Correlation is a symmetric relation between random variables, but causation is not: it says more, it implies an inference from a cause to a consequence. Intuitively, there is no correlation without causation -- this is often referred to as Reichenbach's common cause principle. Given a set of random variables and joint or marginal probabilities, we would like to learn the causal relations between the random variables. A Bayesian network is a directed acyclic graph (DAG) which encodes these dependences while reproducing the observed distribution.

Learning the structure of a classical Bayesian network is a hard task in itself. Since we talk about correlations, it is also natural to ask how we can extend these causal structures to quantum correlations, particularly to instances of nonlocality. Based on a brief review of the literature, three main research directions became apparent in recent years:

- Characterizing quantum correlations by causal structures.
- Extending the d-seperation theorem to address quantum correlations in a Bayesian net.
- Learning classical Bayesian networks by adiabatic or gate quantum computers.

In what follows, some relevant references are collected to each of these directions.

# Quantum correlations and causal structures

Bell inequalities are the traditional way of studying quantum correlations. A recent body of work shows that various extensions of Bell scenarios -- for instance, sequential correlation scenarios [3] -- can be captured by studying causal structures on DAGs.

A seminal work in this area is by Wood and Spekkens (2012), which asserts that while causal discovery algorithms cannot distinguish between correlations that violate Bell inequalities from ones that do not, the violating ones require fine-tuning of the causal parameters [11]. Fine-tuning means that the conditional independences of the model do not hold for variations of the causal-statistical parameters of the model, and the lack of need for fine-tuning is a core assumptions of causal discovery algorithms.

Leifer and Spekkens (2013) replace conditional probabilities with a quantum analogue [5]. To treat spatial and temporal conditioning on random variables in the same framework, they introduce quantum conditional states. In this arrangement, the Hilbert space is a region of space-time, and they define unifying manipulations. Unfortunately the compound state may not be positive. Using the quantum conditional states, the authors introduce a new Bayes' rule.

Likewise, Fritz (2014) starts by defining an event in space-time that can be a source, choice of measurement setting, or measurement [2]. The paper uses this notion to define correlations which are different from the traditional definition. With this generalized concept, the author is able to unify different causal structures and various configurations of Bell tests and various generalizations of them.

Chaves et. al (2014) introduce an entropic framework for identifying latent structures in classical Bayesian networks, and, as a byproduct, adding an additional inequality on the entropies, they also suggest a test for quantum nonlocality [1].

# Extending the d-separation theorem

Papers in this category take a different angle at causal structures in quantum correlation than the ones in the previous section. The focus is on extending the classical d-separation theorem, which gives a graphically intuitive criterion that in a given DAG whether a set of random variables is independent of another one conditioned on a third one.

Henson et al. (2014) extend the criterion to arbitrary general, nonsignalling correlations [4]. They introduce two distinct type of nodes in a DAG, observed and unobserved ones. The former resemble classical nodes, in fact, a DAG with only observed nodes is just a classical Bayesian net. Unobserved are generic resources, for instance, the source of entangled particles in a Bell test. The clever consequence of this generalized DAG is that the d-separation theorem holds in almost identical form to the classical variant. The quantum case of the generalization is similar to Tucci's earlier work on the topic [9].

Pienaar and Brukner (2014) treat all nodes are equal [7]. Instead, they forgo Reichenbach's common cause principle, arguing that its validity is questionable in quantum mechanics. Akin to a classical Bayesian model, the authors define a quantum causal model and a corresponding q-separation criterion, and they prove that this criterion has the same effect as the d-separation in the classical case.

# Learning classical networks

Learning the optimal network that might have produced a given set of data is a major challenge in optimization theory: it is an NP-hard problem. Using classical heuristics, simulated quantum annealing has been suggested [8]. Actual quantum annealing follows a similar procedure to mapping boosting to an annealing process: the learning problem is formulated as a quadratic unconstrained binary optimization problem, which in turn maps to an Ising model that can be solved in an annealing processor [6]. The binary variables become the directed edges between random variables in the DAG: the presence of an directed edge indicates causation. The objective function evaluates how well a particular configuration of the directed edges matches the constraints on the distribution.

There are many potentially good causal structures fitting a distribution, and if a connection between two random variables is important, the corresponding directed edge has a high probability of appearing in most of the good candidates. The optimization is thus a search over possible graphs for high-probability directed edges. The search space is superexponential, but if we have some basic assumptions on the graph structure, we can reduce the size of the search space. For a target directed edge, we want to calculate the probability over the search space. For this summation, Tucci suggested a variant of Grover's search, hence opening up the way to learn classical Bayesian networks in gate quantum computers [10].

# References

- R. Chaves, L. Luft, T. O. Maciel, D. Gross, D. Janzing, and B. Schölkopf, "Inferring Latent Structures via Information Inequalities," in Proceedings of UAI-14, 30th Conference on Uncertainty in Artificial Intelligence, 2014, pp. 112-121.
`@InProceedings{chaves2014inferring, title = {Inferring Latent Structures via Information Inequalities}, author = {Chaves, Rafael and Luft, L. and Maciel, T. O. and Gross, D. and Janzing, D. and Sch\"olkopf, Bernhard}, booktitle = {Proceedings of UAI-14, 30th Conference on Uncertainty in Artificial Intelligence}, year = {2014}, month = {July}, pages = {112--121}, eprint = {http://www.arxiv.org/abs/1407.2256}, file = {chaves2014inferring.pdf:machine_learning/chaves2014inferring.pdf:PDF}, keywords = {Bayes}, location = {Quebec City, Canada}, timestamp = {2014.09.17} }`

- T. Fritz, "Beyond Bell's Theorem II: Scenarios with Arbitrary Causal Structure," arXiv:1404.4812, 2014.
`@Article{fritz2014beyond, title = {Beyond {Bell}'s Theorem {II}: Scenarios with Arbitrary Causal Structure}, author = {Fritz, Tobias}, journal = {arXiv:1404.4812}, year = {2014}, eprint = {http://arxiv.org/abs/1404.4812}, file = {fritz2014beyond.pdf:qpt/fritz2014beyond.pdf:PDF}, keywords = {Bayes}, timestamp = {2014.09.11} }`

- R. Gallego, L. E. Würflinger, R. Chaves, A. Acín, and M. Navascués, "Nonlocality in Sequential Correlation Scenarios," New Journal of Physics, vol. 16, iss. 3, p. 33037, 2014.
`@Article{gallego2014nonlocality, title = {Nonlocality in Sequential Correlation Scenarios}, author = {Gallego, Rodrigo and W\"urflinger, Lars Erik and Chaves, Rafael and Ac\'in, Antonio and Navascu\'es, Miguel}, journal = {New Journal of Physics}, year = {2014}, number = {3}, pages = {033037}, volume = {16}, doi = {10.1088/1367-2630/16/3/033037}, eprint = {http://arxiv.org/abs/1308.0477}, file = {gallego2014nonlocality.pdf:qpt/gallego2014nonlocality.pdf:PDF}, keywords = {Bayesian}, timestamp = {2014.05.25} }`

- J. Henson, R. Lal, and M. F. Pusey, "Theory-Independent Limits on Correlations from Generalised Bayesian Networks," arXiv:1405.2572, 2014.
`@Article{henson2014theory, title = {Theory-Independent Limits on Correlations from Generalised {Bayesian} Networks}, author = {Henson, Joe and Lal, Raymond and Pusey, Matthew F.}, journal = {arXiv:1405.2572}, year = {2014}, eprint = {http://arxiv.org/abs/1405.2572}, file = {henson2014theory.pdf:qpt/henson2014theory.pdf:PDF}, keywords = {Bayes}, timestamp = {2014.09.11} }`

- M. S. Leifer and R. W. Spekkens, "Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference," Physics Review A, vol. 88, p. 52130, 2013.
`@Article{leifer2013towards, title = {Towards a Formulation of Quantum Theory as a Causally Neutral Theory of {Bayesian} Inference}, author = {Leifer, M. S. and Spekkens, Robert W.}, journal = {Physics Review A}, year = {2013}, month = {November}, pages = {052130}, volume = {88}, doi = {10.1103/PhysRevA.88.052130}, eprint = {http://arxiv.org/abs/1107.5849}, file = {leifer2013towards.pdf:qpt/leifer2013towards.pdf:PDF}, issue = {5}, keywords = {Bayes}, publisher = {American Physical Society}, timestamp = {2014.09.11} }`

- B. A. O'Gorman, A. Perdomo-Ortiz, R. Babbush, A. Aspuru-Guzik, and V. Smelyanskiy, "Bayesian Network Structure Learning Using Quantum Annealing," arXiv:1407.3897, 2014.
`@Article{ogorman2014bayesian, title = {Bayesian Network Structure Learning Using Quantum Annealing}, author = {O'Gorman, Bryan A. and Perdomo-Ortiz, Alejandro and Babbush, Ryan and Aspuru-Guzik, Al\'an and Smelyanskiy, Vadim}, journal = {arXiv:1407.3897}, year = {2014}, eprint = {http://www.arxiv.org/abs/1407.3897}, file = {ogorman2014bayesian.pdf:qc/ogorman2014bayesian.pdf:PDF}, keywords = {Bayes}, timestamp = {2014.09.11} }`

- J. Pienaar and Č. Brukner, "A Graph-Separation Theorem for Quantum Causal Models," arXiv:1406.0430, 2014.
`@Article{pienaar2014graph, title = {A Graph-Separation Theorem for Quantum Causal Models}, author = {Pienaar, Jacques and Brukner, {\v{C}}aslav}, journal = {arXiv:1406.0430}, year = {2014}, eprint = {http://arxiv.org/abs/1406.0430}, file = {pienaar2014graph.pdf:qpt/pienaar2014graph.pdf:PDF}, keywords = {Bayesian}, timestamp = {2014.09.15} }`

- I. Sato, K. Kurihara, S. Tanaka, H. Nakagawa, and S. Miyashita, "Quantum Annealing for Variational Bayes Inference," in Proceedings of UAI-09, 25th Conference on Uncertainty in Artificial Intelligence, 2009, pp. 479-486.
`@InProceedings{sato2009quantumannealing, title = {Quantum Annealing for Variational Bayes Inference}, author = {Sato, Issei and Kurihara, Kenichi and Tanaka, Shu and Nakagawa, Hiroshi and Miyashita, Seiji}, booktitle = {Proceedings of UAI-09, 25th Conference on Uncertainty in Artificial Intelligence}, year = {2009}, pages = {479--486}, eprint = {http://arxiv.org/abs/0905.3528}, file = {sato2009quantumannealing.pdf:machine_learning/sato2009quantumannealing.pdf:PDF}, isbn = {978-0-9749039-5-8}, location = {Montreal, Quebec, Canada} }`

- R. T. Tucci, "Factorization of Quantum Density Matrices According to Bayesian and Markov Networks," arXiv:quant-ph/0701201, 2007.
`@Article{tucci2007factorization, title = {Factorization of Quantum Density Matrices According to {Bayesian} and {Markov} Networks}, author = {Tucci, Robert T.}, journal = {arXiv:quant-ph/0701201}, year = {2007}, eprint = {http://arxiv.org/abs/quant-ph/0701201}, file = {tucci2007factorization.pdf:qpt/tucci2007factorization.pdf:PDF}, timestamp = {2014.09.15} }`

- R. R. Tucci, "Quantum Circuit For Discovering from Data the Structure of Classical Bayesian Networks," arXiv:1404.0055, 2014.
`@Article{tucci2014quantum, title = {Quantum Circuit For Discovering from Data the Structure of Classical {Bayesian} Networks}, author = {Tucci, Robert R.}, journal = {arXiv:1404.0055}, year = {2014}, eprint = {http://www.arxiv.org/abs/1404.0055}, file = {tucci2014quantum.pdf:qpt/tucci2014quantum.pdf:PDF}, keywords = {Bayesian}, timestamp = {2014.09.15} }`

- C. J. Wood and R. S. Spekkens, "The Lesson of Causal Discovery Algorithms for Quantum Correlations: Causal Explanations of Bell-Inequality Violations Require Fine-Tuning," arXiv:1208.4119, 2012.
`@Article{wood2012lesson, title = {The Lesson of Causal Discovery Algorithms for Quantum Correlations: Causal Explanations of {Bell}-Inequality Violations Require Fine-Tuning}, author = {Wood, Christopher J. and Spekkens, Robert S.}, journal = {arXiv:1208.4119}, year = {2012}, eprint = {http://arxiv.org/abs/1208.4119}, file = {woods2012lesson.pdf:qpt/woods2012lesson.pdf:PDF}, keywords = {Bayes}, timestamp = {2014.09.11} }`