Fig. 2

Evolution of the flux probability distribution \({\cal P}_k\)(Φ _i −Φ), during the run of the first k = 1, …, 4 steps of the Fourier (red panels) and Kitaev (blue curves) estimation algorithms. The magnetic flux is measured with respect to a reference flux (as explained in the Methods). The actual flux value is shown by a thick black line in the flux-step plane. The Kitaev algorithm starts at a zero delay τ = 0 and a first step returns a broad probability distribution with a single peak centered near the actual flux value. During the run of the Kitaev algorithm this peak narrows down. The Fourier algorithm starts from the Ramsey measurement at large delay τ^(s) = 360 ns, with the first step returning a probability distribution with six out of twelve flux intervals assuming a non-vanishing value. Hence, this first step selects half of the n = τ^(s)/τ₀~12 different flux intervals ΔΦ _m given by Δω _m  = Δω₀ + 2πi/τ _s , m = 0, …, n−1, where Δω _m  ≡ ω₀₁(ΔΦ _m ) − ω_d is the frequency interval corresponding to the flux interval ΔΦ _m , and determines the parity of the yet unknown index m ∈ [0, n − 1] associated with the true flux interval. In the second step, the Fourier algorithm proceeds to a shorter delay and rules out another half of the remaining six intervals. In the next two steps the algorithm discriminates between the remaining three alternatives and ends up with the correct flux interval. The green line at the fourth step displays the probability distribution learned by the standard (classical) procedure during the same number of Ramsey measurements as was required by the quantum procedures. The distributions obtained at the step number 4 for the Kitaev and Fourier estimation algorithms and in the standard (classical) measurement are shown in the inset