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 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.
