167,252 research outputs found
Optimising the Solovay-Kitaev algorithm
The Solovay-Kitaev algorithm is the standard method used for approximating
arbitrary single-qubit gates for fault-tolerant quantum computation. In this
paper we introduce a technique called "search space expansion", which modifies
the initial stage of the Solovay-Kitaev algorithm, increasing the length of the
possible approximating sequences but without requiring an exhaustive search
over all possible sequences. We show that our technique, combined with a GNAT
geometric tree search outputs gate sequences that are almost an order of
magnitude smaller for the same level of accuracy. This therefore significantly
reduces the error correction requirements for quantum algorithms on encoded
fault-tolerant hardware.Comment: 9 page
From approximating to interpolatory non-stationary subdivision schemes with the same generation properties
In this paper we describe a general, computationally feasible strategy to
deduce a family of interpolatory non-stationary subdivision schemes from a
symmetric non-stationary, non-interpolatory one satisfying quite mild
assumptions. To achieve this result we extend our previous work [C.Conti,
L.Gemignani, L.Romani, Linear Algebra Appl. 431 (2009), no. 10, 1971-1987] to
full generality by removing additional assumptions on the input symbols. For
the so obtained interpolatory schemes we prove that they are capable of
reproducing the same exponential polynomial space as the one generated by the
original approximating scheme. Moreover, we specialize the computational
methods for the case of symbols obtained by shifted non-stationary affine
combinations of exponential B-splines, that are at the basis of most
non-stationary subdivision schemes. In this case we find that the associated
family of interpolatory symbols can be determined to satisfy a suitable set of
generalized interpolating conditions at the set of the zeros (with reversed
signs) of the input symbol. Finally, we discuss some computational examples by
showing that the proposed approach can yield novel smooth non-stationary
interpolatory subdivision schemes possessing very interesting reproduction
properties
- …