws
of type 1/f (1/f^2) characteristics of chaotic (integrable) systems, observed
in almost the whole frequency domain. Although we use random matrix theory to
derive these results, they are also valid for semiclassical systems.Comment: 5 pages (Latex), 3 figure
Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity
We present an in-depth investigation of the momentum
space describing point particles coupled to Einstein gravity in three
space-time dimensions. We introduce different sets of coordinates on the group
manifold and discuss their properties under Lorentz transformations. In
particular we show how a certain set of coordinates exhibits an upper bound on
the energy under deformed Lorentz boosts which saturate at the Planck energy.
We discuss how this deformed symmetry framework is generally described by a
quantum deformation of the Poincar\'e group: the quantum double of . We then illustrate how the space of functions on the group
manifold momentum space has a dual representation on a non-commutative space of
coordinates via a (quantum) group Fourier transform. In this context we explore
the connection between Weyl maps and different notions of (quantum) group
Fourier transform appeared in the literature in the past years and establish
relations between them
Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map
We rationalize the somewhat surprising efficacy of the Hadamard transform in
simplifying the eigenstates of the quantum baker's map, a paradigmatic model of
quantum chaos. This allows us to construct closely related, but new, transforms
that do significantly better, thus nearly solving for many states of the
quantum baker's map. These new transforms, which combine the standard Fourier
and Hadamard transforms in an interesting manner, are constructed from
eigenvectors of the shift permutation operator that are also simultaneous
eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal)
symmetry.Comment: Version to appear in J. Phys. A. Added discussions; modified title;
corrected minor error
Transmission amplitudes from Bethe ansatz equations
We consider the Heisenberg spin chain in the presence of integrable spin
defects. Using the Bethe ansatz methodology, we extract the associated
transmission amplitudes, that describe the interaction between the
particle-like excitations displayed by the models and the spin impurity. In the
attractive regime of the XXZ model, we also derive the breather's transmission
amplitude. We compare our findings with earlier relevant results in the context
of the sine-Gordon model.Comment: 27 pages Latex. References adde
Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors
We provide an alternative proof of Wallman's [Quantum 2, 47 (2018)] and
Proctor's [Phys. Rev. Lett. 119, 130502 (2017)] bounds on the effect of
gate-dependent noise on randomized benchmarking (RB). Our primary insight is
that a RB sequence is a convolution amenable to Fourier space analysis, and we
adopt the mathematical framework of Fourier transforms of matrix-valued
functions on groups established in recent work from Gowers and Hatami [Sbornik:
Mathematics 208, 1784 (2017)]. We show explicitly that as long as our faulty
gate-set is close to some representation of the Clifford group, an RB sequence
is described by the exponential decay of a process that has exactly two
eigenvalues close to one and the rest close to zero. This framework also allows
us to construct a gauge in which the average gate-set error is a depolarizing
channel parameterized by the RB decay rates, as well as a gauge which maximizes
the fidelity with respect to the ideal gate-set
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