Matches in UGent Biblio for { ?s ?p Thanks to the cosine-sine decomposition of unitary matrices, an arbitrary quantum circuit, acting on w qubits, can be decomposed into 2^w-1 elementary quantum gates, called controlled V gates. Thanks to the Birkhoff decomposition of doubly stochastic matrices, an arbitrary (classical) reversible circuit, acting on w bits, can be decomposed into 2w-1 elementary gates, called controlled NOT gates. The question arises under which conditions these two synthesis methods are applicable for intermediate cases, i.e. computers based on some group, which simultaneously is a subgroup of the unitary group U(2^w) and a supergroup of the symmetric group S_{2^w}. It turns out that many groups either belong to a class that might have a cosine-sine-like decomposition but no Birkhoff-like decomposition and a second class that might have both decompositions. For an arbitrary group, in order to find out to which class it belongs, it suffices to evaluate a function Phi(m), deduced either from its order (in case of a finite group) or from its dimension (in case of a Lie group). Here m=2^w is the degree of the group.. }
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- aggregation abstract "Thanks to the cosine-sine decomposition of unitary matrices, an arbitrary quantum circuit, acting on w qubits, can be decomposed into 2^w-1 elementary quantum gates, called controlled V gates. Thanks to the Birkhoff decomposition of doubly stochastic matrices, an arbitrary (classical) reversible circuit, acting on w bits, can be decomposed into 2w-1 elementary gates, called controlled NOT gates. The question arises under which conditions these two synthesis methods are applicable for intermediate cases, i.e. computers based on some group, which simultaneously is a subgroup of the unitary group U(2^w) and a supergroup of the symmetric group S_{2^w}. It turns out that many groups either belong to a class that might have a cosine-sine-like decomposition but no Birkhoff-like decomposition and a second class that might have both decompositions. For an arbitrary group, in order to find out to which class it belongs, it suffices to evaluate a function Phi(m), deduced either from its order (in case of a finite group) or from its dimension (in case of a Lie group). Here m=2^w is the degree of the group.".