Online learning of quantum processes

TL;DR

This work establishes a formal framework for online learning of quantum processes, showing that general CPTP channels cannot be online-tomographed efficiently, but two practically relevant classes—gated-complexity-bounded channels and Pauli channels—are online-learnable with polynomial regret and mistake bounds. It extends the online-learning paradigm to non-Markovian multi-time quantum processes and proves shadow-tomography results for Pauli channels and, by twirling, for general channels. The analysis combines multiplicative-weights methods, sequential covering numbers, Bell sampling, and quantum combs to derive learning bounds, while also proving lower bounds and cryptographic hardness results that delineate the landscape. The results open a path toward channel-specific adaptive tomography and highlight both the potential and limits of online learning in quantum information processing, with implications for noise characterization and adaptive diagnostics in quantum devices.

Abstract

Among recent insights into learning quantum states, online learning and shadow tomography procedures are notable for their ability to accurately predict expectation values even of adaptively chosen observables. In contrast to the state case, quantum process learning tasks with a similarly adaptive nature have received little attention. In this work, we investigate online learning tasks for quantum processes. Whereas online learning is infeasible for general quantum channels, we show that channels of bounded gate complexity as well as Pauli channels can be online learned in the regret and mistake-bounded models of online learning. In fact, we can online learn probabilistic mixtures of any exponentially large set of known channels. We also provide a provably sample-efficient shadow tomography procedure for Pauli channels. Our results extend beyond quantum channels to non-Markovian multi-time processes, with favorable regret and mistake bounds, as well as a shadow tomography procedure. We complement our online learning upper bounds with mistake as well as computational lower bounds. On the technical side, we make use of the multiplicative weights update algorithm, classical adaptive data analysis, and Bell sampling, as well as tools from the theory of quantum combs for multi-time quantum processes. Our work initiates a study of online learning for classes of quantum channels and, more generally, non-Markovian quantum processes. Given the importance of online learning for state shadow tomography, this may serve as a step towards quantum channel variants of adaptive shadow tomography.

Figures (6)

• Figure 1: Learning of quantum processes. (a) To learn about the unknown evolution of a quantum system (symbolized by the blue shaded region and represented mathematically by the quantum channel $\mathcal{N}$), we prepare a probe quantum state $\rho$, let it evolve, and then measure it according to the POVM $\{M,\mathbbm{1}-M\}$. We encapsulate this process in the circuit diagram shown in (b). (c) More generally, we can prepare an entangled probe state of two systems, let only one of them evolve, and then jointly measure both systems. Our results apply to this more general class of tests, and also more generally to classes of multi-time quantum processes, in which the unknown evolution could be non-Markovian.
• Figure 2: Extensions of our results to general quantum processes. (a) Going beyond one use of a channel $\mathcal{N}$, as shown in Figure \ref{['fig:channel_tests']}, we may want to learn the value of the channel on tests that make multiple, adaptive uses of the channel. Shown are three independent uses of $\mathcal{N}$, whose Choi representation is $C(\mathcal{N})$. (b) We can similarly perform adaptive tests of a non-Markovian process $\mathcal{N}^{[3]}$, characterized by the blue quantum comb, with Choi representation $C(\mathcal{N}^{[3]})$. The generalized Born rule CDP09 tells us that the outcome probabilities of measurements, or "tests", of quantum channels and multi-time quantum processes can be determined by an analogue of the usual Born rule for quantum states, in which the Choi representation takes the place of the quantum state, and the test is characterized by operators $E^{(i)}$ that are generalizations of effect operators for quantum states. (See Section \ref{['sec-basics_quantum_info']} for details.)
• Figure 3: Multi-time processes with bounded complexity. (a) The basic unit of our multi-time processes with bounded complexity is a process consisting of two two-qubit channels connected by an inaccessible memory system. (b) By collapsing the causal structure of the inputs and outputs of the process in (a), we obtain a three-qubit channel belonging to the set $\mathsf{CPTP}_{3,2}$. (c) An example of a multi-time process obtained by composing (ten of) the basic elements in (a) in a circuit.
• Figure 4: Twirling of multi-time quantum processes. (a) A "time-local" Pauli twirl of a multi-time quantum process with $r$ time steps consists of independently applying a random Pauli channel $\mathcal{P}^{\boldsymbol{w}_k}(\cdot)\coloneqq P^{\boldsymbol{w}_k}(\cdot)P^{\boldsymbol{w}_k\dagger}$, where $\boldsymbol{w}_k\equiv (\boldsymbol{z}_k,\boldsymbol{x}_k)\in\{0,1\}^n\times\{0,1\}^n$, to the input and output of every time step $k\in\{1,2,\dotsc,r\}$. (b) After twirling, the process is characterized by an error-rate probability vector, in the same way as Pauli channels. This error-rate vector can be obtained via time-local Bell measurements, as shown. The outcomes of the measurements are $\boldsymbol{w}_1\equiv(\boldsymbol{z}_1,\boldsymbol{x}_1),\boldsymbol{w}_2\equiv(\boldsymbol{z}_2,\boldsymbol{x}_2),\dotsc,\boldsymbol{w}_r\equiv(\boldsymbol{z}_r,\boldsymbol{x}_r)$.
• Figure 5: (Top) A multi-time quantum process with $r=4$ time steps. The input systems are $A_1,\dotsc,A_4$, the output systems are $B_1,\dotsc,B_4$, and the memory systems are $M_1,M_2,M_3$. (Bottom) Every multi-time process is associated with the channel $\mathcal{N}^{[r]}$, obtained by collapsing the causal ordering of the inputs and outputs.

Theorems & Definitions (102)

• Theorem 1: Online learning channels of bounded complexity---informal
• Theorem 2: Online learning Pauli channels---informal
• Theorem 3: Computational lower bounds for online learning---informal
• Theorem 4: Pauli channel shadow tomography---informal
• Theorem 5: Online learning convex mixtures of known channels---informal
• Theorem 6: Online learning multi-time processes of bounded complexity---informal
• Theorem 7: Online learning of convex mixtures of known multi-time processes---informal
• Theorem 8: Shadow tomography of multi-time processes---informal
• Lemma 9: Regret bound
• proof