Learning Temporal Quantum Tomography

Quantifying and verifying the control level in preparing a quantum state are central challenges in building quantum devices. The quantum state is characterized from experimental measurements, using a procedure known as tomography, which requires a vast number of resources. However, tomography for a quantum device with temporal processing, which is fundamentally different from standard tomography, has not been formulated. We develop a practical and approximate tomography method using a recurrent machine learning framework for this intriguing situation. The method is based on repeated quantum interactions between a system called quantum reservoir with a stream of quantum states. Measurement data from the reservoir are connected to a linear readout to train a recurrent relation between quantum channels applied to the input stream. We demonstrate our algorithms for representative quantum learning tasks, followed by the proposal of a quantum memory capacity to evaluate the temporal processing ability of near-term quantum devices.

Figure 1

Our framework can learn the tomography of a device that is supposed to implement an unknown temporal quantum map $\mathcal{F}$; or emulate a predefined $\mathcal{F}$. (a) A stream of quantum channels $\{{\mathrm{\Omega }}_n\}$ applies to the input stream $\{{\beta }_n\}$ where each channel’s output is determined by finite input history and past channels. The device’s output is a function of input history such as a coherent superposition or a convex mixture of channels. (b) Our framework consists of a quantum reservoir $\mathcal{S}$ interacting with $\{{\beta }_n\}$ with memory effects. The reservoir’s internal state is evolved via CPTP maps ${\mathcal{L}}_{{\beta }_n}$, which transfer the information from the input states to the reservoir. Measurement results in $\mathcal{S}$ are used in a readout layer to reconstruct $\mathcal{F}$.

Figure 2

Temporal tomography for (a) the quantum simple moving average filter and (b) the delayed depolarizing map with $d=5$, $N_m=5$, $N_e=1$, $\alpha =1.0$, $J/B=1.0$, and $\tau B=2.6$ [for (a)] and $\tau B=2.0$ [for (b)]. At each time point, the density matrix is vectorized by stacking the real and imaginary parts, where the range of values is indicated in the color bars.

Figure 3

Tomography errors according to $\tau B$ for $\mathcal{F}({\beta }_n)={\mathrm{\Omega }}_{n-1}({\beta }_{n-1})$ with $\alpha =1.0$ and $J/B=1.0$.

Figure 4

(a) QMC broken down in delays $d$ according to $\tau B$ with $N_e=2$, $N_m=4$ with $\alpha =1.0$ and $J/B=1.0$. (b) The color map of QMC as the function of $\alpha$ and $J/B$ at $\tau B=10.0$.