Learning with data in the form of quantum states is of current interest and has led to a variety of problems that boil down to interaction with the available data via quantum measurement and classical post-processing of observed classical outcomes. In quantum measurement PAC learning, one is given a sequence of unknown, prepared quantum states and classical labels, along with a hypothesis class of candidate measurements. The task is to select a measurement from the hypothesis class that minimizes a fixed notion of error in prediction of the classical labels via measurement of a new state by the selected hypothesis. In this work, we consider the advantage of interacting with the given data in the measurement learning framework using learning rules given by measurements that cannot be implemented using local operations and classical communication (LOCC), as opposed to single-copy learning rules. We provide a construction showing that there exist learning scenarios wherein single-copy learning rules are asymptotically suboptimal compared to optimal ones. We then show that learning rules based on entangled measurements enjoy at most a polynomial sample complexity advantage over single-copy learning rules in the PAC learning setting (under a natural joint measurability covering assumption).
Learning with data in the form of quantum states is of current interest and has led to a variety of problems that boil down to interaction with the available data via quantum measurement and classical post-processing of observed classical outcomes. In quantum measurement PAC learning, one is given a sequence of unknown, prepared quantum states and classical labels, along with a hypothesis class of candidate measurements. The task is to select a measurement from the hypothesis class that minimizes a fixed notion of error in prediction of the classical labels via measurement of a new state by the selected hypothesis. In this work, we consider the advantage of interacting with the given data in the measurement learning framework using learning rules given by measurements that cannot be implemented using local operations and classical communication (LOCC), as opposed to single-copy learning rules. We provide a construction showing that there exist learning scenarios wherein single-copy learning rules are asymptotically suboptimal compared to optimal ones. We then show that learning rules based on entangled measurements enjoy at most a polynomial sample complexity advantage over single-copy learning rules in the PAC learning setting (under a natural joint measurability covering assumption).
Can measurements chosen adaptively, in response to earlier outcomes, be exponentially more informative than measurements fixed in advance? In many quantum settings, adaptivity helps at most polynomially; known exponential advantages require coherent measurements across copies or restricted measurements. We show an exponential advantage measuring one copy at a time, with no quantum memory kept between state copies, even when the nonadaptive learner may perform arbitrary global measurements on each copy. We construct explicit families of multi-qubit states in which a scalar parameter is encoded in one of exponentially many candidate observables, with an auxiliary marker indicating which one. An adaptive protocol identifies the relevant observable and then measures it using only single-qubit measurements and one round of adaptation. At fixed accuracy, it needs logarithmically many copies as the qubit count grows, whereas every nonadaptive protocol needs exponentially many. The same mechanism yields an exponential gap in local asymptotic minimax mean squared error constants, tight up to the exponential base. The states are separable and classical in label-dependent product bases; no entanglement or other nonclassical correlations are required. The gap comes from measurement incompatibility: the candidate observables cannot be measured jointly, so a fixed measurement must divide its information budget among them, while adaptivity selects the informative one. Accordingly, the separation disappears when the candidate observables commute, and cannot occur with fully observed classical samples. Measurement incompatibility thus emerges as a resource for quantum learning, distinct from entanglement and quantum memory, that classical feedback alone can exploit.
We continue the study of the learnability of quantum measurement classes in the setting where the learner is given access only to prepared quantum states, aiming for necessary and sufficient conditions for PAC learnability, along with corresponding sample complexity bounds. In the quantum setting, in contrast with the classical probabilistically observed case, sampled states are perturbed when a quantum measurement is applied, according to the Born rule, so that distinct samples in the training data cannot be arbitrarily reused. We first probe the results from previous works on this setting. We show that the empirical risk defined in previous works and matching the definition in the classical theory can fail to satisfy the uniform convergence property enjoyed in the classical learning setting for classes that we can show to be PAC learnable. Moreover, we show that VC dimension generalization upper bounds in previous work are in many cases infinite, even for measurement classes defined on a finite-dimensional Hilbert space. We then show that, nonetheless, every measurement class defined on a finite-dimensional Hilbert space is PAC learnable via a modification of the ERM rule.
Over decades traditional information theory of source and channel coding advances toward learning and effective extraction of information from data. We propose to go one step further and offer a theoretical foundation for learning classical patterns from quantum data. However, there are several roadblocks to lay the groundwork for such a generalization. First, classical data must be replaced by a density operator over a Hilbert space. Hence, deviated from problems such as state tomography, our samples are i.i.d density operators. The second challenge is even more profound since we must realize that our only interaction with a quantum state is through a measurement which - due to no-cloning quantum postulate - loses information after measuring it. With this in mind, we present a quantum counterpart of the well-known probably approximately correct (PAC) framework. Based on that, we propose a quantum analogous of the Empirical Risk Minimization (ERM) algorithm for learning measurement hypothesis classes. Then, we establish upper bounds on the quantum sample complexity quantum concept classes.
Abstract Suppose we have many copies of an unknown n -qubit state . We measure some copies of using a known two-outcome measurement E 1 , then other copies using a measurement E 2 , and so on. At each stage t , we generate a current hypothesis about the state , using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that , the error in our prediction for the next measurement, is at least at most times. Even in the ‘non-realizable’ setting—where there could be arbitrary noise in the measurement outcomes—we show how to output hypothesis states that incur at most excess loss over the best possible state on the first T measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results—using convex optimization, quantum postselection, and sequential fat-shattering dimension—which have different advantages in terms of parameters and portability.
How many samples of a quantum state are required to learn a complete description of it? As we will see in this thesis, the fine-grained answer depends on the measurements available to the learner, but in general it is at least Ω(d^2/ϵ^2) where d is the dimension of the state and ϵ the trace distance accuracy. Optimal algorithms for this task -- known as quantum state tomography -- make use of powerful, yet highly impractical entangled measurements, where some joint measurement is performed on all copies of the state. What can be accomplished without such measurements, where one must perform measurements on individual copies of the states? \n \nIn Chapter 2 we show a relationship between the recently proposed quantum online learning framework and quantum state tomography. Specifically, we show that tomography can be accomplished using online learning algorithms in a black-box manner and O(d^4/ϵ^4) two-outcome measurements on separate copies of the state. The interpretation of this approach is that the experimentalist uses informative measurements to teach the learner by helping it make "mistakes" on measurements as early as possible. \n \nWe move on to proving lower bounds on tomography in Chapter 3. First, we review a known lower bound for entangled measurements as well as a Ω(d^3/ϵ^2) lower bound in the setting where non-entangled measurements are made non-adaptively, both due to Ref.[18]. We then derive a novel bound of Ω(d^4/ϵ^2) samples when the learner is further restricted to observing a constant number of outcomes (e.g., two-outcome measurements). This implies that the folklore "Pauli tomography" algorithm is optimal in this setting. \n \nUnderstanding the power of adaptive measurements, where measurement choices can depend on previous outcomes, is currently an open problem. In Chapter 4 we present two scenarios in which adapting on previous outcomes makes no difference to the number of samples required. In the first, the learner is limited to adapting on at most o(d^2/ϵ^2) of the previous outcomes. In the second, measurements are drawn from some set of at most exp(O(d)) measurements. In particular, this second lower bound implies that adaptivity makes no difference in the regime of efficiently implementable measurements, in the context of quantum computing. \n \nFinally, we apply the above technique to the problems of classical shadows and shadow tomography to obtain similar lower bounds. Here, one is interested only in determining the expectations of some fixed set of observables. We once again find that, for the worst-case input of observables, adaptivity makes no difference to the sample complexity when considering efficient, non-entangled measurements. As a corollary, we find a straightforward algorithm for shadow tomography is optimal in this setting.
We generalize the PAC (probably approximately correct) learning model to the quantum world by generalizing the concepts from classical functions to quantum processes, defining the problem of \emph{PAC learning quantum process}, and study its sample complexity. In the problem of PAC learning quantum process, we want to learn an ϵ-approximate of an unknown quantum process c∗ from a known finite concept class C with probability 1−δ using samples {(x1,c∗(x1)),(x2,c∗(x2)),…}, where {x1,x2,…} are computational basis states sampled from an unknown distribution D and {c∗(x1),c∗(x2),…} are the (possibly mixed) quantum states outputted by c∗. The special case of PAC-learning quantum process under constant input reduces to a natural problem which we named as approximate state discrimination, where we are given copies of an unknown quantum state c∗ from an known finite set C, and we want to learn with probability 1−δ an ϵ-approximate of c∗ with as few copies of c∗ as possible. We show that the problem of PAC learning quantum process can be solved with O(ϵ2log∣C∣+log(1/δ)) samples when the outputs are pure states and O(ϵ2log3∣C∣(log∣C∣+log(1/δ))) samples if the outputs can be mixed. Some implications of our results are that we can PAC-learn a polynomial sized quantum circuit in polynomial samples and approximate state discrimination can be solved in polynomial samples even when concept class size ∣C∣ is exponential in the number of qubits, an exponentially improvement over a full state tomography.