The task of learning a probability distribution from samples is ubiquitous across the natural sciences. The output distributions of local quantum circuits are of central importance in both quantum advantage proposals and a variety of quantum machine learning algorithms. In this work, we extensively characterize the learnability of output distributions of local quantum circuits. Firstly, we contrast learnability with simulatability by showing that Clifford circuit output distributions are efficiently learnable, while the injection of a single T gate renders the density modeling task hard for any depth d=n^{Ω(1)}. We further show that the task of generative modeling universal quantum circuits at any depth d=n^{Ω(1)} is hard for any learning algorithm, classical or quantum, and that for statistical query algorithms, even depth d=ω[log(n)] Clifford circuits are hard to learn. Our results show that one cannot use the output distributions of local quantum circuits to provide a separation between the power of quantum and classical generative modeling algorithms, and therefore provide evidence against quantum advantages for practically relevant probabilistic modeling tasks.
The task of learning a probability distribution from samples is ubiquitous across the natural sciences. The output distributions of local quantum circuits are of central importance in both quantum advantage proposals and a variety of quantum machine learning algorithms. In this work, we extensively characterize the learnability of output distributions of local quantum circuits. Firstly, we contrast learnability with simulatability by showing that Clifford circuit output distributions are efficiently learnable, while the injection of a single T gate renders the density modeling task hard for any depth d=n^{Ω(1)}. We further show that the task of generative modeling universal quantum circuits at any depth d=n^{Ω(1)} is hard for any learning algorithm, classical or quantum, and that for statistical query algorithms, even depth d=ω[log(n)] Clifford circuits are hard to learn. Our results show that one cannot use the output distributions of local quantum circuits to provide a separation between the power of quantum and classical generative modeling algorithms, and therefore provide evidence against quantum advantages for practically relevant probabilistic modeling tasks.
There is currently a large interest in understanding the potential advantages quantum devices can offer for probabilistic modelling. In this work we investigate, within two different oracle models, the probably approximately correct (PAC) learnability of quantum circuit Born machines, i.e., the output distributions of local quantum circuits. We first show a negative result, namely, that the output distributions of super-logarithmic depth Clifford circuits are not sample-efficiently learnable in the statistical query model, i.e., when given query access to empirical expectation values of bounded functions over the sample space. This immediately implies the hardness, for both quantum and classical algorithms, of learning from statistical queries the output distributions of local quantum circuits using any gate set which includes the Clifford group. As many practical generative modelling algorithms use statistical queries -- including those for training quantum circuit Born machines -- our result is broadly applicable and strongly limits the possibility of a meaningful quantum advantage for learning the output distributions of local quantum circuits. As a positive result, we show that in a more powerful oracle model, namely when directly given access to samples, the output distributions of local Clifford circuits are computationally efficiently PAC learnable by a classical learner. Our results are equally applicable to the problems of learning an algorithm for generating samples from the target distribution (generative modelling) and learning an algorithm for evaluating its probabilities (density modelling). They provide the first rigorous insights into the learnability of output distributions of local quantum circuits from the probabilistic modelling perspective.
In this work, we show that learning the output distributions of brickwork random quantum circuits is average-case hard in the statistical query model. This learning model is widely used as an abstract computational model for most generic learning algorithms. In particular, for brickwork random quantum circuits on n qubits of depth d , we show three main results:– At super logarithmic circuit depth d=ω(log(n)) , any learning algorithm requires super polynomially many queries to achieve a constant probability of success over the randomly drawn instance.– There exists a d=O(n) , such that any learning algorithm requires Ω(2n) queries to achieve a O(2−n) probability of success over the randomly drawn instance.– At infinite circuit depth d→∞ , any learning algorithm requires 22Ω(n) many queries to achieve a 2−2Ω(n) probability of success over the randomly drawn instance.As an auxiliary result of independent interest, we show that the output distribution of a brickwork random quantum circuit is constantly far from any fixed distribution in total variation distance with probability 1−O(2−n) , which confirms a variant of a conjecture by Aaronson and Chen.
Quantum generative modeling (QGM) relies on preparing quantum states and generating samples from these states as hidden---or known---probability distributions. As distributions from some classes of quantum states (circuits) are inherently hard to sample classically, QGM represents an excellent test bed for quantum supremacy experiments. Furthermore, generative tasks are increasingly relevant for industrial machine learning applications, and thus QGM is a strong candidate for demonstrating a practical quantum advantage. However, this requires that quantum circuits are trained to represent industrially relevant distributions, and the corresponding training stage has an extensive training cost for current quantum hardware in practice. In this work, we propose protocols for classical training of QGMs based on circuits of the specific type that admit an efficient gradient computation, while remaining hard to sample. In particular, we consider instantaneous quantum polynomial (IQP) circuits and their extensions. Showing their classical simulability in terms of the time complexity, sparsity, and anticoncentration properties, we develop a classically tractable way of simulating their output probability distributions, allowing classical training to a target probability distribution. The corresponding quantum sampling from IQPs can be performed efficiently, unlike when using classical sampling. We numerically demonstrate the end-to-end training of IQP circuits using probability distributions for up to 30 qubits on a regular desktop computer. When applied to industrially relevant distributions this combination of classical training with quantum sampling represents an avenue for reaching advantage in the noisy intermediate-scale quantum (NISQ) era.
Demonstrating quantum advantage in machine learning tasks requires navigating a complex landscape of proposed models and algorithms. To bring clarity to this search, we introduce a framework that connects the structure of parametrized quantum circuits to the mathematical nature of the functions they can actually learn. Within this framework, we show how fundamental properties, like circuit depth and non-Clifford gate count, directly determine whether a model's output leads to efficient classical simulation or surrogation. We argue that this analysis uncovers common pathways to dequantization that underlie many existing simulation methods. More importantly, it reveals critical distinctions between models that are fully simulatable, those whose function space is classically tractable, and those that remain robustly quantum. This perspective provides a conceptual map of this landscape, clarifying how different models relate to classical simulability and pointing to where opportunities for quantum advantage may lie.
We study the classical simulatability of commuting quantum circuits with n input qubits and O(log n) output qubits, where a quantum circuit is classically simulatable if its output probability distribution can be sampled up to an exponentially small additive error in classical polynomial time. First, we show that there exists a commuting quantum circuit that is not classically simulatable unless the polynomial hierarchy collapses to the third level. This is the first formal evidence that a commuting quantum circuit is not classically simulatable even when the number of output qubits is exponentially small. Then, we consider a generalized version of the circuit and clarify the condition under which it is classically simulatable. Lastly, we apply the argument for the above evidence to Clifford circuits in a similar setting and provide evidence that such a circuit augmented by a depth-1 non-Clifford layer is not classically simulatable. These results reveal subtle differences between quantum and classical computation.
We study the classical hardness of learning the output distribution from instantaneous quantum polynomial-time circuits with a varying density of two-qubit gates. We first investigate the complexity phases relevant to simulating the output distribution. In addition to a known parameter regime for anticoncentration, where the classical simulation is believed to be infeasible, we identify a novel parameter condition such that the classically simulability of the circuit can be proven rigorously. Next, we explore the classical learnability of the output distribution. We prove that a circuit instance whose output distribution is not classically learnable exists, assuming the hardness of learning parity with noise. We numerically study this phenomenon deeply using a classical energy-based model, where we find that the classical model fails to learn the output distribution even when the density of two-qubit gates is much smaller than that for the anticoncentration. We argue that this is mainly due to the fact that an accurate classical description of the output distribution requires exponentially large amount of resources.