Differentiable Logical Programming for Quantum Circuit Discovery and Optimization

TL;DR

This work introduces Differentiable Logical Programming (DLP) to quantum circuit design by recasting discrete circuit structure search as differentiable optimization over continuous gate switches on a circuit scaffold. Gate inclusion is controlled through learnable logits mapped to switches with either linear or geodesic interpolation, and the design is guided by differentiable axioms for fidelity, energy, and simplicity, enabling end-to-end optimization of both structure and parameters. The authors provide a theoretical analysis of the linear relaxation as a surrogate, including norm deviation bounds and gradient stability, and enhance tractability with curriculum learning and Hierarchical Synthesis to scale to larger systems. Empirically, DLP demonstrates de novo QFT circuit discovery, Trotter-step optimization, noise-resilient learning under shot noise, hardware-aware topology adaptation, and real-time adaptation to hardware failures on IBM qubits, including substantial fidelity gains and much shallower circuits when hardware constraints are honored. Overall, DLP offers a flexible, neuro-symbolic framework that unifies circuit synthesis, compilation, and hardware-aware optimization under a gradient-based paradigm with broad potential for NISQ and early fault-tolerant quantum computing.

Abstract

Designing high-fidelity quantum circuits remains challenging, and current paradigms often depend on heuristic, fixed-ansatz structures or rule-based compilers that can be suboptimal or lack generality. We introduce a neuro-symbolic framework that reframes quantum circuit design as a differentiable logic programming problem. Our model represents a scaffold of potential quantum gates and parameterized operations as a set of learnable, continuous ``truth values'' or ``switches,'' $s \in [0, 1]^N$. These switches are optimized via standard gradient descent to satisfy a user-defined set of differentiable, logical axioms (e.g., correctness, simplicity, robustness). We provide a theoretical formulation bridging continuous logic (via T-norms) and unitary evolution (via geodesic interpolation), while addressing the barren plateau problem through biased initialization. We illustrate the approach on tasks including discovery of a 4-qubit Quantum Fourier Transform (QFT) from a scaffold of 21 candidate gates. We also report a hardware-aware adaptation experiment on the 133-qubit IBM Torino processor, where the method improved fidelity by 59.3 percentage points in a localized routing task while adapting to hardware failures.

Figures (15)

• Figure 1: Conceptual overview of the Differentiable Logical Programming framework for quantum circuit design. The entire process, from the logical axioms to the circuit structure, is connected by differentiable operations, allowing for end-to-end optimization using standard gradient-based methods. This workflow unifies discrete structural search and continuous parameter optimization.
• Figure 2: Circuit fidelity over training. Circuit fidelity ($\mathcal{T}_{\text{fid}}$) over training epochs for varying noise levels $\sigma$. The system consistently converges to high fidelity even under significant noise ($\sigma=0.5$).
• Figure 3: Simplicity score over training. Simplicity score ($\mathcal{T}_{\text{simp}}$) over training epochs. The convergence to a specific score indicates the pruning of redundant gates across all noise levels.
• Figure 4: Initial scaffold configuration. Initial scaffold input with all gates activated. This represents the starting point for the pruning task.
• Figure 5: Discovered circuit topology. Discovered circuit topology for noise level $\sigma=0.5$. The framework correctly identifies the 2nd-order Trotter decomposition.

Theorems & Definitions (1)

• proof