Fig. 3 | Nature Communications

Fig. 3

From: Variational consistent histories as a hybrid algorithm for quantum foundations

Fig. 3

Consistency of three-time histories for a spin-1/2 particle in a magnetic field, with initial state ρ = | + 〉〈 + |. The full-trace cost landscape, C(ϕ1, ϕ2), is plotted as a function of the azimuths, ϕ1 and ϕ2, of the first and second projection bases, which we constrained to the xy plane of the Bloch sphere. The point (0, 0) corresponds to both projections being along the x axis. Consistency is expected everywhere along certain vertical lines (ϕ1 = 2 +  rad), as they correspond to histories where the initial state is one of the projectors after the first time step, so there are no branches to interfere in the second time step. In addition, some slope-one lines (ϕ2 = ϕ1 + (2 + )rad) should be consistent, as they correspond to histories where the second projectors are the same as the first after time evolution, so no interference occurs in the second time step. Indeed, one can see valleys in the cost landscapes for these vertical and slope-one lines, when the cost is quantified on a simulator a and on the ibmqx5 quantum computer b. It is noteworthy that negative cost values are possible due to finite statistics. The white “x” symbols in b mark some of the non-unique minima that the VCH algorithm found

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