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:

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

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