Learnability of the output distributions of local quantum circuits

Hinsche, Marcel; Ioannou, Marios; Nietner, A.; Haferkamp, Jonas; Quek, Yihui; Hangleiter, Dominik; Seifert, Jean-Pierre; Eisert, Jens; Sweke, Ryan

doi:10.48550/arxiv.2110.05517

Cited by 9 publications

(11 citation statements)

References 18 publications

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“…This is analogous to the classical setting where parity functions are hard to learn in the noisy SQ setting, but efficient to learn using simple linear regression [45]. Similarly, the related work of [41] showed that output distributions of Clifford circuits can be hard to learn using statistical queries, but efficient using a technique that resorts to linear regression on a matrix formed from samples of the overall distribution. More loosely, our results provide support to the basic maxim that algorithms which apply too broadly will work very rarely [80].…”

Section: Hardness Of Sq Learning Variational Function Classesmentioning

confidence: 84%

“…Recent results have shown that certain fundamental and rather simple classes of quantum "functions" are hard to learn in the SQ setting. Namely, (classical) output distributions of locally constructed quantum states [41] and the set of Clifford circuits [37] are hard to learn given properly chosen statistical query oracles. Following these results, we show that simple classes of functions generated by variational circuits are also exponentially difficult to learn in the SQ settings we consider.…”

Section: Quantum Statistical Query Modelsmentioning

confidence: 99%

“…Two very recent works have studied the SQ hardness of learning data generated by quantum circuits. First, [41] analyze the hardness of learning the output distribution of clifford circuits and stabilizer states showing that these distributions are hard to learn using classical Boolean SQ oracles. Nevertheless, when given samples from the Boolean hypercube of the distribution, they provide an efficient algorithm based on linear regression to determine the stabilizer state underlying the distribution.…”

Section: Correlational Statistical Query Modelmentioning

confidence: 99%

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Beyond Barren Plateaus: Quantum Variational Algorithms Are Swamped With Traps

Anschuetz¹,

Kiani²

2022

Preprint

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One of the most important properties of classical neural networks is how surprisingly trainable they are, though their training algorithms typically rely on optimizing complicated, nonconvex loss functions. Previous results have shown that unlike the case in classical neural networks, variational quantum models are often not trainable. The most studied phenomenon is the onset of barren plateaus in the training landscape of these quantum models, typically when the models are very deep. This focus on barren plateaus has made the phenomenon almost synonymous with the trainability of quantum models. Here, we show that barren plateaus are only a part of the story. We prove that a wide class of variational quantum models-which are shallow, and exhibit no barren plateus-have only a superpolynomially small fraction of local minima within any constant energy from the global minimum, rendering these models untrainable if no good initial guess of the optimal parameters is known. We also study the trainability of variational quantum algorithms from a statistical query framework, and show that noisy optimization of a wide variety of quantum models is impossible with a sub-exponential number of queries. Finally, we numerically confirm our results on a variety of problem instances. Though we exclude a wide variety of quantum algorithms here, we give reason for optimism for certain classes of variational algorithms and discuss potential ways forward in showing the practical utility of such algorithms.

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Section: Hardness Of Sq Learning Variational Function Classesmentioning

confidence: 84%

Section: Quantum Statistical Query Modelsmentioning

confidence: 99%

Section: Correlational Statistical Query Modelmentioning

confidence: 99%

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Beyond Barren Plateaus: Quantum Variational Algorithms Are Swamped With Traps

Anschuetz¹,

Kiani²

2022

Preprint

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“…Our proposal bridges the gap between two important research efforts. The first is the known rigorous and theoretical standpoint, which usually lacks immediate real-world application (see e.g., [34,35]). Conversely, the other uses state-of-the-art real-world datasets, where only heuristics and approximate metrics can be proposed, but where the complexity of the models and tasks at hand blur any definite conclusions about the model generalization capacity.…”

Section: Introductionmentioning

confidence: 99%

Evaluating Generalization in Classical and Quantum Generative Models

Gili¹,

Mauri²,

Perdomo-Ortiz³

2022

Preprint

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Defining and accurately measuring generalization in generative models remains an ongoing challenge and a topic of active research within the machine learning community. This is in contrast to discriminative models, where there is a clear definition of generalization, i.e., the model's classification accuracy when faced with unseen data. In this work, we construct a simple and unambiguous approach to evaluate the generalization capabilities of generative models. Using the sample-based generalization metrics proposed here, any generative model, from state-of-the-art classical generative models such as GANs to quantum models such as Quantum Circuit Born Machines, can be evaluated on the same ground on a concrete well-defined framework. In contrast to other sample-based metrics for probing generalization, we leverage constrained optimization problems (e.g., cardinality constrained problems) and use these discrete datasets to define specific metrics capable of unambiguously measuring the quality of the samples and the model's generalization capabilities for generating data beyond the training set but still within the valid solution space. Additionally, our metrics can diagnose trainability issues such as mode collapse and overfitting, as we illustrate when comparing GANs to quantum-inspired models built out of tensor networks. Our simulation results show that our quantum-inspired models have up to a 68× enhancement in generating unseen unique and valid samples compared to GANs, and a ratio of 61:2 for generating samples with better quality than those observed in the training set. We foresee these metrics as valuable tools for rigorously defining practical quantum advantage in the domain of generative modeling.

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“…We further study the classical learnability of the outcome distribution by introducing a tractability measure based on the classical parent Hamiltonian of the output distribution as a coherent Gibbs state. In contrast to other recent studies [19][20][21], which are mostly based on theoretical tools such as probably approximately correct (PAC) learning framework that is of limited relevance to neural network simulations [22,23], our measure is closely related to classical energy-based models. Combining insight from the tractability measure and from training an energy-based model, we argue that classical learnability undergoes a transition but earlier than the KL divergence (see Fig.…”

mentioning

confidence: 98%

Complexity phase transitions in instantaneous quantum polynomial-time circuits

Park¹,

Kastoryano²

2022

Preprint

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We study a subclass of the Instantaneous Quantum Polynomial-time (IQP) circuit with a varying density of two-qubit gates. We identify two phase transitions as a function of the gate density. At the first transition, the coherent Gibbs representation of the output state transitions from quasilocal to extended, and becomes completely non-local after the second transition, resembling that of Haar random states. We introduce several new diagnostics to argue that learning the distribution after the first transition already becomes classically intractable, even though its output distribution is far from random. Our work thus opens a novel pathway for exploring quantum advantage in more practical setups whose outcome distributions are more structured than random states.

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Learnability of the output distributions of local quantum circuits

Cited by 9 publications

References 18 publications

Beyond Barren Plateaus: Quantum Variational Algorithms Are Swamped With Traps

Beyond Barren Plateaus: Quantum Variational Algorithms Are Swamped With Traps

Evaluating Generalization in Classical and Quantum Generative Models

Complexity phase transitions in instantaneous quantum polynomial-time circuits

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