Probing Operator Spreading via Floquet Engineering in a Superconducting Circuit

Operator spreading, often characterized by out-of-time-order correlators (OTOCs), is one of the central concepts in quantum many-body physics. However, measuring OTOCs is experimentally challenging due to the requirement of reversing the time evolution of systems. Here we apply Floquet engineering to investigate operator spreading in a superconducting 10-qubit chain. Floquet engineering provides an effective way to tune the coupling strength between nearby qubits, which is used to demonstrate quantum walks with tunable couplings, reversed time evolution, and the measurement of OTOCs. A clear light-cone-like operator propagation is observed in the system with multiple excitations, and has a nearly equal velocity as the single-particle quantum walk. For the butterfly operator that is nonlocal (local) under the Jordan-Wigner transformation, the OTOCs show distinct behaviors with (without) a signature of information scrambling in the near integrable system.

Figure 1

Device and quantum walk with tunable coupling. (a) Optical micrograph of the superconducting processor containing 10 transmon qubits arranged into a chain. Each qubit has a microwave line for the $XY$ control, a flux bias line for the $Z$ control, and a readout resonator for measurement. The NN qubits are coupled capacitively with almost the same strength [49]. The parameters of the device are presented in the Supplemental Material in detail [49]. (b) Experimental results of excitation oscillation between $Q_1$ and $Q_2$. The two qubits are prepared in the initial state of $|01⟩$, and ac magnetic flux is applied on $Q_1$ with an amplitude $ϵ$. The color bar represents the expectation value of the photon number of $Q_1$. (c) Experimental coupling strength $g_1^{\mathrm{eff}}$ versus ac magnetic flux amplitude $ϵ$, which satisfies $g_j^{\mathrm{eff}}/g_1\approx {\mathcal{J}}_0(ϵ/\nu )$. Two arrows indicate the values, ${ϵ}_a$ and ${ϵ}_b$, used in the measurement of OTOCs for the forward and backward time evolution, respectively. Single-photon quantum walks for (d) $ϵ/2\pi =120$, (e) $ϵ/2\pi =288.6$, and (f) $ϵ/2\pi =400\mathrm{MHz}$, observed on $Q_2\text{−}{Q}_{10}$ with the initial state $|000010000⟩$. (g) Normalized group velocity $v/v_0$ of photon propagation versus $ϵ/\nu$, where $v_0=117\pm 4\mathrm{sites}/\mu \mathrm{s}$ corresponds to the value with $ϵ=0$ [49]. The error bar is estimated by fitting errors. Each point is the average result of 8000 single-shot readouts.

Figure 2

Reversed time evolution of the 10-qubit chain with the initial state $|1010101010⟩$, realized by using ${ϵ}_a$ and ${ϵ}_b$ for the time before and after 125 ns, respectively. Symbols are the experimental results. Solid and dashed lines are the calculated results based on the Hamiltonian in Eq. (2) by considering three and two energy levels for each qubit, respectively.

Figure 3

(a) Experimental results of $C_j^{\mathrm{ZZ}}$, with the initial state $|0101010101⟩$ and operators $\widehat{W}={\widehat{\sigma }}_{10}^z$, $\widehat{V}={\widehat{\sigma }}_j^z$ ($j=1,\dots ,10$). The postselection is applied due to the $U(1)$ symmetry, i.e., only the single-shot results that have the same number of bosons as the initial state are considered. (b) Numerical results considering ZZ interaction between NN qubits. (c) Comparison between experimental $C_j^{\mathrm{ZZ}}$ (symbols) and two calculated results using ${\widehat{H}}_{\mathrm{eff}}$ (dashed lines) and considering ZZ interaction (solid lines).

Figure 4

(a) Experimental results of $C_j^{\mathrm{XX}}$ with the initial state $|++++++++++⟩$ and operators $\widehat{W}={\widehat{\sigma }}_{10}^x$, $\widehat{V}={\widehat{\sigma }}_j^x$ ($j=1,\dots ,10$). (b) Numerical results considering the qubit third energy level. (c) Comparison between experimental $C_j^{\mathrm{XX}}$ (symbols) and two numerical results using two (dashed lines) and three (solid lines) qubit levels.