# Quantum Circuit Components for Cognitive Decision-Making

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## Abstract

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## 1. Introduction

## 2. Human Violations of Classical Probability

In a Gallup poll conducted on 6–7 September 1997, half of the 1002 respondents were asked the following pair of questions: “Do you generally think Bill Clinton is honest and trustworthy?” and subsequently, the same question about Al Gore. The other half of the respondents answered exactly the same questions but in the opposite order.… The results of the poll exhibited a striking order effect. In the non-comparative context, Clinton received a 50% agreement rate and Gore received 68%, which shows a large gap of 18%. However, in the comparative context, the agreement rate for Clinton increased to 57% while for Gore, it decreased to 60%.

- Here, the non-comparative context means asking “Is X trustworthy?” on its own, whereas the comparative context means asking “Is Y trustworthy?” first, followed by “Is X trustworthy?”

- ‘Betray’ and attest that the other prisoner is a partner in crime, or,
- ‘Cooperate’ by not implicating the other prisoner in the crime.

For example, people are willing to pay an average of $26 for a $50 gift certificate, but only $16 for a lottery that pays either a $50 or $100 gift certificate, with equal probability.

## 3. Quantum Probability Models for Cognitive Behavior

- The states of systems are represented by vectors.
- Outcomes of ‘measurements’ (for example, answers to questions) are modeled by projecting state vectors onto eigenvectors representing different outcomes.
- The probability of a particular choice or answer is determined by the square of the scalar product (hence the angle) between the system’s state and the outcome state.
- A system can be in a superposition of various different states.
- Contributions to these superpositions can interfere (constructively or destructively). This sometimes represents cognitive couplings between events.
- Measuring the system forces it to ‘choose’ or collapse into a specific output state.

#### 3.1. Example with Order Effects

#### 3.2. Example with Disjunction Effects

If choice is based on reasons, then the unknown condition has two good reasons. Somehow these two good reasons cancel out to produce no reason at all! [22] (p. 267).

- Similar famous problems introduced by Tversky and Shafir [15] include the Vacation Problem (participants are more likely to book a vacation if the result of an important exam is known rather than unknown, whether the known result is pass or fail) and the Two-Stage Gambling problem (participants are more likely to gamble a second time if they know the outcome of a first gamble, whether or not they won or lost).

#### 3.3. Bayesian Theory and the Cognitive Relevance of Classical Logic

#### 3.4. Quantum Cognition and Physics

## 4. Quantum Computing and Quantum Circuits Introduction

#### 4.1. Why Bring Quantum Cognition and Quantum Computing Together?

#### 4.2. A Brief Primer on Qubits, Quantum Gates and Quantum Circuits

## 5. Quantum Circuits for Order Effects

#### Conditions on Probabilities in Order Effect Circuits

- In quantum theory, we can have $P(A\phantom{\rule{4pt}{0ex}}\&\mathrm{then}\phantom{\rule{4pt}{0ex}}B)>P\left(B\right)$, because projecting onto A and then onto B can give a higher probability than projection directly onto B.
- However, it has to be the case that $P(A\phantom{\rule{4pt}{0ex}}\&\mathrm{then}\phantom{\rule{4pt}{0ex}}B)<P\left(A\right)$ (so this model does not allow double conjunction fallacies).
- Since $P(A\phantom{\rule{4pt}{0ex}}\&\mathrm{then}\phantom{\rule{4pt}{0ex}}B)\ne P(B\phantom{\rule{4pt}{0ex}}\&\mathrm{then}\phantom{\rule{4pt}{0ex}}A)$, exchanging the orders of A and B terms in these conjunction expressions can lead to different outcomes.

## 6. Extending the Order Effects Model with Subjective Bias Activation

If an indeterminate cue influences beliefs (via a U-gate operation), evaluation of the cue should affect subsequent evaluation of beliefs and information about the criterion should affect beliefs about the cues [48].

`‘if … then …’`clause that changes behavior based on these parameters. This approach is more like a user asking explicitly “Based upon my prior experiences, am I likely to be engaged by this content?” (which in a quantum circuit would correspond to an explicit measurement step). Part of the attraction of the alternative quantum model in Figure 7 is that these correlations and choices can be modeled implicitly, which allows for unexpected correlations and behaviors which could be characterized as more instinctive. (In artificial intelligence terms, the classical conditional implements a tiny rule-based expert-system, and the quantum circuit implements a tiny quantum neural network.) Additionally, in a quantum framework, asking intermediate questions has the potential to alter the underlying mental states, in a specific way, something which does not occur naturally in classical approaches [50].

## 7. Quantum Circuits for Disjunction Effects

- A method to set the probability of a single event.
- A method to connect events saying that the outcome of a particular event may make an output of a subsequent event more or less likely.
- A method to ‘entangle’ events so that states representing different potential events can interfere with one another, including interference between incompatible outcomes.
- A method to ‘measure’ events, to model what happens when we learn the outcome of one of the hitherto unknown events and remove the possibility of other outcomes.

#### 7.1. Connecting Dependent Events

#### 7.2. Interference between Unknown Outcomes

#### 7.3. Measuring an Outcome and Removing Interference

## 8. Developing and Running on Quantum Hardware

## 9. Related and Further Work

## 10. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Non-commutative projections model the order effect in the Clinton–Gore scenario. We see that the projection of the |0〉 vector onto the Clinton axis gives a point further from the origin if we first project onto the Gore axis (

**right**) rather than if we just project the |0〉 vector onto the Clinton axis (

**left**). These diagrams show only the quadrant with positive real coordinates, so if the Gore axis is at angle $\theta $ above the horizontal, it appears at the point $cos\left(\theta \right)|0\rangle +sin(\theta \left)\right|1\rangle $. In reality, the coordinates can be any complex numbers $\alpha $ and $\beta $ such that $|{\alpha}^{2}|+|{\beta}^{2}|=1$.

**Figure 2.**Bloch sphere representation of a qubit. (From https://en.wikipedia.org/wiki/Bloch_sphere, Creative Commons CC BY-SA 3.0 license. Accessed on 14 March 2023).

**Figure 3.**Basic quantum logic gate diagrams used throughout these examples. A single-qubit rotation gate manipulates the superposition of |0〉 and |1〉 states for the qubit. The two-qubit CNOT gate (right) entangles two qubits (the top qubit is the control qubit and the bottom is the target qubit). The swap gate swaps the states of the two qubits. The measurement operator measures the qubit’s value and stores it in the given classical bit.

**Figure 7.**Order effect circuit with an extra qubit ${q}_{2}$ that controls whether or not the participant is asked the first question.

**Figure 8.**Circuit for setting conditional probability. Note that the white circle means ‘if this qubit is in state |0〉’ and the black circle means ’if this qubit is in state |1〉’.

**Figure 11.**Circuit for simulating interference between unknown outcomes. The ‘H’ gate is a Hadamard gate which maps the state |0〉 to a superposition state $\frac{1}{\sqrt{2}}\left(\right|0\rangle +\left|1\rangle \right)$, and |1〉 to the state $\frac{1}{\sqrt{2}}\left(\right|0\rangle -\left|1\rangle \right)$.

**Figure 12.**Different output probabilities for the target qubit as a function of the phase angle $\phi $, when the gates before and after the ${R}_{z}\left(\phi \right)$ operation of Figure 11 are changed.

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Widdows, D.; Rani, J.; Pothos, E.M.
Quantum Circuit Components for Cognitive Decision-Making. *Entropy* **2023**, *25*, 548.
https://doi.org/10.3390/e25040548

**AMA Style**

Widdows D, Rani J, Pothos EM.
Quantum Circuit Components for Cognitive Decision-Making. *Entropy*. 2023; 25(4):548.
https://doi.org/10.3390/e25040548

**Chicago/Turabian Style**

Widdows, Dominic, Jyoti Rani, and Emmanuel M. Pothos.
2023. "Quantum Circuit Components for Cognitive Decision-Making" *Entropy* 25, no. 4: 548.
https://doi.org/10.3390/e25040548