Result and analysis

In a typical experimental run, the training goal can be achieved with an average of 3% of contingent fluctuation. That is, in 3 out of a hundred tests, the system gives a wrong answer according to the training table. This is due to the statistical nature of quantum mechanics. If a threshold is applied to the output ensemble, an accuracy of 100% can be achieved. In this sense, a quantum mechanical architecture ``implements'' a prototypical everyday non-monotonic reasoning that has access to information from a temporal and spatial vantage point.

However, a quantum mechanical architecture can offer richer structures. For example, if the input states are prepared in such a way that the arguments (phases) of the two input qubits' coefficients vary independently, while the absolute value of every qubit remains the same as in the training process (since the coefficient of either the assertion eigenstate or the refutation eigenstate of a particular qubit must be zero, we have only two independent phase parameters instead of four), we find that the deviation from the targeted output can be very large. Specifically, the input state is prepared as:

$${e^{i\theta_1}\over {\sqrt 2}} {\left\vert x_1 \right\rangle... ...{e^{i\theta_2}\over {\sqrt 2}} {\left\vert x_2 \right\rangle}.$$ (6.7)

where $x_1 \in \{p+,p-\}$; $x_2 \in \{r+,r-\}$. The deviation from the targeted output is shown in Figure 6.8, in which the error (deviation to classical result) is defined as in Equation (6.3). The more interesting situations are when $(p,r)=(\mathbb{T},\mathbb{F})$ and $(p,r)=(\mathbb{T},\mathbb{T})$ (see the second and the third graphics of the first row in Figure 6.8). As can be seen in the figure, the intuitive valid argument can be implemented only in a narrow area along the diagonal (when $\theta_1 \approx \theta_2$). On the other hand, in a situation where $(p,r)=(\mathbb{F},\mathbb{T})$, the output remains nevertheless largely unknown. In a situation where $(p,r)=(\mathbb{F}, \mathbb{F})$, the architecture seems to be quite stable in the output ($q$ is unknown).

Figure 6.8: The deviation from the targets due to phase-difference in preparing the input state ( $\theta _1, \theta _2$)

It is common in everyday reasoning that the input is not ``well-behaved.'' That is, sometimes we cannot be sure of how ``true'' the antecedents are. In a quantum mechanical framework, this situation can be easily represented by a mixed state of affairs. For example, if proposition $p$ is known to be true but $r$ is refuted to a certain degree, the result should become somewhat uncertain as well. Specifically, in this situation the input state can be prepared as follows

$${\vert p+ \rangle + \rho e^{i\theta} \vert r- \rangle \over 1+\rho^2},$$

where $\rho \in \mathbb{R}$ with $0 < \rho \le 1$. (The denominator $1+\rho^2$ is introduce so that the input state is normalized.) The deviation to the targeted output is shown in Figure 6.9. As can be seen in the figure, proposition $q$ remains largely asserted if the phase of $p$ and that $r$ are far away from $\pi$. However, if the phase difference between $p$ and $r$ happens to be near $\pi$ and $\rho$ is near 1, the output is switched ( $r \approx \mathbb{F}$).

Figure 6.9: Relationship between the argument ($\theta$) / absolute value ($\rho$) of the refuted second antecedent and the output

On the other hand, if proposition $p$ is known to be true but $r$ is asserted to a certain degree, we expect that $q$ will become somewhat ``fuzzy'' as well. Specifically, the input state is prepared as follows

$${\vert p+ \rangle + \rho e^{i\theta} \vert r+ \rangle \over 1+\rho^2},$$

where $\rho \in \mathbb{R}$ with $0 < \rho \le 1$. If the situation is correspondingly prepared, the relative probability of the output state is shown in Figure 6.10. It looks somewhat like the complementary picture of Figure 6.9, but a close comparison with Figure 6.10 shows that this is not the case. This should come with no surprise, for in many everyday arguments, we do not treat a statement that is to a certain degree refuted as the logical complement of a statement that is to a certain degree asserted, especially when we are not sure whether the statement stands. As is shown in this example, a quantum architecture also does not treat statements this way.

Figure 6.10: Relationship between the argument ($\theta$) / absolute value ($\rho$) of the asserted second antecedent and the output

Another example is when $p$ is known to be true but $r$ is both asserted and refuted to a certain degree at the same time. In this situation, we, intuitively, treat the ``truth-value'' of $r$ as a complementary state of refutation and assertion. To pursue this issue further, the input state of affair can be represented as

$${\vert p+ \rangle + \rho e^{i\theta} \vert r+ \rangle + (1-\rho) e^{i\theta} \vert r- \rangle \over 1+\rho^2+(1-\rho)^2},$$

where $\rho \in \mathbb{R}$ with $0 < \rho \le 1$. The deviation from the targeted output is shown in Figure 6.11. This figure shows the complexity of this situation. Specifically, there is a semi-``equipotential'' contour of $\rho-\theta$ graph, on which the increase of $\rho$ (i.e. assertion of $r$ is counteracted by the increase of $\theta$).

Figure 6.11: Relationship between the argument ($\theta$) / absolute value ($\rho$) of the second antecedent when it is both asserted and refuted at the same time, and the output