Abstract. Quantum learning of a unitary transformation estimates a quantum channel in a process similar to process tomography. The classical counterpart of this goal, finding an unknown function, is regression, although the methodology hardly resembles the outline of classical algorithms. To gain a better understanding what such a methodology means to learning theory, we anchor it to the familiar concepts of active learning and transduction. Learning the unitary translates to storing it in quantum memory, but the procedure also requires an optimal maximally entangled input state; this resembles active learning. The retrieval strategy is a blend of inductive and transductive learning.
Abstract. Quantum learning of a unitary transformation estimates a quantum channel in a process similar to process tomography. The classical counterpart of this goal, finding an unknown function, is regression, although the methodology hardly resembles the outline of classical algorithms. To gain a better understanding what such a methodology means to learning theory, we anchor it to the familiar concepts of active learning and transduction. Learning the unitary translates to storing it in quantum memory, but the procedure also requires an optimal maximally entangled input state; this resembles active learning. The retrieval strategy is a blend of inductive and transductive learning.
This chapter delves deeply into quantum process tomography (QPT) and regression: two critical approaches in the field of processing quantum information. The process of characterizing and estimating unknown quantum processes executed on quantum systems is known as quantum process tomography, while regression aims to find optimal parameters for input states to achieve desired unitary transformations. The chapter begins by introducing the concept of channel-state duality, which establishes a fundamental connection between quantum channels and quantum states. This duality enables the analysis of quantum processes by examining the input and output states, providing insights into their properties and behavior. Quantum process tomography is then discussed in detail, covering various tomographic methods and strategies used to reconstruct unknown quantum processes. Then the mathematical foundations, experimental techniques, and statistical analysis involved in quantum process tomography are explored to provide a comprehensive understanding of the field. Next, the chapter delves into the role of groups, the unitary group, and compact lie groups in the study of quantum processes. These mathematical concepts provide a framework for understanding the structure and symmetries of quantum processes, and representation theory is introduced as a powerful tool for decomposing and analyzing quantum processes into irreducible representations. The concept of storage of unitary and parallel application operations is crucial for the practical implementation of quantum processes insights about techniques such as parallelization and matrix factorization, which optimize the computational resources required for large-scale quantum systems, enabling efficient and scalable execution of quantum processes. The chapter also addresses the concept of finding optimal input states for learning quantum processes, adaptive measurement, and statistical optimization techniques, which enables researchers to identify the input states that provide the most informative data for accurate process estimation. Finally, the chapter concentrates on establishing the unitary operation and defining the parameterization for the initial state to achieve desired transformations. The effective learning criteria and quantum machine learning (QML) for leveraging quantum states provides advantages of QML over classical methods. In summary, this chapter provides a comprehensive exploration of quantum process tomography and regression. By leveraging channel-state duality, representation theory, and optimization techniques, researchers and practitioners can accurately characterize and estimate quantum processes. It also offers insights into the mathematical foundations, experimental methodologies, and practical applications of these techniques, contributing to the advancement of quantum information processing and technology.
Quantum process tomography, the task of learning an unknown quantum channel from black-box access, is a central problem in quantum information. In this setting, protocols with quantum memory can coherently store and jointly process quantum information obtained from multiple channel uses, whereas protocols without quantum memory must measure after each use and retain only a classical transcript of the measurement outcomes. A fundamental open question is whether quantum memory provides a query-complexity advantage even when protocols without quantum memory may adapt their experiments based on all previous outcomes with unbounded classical computational power. In this work, we show that it does. We determine the optimal query complexity of quantum process tomography without quantum memory up to a constant factor to be Θ(din3dout3/ε2), where din and dout are the channel input and output dimensions, respectively, and ε is the target diamond-norm accuracy. More precisely, we prove that any incoherent protocol for this task, including adaptive protocols, requires Ω(din3dout3/ε2) queries, even when each channel use may be assisted by arbitrary fresh ancilla, and we present a non-adaptive, ancilla-free incoherent protocol achieving the matching upper bound O(din3dout3/ε2). Our results thereby generalize the optimal sample-complexity bounds for single-copy state tomography, recovered as the special case din=1. By contrast, coherent protocols with quantum memory achieve query complexity Θ(din2dout2/ε2). Hence, our results establish a rigorous learning separation between quantum process tomography with and without quantum memory.
We address the problem of learning an unknown unitary transformation from a finite number of examples. The problem consists in finding the learning machine that optimally emulates the examples, thus reproducing the unknown unitary with maximum fidelity. Learning a unitary is equivalent to storing it in the state of a quantum memory (the memory of the learning machine) and subsequently retrieving it. We prove that, whenever the unknown unitary is drawn from a group, the optimal strategy consists in a parallel call of the available uses followed by a ``measure-and-rotate'' retrieving. Differing from the case of quantum cloning, where the incoherent ``measure-and-prepare'' strategies are typically suboptimal, in the case of learning the ``measure-and-rotate'' strategy is optimal even when the learning machine is asked to reproduce a single copy of the unknown unitary. We finally address the problem of the optimal inversion of an unknown unitary evolution, showing also in this case the optimality of the ``measure-and-rotate'' strategies and applying our result to the optimal approximate realignment of reference frames for quantum communication.
Quantum process tomography (QPT) is a fundamental tool for fully characterizing quantum systems. It relies on querying a set of quantum states as input to the quantum process. Previous QPT methods typically employ a straightforward strategy for randomly selecting quantum states, overlooking differences in informativeness among them. In this work we propose a general active learning (AL) framework that adaptively selects the most informative subset of quantum states for reconstruction. We design and evaluate various AL algorithms and provide practical guidelines for selecting suitable methods in different scenarios. In particular, we introduce a learning framework that leverages the widely used variational quantum circuits to perform the QPT task and integrate our AL algorithms into the query step. We demonstrate our algorithms by reconstructing the unitary quantum processes resulting from random quantum circuits with up to seven qubits. Numerical results show that our AL algorithms achieve significantly improved reconstruction and the improvement increases with the size of the underlying quantum system. Our work opens new avenues for further advancing existing QPT methods.
The subject of this thesis, quantum estimation theory, is inspired by recent developments, both practical and theoretical, in the new field of quantum information and computation. The main task of quantum estimation theory, the subject of this thesis, is that given many identical copies of an unknown quantum system one wants to guess or estimate what the system is. This is done by performing an measurement and analyzing the data obtained from it.
A quantum state is represented by a density matrix. In some cases one does not need to know the whole density matrix, sometimes a partial characterization is enough. Chapter 2 of this thesis is about optimally estimating the spectrum, i.e. the eigenvalues, of a density matrix. This is useful, for example, for estimating the entanglement of a bipartite pure state. The results of chapter 2 tell us that one can optimally estimate bipartite entanglement with LOCC measurements on one of the two parties.
Unitary operations transform density matrices to density matrices. They are the gates of quantum information processing. Suppose one has one such unitary operation but one does not know what it does. In order to characterize it one needs to know how it transforms any input state. In order to know how a particular state is transformed, one can prepare that state, feed it to the unitary many times, and measure the output. One state is, of course, not enough. It turns out that the unitary can be completely characterized with a finite set of states, this is known as quantum process tomography. Unfortunately, in many cases it is difficult to have available in the lab all the input states needed to perform quantum process tomography. It is here where entanglement comes to the rescue. One pair of maximally entangled qubits, where one of them is used as input for the unitary gate and the other is left untouched, is sufficient. All the information about the unknown gate is contained in the joint output state. This is the problem treated in chapter 3. Now the output state needs to be measured. Again, one could measure the two qubits separately (LOCC) or jointly (collective measurement). Is there anything to gain by performing joint measurements? The answer is yes, optimal collective measurements are at least 3 times more accurate than any LOCC measurement.
Chapter 4 treats the problem of estimating commuting unitary gates. There it is shown that this extra information makes it unnecessary to use an entangled input state and to use collective measurements at the output.
In chapters 3 and 4 it is assumed that only one copy of the unitary gate is available. If, however, more than one copy (say N ) are available at the same time one could use an N-fold entangled state as input. If no entanglement is used, classical statistics tells us that the error of the estimation is proportional to 1/√N. In the literature, it has been shown that by using an entangled state in the input one can achieve an error proportional to 1/N. However, this has been shown only for qubit gates. In chapter 5 important steps are taken in the direction of showing the same thing for a general unitary gate, but this remains an open problem.
Full tomography of an unknown quantum evolution is resource-intensive and often unnecessary when the goal is only to predict selected properties. This motivates the study of classical shadow estimation of unitary channels (CSEU), a task in which one queries an unknown d-dimensional unitary U and stores classical data that can later be used to predict expectation values tr[OUρU†] for arbitrary input states ρ and observables O. Here we propose a parallel and non-adaptive CSEU protocol that uses O(min{dε−2,d3/2ε−1}) queries to U when the observables have constant effective size. This achieves Heisenberg scaling with respect to the additive error ε and is query-optimal, as we prove a matching lower bound Ω(min{dε−2,d3/2ε−1}) for the query complexity, which remains valid even when stronger access to the unknown unitary is available. Our CSEU protocol is a versatile primitive with applications to several quantum learning tasks. In particular, it yields a near-optimal algorithm for learning an arbitrary Hamiltonian from its real-time evolution in the high-precision regime and recovers the best-known bounds for several tasks in a unified way.