It's too late for Valentine's Day, and too early for April Fool's Day, so when I first saw the paper a paper titled "Quantum Dating Market, I wasn't quite sure what to make of it. But in the spirit of William Orem's post, "Quantum of Love," I decided to take a look.
It turns out that the paper deals with the well-known "Stable Marriage Problem." From Wikipedia's entry on the topic: "Given 'n' men and 'n' women, where each person has ranked all members of the opposite sex with a unique number between '1' and 'n' in order of preference, marry the men and women off such that there are no two people of opposite sex who would both rather have each other than their current partners. If there are no such people, all the marriages are "stable"."
In the quantum dating market paper, O.G. Zabaleta and C.M. Arizmendi use a quantum algorithm to attack the problem rather than the traditional classical algorithm.
To better appreciate this result, you might want to look at some of these sources to better understand the needed ingredients.
Game theory: Wikipedia and Stanford Encyclopedia of Philosophy
Quantum Game Theory: Wikipedia, "An invitation to Quantum Game Theory" and "Quantum Game Theory (AMS Notices)"
Game Theory (classical, non-quantum), roughly speaking, deals with the following problem: situations in which strategical interactions between rational players yields results with respect to the preferences chosen by said players.
Speaking a bit more mathematically, let us define a game with n people (i.e., players) through the following 2 properties,
1. 'n' sets
[equation]S_{i}, \qquad i=1, \,\ldots\, , \, n ;[/equation]
2. 'n' real-valued functions
[equation] P_{i}\qquad : \qquad S_{1}\times\,\dots\,\times S_{n} \rightarrow\mathbb{R}\; , \qquad i = 1,\,\ldots\, ,\, n .[/equation]
The set Si is called the "Strategy Space" of the i-th player, and the function Pi is called the "Payoff Function" of the i-th player.
This formulation is generic enough to model almost any concrete problem of strategic interactions: the Si are the available actions to player (we imagine the each player must choose an action); the actions have some consequence and Pi measures what player measures as this consequence.
Given the above, we can try and define what is a "Quantum Game": naïvely speaking, a quantum game is one in which each player implements a _mixed_ strategy, what requires that the Strategy Space be _expanded_. Thus, in a quantum game, the player can choose a strategy that is a _linear combination_ of the classical strategies,
[equation]S = \alpha_{1}\; S_{1} + \,\dots\, + \alpha_{n}\; S_{n}[/equation]
such that,
[equation]\sum_{i=1}^{n} \alpha_{i} = 1 \; .[/equation]
However, this linear combination only captures one of the characteristics present in a quantum game: there is still another one which is relevant in this problem: quantum entanglement. Therefore, at the end of the day, the final result of a quantum game is different from what would be obtained through the use of a mixed strategy, once quantum entanglement could lead to very non-trivial results.
In the paper, the authors apply Grover's algorithm, which is a quantum algorithm, to the Stable Marriage Problem. The power of Grover's Algorithm is that it performs an unordered database search in
[equation]O\bigl(\sqrt{n}\bigr)[/equation]
rather than the classical [equation]O(N)[/equation]
Making a loose analogy (a free interpretation of the meaning of a quantum strategy for the dating market), essentially, the paper says that a quantum strategy is more efficient to solve the SMP. Well, this implies that a certain male player would choose to date several female players (with a certain probability to each female player) at the same time, repeating this process several times, until an "equilibrium state" would be found, i.e., until the male player found its "better half".
However, one of the possible outcomes of the quantum entanglement would be that in which all female players decide, at the _same time_, not to date the male player anymore. So perhaps not such a good romantic outcome after all.