223 research outputs found
Logics between classical reversible logic and quantum logic
Classical reversible logic and quantum computing share the common feature that all computations are reversible, each result of a computation can be brought back to the initial state without loss of information
The Birkhoff theorem for unitary matrices of prime dimension
The Birkhoff's theorem states that any doubly stochastic matrix lies inside a
convex polytope with the permutation matrices at the corners. It can be proven
that a similar theorem holds for unitary matrices with equal line sums for
prime dimensions
The Birkhoff theorem for unitary matrices of prime-power dimension
The unitary Birkhoff theorem states that any unitary matrix with all row sums
and all column sums equal unity can be decomposed as a weighted sum of
permutation matrices, such that both the sum of the weights and the sum of the
squared moduli of the weights are equal to unity. If the dimension~ of the
unitary matrix equals a power of a prime , i.e.\ if , then the
Birkhoff decomposition does not need all possible permutation matrices, as
the epicirculant permutation matrices suffice. This group of permutation
matrices is isomorphic to the general affine group GA() of order only
Scaling a unitary matrix
The iterative method of Sinkhorn allows, starting from an arbitrary real
matrix with non-negative entries, to find a so-called 'scaled matrix' which is
doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and
with all line sums equal to 1. We conjecture that a similar procedure exists,
which allows, starting from an arbitrary unitary matrix, to find a scaled
matrix which is unitary and has all line sums equal to 1. The existence of such
algorithm guarantees a powerful decomposition of an arbitrary quantum circuit.Comment: A proof of the conjecture is now provided by Idel & Wolf
(http://arxiv.org/abs/1408.5728
The block-ZXZ synthesis of an arbitrary quantum circuit
Given an arbitrary unitary matrix , a powerful matrix
decomposition can be applied, leading to four different syntheses of a
-qubit quantum circuit performing the unitary transformation. The
demonstration is based on a recent theorem by F\"uhr and Rzeszotnik,
generalizing the scaling of single-bit unitary gates () to gates with
arbitrary value of~. The synthesized circuit consists of controlled 1-qubit
gates, such as NEGATOR gates and PHASOR gates. Interestingly, the approach
reduces to a known synthesis method for classical logic circuits consisting of
controlled NOT gates, in the case that is a permutation matrix.Comment: Improved (non-sinkhorn) algorithm to obtain the proposed circui
The decomposition of an arbitrary unitary matrix into signed permutation matrices
Birkhoff's theorem tells that any doubly stochastic matrix can be decomposed
as a weighted sum of permutation matrices. A similar theorem reveals that any
unitary matrix can be decomposed as a weighted sum of complex permutation
matrices. Unitary matrices of dimension equal to a power of~2 (say )
deserve special attention, as they represent quantum qubit circuits. We
investigate which subgroup of the signed permutation matrices suffices to
decompose an arbitrary such matrix. It turns out to be a matrix group
isomorphic to the extraspecial group {\bf E} of order
. An associated projective group of order equally suffices.Comment: 4th paper in a series of Birkhoff decompositions for unitary matrices
[(1) arXiv:1509.08626; (2) arXiv:1606.08642; (3) arXiv:1812.08833
The Birkhoff theorem for unitary matrices of arbitrary dimensions
It was shown recently that Birkhoff's theorem for doubly stochastic matrices
can be extended to unitary matrices with equal line sums whenever the dimension
of the matrices is prime. We prove a generalization of the Birkhoff theorem for
unitary matrices with equal line sums for arbitrary dimension.Comment: This manuscript presents a proof for the general unitary birkhoff
theorem, conjectured in arXiv:1509.0862
A unified approach to quantum computation and classical reversible computation
The design of a quantum computer and
the design of a classical computer can be based
on quite similar circuit designs.
The former is based on the subgroup structure of the infinite group of unitary matrices, whereas
the latter is based on the subgroup structure of the finite group of permutation matrices.
Because these two groups display similarities as well as differences,
the corresponding circuit designs are comparable but not identical
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