Bayesian Collapse in Quantum Monadics

In traditional quantum computing, collapse is treated as a final, probabilistic event: a measurement causes a superposed state to resolve into a definite one. But in Monadics—a quantum-aware computational paradigm—we take collapse seriously as a computational operation, not just a side effect.

This post introduces Bayesian collapse as a fundamental extension to quantum computation using Qiskit, exploring how inference, not randomness, can govern the evolution and selection of quantum states. This forms a crucial part of Monadics, where monadic structure is fused with quantum behavior to simulate inference, feedback, and conscious computation.

Quantum Computation: A Brief Primer

Quantum computers operate on qubits, which exist in linear superpositions:

$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \quad |\alpha|^2 + |\beta|^2 = 1$

The computation proceeds via unitary gates (e.g., Hadamard, CNOT), transforming the quantum state. Eventually, a measurement collapses the state, selecting a classical outcome based on the probability amplitudes:

$P(0) = |\alpha|^2, \quad P(1) = |\beta|^2$

This traditional collapse is irreversible and is usually deferred to the end of a quantum circuit.

Quantum State Evolution

Mid-Circuit Collapse

Modern quantum programming tools like Qiskit allow mid-circuit measurement, enabling the system to:

• Observe and collapse part of the quantum state mid-execution
• Use classical logic (if conditions) to control subsequent operations

Example:

python
qc = QuantumCircuit(2, 1)
qc.h(0)
qc.measure(0, 0)
qc.x(1).c_if(0, 1)  # Apply X gate on qubit 1 if qubit 0 collapses to 1

This introduces a primitive form of feedback. However, it is still rooted in random collapse.

Mid-Circuit Measurement Flow

From Randomness to Inference: Bayesian Collapse

In Monadics, collapse is not mere randomness—it is inference. We introduce Bayesian reasoning into the collapse process, allowing the system to prefer certain outcomes based on prior beliefs.

Given a superposed state:

$|\psi\rangle = \sum_{\omega \in \Omega} \alpha_\omega |\omega\rangle$

We reinterpret the amplitude squared, (|\alpha_\omega|^2), as the likelihood of outcome (\omega). We then combine this with a prior distribution over the possible outcomes to compute a posterior via Bayes' Rule:

$P(\omega|D) = \frac{P(D|\omega) \cdot P(\omega)}{\sum_{\omega'} P(D|\omega') \cdot P(\omega')}$

Where:

• (P(\omega)) is the prior belief about outcome (\omega)
• (P(D|\omega)) is the likelihood (based on quantum amplitudes)
• (P(\omega|D)) is the posterior—how much we believe (\omega) is the correct collapse, given the data

We then sample from the posterior, not the raw quantum amplitudes.

Bayesian Collapse Process

Qiskit Implementation

Here's how Bayesian collapse is implemented in Qiskit:

python
from qiskit import QuantumCircuit, Aer, execute
import random

# Step 1: Simulate state
qc = QuantumCircuit(1)
qc.h(0)  # Create superposition
backend = Aer.get_backend('statevector_simulator')
result = execute(qc, backend).result()
statevector = result.get_statevector()
likelihoods = [abs(amplitude)**2 for amplitude in statevector]

# Step 2: Define prior
prior = [0.9, 0.1]  # Example: strongly believe in |0⟩

# Step 3: Compute posterior
evidence = sum(l * p for l, p in zip(likelihoods, prior))
posterior = [(l * p) / evidence for l, p in zip(likelihoods, prior)]

# Step 4: Sample collapse outcome
collapsed_state = random.choices([0, 1], weights=posterior)[0]

This turns collapse into a belief-weighted decision, modeling inference and even attention bias.

Monadics: Bayesian Collapse as Monadic Bind

We can model Bayesian collapse in Haskell using monadic structure. Consider this:

haskell
data QState a = Superposed [(a, Complex Double)]  -- Superposed state
              | Collapsed a                       -- After collapse

collapseBayesian :: QState a -> (a -> QState b) -> (a -> Double) -> QState b
collapseBayesian (Superposed amps) f prior =
  let likelihoods = map (\(_, amp) -> magnitude amp ** 2) amps
      states      = map fst amps
      priors      = map prior states
      evidence    = sum $ zipWith (*) likelihoods priors
      posterior   = zipWith (\l p -> (l * p) / evidence) likelihoods priors
      chosen      = sampleFromDistribution states posterior
  in f chosen

This collapseBayesian function behaves as a Bayesian monadic bind, weighting future computation by informed collapse.

Monadic Collapse Chain

haskell
collapseBayesian :: QState a -> (a -> QState b) -> (a -> Double) -> QState b

-- Monadic do-notation example:
quantumProcess = do
  state ← superposed
  prior ← getPrior  
  collapsed ← bayesianCollapse state prior
  result ← continuation collapsed
  return result

Collapse as Reasoning

In Monadics, collapse is computation. With Bayesian reasoning added:

• Collapse selects outcomes based on prior knowledge
• Circuits can adapt mid-execution, simulating intelligent inference
• Collapse becomes controlled resolution of superposition, not a passive measurement

This approach aligns collapse with decision-making, control, and consciousness.

Conscious Collapse Applications

This mirrors biological cognition: observation affects state, state affects future inference.

Conscious Collapse Architecture

Summary

We have extended Monadics with a Bayesian model of collapse, replacing randomness with inference. Using Qiskit, we demonstrate how mid-circuit measurement can be modified to reflect posterior-weighted decisions.

This allows us to treat quantum computation not merely as a system of gates and probabilities, but as a compositional reasoning system—a step closer to artificial consciousness.