We study a simple-harmonic-oscillator quantum computer solving oracle decision problems. We show that such computers can perform better by using nonorthogonal Gaussian wave functions rather than orthogonal top-hat wave functions as input to the infor
mation encoding process. Using the Deutsch-Jozsa problem as an example, we demonstrate that Gaussian modulation with optimized width parameter results in a lower error rate than for the top-hat encoding. We conclude that Gaussian modulation can allow for an improved trade-off between encoding, processing and measurement of the information.
We present efficient quantum algorithms for simulating time-dependent Hamiltonian evolution of general input states using an oracular model of a quantum computer. Our algorithms use either constant or adaptively chosen time steps and are significant
because they are the first to have time-complexities that are comparable to the best known methods for simulating time-independent Hamiltonian evolution, given appropriate smoothness criteria on the Hamiltonian are satisfied. We provide a thorough cost analysis of these algorithms that considers discretizion errors in both the time and the representation of the Hamiltonian. In addition, we provide the first upper bounds for the error in Lie-Trotter-Suzuki approximations to unitary evolution operators, that use adaptively chosen time steps.
We establish a framework for oracle identification problems in the continuous variable setting, where the stated problem necessarily is the same as in the discrete variable case, and continuous variables are manifested through a continuous representa
tion in an infinite-dimensional Hilbert space. We apply this formalism to the Deutsch-Jozsa problem and show that, due to an uncertainty relation between the continuous representation and its Fourier-transform dual representation, the corresponding Deutsch-Jozsa algorithm is probabilistic hence forbids an exponential speed-up, contrary to a previous claim in the literature.
