Quantifying information scrambling via classical shadow tomography on programmable quantum simulators

We develop techniques to probe the dynamics of quantum information and implement them experimentally on an IBM superconducting quantum processor. Our protocols adapt shadow tomography for the study of time-evolution channels rather than of quantum states and rely only on single-qubit operations and measurements. We identify two unambiguous signatures of quantum information scrambling, neither of which can be mimicked by dissipative processes, and relate these to many-body teleportation. By realizing quantum chaotic dynamics in experiment, we measure both signatures and support our results with numerical simulations of the quantum system. We additionally investigate operator growth under this dynamics and observe behavior characteristic of quantum chaos. As our methods require only a single quantum state at a time, they can be readily applied on a wide variety of quantum simulators.

Figure 1

(a) Representation of the operator state ${\rho }_{\mathrm{op}}\left(t\right)$ [Eq. (1)]. Each qubit in $Q_{\mathrm{out}}$ is prepared in a maximally entangled state (black dots) with the corresponding qubit $Q_{\mathrm{in}}$, before being time evolved under the channel ${\mathcal{N}}_t$. (b) Illustration of the Hayden-Preskill protocol [5]. An unknown quantum state $|\psi \rangle$ is used as an input to a small subregion $A$, while the remaining qubits $B$ are prepared in a maximally entangled state with a set of ancillas $B^{\prime }$ (circled). If the channel is perfectly scrambling then $|\psi \rangle$ can be reconstructed using the ancillas combined with a subset of output qubits $C$ of the same size as $A$, regardless of which qubits are in $C$ (qubits in $D$ are discarded). Formally, the final state of the ancillas combined with the outputs $C$ depends on the input state to $A$ through the channel ${\mathcal{N}}_t^{A\to B^{\prime }C}$ (see the text for details).

Figure 2

Illustration of experimental protocol to measure operator-space entanglement of a quantum channel ${\mathcal{N}}_t$ in a system with $N=4$ qubits. The single-qubit unitaries $U_j$ and $V_j$ are drawn independently at random from the discrete gate sets described in the text. Once the measurement outcomes ${\widehat{b}}_j$ are known, one can construct a snapshot of the doubled state ${\rho }_{\mathrm{op}}\left(t\right)$ using Eq. (7) and then repeat $M$ times with different unitaries.

Figure 3

(a) Qubit layout and connectivity of ibm_lagos. Purple circles represent the five qubits used for the experiments detailed in the text. (b) Circuit design for the chaotic unitary ${\mathcal{N}}_t$, with $t=4$ time steps shown. Each single-qubit gate (colored boxes) is independently sampled from the four gates $W_{1,...,4}$ (see the text for details).

Figure 4

Rényi mutual information [Eq. (2)], with $A=\left\{1\right\}$ and $C=\left\{j_C\right\}$. (a) Dashed lines indicate the exact value without noise or sampling error and points are estimations obtained using shadow postprocessing methods on data from numerical simulations of the full circuit (Fig. 2) without noise. The deviations between these two values can be used to estimate the typical size of the sampling errors that arise from the shadow tomography protocol. (b) Results obtained from ibm_lagos. Solid lines are to guide the eye. The region above the threshold $I^{\left(2\right)}(A:BC)>1$ is shaded green (see Sec. 2b for details).

Figure 5

Logarithm of the ratio $R_{A:BC}=p_{2,A:BC}^2/p_{3,A:BC}$, where $A=\left\{1\right\}$ and $C=\left\{j_C\right\}$. Data are presented as in Fig. 4 for results from (a) simulation and (b) ibm_lagos. The region above the threshold ${log}_2{R}_{A:BC}>0$ is shaded green (see Sec. 2b for details).

Figure 6

Evolution of the $k$-locality of time-evolved operators, as quantified by $D_k^C$ [Eq. (6)]. Specifically, we plot the cumulative weight ${\sum }_{l\le k}{D}_l^C$, which measures the total weight of the time-evolved operator acting nontrivially on at most $k$ qubits, averaged over all nontrivial initial operators with support on $C$. We fix $C=\left\{3\right\}$, the central qubit in Fig. 3. The shaded areas and dashed lines indicate the exact values without sampling error or noise. Markers indicate shadow tomographic estimates calculated from the data sets obtained from (a) noise-free simulations and (b) the quantum device. The former are affected by sampling error only, while the latter are affected by both sampling error and noise.

Figure 7

(a) Conventional shadow tomography on the operator state ${\rho }_{\mathrm{op}}\left(t\right)=({\text{id}}_{\mathrm{in}}\otimes {\mathcal{N}}_t)\left[\mathrm{\Phi }\right]$. The distribution of unitaries $U$ and $V$ and measurement outcomes $\widehat{a}$ and $\widehat{b}$ are the same as that of (b) a hybrid classical-quantum process, where $\widehat{a}$ are sampled from a uniform distribution and then used as the input for a quantum circuit.

Figure 8

Configuration of the cnot layers in the model of chaotic dynamics that uses all seven qubits of ibm_lagos. The pattern of cnot gates repeats every three time steps. Small orange circles denote the control qubits and large orange circles with a plus are the target qubits. Note that the numbering of the qubits (indicated in the leftmost panel) differs from that used in the text.

Figure 9

Color plot of the second Rényi mutual information $I^{\left(2\right)}(A:BC)$ for the circuit model of dynamics described in Appendix pp4, using all seven qubits of ibm_lagos. We fix $A=\left\{1\right\}$ [the bottom right qubit in Fig. 3] and $C=\left\{j_C\right\}$, where $j_C$ is varied. (a) Results obtained from noiseless classical simulations of the time evolution ${\mathcal{N}}_t$ and the shadow protocol. By averaging the deviation of these data from the exact value of $I^{\left(2\right)}(A:BC)$ for the circuit in question, we obtain an estimate of the statistical fluctuations due to the shadow protocol of 0.07. (b) Data obtained from ibm_lagos.