Compile-Time Simplification of Classically Controlled Operations in Dynamic Circuits

Abstract: Dynamic circuits use real-time outcomes of mid-circuit measurements, processed by a classical controller, to adapt subsequent operations during circuit execution. This additional flexibility over static circuits comes at a price. Mid-circuit measurements are typically slower and noisier than unitary gates. Furthermore, classical feedforward requires exchanging information between the quantum processor (QPU) and the classical controller, introducing latency that erodes the practical performance of dynamic circuits. We propose a compile-time optimization framework that reduces the use of classical controls in dynamic circuits while preserving their semantics. At its core, the framework uses a static analysis that symbolically executes the circuit by propagating classical information alongside the quantum state. By combining this classical-quantum information with the Probabilistic Circuit Model extended with probabilistic controls that emulate classical feedforward, we obtain an intermediate probabilistic representation of the dynamic circuit. In this representation, mid-circuit measurements and classically controlled operations can be removed or rewritten as purely unitary operations and probabilistic components. Compared to existing compile-time optimizations that target only mid-circuit measurements, our method applies to a broader class of dynamic circuits expressible in modern quantum programming languages. We evaluated our framework on randomly generated dynamic circuits, achieving about 50% classical feedforward reduction and even higher reductions in favorable settings.

• The paper's main contribution is a framework that combines Quantum Constant Propagation with a novel Classical Constant Propagation to simplify classically controlled operations.
• It employs a two-phase pipeline—symbolic simulation followed by probabilistic instantiation—to achieve up to 80% reduction in classical feedforward operations.
• Empirical validation on random circuits and a GHZ case study demonstrates significant potential for reducing latency and error rates in NISQ-era hardware.

Compile-Time Simplification of Classically Controlled Operations in Dynamic Circuits

Motivation and Background

Dynamic quantum circuits augment static quantum circuits with classical feedforward: operations can depend on mid-circuit measurement results, allowing for flexible control flow. Such dynamism, supported by hardware and high-level quantum programming languages (OpenQASM 3.0, Qiskit, Cirq), is essential for a wide range of applications—quantum error correction, distributed quantum computing, qubit reuse, and teleportation protocols. However, in current architectures, mid-circuit measurements and resets introduce significant latency and error, mainly due to slow, noisy readout and required classical feedback between quantum and classical controllers [mcm_hw_1, mcm_hw_error]. Additionally, modern quantum software stacks promote explicit classical control constructs, making programming more natural but also increasing the likelihood of redundant or suboptimal classical control.

Prior optimization frameworks focused primarily on simplifying or eliminating mid-circuit measurements where possible, typically leveraging static analysis techniques such as Quantum Constant Propagation (QCP) [qcp] and modeling stochasticity via a Probabilistic Circuit Model (PCM) [big_prob, pcm]. However, these approaches have only limited support for circuits utilizing complex patterns of classical control, as they do not track the evolution of the classical register during symbolic simulation.

Contributions and Methodology

This paper introduces a comprehensive framework for compile-time optimization of classically controlled operations in dynamic quantum circuits (2605.28439). The key components are:

• Propagation of Classical and Quantum Information: The framework concurrently propagates quantum and classical register abstractions during symbolic execution, using QCP for quantum state analysis and a newly introduced Classical Constant Propagation (CCP) for classical register analysis.
• PCM Extension with Probabilistic Controls: It extends the PCM to represent not just random quantum operations but also abstract compile-time classical bits—probabilistic controls—linked to outcomes of measurements conceptualized at compile time.
• Probabilistic Gate-Control Binding: Each mid-circuit measurement replaced during optimization is associated with a probabilistic gate and a corresponding probabilistic control, ensuring that classical-quantum correlations are accurately represented at the stochastic intermediate level.
• Condition Simplification: Boolean conditions in classically controlled operations are recursively simplified as much as possible at compile time, exploiting the latest available abstract information for each classical bit.

Two-Phase Pipeline: The approach proceeds in two phases:

1. Phase I (Symbolic Simulation and Rewriting): The dynamic circuit is analyzed instruction-by-instruction. Dynamic operations (mid-circuit measurements, resets, and conditionals) are eliminated, reduced, or rewritten into probabilistic and/or statically determined operations where possible, guided by the propagated abstractions. When quantum or classical information is indecisive, dynamic components are left unchanged.
2. Phase II (Probabilistic Instantiation): Each execution of the optimized circuit corresponds to sampling all probabilistic components, resolving all probabilistic controls, and potentially specializing remaining conditionals further.

The paper provides detailed formalization of the models, rewriting rules, and abstraction update rules, including correctness proofs by induction over the compilation rules.

Results and Numerical Evaluation

Random Circuit Benchmarks

The authors evaluate their framework on a substantial dataset of randomly generated dynamic circuits of varying widths and depths, using Qiskit for circuit representation.

Key empirical results:

• Classical feedforward reduction: On circuits with 80–100 qubits and moderate depth (fixed at 250), the method removes up to 80% of classically controlled operations, driven by improved classical information retention in wider circuits.
• For fixed 25-qubit circuits, increasing depth from 100 to 1000 reduces removal rates modestly (stabilizing within a 35%–50% range), attributed to increased persistent entanglement.
• The optimization is conservative: only operations proven redundant or instantiable at compile-time are counted as removed. Further simplification may occur probabilistically in Phase II.
• Latency and overhead implications: Reductions in classical feedback translate directly into latency and error overhead mitigation, particularly critical on current hardware with slow, noisy measurement channels.

Case Study: GHZ Preparation Circuit

A concrete case study applies the framework to a dynamic-circuit-based GHZ preparation algorithm [ghz]. The method demonstrates its ability to eliminate controlled operations whose classical guards become fully resolvable via compile-time analysis, while conservatively preserving quantum/dynamic components for which information is insufficient due to entanglement or loss of trackable state.

The construction maintains semantic equivalence, ensuring that the optimized circuit samples from the same probabilistic execution path distribution as the original.

Practical and Theoretical Implications

Practical implications:

• The framework directly addresses an ongoing bottleneck in dynamic circuit execution for NISQ-era hardware by compile-time removal or simplification of classically controlled operations that otherwise introduce latency and errors.
• It provides a rigorous optimization mechanism compatible with realistic, high-level quantum programming patterns, making it broadly applicable and ready for integration with leading quantum software stacks.
• Empirical results suggest significant overhead reductions are possible even with randomized circuits, and more so in algorithmic benchmarks with regular structure and repeated patterning of control/measurement.

Theoretical implications and future directions:

• This work pushes the boundary on what is possible with static symbolic analysis bridging quantum and classical registers. When combined with further model-based reasoning (e.g., cost models for target hardware, error models for measurement or resets), one could tune the aggressiveness of compilation specifically for targeted noise and latency characteristics.
• The probabilistic intermediate form (PCM plus controls) offers a powerful abstraction for further program analysis, formal verification, or hybrid classical-quantum simulation.
• Future work could further refine condition simplification through path-sensitive or context-sensitive abstraction, or leverage SMT-based reasoning for more complex conditions.
• Optimizations could be made hardware-aware by incorporating models of mid-circuit measurement and feedback time, gate fidelities, or qubit connectivity.

Conclusion

This paper presents a rigorous, formal, and empirically validated framework for the compile-time simplification of classically controlled operations in dynamic quantum circuits (2605.28439). By simultaneously abstracting quantum and classical registers during symbolic execution and introducing a refined probabilistic intermediate representation, the approach achieves substantial reductions in runtime classical feedforward while preserving circuit semantics. These advances will directly impact both the efficiency and feasibility of near-term dynamic quantum algorithms and inform future research in quantum program optimization, verification, and hardware/software co-design.