Abstract
As a compact representation of joint probability distributions over a dependence graph of random variables, and a tool for modelling and reasoning in the presence of uncertainty, Bayesian networks are of great importance for artificial intelligence to combine domain knowledge, capture causal relationships, or learn from incomplete datasets. Known as a NP-hard problem in a classical setting, Bayesian inference pops up as a class of algorithms worth to explore in a quantum framework. This paper explores such a research direction and improves on previous proposals by a judicious use of the utility function in an entangled configuration. It proposes a completely quantum mechanical decision-making process with a proven computational advantage. A prototype implementation in Qiskit (a Python-based program development kit for the IBM Q machine) is discussed as a proof-of-concept.
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Notes
Notice, that the use of the density matrix notation with projectors is equivalent to the description of the measurement in the Dirac notation:
$$\begin{aligned} \langle {Var_1=true,Var_2=true,\ldots }|{\varPsi '}\rangle \equiv Tr(P_0 * \rho _{\varPsi '}) \end{aligned}$$(12)The matrix density notation was not required, as we are not dealing with mixed quantum states. However, it helped, later on, to expose the main ideas more clearly.
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This work is financed by the ERDF–European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation–COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT, within project POCI-01-0145-FEDER-030947. The first author was further supported by project NORTE-01-0145-FEDER-000037, funded by Norte Portugal Regional Operational Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement.
Appendix
Appendix
1.1 A Complexity Comparison
The decision-making processes we aim at comparing require inequality (50) to be satisfied. It assures that the decision maker chooses with certainty the best action.
Thus, to compare Process A and Process B it is necessary to consider all terms that are different. Therefore, the error term \(\delta _a\) for Process A is related to directly sampling values for the expected utilities, while in Process B the expected utility is determined indirectly. For this reason, in Process B it is necessary to apply error propagation rules:
Before applying error propagation to this equation, we need to normalize it so that EU(a|e)/k is equal to P(a).
where the normalization function (F(r)) is expressed as,
Here, again, the mean value of U(r) is used:
Expressing the equation that determines the error term \(\delta _a\) as a function of the error term \(\delta _b\) yields
Using Eq. (55) we obtain:
Then, assuming that \(\delta _b\) is similar, which is in favor of Process B because it minimizes the \(\delta _a\) term:
yielding,
With the relation between the error terms determined, it is possible to compare the difference of the computational effort involved in both processes, assuming again the mean terms for the probabilities:
Let us call the term on the right,
Using 59,
also,
and,
From Inglot (2010) we obtain a lower bound for \(A_{\alpha ,k}\). Although these terms are different for distinct values of \(\alpha\), we consider the one where \(\alpha\) is not leaning to zero too fast. Thus,
With this equation it is possible to define a better value for the difference between the computational efforts,
As
and,
it is possible to approximate the expression to
Writing the term of \(\delta _a\) as a function of its dimension and a factor that adjusts the precision,
we obtain,
Rewriting this the expression as,
Because,
for any value of the composing variables, then,
we prove that the relation between Process A and B is under bounded by,
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de Oliveira, M., Barbosa, L.S. Quantum Bayesian Decision-Making. Found Sci 28, 21–41 (2023). https://doi.org/10.1007/s10699-021-09781-6
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DOI: https://doi.org/10.1007/s10699-021-09781-6