8,281 research outputs found
Computing a logarithm of a unitary matrix with general spectrum
We analyze an algorithm for computing a skew-Hermitian logarithm of a unitary
matrix. This algorithm is very easy to implement using standard software and it
works well even for unitary matrices with no spectral conditions assumed.
Certain examples, with many eigenvalues near -1, lead to very non-Hermitian
output for other basic methods of calculating matrix logarithms. Altering the
output of these algorithms to force an Hermitian output creates accuracy issues
which are avoided in the considered algorithm.
A modification is introduced to deal properly with the -skew symmetric
unitary matrices. Applications to numerical studies of topological insulators
in two symmetry classes are discussed.Comment: Added discussion of Floquet Hamiltonian
Approximate locality for quantum systems on graphs
In this Letter we make progress on a longstanding open problem of Aaronson
and Ambainis [Theory of Computing 1, 47 (2005)]: we show that if A is the
adjacency matrix of a sufficiently sparse low-dimensional graph then the
unitary operator e^{itA} can be approximated by a unitary operator U(t) whose
sparsity pattern is exactly that of a low-dimensional graph which gets more
dense as |t| increases. Secondly, we show that if U is a sparse unitary
operator with a gap \Delta in its spectrum, then there exists an approximate
logarithm H of U which is also sparse. The sparsity pattern of H gets more
dense as 1/\Delta increases. These two results can be interpreted as a way to
convert between local continuous-time and local discrete-time processes. As an
example we show that the discrete-time coined quantum walk can be realised as
an approximately local continuous-time quantum walk. Finally, we use our
construction to provide a definition for a fractional quantum fourier
transform.Comment: 5 pages, 2 figures, corrected typ
Quantitative K-Theory Related to Spin Chern Numbers
We examine the various indices defined on pairs of almost commuting unitary
matrices that can detect pairs that are far from commuting pairs. We do this in
two symmetry classes, that of general unitary matrices and that of self-dual
matrices, with an emphasis on quantitative results. We determine which values
of the norm of the commutator guarantee that the indices are defined, where
they are equal, and what quantitative results on the distance to a pair with a
different index are possible. We validate a method of computing spin Chern
numbers that was developed with Hastings and only conjectured to be correct.
Specifically, the Pfaffian-Bott index can be computed by the "log method" for
commutator norms up to a specific constant
Efficient Algorithms for Universal Quantum Simulation
A universal quantum simulator would enable efficient simulation of quantum
dynamics by implementing quantum-simulation algorithms on a quantum computer.
Specifically the quantum simulator would efficiently generate qubit-string
states that closely approximate physical states obtained from a broad class of
dynamical evolutions. I provide an overview of theoretical research into
universal quantum simulators and the strategies for minimizing computational
space and time costs. Applications to simulating many-body quantum simulation
and solving linear equations are discussed
Logarithmic link smearing for full QCD
A Lie-algebra based recipe for smoothing gauge links in lattice field theory
is presented, building on the matrix logarithm. With or without hypercubic
nesting, this LOG/HYL smearing yields fat links which are differentiable w.r.t.
the original ones. This is essential for defining UV-filtered ("fat link")
fermion actions which may be simulated with a HMC-type algorithm. The effect of
this smearing on the distribution of plaquettes and on the residual mass of
tree-level O(a)-improved clover fermions in quenched QCD is studied.Comment: 29 pages, 7 figures; v2: improved text, includes comparison of
APE/EXP/LOG with optimized parameters, 3 references adde
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