We have investigated the negative-parity states and electromagnetic transitions in $^{151,153}$Ho and $^{151,153}$Dy within the framework of the interacting boson fermion model 2 (IBFM-2). Spin assignments for some states with uncertain spin are made
based on this calculation. Calculated excitation energies, electromagnetic transitions and branching ratios are compared with available experimental data and a good agreement is obtained. The model wave functions have been used to study $beta$-decays from Ho to Dy isotones, and the calculated $log ft$ values are close to the experimental data.
In this Letter, we present two analytic expressions that most generally simulate $n$-qubit controlled-$U$ gates with standard one-qubit gates and CNOT gates using exponential and polynomial complexity respectively. Explicit circuits and general expre
ssions of decomposition are derived. The exact numbers of basic operations in these two schemes are given using gate counting technique.
Quantum computer possesses quantum parallelism and offers great computing power over classical computer cite{er1,er2}. As is well-know, a moving quantum object passing through a double-slit exhibits particle wave duality. A quantum computer is static
and lacks this duality property. The recently proposed duality computer has exploited this particle wave duality property, and it may offer additional computing power cite{r1}. Simply put it, a duality computer is a moving quantum computer passing through a double-slit. A duality computer offers the capability to perform separate operations on the sub-waves coming out of the different slits, in the so-called duality parallelism. Here we show that an $n$-dubit duality computer can be modeled by an $(n+1)$-qubit quantum computer. In a duality mode, computing operations are not necessarily unitary. A $n$-qubit quantum computer can be used as an $n$-bit reversible classical computer and is energy efficient. Our result further enables a $(n+1)$-qubit quantum computer to run classical algorithms in a $O(2^n)$-bit classical computer. The duality mode provides a natural link between classical computing and quantum computing. Here we also propose a recycling computing mode in which a quantum computer will continue to compute until the result is obtained. These two modes provide new tool for algorithm design. A search algorithm for the unsorted database search problem is designed.
