Testing complementarity on a transmon quantum processor

We propose quantum circuits to test interferometric complementarity using symmetric two-way interferometers coupled to a which-path detector. First, we consider the two-qubit setup in which the controlled transfer of path information to the detector subsystem depletes interference on the probed subspace, testing the visibility-distinguishability tradeoff via minimum-error state discrimination measurements. Next, we consider the quantum eraser setup, in which reading out path information in the right basis recovers an interference pattern. These experiments are then carried out in an IBM superconducting transmon processor. A detailed analysis of the results is provided. Despite finding good agreement with theory at a coarse level, we also identify small but persistent systematic deviations preventing the observation of full particlelike and wavelike statistics. We understand them by carefully modeling two-qubit gates, showing that even small coherent errors in their implementation preclude the observation of Bohr's strong formulation of complementarity.

Figure 1

(a) Single-qubit circuit equivalent to a Mach-Zehnder interferometer. (b) Same setup, coupling a second qubit, $q_d$, to the interferometer qubit, $q_i$, to act as a which-path detector. The readout basis is chosen by defining $P_i^{†}$ and $P_d^{†}$ before the $Z$ basis measurement. Setting $P_i^{†}=H$ and varying the value of $\alpha$, interference on $q_i$ can be depleted, independently of reading out $q_d$. Setting $P_d^{†}=O_{\alpha }^{†}$ maximizes which-path information retrieval. Making $P_d^{†}=\mathbb{1}$ instead erases path information rendering detection outcomes useless while allowing full-visibility interference patterns on $q_i$ to be recovered. Both measurements occur simultaneously. A delayed-choice experiment would probe the detector after the interferometer.

Figure 2

In-silico pure-state simulations obtained with two families of circuits of the type of Fig. 1, executed with 8192 shots for each of 10 201 pairs of values $\alpha ,\varphi \in [0,2\pi ]$. The first family consists of the circuit with $P_i^{†}=H$ and $P_d^{†}=O_{\alpha }^{†}$. The top right panel shows the measured $\langle X\rangle$ on $q_i$ for all these circuits. The left side panel illustrates a subset of the data for six different values of $\alpha$: actual results (dots with $\pm 2{\sigma }_{\langle X\rangle }$ errorbar) are compared against the theoretical expectation given by Eq. (14) (solid lines). The third panel shows the results for visibility and distinguishability; data in blue are obtained from the second family of circuits, where the circuit in Fig. 1 is executed with $P_i^{†}=\mathbb{1}$ and $P_d^{†}=O_{\alpha }^{†}$, and Eqs. (10) and (11) are employed to extract $\mathcal{D}$.

Figure 3

Actual physical circuit implemented to reproduce Fig. 1 in terms of the elementary gateset of the device. Final gates are defined for each readout basis according to the correspondence: $P_i^{†}=H\Rightarrow G_i=U_2(0,\pi ), P_i^{†}=\mathbb{1}\Rightarrow G_i=\mathbb{1}, P_d^{†}=O_{\alpha }^{†}\Rightarrow G_d=U_2(0,\frac{3\pi }{2})$, and $P_d^{†}=\mathbb{1}\Rightarrow G_d=U_1(\alpha /2)$.

Figure 4

Experimental results for the circuit in Fig. 1. The $\overline{X}$ estimations are represented with almost imperceptible $\pm 3{\sigma }_{\overline{X}}$ confidence intervals and adjust very well to the properly constructed Eq. (25) model, as measured by ${\chi }_{\nu }^2$. This experimental model is also very close to the pure state model, Eq. (3).

Figure 5

Experimental results for the two families of circuits in Fig. 1 in which the detector qubit is measured in the optimal basis and the interferometer qubit is measured both in the $X$ and the $Z$ basis. Experimental data for $q_i$ on the left side panel correspond to $\overline{X}\pm 2{\sigma }_{\overline{X}}$. Each of these panels is equivalent to those described in Fig. 2.

Figure 6

Residual error maps $z_j-{\widehat{z}}_j$ for the second experiment. On the left, the residuals of the fit to the $\overline{X}$ model of the first family of circuits are shown. On the right, those of the fit to the $\overline{\mathcal{D}}$ model, obtained with the second family of circuits, are shown. The colorbars are marked with the lowest and highest recorded values.

Figure 7

Experimental results for the quantum eraser circuit. The expectation values of the $X$ operator on the first qubit broken down by the outcome of the detector are shown, with $\overline{X_0}\pm 2{\sigma }_{{\overline{X}}_0}$ ($\overline{X_1}\pm 2{\sigma }_{{\overline{X}}_1}$) in the top (bottom) row. Right-side panels plot the full set of results for the tested $(\varphi ,\alpha )$ values, while the left-side panels show a selection for six different $\alpha$.

Figure 8

Exact visibility and distinguishability plots for the different circuits encoded in Fig. 3, with the cnot gate replaced by a ${\text{bcnot}}_5$ with example bias values. In all of the panels, ${\beta }_1=0.06, {\beta }_3=-0.05, {\beta }_4=-0.07$, and ${\beta }_5=0.06$. While ${\beta }_2=0.09$ on the left-side panels, it is drastically increased to 0.3 on the right-side ones to emphasize the effect.