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Behavior-Driven Quantum Refactoring via Emulation-Guided LLM Training

Abstract

This document proposes a practical and technically grounded framework for automating the refactoring of classical computational models into quantum-compatible representations using large language models (LLMs). Rather than relying on symbolic rewriting or theoretical supervision, this approach utilizes quantum emulation feedback to iteratively guide the generation of quantum circuits. The model is rewarded based on behavioral fidelity to classical outputs, allowing it to learn to approximate functionality under quantum constraints. To address the complexity and feasibility concerns associated with quantum data encoding, noisy measurement, and circuit validation, this method integrates structural constraints, hybrid validation techniques, and performance-aware reward shaping.

  1. Motivation

Classical-to-quantum translation is constrained by the requirements of unitarity, reversibility, and probabilistic measurement. Existing compilers for quantum systems, such as Scaffold or RevKit, require structured symbolic inputs and do not generalize well to arbitrary classical functions. Manual derivation of quantum logic is labor-intensive and prone to error propagation as circuit complexity increases. LLMs, by contrast, can generalize across diverse input patterns and suggest multiple functionally distinct yet semantically equivalent circuit configurations.

This proposal introduces a dynamic, data-driven training loop where LLM-generated circuits are evaluated using quantum simulators acting as behavioral validators. The loop is driven not by token-level losses but by emulated execution feedback, transforming the synthesis problem into a continuous optimization process governed by output match probability. The aim is to align statistical measurement outcomes from quantum circuits with deterministic classical reference values over a constrained input space.

  1. Methodology

2.1 Input and Preprocessing

Each training sample is defined as a tuple (f, X, Y), where f: X -> Y is a classical function f defined over binary inputs X and corresponding outputs Y. For example, f(x, y) = (x ^ y) & (x + 3). The classical function is converted into a canonical truth table or decision diagram and used to precompute a corpus of input-output mappings. Input vectors are encoded as basis states |x⟩ using a fixed register width (n qubits for n-bit inputs). Optional pre-encoding layers using parameterized unitary matrices U(x) can be introduced to reduce depth.

2.2 Circuit Generation and Constraints

The LLM acts as a code generator over a constrained token vocabulary consisting of quantum operations such as {X, CX, H, T, T†, U3, CRZ, SWAP, measure, reset}, qubit register references, and control syntax. Token sequences are decoded using beam search with syntax-aware masking. Circuits are post-processed with unitarity checks and static flow analysis to ensure reversibility. A grammar specification (e.g. using PEG or context-free BNF) restricts invalid sequences.

To encode multi-bit outputs, ancilla registers are allocated. Logical output bits are decoded from measurement results using a classical post-processing function M̂: {|ψ⟩} → {0,1}^k based on measurement frequency histograms. A key constraint is that the circuit must produce output states with high fidelity to the classical truth table when averaged over multiple simulations.

2.3 Feedback via Emulation

Given a candidate quantum circuit C and classical function f, the training loop executes the following:

  1. For a batch of input vectors X_batch ⊆ X:
  2. Prepare input states |x_i⟩
  3. Apply generated circuit C to each |x_i⟩
  4. Run simulator backend (e.g. Qiskit Aer or Cirq) for N shots per input
  5. Extract most probable output from measured bitstring distribution
  6. Compare to ground-truth outputs y_i = f(x_i) using fidelity metric F
  7. Compute reward R = Σ_i F(ŷ_i, y_i) – α * (circuit_cost(C))
  8. Update model parameters via policy gradient or genetic programming

Reward shaping includes penalties for gate depth, nonlocal gate usage, and ancilla count. Optional approximators, such as tensor contraction backends or matrix product state (MPS) simulations, are used to validate deeper circuits infeasible to simulate exhaustively.

  1. Advantages and Limitations

This architecture allows partial quantumization of classical functions and supports incremental refactoring strategies. The LLM can learn to isolate quantum-feasible subgraphs (e.g. XOR logic, reversible counters) and construct functionally equivalent circuits. It also enables approximate circuit discovery when symbolic mappings are unavailable or computationally intractable.

However, current emulators scale poorly beyond 30 qubits. To mitigate this, we adopt dimension-aware sampling (e.g. training on reduced-width slices of logic), hybrid symbolic gates, and low-rank approximations of unitary operators. Fidelity degradation due to measurement noise is corrected via statistical debiasing (bootstrap aggregation, histogram sharpening).

  1. Optimization Strategies

Key enhancements include:

  • Hybrid training using a symbolic teacher model for early-stage convergence
  • Reinforcement learning with entropy-regularized policy updates
  • Dynamic token masking to eliminate invalid gate sequences mid-decoding
  • Multi-resolution input encoding (e.g. Gray code, parity features) to reduce model confusion
  • Curriculum learning: start with 1-2 qubit circuits and scale up as the model stabilizes

An optional post-training pruning stage applies template matching and algebraic simplification to remove redundant gate pairs (e.g. adjacent H gates, cancelable phase shifts).

  1. Conclusion

This approach reframes quantum code synthesis as a behavioral imitation task, with emulation feedback guiding an LLM to produce executable quantum circuits. By integrating structural constraints, approximate validation, and cost-aware optimization, it creates a scalable framework for quantum refactoring. Although bounded by simulator and hardware limits, this method allows rapid prototyping of refactorable logic under quantum constraints and may seed the next generation of self-improving quantum compilation systems.