Practical learning of multi-time statistics in open quantum systems

TL;DR

The theoretical tools of classical shadow tomography are generalised to the temporal domain to explore multi-time phenomena to efficiently learn the features of multi-time processes such as correlated error rates, multi-time non-Markovianity, and temporal entanglement.

Abstract

Randomised measurements can efficiently characterise many-body quantum states by learning the expectation values of observables with low Pauli weights. In this paper, we generalise the theoretical tools of classical shadow tomography to the temporal domain to explore multi-time phenomena. This enables us to efficiently learn the features of multi-time processes such as correlated error rates, multi-time non-Markovianity, and temporal entanglement. We test the efficacy of these tools on a noisy quantum processor to characterise its noise features. Implementing these tools requires mid-circuit instruments, typically slow or unavailable in current quantum hardware. We devise a protocol to achieve fast and reliable instruments such that these multi-time distributions can be learned to a high accuracy. This enables a compact matrix product operator representation of large processes allowing us to showcase a reconstructed 20-step process (whose naive dimensionality is that of a 42-qubit state). Our techniques are pertinent to generic quantum stochastic dynamical processes, with a scope ranging across condensed matter physics, quantum biology, and in-depth diagnostics of noisy intermediate-scale quantum devices.

Figures (7)

• Figure 1: Circuit diagrams of the conventional and generalised Choi-Jamiolkowski isomorphism. a Two-time processes are represented by quantum channels; their Choi state is given by the channel acting on one half of a Bell pair. b many-time processes are represented by process tensors; their Choi state is given for a $k$-step process by swapping in one half of $k$ Bell pairs at different times. The non-Markovian correlations are mapped onto spatial correlations in the Choi state. The marginals of the process tensor are quantum channels, as well as the average initial state.
• Figure 2: a We refer to the Choi states of quantum processes as many-time states, which are isomorphic to many-body states modulo causality conditions. Hence, we can employ many of the tools developed to study many-body states to study quantum stochastic processes. b A many-time non-Markovian process is represented by a process tensor, which is a multitime density matrix that contracts with a sequence of instruments to yield a final state for the system. c A $k$-step process tensor can also be seen as a sequence of correlated CPTP maps between different points in time, plus the effect due to the initial state. Output legs $\mathfrak{o}_l$ are mapped by a control operation $\mathcal{A}_l$ to the next input leg $\mathfrak{i}_{l+1}$.
• Figure 3: Instrument design for complete multi-time statistics. a Using a structured system-ancilla interaction, we construct a restricted process tensor with local unitaries to realise a locally-controllable instrument. This gives an exact characterisation of the effective instrument $\mathcal{A}_x(\theta,\phi,\lambda)$ for any choice of parameter values. Importantly, as per Eq. \ref{['eq:PTM-vals']}, these instruments will access the full trace-decreasing and non-unital subspace of superoperator space. b Some sample basis instruments implemented using this method on ibm_perth written in the Pauli basis, showing IC controllability including non-unital and trace-decreasing directions. These PTMs demonstrate that well-conditioned IC instrument sets can be achieved.c An illustration of how these instrument sets can be used to probe multi-time statistics. Each multi-time instrument is realised with the circuit as shown -- by applying the locally-modulated interaction with the ancilla, measuring (and recording) the ancilla, and subsequently resetting it. The different instruments applied capture temporal quantum correlations from an arbitrarily strong $SE$ interaction.
• Figure 4: The two-map (four-body) marginals of a system enduring naturally idle dynamics on ibm_perth. a We compute both the negativity between the two steps and the trace-distance between the maps and the product of their marginals. b We compute the joint overlap of correlated Pauli errors (their Choi states): $\langle \hat{P}_i\hat{P}_j\rangle - \langle \hat{P}_i\rangle\langle \hat{P}_j\rangle$ for $ZZ$ and $XX$ as well as c$ZX$ and $XZ$.
• Figure 5: Graphical illustration of a reconstructed many-body process tensor from its marginals. a We construct the process tensor as a finitely correlated state by stitching together $\ell$ step marginals estimated from classical shadows. Here is an example taking $\ell=3$. Note that the overline denotes a pseudoinverse of the matrix. Orange legs are free indices, and grey legs are contracted indices between process tensor marginals. b The resulting many-time representation after condensing down the above, and appropriately splitting sites at the start and finish. The $\lambda_j$ indicate summation over the indices given in the marginal pseudoinverses.