# Specification 014: Quantum Computational Mechanics

## Overview

This specification outlines the path from classical computational mechanics to quantum computational mechanics, with the goal of understanding quantum emergence and structure in quantum processes.

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## Motivation

Classical computational mechanics answers: *What is the minimal information from the past needed to optimally predict the future?*

Quantum computational mechanics asks the same question for quantum systems:
- Quantum correlations (entanglement) between past and future
- Superposition of causal states
- Measurement-induced structure
- Emergence of classical structure from quantum dynamics

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## Prerequisites

### Classical Foundations (Current Work)

| Prerequisite | Status | Notes |
|--------------|--------|-------|
| CSSR implementation | ✅ Done | Core inference working |
| Complexity measures (Cμ, hμ, E) | ✅ Done | Analysis module |
| Mixed-state inference | 📝 Planned | Spec 010: HE-2 |
| Crypticity computation | 📝 Planned | Hidden information |
| Bidirectional machines | 📝 Planned | Time-reversal |

### Mathematical Background

| Topic | Relevance |
|-------|-----------|
| Density matrices | Quantum states |
| POVMs | Generalized measurements |
| Quantum channels | Dynamics |
| Quantum mutual information | Correlations |
| Quantum conditional entropy | Prediction |

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## Key Concepts

### 1. Quantum Causal States

**Classical**: Equivalence classes of histories with same predictive distribution
$$\sigma(\overleftarrow{x}) = [\overleftarrow{x}]_\sim \text{ where } \overleftarrow{x} \sim \overleftarrow{y} \iff P(\overrightarrow{X}|\overleftarrow{x}) = P(\overrightarrow{X}|\overleftarrow{y})$$

**Quantum**: Equivalence classes with same quantum predictive state
$$\sigma_Q(\overleftarrow{x}) = [\overleftarrow{x}]_\sim \text{ where } \overleftarrow{x} \sim \overleftarrow{y} \iff \rho_{\overrightarrow{X}|\overleftarrow{x}} = \rho_{\overrightarrow{X}|\overleftarrow{y}}$$

### 2. Quantum Statistical Complexity

$$C_\mu^Q = S(\rho_\mathcal{S})$$

where $S$ is von Neumann entropy and $\rho_\mathcal{S}$ is the density matrix over quantum causal states.

### 3. Quantum Excess Entropy

$$E_Q = I_Q(\overleftarrow{X}; \overrightarrow{X})$$

Quantum mutual information between past and future.

### 4. Quantum Crypticity

$$\chi_Q = C_\mu^Q - E_Q$$

Quantum information stored but not transmitted.

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## Research Questions

### Foundational

1. **Existence**: Do quantum causal states always exist? Under what conditions?
2. **Uniqueness**: Is there a unique minimal quantum representation?
3. **Computability**: Can we infer quantum causal states from measurement data?

### Structural

4. **Classical limit**: How do quantum causal states reduce to classical ones?
5. **Entanglement role**: How does entanglement contribute to structure?
6. **Measurement effects**: How does measurement choice affect inferred structure?

### Emergent

7. **Quantum-to-classical**: When does classical structure emerge from quantum dynamics?
8. **Decoherence**: How does decoherence affect quantum complexity?
9. **Thermodynamics**: Connection to quantum thermodynamic costs?

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## Phased Approach

### Phase 1: Classical Completion (Current)
- Complete CSSR and analysis
- Implement alternative algorithms
- Validate on known processes

### Phase 2: Mixed States & Crypticity
- Implement mixed-state inference
- Compute crypticity for classical processes
- Understand hidden information

### Phase 3: Quantum Simulation
- Simulate quantum processes classically
- Define quantum causal states for toy models
- Implement quantum complexity measures

### Phase 4: Quantum Inference
- Develop quantum state tomography integration
- Infer quantum causal states from measurement data
- Handle finite-sample effects

### Phase 5: Quantum Emergence
- Study quantum-to-classical transitions
- Characterize emergence via complexity
- Connect to quantum thermodynamics

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## Toy Models

### Model 1: Quantum Random Walk

Discrete-time quantum walk on a line. Compare:
- Quantum causal states of position measurement outcomes
- Classical causal states of the measurement record
- Complexity vs classical random walk

### Model 2: Repeated Interaction Model

System interacting with stream of ancillas:
- Each ancilla measured after interaction
- Natural "past" (measured) and "future" (unmeasured)
- Study how system state encodes predictive information

### Model 3: Quantum Cellular Automata

Unitary evolution on 1D chain:
- Partition into past/future light cones
- Quantum mutual information across cut
- Emergence of effective classical dynamics

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## Key References

### Crutchfield Group

1. **Riechers, P.M. & Crutchfield, J.P. (2021)**. "Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction"

2. **Aghamohammadi, C. et al. (2018)**. "Extreme Quantum Memory Advantage for Rare-Event Sampling"

3. **Mahoney, J.R. et al. (2016)**. "Quantum Information in the Randomness of Quantum Processes"

### Quantum Foundations

4. **Binder, F.C. et al. (2018)**. "Practical Unitary Simulator for Non-Markovian Complex Processes"

5. **Thompson, J. et al. (2018)**. "Causal Asymmetry in a Quantum World"

### Information Theory

6. **Cerf, N.J. & Adami, C. (1997)**. "Quantum Extension of Conditional Probability"

7. **Wilde, M.M. (2017)**. "Quantum Information Theory" (textbook)

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## Implementation Roadmap

```
Q1 2026: Classical completion
├── Alternative algorithms (CSM, Spectral)
├── Mixed-state inference
└── Crypticity computation

Q2 2026: Quantum foundations
├── Quantum process simulation
├── Define quantum complexity measures
└── Toy model implementations

Q3 2026: Quantum inference
├── Measurement data integration
├── Quantum causal state inference
└── Validation on simulated data

Q4 2026: Emergence studies
├── Quantum-to-classical transitions
├── Decoherence effects
└── Thermodynamic connections
```

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## Success Criteria

1. **Paper 1**: Classical emic framework (current trajectory)
2. **Paper 2**: Quantum extensions and toy models
3. **Paper 3**: Quantum emergence and applications
