Quantum Sequence Samplers

---
name: quantum-sequence-samplers
description: "Quantum sequence models for learning and generating stochastic processes. Quantum circuits that generate coherent superpositions of stochastic processes enable quantum-accelerated risk analysis, importance sampling, and DNA sequencing. Addresses the challenge of encoding classical stochastic processes into quantum states efficiently."
arxiv_id: "2603.24069"
published: "2026-03-24"
tags: [quantum-machine-learning, stochastic-processes, quantum-sampling, sequence-models, risk-analysis, importance-sampling]
---

# Quantum Sequence Samplers for Stochastic Processes

Methodology from arXiv:2603.24069 (March 2026). Quantum circuits that generate coherent superpositions of stochastic processes for quantum-accelerated downstream tasks.

## Core Problem

Many quantum-accelerated tasks (risk analysis, importance sampling, DNA sequencing) require preparing quantum states that encode stochastic processes. Classical sampling methods generate sequential trajectories one at a time. Quantum approaches can generate **coherent superpositions** of many trajectories simultaneously, enabling amplitude amplification and quantum speedups.

The challenge: efficiently encoding classical stochastic process statistics into quantum circuit parameters while preserving the temporal correlations of the process.

## Methodology: Quantum Sequence Models

### Architecture

A Quantum Sequence Model (QSM) maps a stochastic process specification to a quantum circuit:

```
Process parameters (μ, σ, transition matrix) → Quantum Circuit → Superposition of trajectories
```

### Key Components

1. **State Encoding**: Map process state space to computational basis states
   - For discrete processes: direct mapping to |x₁⟩|x₂⟩...|x_T⟩
   - For continuous processes: discretization + amplitude encoding

2. **Temporal Correlation Encoding**: Use entangling gates to capture Markov/non-Markov dependencies
   ```
   |ψ⟩ = Σ_{x₁,...,x_T} √P(x₁,...,x_T) |x₁,...,x_T⟩
   ```
   where P is the joint probability of the stochastic process.

3. **Coherent Superposition**: The circuit prepares a superposition weighted by √P, enabling:
   - Amplitude amplification for rare event sampling
   - Quantum Monte Carlo with O(1/ε) vs O(1/ε²) convergence
   - Parallel trajectory generation

### Circuit Construction Patterns

#### Pattern 1: Markov Process Encoding

For a Markov chain with transition matrix P:

```python
def markov_quantum_sampler(transition_matrix, n_steps, n_qubits_per_step):
    """
    Build quantum circuit for Markov chain trajectory superposition.
    
    Args:
        transition_matrix: P[i,j] = P(X_{t+1}=j | X_t=i)
        n_steps: trajectory length T
        n_qubits_per_step: qubits to encode each state
    
    Circuit structure:
        |0⟩^⊗n --[Init P(X₀)]--●--●--...--●
        |0⟩^⊗n ---------------⊕--●--...--●
        |0⟩^⊗n -------------------⊕--...--●
        ...
    """
    # Step 1: Prepare initial distribution
    state_prep = amplitude_encoding(transition_matrix[0])
    
    # Step 2: For each time step, apply controlled rotation
    # conditioned on previous state
    for t in range(1, n_steps):
        for i in range(n_states):
            # Controlled rotation: if |i⟩ at time t-1,
            # prepare P(i,·) at time t
            apply_controlled_state_prep(
                control_qubits=state_at(t-1),
                target_qubits=state_at(t),
                distribution=transition_matrix[i]
            )
```

#### Pattern 2: Quantum Amplitude Estimation Integration

```python
def quantum_monte_carlo(sampler_circuit, observable, epsilon):
    """
    Quantum Monte Carlo using amplitude estimation.
    Converges in O(1/epsilon) vs classical O(1/epsilon²).
    """
    # A = sampler_circuit encodes |ψ⟩ = Σ √p_x |x⟩
    # Q = Grover-like operator for amplitude estimation
    # Estimate E[f(X)] with ε precision using O(1/ε) queries
    
    n_iterations = int(np.pi / (4 * epsilon))
    for _ in range(n_iterations):
        apply_grover_iteration(sampler_circuit, observable)
    
    return measure_phase_estimation()
```

## Applications

### 1. Financial Risk Analysis
- VaR/CVaR estimation with quantum speedup
- Portfolio risk under correlated stochastic processes
- Credit risk modeling with quantum sequence samplers

### 2. DNA Sequence Analysis
- Generate superpositions of mutation pathways
- Estimate probabilities of rare genetic events
- Quantum-enhanced sequence alignment

### 3. Physics Simulation
- Path integral Monte Carlo with quantum trajectories
- Stochastic differential equation solutions
- Rare event sampling in high-dimensional systems

## Key Advantages

1. **Quadratic speedup**: Amplitude estimation gives O(1/ε) convergence vs O(1/ε²)
2. **Coherent processing**: Superpositions enable quantum interference effects
3. **Parallel trajectories**: All possible trajectories exist simultaneously in superposition
4. **Composable**: Sampler circuits can be composed with other quantum algorithms

## Implementation Requirements

- **Circuit depth**: O(T × n_qubits) for T-step process
- **Controlled rotations**: O(n_states × T) controlled operations
- **Classical preprocessing**: Transition matrix decomposition for efficient gate synthesis

## Pitfalls

1. **State space explosion**: Discrete processes with large state spaces require many qubits
2. **Gate decomposition overhead**: Arbitrary state preparation requires O(2^n) gates in worst case
3. **Noise sensitivity**: Deep circuits for long trajectories are NISQ-limited
4. **Classical data loading bottleneck**: Encoding process parameters into quantum form may negate speedup

## Activation

quantum sequence sampler, stochastic process quantum encoding, quantum Monte Carlo, amplitude estimation sampling, quantum risk analysis, quantum trajectory generation, DNA quantum sequencing, quantum importance sampling, 量子随机过程采样

## References

- **Paper**: "Learning Quantum-Samplers for Stochastic Processes with Quantum Sequence Models"
- **arXiv**: 2603.24069
- **Published**: March 2026