2,255 research outputs found
Approximation by Quantum Circuits
In a recent preprint by Deutsch et al. [1995] the authors suggest the
possibility of polynomial approximability of arbitrary unitary operations on
qubits by 2-qubit unitary operations. We address that comment by proving
strong lower bounds on the approximation capabilities of g-qubit unitary
operations for fixed g. We consider approximation of unitary operations on
subspaces as well as approximation of states and of density matrices by quantum
circuits in several natural metrics. The ability of quantum circuits to
probabilistically solve decision problem and guess checkable functions is
discussed. We also address exact unitary representation by reducing the upper
bound by a factor of n^2 and by formalizing the argument given by Barenco et
al. [1995] for the lower bound. The overall conclusion is that almost all
problems are hard to solve with quantum circuits.Comment: uuencoded, compressed postscript, LACES 68Q-95-2
On the Limits of Gate Elimination
Although a simple counting argument shows the existence of Boolean functions of exponential circuit complexity, proving superlinear circuit lower bounds for explicit functions seems to be out of reach of the current techniques. There has been a (very slow) progress in proving linear lower bounds with the latest record of 3 1/86*n-o(n). All known lower bounds are based on the so-called gate elimination technique. A typical gate elimination argument shows that it is possible to eliminate several gates from an optimal circuit by making one or several substitutions to the input variables and repeats this inductively. In this note we prove that this method cannot achieve linear bounds of cn beyond a certain constant c, where c depends only on the number of substitutions made at a single step of the induction
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