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Prompt: Read the paper by Say and Yakaryılmaz and theorize a computer program (in the terms of the paper) whose task is to prove the existence or inexistence of God, with one bit of information it receives from its future tells it that the God indeed exists. Write out this scenario and discuss with respect to the Pascalian wager!
(Computation with narrow CTCs A. C. Cem Say, Abuzer Yakaryilmaz)

To address the task, we can create a theoretical computer program based on the model described by Say and Yakaryılmaz, specifically the MCTC1 model which utilizes a single classical bit of information from the future (in the form of a closed timelike curve or CTC) to perform its computation.

Theoretical Computer Program

Objective

To determine the existence or inexistence of God with the help of a single bit of information from the future.

Program Structure

1. Input: The program starts with the input question: “Does God exist?”
2. Initialization: The program initializes two states:
   • State S0: Represents the non-existence of God.
   • State S1: Represents the existence of God.
3. CTC Bit: The program has access to a single bit from the future, which we will call the CTC bit. This bit will be 0 if the program, after completing its execution, determines that God does not exist, and 1 if it determines that God does exist.
4. Transition Logic:
   • The program enters a loop where it tries to evaluate evidence and arguments regarding the existence of God.
   • If it finds evidence that suggests God exists, it transitions to state S1.
   • If it finds evidence against the existence of God, it transitions to state S0.
5. CTC Check: At the end of the loop, the program checks the CTC bit.
   • If the bit is 1 (indicating that God exists), the program re-evaluates its logic to align with this information and confirms the existence of God.
   • If the bit is 0 (indicating that God does not exist), the program re-evaluates its logic to align with this information and confirms the non-existence of God.

Example Scenario

1. Initialization: The program receives the question and initializes with states S0 and S1.
2. Evaluation Loop: The program begins evaluating various philosophical, theological, and scientific arguments.
3. Future Information: Midway through its computation, the program receives the CTC bit from the future. Let’s assume the bit is 1.
4. Re-Evaluation: Upon receiving the bit, the program re-evaluates all previous arguments in light of the information that God exists.
5. Output: The program confirms that God exists and outputs this result.

Discussion with Respect to Pascal’s Wager

Pascal’s Wager argues that it is a better “bet” to believe in God, because if God exists, the believer gains infinitely (eternal happiness), and if God does not exist, the loss is finite (minor earthly pleasures or losses). Here’s how this theoretical program relates to Pascal’s Wager:

• Alignment with the Wager: The program, by receiving the CTC bit that tells it the outcome (existence of God), effectively skips the “betting” part and directly reaches the conclusion that would align with the highest payoff in Pascal’s terms.
• Future Information as Assurance: The use of the CTC bit ensures that the decision made by the program is based on assured information from the future, which removes the uncertainty that Pascal’s Wager tries to navigate.
• Theoretical Implication: If such a program could exist and function correctly, it would eliminate the need for Pascal’s Wager entirely, as the existence of God could be determined with certainty based on future information feedback.

In summary, while Pascal’s Wager operates under uncertainty and risk management, our theoretical program uses future knowledge (via the CTC bit) to achieve a definitive answer regarding the existence of God. This scenario pushes the boundaries of classical computational theory and philosophical inquiry into new realms of time travel and information theory.

Prompt: Read the paper by Say and O’Donnell and adapt the same example to qubit!
(The weakness of CTC qubits and the power of approximate counting Ryan O’Donnell, A. C. Cem Say)

To adapt the example from Say and Yakaryılmaz to the context of qubits as discussed in the paper by Say and O’Donnell, we can frame the scenario in terms of quantum computing with closed timelike curves (CTCs).

Scenario Using Qubits

Problem Definition

We want to design a quantum algorithm within the BQPQCTC[w] model that decides a language L with high probability using a w-qubit CTC register. The goal is to determine whether God exists based on a bit of information from the future indicating that God exists.

Quantum Algorithm

Initialization:

1. Prepare a quantum circuit CCC that includes a w-qubit CTC register and additional qubits in the causality-respecting (CR) register.
2. Initialize the CR register qubits to ∣0⟩ and end with measurement gates.

CTC Register:

1. The w-qubit CTC register is initialized to an arbitrary stationary mixed state ρ.
2. According to the model, the CTC register interacts with the quantum channel defined by the circuit CCC and samples from its stationary state.

Quantum Channel Construction:

1. Design the quantum channel such that it implements the decision problem: detecting whether the input bit (indicating God’s existence) is part of a valid state indicating “God exists.”

Output Measurement:

1. The last qubit of the CR register is designated as the output bit.
2. Measure the output bit to determine whether the input bit indicated the existence of God.

Theoretical Considerations

The model ensures that the quantum algorithm works with high probability (at least 2/32/32/3) irrespective of the initial state ρ chosen for the CTC register.

Discussion with Respect to Pascalian Wager

Pascal’s Wager argues that it is rational to believe in God because the potential benefits (eternal bliss) vastly outweigh the finite costs of belief. In the context of quantum computing:

Risk and Reward:

• The quantum algorithm designed with a bit from the future adds a probabilistic assurance to Pascal’s argument. If the bit indicates God’s existence, the algorithm’s output guides the rational choice (analogous to choosing belief in Pascal’s Wager).
• In the quantum scenario, the algorithm’s high probability of correctness can be seen as a computational analog to the infinite expected value Pascal argued for belief in God.

Quantum Uncertainty:

• The inherent uncertainty and probabilistic nature of quantum measurements can be paralleled with the uncertainty in belief without direct evidence.
• The CTC model, allowing for time-traveling qubits, introduces a unique dimension where information from the future (about God’s existence) directly impacts present decisions, offering a quantum twist to Pascal’s philosophical wager.

By leveraging the BQPQCTC[w] model, we adapt the decision-making scenario to quantum computing, providing a rigorous computational framework for examining Pascal’s Wager in the presence of quantum information about God’s existence.

References

• Say, A.C.C., Yakaryılmaz, A. “Quantum Finite Automata: A Modern Introduction.” This reference provides a foundational understanding of the principles adapted here to the quantum context with CTCs.
• Say, A., O’Donnell, R. “Computational Models with Quantum Closed Timelike Curves.” This paper elaborates on the BQPQCTC[w] model and its applications in decision problems.