Strengthening weak measurements of qubit out-of-time-order correlators

For systems of controllable qubits, we provide a method for experimentally obtaining a useful class of multitime correlators using sequential generalized measurements of arbitrary strength. Specifically, if a correlator can be expressed as an average of nested (anti)commutators of operators that square to the identity, then that correlator can be determined exactly from the average of a measurement sequence. As a relevant example, we provide quantum circuits for measuring multiqubit out-of-time-order correlators using optimized control-$Z$ or $\mathit{ZX}$-90 two-qubit gates common in superconducting transmon implementations.

Figure 1

(a) Quantum circuit using an optimized control-$Z$ (cz) entangling gate to implement the generalized $\widehat{A}$ measurement ${\widehat{M}}_{\varphi ,a}^{\left(A\right)}=[cos(\varphi /2)\widehat{1}-{(-1)}^{a}sin(\varphi /2)\widehat{A}]/\sqrt{2}$. The (potentially $n$-qubit) unitary gate ${\widehat{U}}_A$ is chosen so that ${\widehat{U}}_A\widehat{Z}{\widehat{U}}_A^{†}=\widehat{A}$ on the target qubits. The $y$-rotation gate ${\widehat{R}}_y\left(\phi \right)=exp[-i(\phi /2)\widehat{Y}]$ rotates the ancilla qubit through an angle $\phi$ in the $xz$ plane of the Bloch sphere. (b) Bloch-$xz$-plane detail of the ancilla evolution, showing each possible ancilla state in the entangled superposition as a distinct colored arrow. The ancilla $z$-measurement result $a=0,1$ is correlated with the eigenstates of the observable $\widehat{A}$, with perfect correlation when $\varphi =\pi /2$. This correlation results in a partial collapse into the $\widehat{A}$ eigenstates. For any correlation strength $\varphi$, the observable's expectation value can be determined empirically by averaging the scaled values ${\alpha }_{\varphi ,a}={(-1)}^{a+1}/sin\varphi$ due to the operator identity ${\sum }_a{\alpha }_{\varphi ,a}{\widehat{M}}_{\varphi ,a}^{†\left(A\right)}{\widehat{M}}_{\varphi ,a}^{\left(A\right)}=\widehat{A}$.

Figure 2

(a) Quantum circuit using a cz gate to implement the noninformative generalized $\widehat{A}$ measurement ${\widehat{N}}_{\varphi ,a}^{\left(A\right)}=[cos(\varphi /2)\widehat{1}+i{(-1)}^{a}sin(\varphi /2)\widehat{A}]/\sqrt{2}$, for comparison with Fig. 1. The only difference is the added $x$-rotation gate ${\widehat{R}}_x(\pi /2)=exp[-i(\pi /4)\widehat{X}]$ that rotates the ancilla qubit through an angle $\pi /2$ in the $yz$ plane. (b) Bloch-$xz$-plane detail of the ancilla evolution. The added rotation moves the $\widehat{A}$ correlation to the $xy$ plane, so the $z$-measurement result $a=0,1$ is no longer informative. Despite the lack of correlation, each result $a$ enacts a conditional unitary, generated by $\widehat{A}$, on the target.

Figure 3

(a) Quantum circuit using an optimized ${\widehat{ZX}}_{90}$ ($\mathit{ZX}$-90) entangling gate to implement the generalized $\widehat{A}$ measurement ${\widehat{M}}_{\varphi ,a}^{\left(A\right)}$, for contrast with Fig. 1. (b) Bloch-$xz$-plane detail of the ancilla evolution.

Figure 4

(a) Quantum circuit using a $\mathit{ZX}$-90 gate to implement the noninformative generalized $\widehat{A}$ measurement ${\widehat{N}}_{\varphi ,a}^{\left(A\right)}$, for comparison with Fig. 2. (b) Bloch-$xz$-plane detail of the ancilla evolution.

Figure 5

Quantum circuit for measuring the time-ordered correlator ${\langle \widehat{B}\left(t\right)\widehat{A}\rangle }_{{\rho }_S}$, with $\widehat{B}\left(t\right)={\widehat{U}}_t^{†}\widehat{B}{\widehat{U}}_t$. The operators $\widehat{A}$ and $\widehat{B}$ may act on any distinct combinations of the $n$ qubits. Using the generalized measurement procedures of any strength from Figs. 1 and 3, this circuit yields the distribution of results $P(a,b)$, with $a,b\in \{0,1\}$. Averaging this distribution yields ${\sum }_{a,b}{\alpha }_{{\varphi }_a,a}{\alpha }_{{\varphi }_b,b}P(a,b)=\text{Re}{\langle \widehat{B}\left(t\right)\widehat{A}\rangle }_{{\rho }_S}$, with ${\alpha }_{{\varphi }_a,a}={(-1)}^{1+a}/sin{\varphi }_a$ and similar for $b$. Replacing the first measurement with ${\widehat{N}}_{{\varphi }_a,a}^{\left(A\right)}$ from Figs. 2 and 4 and performing the same weighted average of results yields $\text{Im}{\langle \widehat{B}\left(t\right)\widehat{A}\rangle }_{{\rho }_S}$.

Figure 6

Quantum circuit for measuring the out-of-time-ordered correlator $F\left(t\right)={\langle \widehat{B}\left(t\right)\widehat{A}\widehat{B}\left(t\right)\widehat{A}\rangle }_{{\rho }_S}$, with $\widehat{B}\left(t\right)={\widehat{U}}_t^{†}\widehat{B}{\widehat{U}}_t$. Similarly to Fig. 5, this circuit yields the distribution of results $P(a,b,a^{\prime },b^{\prime })$, with $a,b,a^{\prime },b^{\prime }\in \{0,1\}$. Averaging this distribution produces ${\sum }_{a,b,a^{\prime },b^{\prime }}{\alpha }_{{\varphi }_a,a}{\alpha }_{{\varphi }_b,b}{\alpha }_{{\varphi }_{a^{\prime }},a^{\prime }}{\alpha }_{{\varphi }_{b^{\prime }},b^{\prime }}P(a,b,a^{\prime },b^{\prime })=[1+\text{Re}F\left(t\right)]/2$, with ${\alpha }_{{\varphi }_a,a}={(-1)}^{1+a}/sin{\varphi }_a$ and similar for $b,a^{\prime },b^{\prime }$. Replacing the first measurement with ${\widehat{N}}_{{\varphi }_a,a}^{\left(A\right)}$ and performing the same weighted average of results yields $\text{Im}F\left(t\right)/2$.