Fig. 4: Implementations of generalised Deutsch-Jozsa and Bernstein-Vazirani algorithms in quaternary.
a Quantum logical circuit for implementing the d-ary Deutsch-Jozsa and Bernstein-Vazirani algorithms. This circuit can be implemented by the scheme in Fig. 1a, b with an exchange of the x and y registers. The task of the d-ary Deutsch-Jozsa algorithm is to determine an unknown multi-value function f: {0, 1,..., d−1}n → {0, 1,..., d − 1} is either constant or balanced, while that of the d-ary Bernstein-Vazirani algorithm is to compute the close expression of a multi-value affine function f: A0 ⊕ A1x1. . . ⊕ Anxn, using only a single call of quantum oracle. When d equals to 2, the two algorithms return to the original Deutsch’s algorithms. The key part is the implementation of f(x) ⊕ dy by the MVCU gate. The outcome of the algorithms is measured in the computation basis of the x-register states. b–i Measured probability distributions (normalised coincidence counts) of the x-register in the computational basis. Results in b–h demonstrate that the d-ary Deutsch-Jozsa algorithm allows the determination of whether f(x) is constant (b) or balanced (c–h). Results in b, c, i, h show the d-ary Bernstein-Vazirani algorithm can determine the expression of affine functions f: b, f(x) is constant and A1=0; c, f(x) is affine and A1=1; i, f(x) is affine and A1=2; h, f(x) is affine and A1=3; Dotted boxes in (b--i) refer to theoretical probability distributions. Experimental probability distributions (coloured bars) are obtained from photon coincidences, which are accumulated by 20s per measurement. The classical fidelity Fc presents the success probability of each measurement. In order to make the small error bars visible in the plots, they are plot by ± 3σ. The values in parentheses refer to ± 1σ uncertainty. All error bars are estimated from photon Poissonian statistics.
