The exponential scaling of the wave function is a fundamental property of quantum systems with far reaching implications in our ability to process quantum information. A problem where these are particularly relevant is quantum state tomography. State tomography, whose objective is to obtain an approximate description of a quantum system, can be analysed in the framework of computational learning theory. In this model, Aaronson (2007) showed that quantum states are Probably Approximately Correct (PAC)-learnable with sample complexity linear in the number of qubits. However, it is conjectured that in general quantum states require an exponential amount of computation to be learned. Here, using results from the literature on the efficient classical simulation of quantum systems, we show that stabiliser states are efficiently PAC-learnable. Our results solve an open problem formulated by Aaronson (2007) and establish a connection between classical simulation of quantum systems and efficient learnability.
The exponential scaling of the wave function is a fundamental property of quantum systems with far reaching implications in our ability to process quantum information. A problem where these are particularly relevant is quantum state tomography. State tomography, whose objective is to obtain an approximate description of a quantum system, can be analysed in the framework of computational learning theory. In this model, Aaronson (2007) showed that quantum states are Probably Approximately Correct (PAC)-learnable with sample complexity linear in the number of qubits. However, it is conjectured that in general quantum states require an exponential amount of computation to be learned. Here, using results from the literature on the efficient classical simulation of quantum systems, we show that stabiliser states are efficiently PAC-learnable. Our results solve an open problem formulated by Aaronson (2007) and establish a connection between classical simulation of quantum systems and efficient learnability.
We study classical and quantum learning algorithms with access to data produced by a quantum process. First, we consider the problem of learning quantum states and, in the framework of the probably approximately correct (PAC) model, prove that stabiliser states are efficiently learnable. Second, we introduce a generative model based on artificial neural networks capable of finding efficient representations of quantum states and assess its performance on states with varying levels of complexity. Third, we discuss the time complexity of classical and quantum learning algorithms and prove that Boolean functions in disjunctive normal form are efficiently quantum PAC learnable under product distributions.
Learning an unknown n-qubit quantum state ρ is a fundamental challenge in quantum computing. Information-theoretically, it is known that tomography requires exponential in n many copies of ρ to estimate it up to trace distance. Motivated by computational learning theory, Aaronson et al. introduced many (weaker) learning models: the PAC model of learning states (Proceedings of Royal Society A'07), shadow tomography (STOC'18) for learning "shadows" of a state, a model that also requires learners to be differentially private (STOC'19) and the online model of learning states (NeurIPS'18). In these models it was shown that an unknown state can be learned "approximately" using linear-in-n many copies of rho. But is there any relationship between these models? In this paper we prove a sequence of (information-theoretic) implications from differentially-private PAC learning, to communication complexity, to online learning and then to quantum stability. Our main result generalizes the recent work of Bun, Livni and Moran (Journal of the ACM'21) who showed that finite Littlestone dimension (of Boolean-valued concept classes) implies PAC learnability in the (approximate) differentially private (DP) setting. We first consider their work in the real-valued setting and further extend their techniques to the setting of learning quantum states. Key to our results is our generic quantum online learner, Robust Standard Optimal Algorithm (RSOA), which is robust to adversarial imprecision. We then show information-theoretic implications between DP learning quantum states in the PAC model, learnability of quantum states in the one-way communication model, online learning of quantum states, quantum stability (which is our conceptual contribution), various combinatorial parameters and give further applications to gentle shadow tomography and noisy quantum state learning.
We consider the problem of learning stabilizer states with noise in the Probably Approximately Correct (PAC) framework of Aaronson (2007) for learning quantum states. In the noiseless setting, an algorithm for this problem was recently given by Rocchetto (2018), but the noisy case was left open. Motivated by approaches to noise tolerance from classical learning theory, we introduce the Statistical Query (SQ) model for PAC-learning quantum states, and prove that algorithms in this model are indeed resilient to common forms of noise, including classification and depolarizing noise. We prove an exponential lower bound on learning stabilizer states in the SQ model. Even outside the SQ model, we prove that learning stabilizer states with noise is in general as hard as Learning Parity with Noise (LPN) using classical examples. Our results position the problem of learning stabilizer states as a natural quantum analogue of the classical problem of learning parities: easy in the noiseless setting, but seemingly intractable even with simple forms of noise.
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.
The complete learning of any unknown many-body systems requires a number of samples exponentially in the particle number. Several works consider subclasses of physical meaningful states such as stabilizer states or high-temperature Gibbs states that can be learned in polynomial sample complexity. Others consider to learn states under special assumptions, such as PAC learning and shadow tomography. In this work, we consider learning many-body systems that can be approximately represented by restricted Boltzmann machines (RBMs). We exhibit robustness results for efficient provable two-hop neighborhood learning method for ferromagnetic and locally consistent RBMs. We consider the L p norm as a distance measure, including both total variation distance and max-norm distance in the limit. We show that various quantum states can be learned with a sample complexity exponentially better than the naive state tomography. Our results establish a broader theoretical framework for the efficient learning of complex quantum many-body systems.
The probably approximately correct (PAC) model [Leslie G. Valiant, 1984] is a well studied model in classical learning theory. Here, we generalize the PAC model from concepts of Boolean functions to quantum channels, introducing PAC model for learning quantum channels, and give two sample efficient algorithms that are analogous to the classical "Occam’s razor" result [Blumer et al., 1987]. The classical Occam’s razor algorithm is done trivially by excluding any concepts not compatible with the input-output pairs one gets, but such an approach is not immediately possible with a concept class of quantum channels, because the outputs are unknown quantum states from the quantum channel. To study the quantum state learning problem associated with PAC learning quantum channels, we focus on the special case where the channels all have constant output. In this special case, learning the channels reduce to a problem of learning quantum states that is similar to the well known quantum state discrimination problem [Joonwoo Bae and Leong-Chuan Kwek, 2017], but with the extra twist that we allow ε-trace-distance-error in the output. We call this problem Approximate State Discrimination, which we believe is a natural problem that is of independent interest. We give two algorithms for learning quantum channels in PAC model. The first algorithm has sample complexity O((log|C| + log(1/ δ))/(ε²)), but only works when the outputs are pure states, where C is the concept class, ε is the error of the output, and δ is the probability of failure of the algorithm. The second algorithm has sample complexity O((log³|C|(log|C|+log(1/ δ)))/(ε²)), and work for mixed state outputs. 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 the size of the input set is exponential in the number of qubits, exponentially better than a naive state tomography.