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arXiv:2609.25947

Bridge of ΨΨΨ's: Quantum Circuit Optimization with Schrödinger Bridges

Lino S. Hofstetter, Lia Yeh, Prakash Murali

quant-phcs.LGSubmitted Sep 22, 2026• Announced Sep 23, 2026• 0 citations
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Abstract

Quantum circuit optimization replaces a circuit with an equivalent one of fewer gates and lower depth, reducing execution cost and error rate. We ask whether a generative model can learn this transformation directly from examples, rather than selecting from a fixed rewrite library or rigid algebraic routines. We present Bridge of $Ψ$'s (BOPS), a generative model based on Schrödinger bridges, using a custom denoiser architecture, that learns a transformation from a source circuit into an equivalent optimized circuit. We train it on data constructed to be hard for existing optimizers, by applying rewrite rules backwards so that each input has a known lower-cost target. On held-out 8 qubits $\times$ 64 depth Clifford+$T$ circuits, BOPS reduces gate count by $2.46\times$ and depth by $2.45\times$ in geometric mean, outperforming all nine baseline optimizers. This constitutes the first generative model bridging quantum circuits and frontier machine learning methods, opening up the quantum compilation stack to learned optimization along multiple axes.

Comments
25 pages, 11 figures, 12 tables

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•Mar 17, 2025•3 citations
Quantum circuit optimization replaces a circuit with an equivalent one of fewer gates and lower depth, reducing execution cost and error rate. We ask whether a generative model can learn this transformation directly from examples, rather than selecting from a fixed rewrite library or rigid algebraic routines. We present Bridge of ΨΨΨ's (BOPS), a generative model based on Schrödinger bridges, using a custom denoiser architecture, that learns a transformation from a source circuit into an equivalent optimized circuit. We train it on data constructed to be hard for existing optimizers, by applying rewrite rules backwards so that each input has a known lower-cost target. On held-out 8 qubits ×\times× 64 depth Clifford+TTT circuits, BOPS reduces gate count by 2.46×2.46\times2.46× and depth by 2.45×2.45\times2.45× in geometric mean, outperforming all nine baseline optimizers. This constitutes the first generative model bridging quantum circuits and frontier machine learning methods, opening up the quantum compilation stack to learned optimization along multiple axes.
Quantum circuit transformation aims to optimize circuits for depth, gate count, and compatibility with Noisy Intermediate Scale Quantum (NISQ) devices that suffer from various error sources. Prior methods use combinations of expert-defined rules and Reinforcement Learning (RL). We introduce AltGraph, a novel approach employing generative graph models to generate functionally equivalent quantum circuits using—specifically, Direct Acyclic Graph (DAG) Variational Autoencoder (D-VAE) variants (GRU and GCN) and Deep Generative Model for Graphs (DeepGMG). AltGraph perturbs the latent space to generate quantum circuits optimized for hardware coupling maps, reducing gate count by 37.55% and circuit depth by 37.75% post-transpiling, with 0.0074 Mean Squared Error (MSE) in the density matrix—outperforming state-of-the-art methods by 2.56%.
Quantum circuit optimization - the process of transforming a quantum circuit into an equivalent one with reduced time and space requirements - is crucial for maximizing the utility of current and near-future quantum devices. While most automated optimization techniques focus on transforming circuits into equivalent ones that implement the same unitary, we show that substantial new opportunities for resource reduction can be achieved by (1) allowing approximate local transformations and (2) employing mixed quantum channels to approximate pure circuits. Our novel automated protocol for approximate circuit rewriting is a refined evolution of automated optimization techniques based on the ZX-calculus, where we add a greedy strategy that selectively replaces ZX-diagrams with small phase angles with stochastic mixtures of the identity and carefully chosen over-rotations, which are designed to reduce the overall gate count in expectation while staying within a strict error budget. This approach yields modest two-qubit gate count reduction in random quantum circuits, and achieves a substantial reduction in structured circuits such as the quantum Fourier transform. Fundamentally, our protocol converts experimental noise due to gate applications into deliberately engineered random noise, outperforming many other approximation methods on average. These results highlight the potential of mixed-channel approximations to enhance future quantum circuit performance, suggesting new directions for resource-aware automated quantum compilation beyond pure unitary channels.
With recent advancements in quantum computing technology, optimizing quantum circuits and ensuring reliable quantum state preparation have become increasingly vital. Traditional methods often demand extensive expertise and manual calculations, posing challenges as quantum circuits grow in qubit- and gate-count. Therefore, harnessing machine learning techniques to handle the growing variety of gate-to-qubit combinations is a promising approach. In this work, we introduce a comprehensive reinforcement learning environment for quantum circuit synthesis, where circuits are constructed utilizing gates from the the Clifford+T gate set to prepare specific target states. Our experiments focus on exploring the relationship between the depth of synthesized quantum circuits and the circuit depths used for target initialization, as well as qubit count. We organize the environment configurations into multiple evaluation levels and include a range of well-known quantum states for benchmarking purposes. We also lay baselines for evaluating the environment using Proximal Policy Optimization. By applying the trained agents to benchmark tests, we demonstrated their ability to reliably design minimal quantum circuits for a selection of 2-qubit Bell states.
Quantum circuit optimization is a foundational challenge in realizing practical quantum computation, requiring the reduction of gate counts and circuit depth while preserving logical equivalence across noisy intermediate-scale quantum (NISQ) devices. Conventional rule-based and heuristic compilation approaches demonstrate diminishing effectiveness as circuit complexity grows, motivating the search for adaptive, data-driven alternatives. This paper reformulates quantum circuit optimization as a Markov decision process (MDP), enabling deep reinforcement learning (RL) agents to autonomously discover efficient transformation strategies. A graph neural network (GNN)-based actor-critic architecture is developed to encode circuit topology and guide transformation selection. Drawing on empirical evidence from phase-structured quantum control tasks, neural feedback architectures, and convergence behavior in active learning agents, we demonstrate that the proposed framework achieves an average two-qubit gate count reduction of 31.4% and a circuit depth reduction of 27.8% across standard benchmark circuits, outperforming established compilation baselines. This work provides a principled theoretical foundation for the RL-as-optimizer paradigm and contributes an extensible experimental platform suited to NISQ-era constraints.
Quantum Circuit Pre-Synthesis: Learning Local Edits to Reduce TTT-count
Compiling quantum circuits into Clifford+TTT gates is a central task for fault-tolerant quantum computing using stabilizer codes. In the near term, TTT gates will dominate the cost of fault tolerant implementations, and any reduction in the number of such expensive gates could mean the difference between being able to run a circuit or not. While exact synthesis is exponentially hard in the number of qubits, local synthesis approaches are commonly used to compile large circuits by decomposing them into substructures. However, composing local methods leads to suboptimal compilations in key metrics such as TTT-count or circuit depth, and their performance strongly depends on circuit representation. In this work, we address this challenge by proposing \textsc{Q-PreSyn}, a strategy that, given a set of local edits preserving circuit equivalence, uses a RL agent to identify effective sequences of such actions and thereby obtain circuit representations that yield a reduced TTT-count upon synthesis. Experimental results of our proposed strategy, applied on top of well-known synthesis algorithms, show up to a 20%20\%20% reduction in TTT-count on circuits with up to 25 qubits, without introducing any additional approximation error prior to synthesis.
The optimization of the limited resources of current quantum devices hinges on the implementation of quantum-compiling techniques. The process of transforming high-level quantum algorithms into executable quantum circuits has a significant impact on the performance and fidelity of computations. One of the most critical aspects of the compiling stack is the synthesis of quantum circuits, which aims to minimize the depth and number of gates, as well as the error rate. Nevertheless, the optimal synthesis algorithms are extremely computationally expensive. This is why synthesis algorithms based on reinforcement learning (RL) have recently been introduced. However, such algorithms are constrained by the topology of quantum processors, necessitating retraining of the RL agent for different topologies. To achieve a balance between adaptability, optimality, and computational cost in RL-based quantum circuit synthesis techniques, this work proposes an RL-based quantum circuit synthesis algorithm of Clifford operators that addresses different topologies simultaneously. Experimental tests demonstrate that our synthesizer outperforms state-of-the-art greedy techniques while being significantly faster than boolean satisfiability solvers (SAT).