qitd322 - qitd322 Unitary Dynamics and Quantum Circuits...

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Unformatted text preview: qitd322 Unitary Dynamics and Quantum Circuits Robert B. Griffiths Version of 23 January 2012 Contents 1 Unitary Dynamics 1 1.1 Time development operator T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Particular cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Quantum Circuits 2 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 One qubit gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Controlled-NOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Other two-qubit gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 More complicated circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 References: CQT = Consistent Quantum Theory by Griffiths (Cambridge, 2002), Ch. 7. QCQI = Quantum Computation and Quantum Information by Nielsen and Chuang (Cambridge, 2000), Secs. 2.2.2, 4.2, 4.3 1 U n i t a r y D y n a m i c s 1.1 Time development operator T In general the time development of a quantum system is a stochastic (i.e., random) process governed by probabilities. However, there is a deterministic unitary dynamics which is of interest in itself, and which is also used to calculate the probabilities for the stochastic dynamics. Start with the Schr¨ odinger equation i d dt | ψ t = H | ψ t , (1) where t is the time, H = H † the Hamiltonian or energy operator. • Given an initial state | ψ at t = 0 there is (provided H satisfies certain properties) a unique solution to the differential equation, both for t > 0 and for t < 0. • Strictly speaking, the Schr¨ odinger equation only applies to an isolated system, one that does not interact with its surroundings or environment. However, it can also be applied, at least as a good approximation, in circumstances in which this interaction is weak or slowly varying or chosen in some way that does not decohere the system of interest. In such cases the Hamiltonian H is still hermitian, but can depend upon the time. A solution | ψ t to Schr¨ odinger’s equation satisfies: | ψ t = T ( t ,t ) | ψ t (2) for any pair of times t and t , where T ( t ,t ) is the time development operator ; in fact a collection of operators labeled by the two times t and t . 1 • Note that the single unitary operator T ( t 1 ,t ), with t and t 1 given, will serve to integrate Schr¨ odinger’s equation from t to t 1 for any initial state | ψ in the Hilbert space. That is, given the starting state | ψ at t , T ( t 1 ,t ) | ψ will be the corresponding final state at time t 1 ....
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qitd322 - qitd322 Unitary Dynamics and Quantum Circuits...

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