Optimizing measurement strengths for qubit quasiprobabilities behind out-of-time-ordered correlators

Out-of-time-ordered correlators (OTOCs) have been proposed as a tool to witness quantum information scrambling in many-body system dynamics. These correlators can be understood as averages over nonclassical multitime quasiprobability distributions (QPDs). These QPDs have more information and their nonclassical features witness quantum information scrambling in a more nuanced way. However, their high dimensionality and nonclassicality make QPDs challenging to measure experimentally. We focus on the topical case of a many-qubit system and show how to obtain such a QPD in the laboratory using circuits with three and four sequential measurements. Averaging distinct values over the same measured distribution reveals either the OTOC or parameters of its QPD. Stronger measurements minimize experimental resources despite increased dynamical disturbance.

Figure 1

Sequential measurement circuit. Repeated circuit realizations yield the joint distribution $P_{{\varphi }_a,{\varphi }_b,{\varphi }_{a^{\prime }},{\varphi }_{b^{\prime }}}^{A,B,A,B}(a,b,a^{\prime },b^{\prime })$ of ancilla-qubit outcomes. Averaging this distribution with strategic values (see main text) yields the multiqubit out-of-time-ordered correlator $F\left(t\right)=〈B\left(t\right)AB\left(t\right)A〉$ and eight correlators that determine its corresponding quasiprobability distribution ${\tilde{p}}_t$. (a) Averaging the one-measurement subcircuit (red, dashed) produces $\langle A\rangle$. (b) Averaging the two-measurement subcircuit (blue, dotted) produces $\text{Re}\langle B\left(t\right)A\rangle$. (c) Averaging the three-measurement subcircuit (green, dot-dashed) produces $\langle B\left(t\right)\rangle , \langle AB\left(t\right)A\rangle$, and $\text{Re}\langle B\left(t\right)AB\left(t\right)A\rangle$. (d) Averaging the four-measurement circuit produces $\langle B\left(t\right)AB\left(t\right)\rangle$. To obtain the final two correlators $\text{Im}\langle B\left(t\right)A\rangle$ and $\text{Im}\langle B\left(t\right)AB\left(t\right)A\rangle$ that determine ${\tilde{p}}_t$, the first informative measurement ${\widehat{M}}_{{\varphi }_a,a}^{\left(A\right)}$ should be replaced with a noninformative measurement ${\widehat{N}}_{{\varphi }_a,a}^{\left(A\right)}$ (see text for details), and the last measurement may be omitted.