Article

# Time-optimal synthesis of unitary transformations in coupled fast and slow qubit system

Physical Review A (Impact Factor: 3.04). 09/2007; DOI: 10.1103/PhysRevA.77.032332

Source: arXiv

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**ABSTRACT:**We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with equal coupling $J$ plus a local magnetic field acting on the intermediate qubit 2. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the $\mathrm{CNOT}(1, 3)$ between the indirectly coupled qubits 1 and 3 is $T=\sqrt{3/2} J^{-1}$, i.e. faster than a previous estimate based on a similar Hamiltonian and the assumption of local unitaries with zero time cost. Furthermore, performing a simple Walsh-Hadamard rotation in the Hlibert space of qubit 3 shows that the time-optimal synthesis of the $\mathrm{CNOT}^{\pm}(1, 3)$ (which acts as the identity when the control qubit 1 is in the state $\ket{0}$, while if the control qubit is in the state $\ket{1}$ the target qubit 3 is flipped as $\ket{\pm}\rightarrow \ket{\mp}$) also requires the same time $T$. Comment: 9 pagesJournal of Physics A Mathematical and Theoretical 09/2010; · 1.77 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Experiments in coherent nuclear and electron magnetic resonance and optical spectroscopy correspond to control of quantum-mechanical ensembles, guiding them from initial states to target states by unitary transformations. The control inputs (pulse sequences) that accomplish these unitary transformations should take as little time as possible so as to minimize the effects of relaxation and decoherence, and to optimize the sensitivity of the experiments. Here, we give an efficient synthesis of a class of unitary transformations on a three coupled spin-1/2 system with equal Ising coupling strengths. We show a significant time saving compared with conventional methods.Physical Review A 12/2011; 84(6). · 3.04 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We study the quantum brachistochrone evolution for a system of two spins-1/2 describing by an anisotropic Heisenberg Hamiltonian without $zx$, $zy$ interecting couplings in magnetic field directed along z-axis. This Hamiltonian realizes quantum evolution in two subspaces spanned by $|\uparrow\uparrow>$, $|\downarrow\downarrow>$ and $|\uparrow\downarrow>$, $|\downarrow\uparrow>$ separately and allows to consider brachistochrone problem on each subspace separately. Using operator of evolution for this Hamiltonian we generate quantum gates, namely an entanler gate, $SWAP$ gate, $iSWAP$ gate. We also show that the time required for the generation of an entangler gate and $iSWAP$ gate is minimal from all possible.Journal of Physics A Mathematical and Theoretical 11/2012; 46(15). · 1.77 Impact Factor

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