Timeoptimal twoqubit gate simulation is impractical Henry L

Time-optimal two-qubit gate simulation is impractical Henry L. Haselgrove 1, 2, 3, Michael A. Nielsen 1, 2, and Tobias J. Osborne 1, 4 1 Department of Physics, The University of Queensland, QLD 4072, Australia. 2 Institute for Quantum Information, California Institute of Technology, Pasadena CA 91125, USA. 3 Information Sciences Laboratory, Defence Science and Technology Organisation, Edinburgh 5111, Australia. 4 School of Mathematics, University of Bristol, University Walk, Bristol BS 8 1 TW, United Kingdom. Introduction Consider a system of two qubits, where the evolution is governed by both a fixed entangling Hamiltonian, and by fast local control operations. We know that any two qubit unitary can be simulated by this system. What happens when we try to achieve a time optimal simulation? We show that for most Hamiltonians, the time optimal simulation requires infinitely many steps of evolution, each infinitesimally small. Two-qubit simulation The simulation of a two qubit unitary U using the Hamiltonian H and fast local unitaries, is written as: Finite steps A time optimal simulation using a lazy Hamiltonian must have infinitely many time steps. If we use only a finite number of time steps, how much longer will the simulation take? RESULT: When a lazy Hamiltonian is used to simulate a CNOT gate*, with time steps fixed at length , the penalty in total simulation time (compared with optimal) is O( 2) (not too bad!) The simulation time is ts=t 1+t 2+t 3+ A time optimal simulation is one that minimises ts. Self simulation How efficiently can a Hamiltonian be used to simulate its own action e i. Ht? That is, what is the minimum possible value of t 1+t 2+t 3+ in the following circuit? Example, when H=0. 1 X X + I Z: Total simulation time optimal (infinte # time steps) Number of simulation steps *using the CNOT simulation scheme of [3] DEFINE H(t) to be the minimum possible time to simulate e i. Ht using H and local unitaries. The function is calculated with help from [2]. The “lazy” Hamiltonian We say that an entangling two qubit Hamiltonian H is lazy if H(t)<t for all t>0 THEOREM: Whenever a lazy Hamiltonian is used in a time optimal simulation of a nonlocal unitary, the simulation has infinitely many time steps. For proof see [1] Conditions for laziness We parameterise the Hamiltonian as follows: where 1> 2> 3. We have shown[1] that H is lazy whenever neither of these three conditions hold: So, most two qubit Hamiltonians are in fact lazy! Example, H=2 X X + Y Y + 5 I Z + 5 X I [1] H. L. Haselgrove, M. A. Nielsen, and T. J. Osborne, ar. Xiv: quant ph/0303070 is lazy (see graph) ar. Xiv: quant ph/0205100 (2002). [2] K. Hammerer, G. Vidal, J. I. Cirac, Phys. Rev. A 66, 062321 (2002) [3] M. J. Bremner et. al. , Phys. Rev. Lett. 89, 247902 (2002) ar. Xiv: quant ph/0207072 (2002).