⚛️ quantum physics

Propaq: A Python package for Heisenberg Propagation

This paper introduces Propaq, a high-performance and flexible Python package for classically simulating quantum circuits via Heisenberg propagation, which integrates with major quantum software ecosystems and offers advanced features like arbitrary basis simulation, hybrid methods, and detailed performance logging.

Original authors: Hrishikesh Belagali, Ryan LaRose

Published 2026-09-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Hrishikesh Belagali, Ryan LaRose

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand the work of Hrishikesh Belagali and Ryan LaRose, one must first grasp the fundamental challenge facing modern physics: how to predict the behavior of quantum machines without actually building them. Quantum computers promise to solve problems that are impossible for today's supercomputers, but to know if they will succeed, scientists need a way to test them on ordinary machines first. This is where classical simulation comes in. In a standard approach, a computer tries to track the state of every particle in a system as it changes, a task that becomes impossibly heavy as the system grows. However, there is a different way to look at the problem. Instead of watching the particles move forward, one can imagine the question itself traveling backward through the machine. This is known as Heisenberg propagation. Rather than calculating the final state of a quantum system, this method takes the measurement we want to make and traces it backward through the circuit to see how it interacts with the starting point. It is a powerful trick that allows researchers to simulate complex quantum behaviors on classical hardware, provided they can manage the explosion of mathematical terms that occurs during the process.

In their new paper, the researchers introduce a software tool called Propaq, designed to make this backward-tracing method faster, more flexible, and easier to use. The team built Propaq using a programming language known for its speed and safety, then wrapped it in a user-friendly interface that works with the major quantum software packages scientists already use. The result is a program that not only matches the performance of the fastest existing tools but often surpasses them. In tests involving the simulation of magnetic materials and electron interactions, Propaq calculated the expected outcomes significantly faster than its competitors, sometimes by a factor of one hundred. This speed is crucial because it allows researchers to simulate larger systems and more complex scenarios than was previously possible, giving them a clearer picture of what future quantum computers might actually achieve.

What sets Propaq apart is not just its speed, but its ability to adapt to different ways of thinking about quantum mechanics. While most tools force users to work within a single, rigid framework, Propaq allows scientists to choose the mathematical language, or basis, that best fits their specific problem. The researchers demonstrated this by successfully simulating a model of magnetic spins using a custom mathematical system that other software could not handle. This flexibility means that scientists are no longer limited to the standard tools; they can invent new ways to represent quantum systems if it helps them get a better answer. Furthermore, the software includes a hybrid mode that splits the work between two different simulation strategies. It can push the state of the system forward for part of the journey and then pull the measurement backward for the rest. This approach offers a clever trade-off, allowing researchers to save memory or time depending on the specific problem they are trying to solve.

The tool also addresses the messy reality of real-world quantum computers, which are prone to errors and noise. Propaq can simulate these imperfections directly, showing how noise affects the results and even helping scientists estimate what the perfect, error-free answer would be. By running simulations with different levels of noise and then mathematically projecting the results back to a noise-free state, the software provides a way to clean up data without needing a perfect machine. To help researchers understand exactly how their simulations are performing, the software keeps a detailed log of every step, tracking how many terms are being calculated, how much memory is being used, and how much accuracy is being lost when the computer has to discard less important information to keep the process moving. This transparency allows scientists to fine-tune their settings and understand the limits of their simulations.

Ultimately, the work presented in this paper is about giving researchers better tools to explore the boundaries of quantum computing. By making Heisenberg propagation faster and more adaptable, Propaq enables scientists to test new algorithms, design better error-correction methods, and understand the true potential of quantum advantage. The researchers have made the software freely available, inviting the community to use it for their own investigations. As quantum technology moves from theory to practice, tools like this will be essential for verifying that the machines we build are doing exactly what we expect them to do, ensuring that the leap into the quantum future is built on a foundation of solid, verified understanding.

Technical Summary: Propaq – A Python Package for Heisenberg Propagation

Problem Statement

Classical simulation of quantum circuits is essential for benchmarking algorithms, developing error mitigation techniques, and theoretically understanding the boundaries of quantum advantage. While the Schrödinger picture (evolving the state vector) is common, it often suffers from exponential memory scaling. Heisenberg propagation offers an alternative by evolving observables backward through the circuit while keeping the state fixed. However, existing software implementations for Heisenberg propagation (such as Pauli or Majorana propagation) often lack flexibility in basis selection, hybrid simulation capabilities, and detailed analysis tools for truncation and noise. Furthermore, while techniques like sparse Pauli dynamics and Majorana propagation have gained traction, there is a need for a performant, flexible, and extensible Python package that supports arbitrary bases, hybrid Schrödinger-Heisenberg approaches, and robust logging for research and development.

Methodology

The authors introduce Propaq, a Python package implemented with a high-performance Rust backend and a Python frontend. The core methodology relies on Heisenberg propagation, where an observable $O$ is back-propagated through a circuit $C$ to compute the expectation value $\text{Tr}[(C^\dagger O C)\rho]$ .

Key methodological components include:

Arbitrary Basis Support: Unlike many existing tools restricted to Pauli or Majorana bases, Propaq allows users to define custom operator bases by subclassing abstract classes. This requires implementing specific methods for basis element products, branching under circuit operations, operator weight calculation, diagonal expectations, and gate representation in terms of basis generators.
Hybrid Simulation: Propaq implements a hybrid Schrödinger-Heisenberg algorithm. The circuit is partitioned at index $k$ ; the initial state is evolved forward to $k$ using a Schrödinger method (currently supporting Matrix Product States), and the observable is back-propagated from the end of the circuit to $k$ using Heisenberg propagation. The final result is computed via overlap calculations between the forward-evolved state and the back-propagated observable.
Truncation and Noise: To manage the exponential growth of terms, Propaq employs truncation strategies (coefficient, weight, and term budget) and supports noisy simulation. Noise channels, such as depolarizing noise, are implemented to damp coefficients, potentially reducing simulation complexity.
Optimizations:
- Clifford Deferral: Clifford gates are deferred using a stabilizer tableau, applying their adjoint action in bulk only when a non-Clifford gate is encountered.
- Symbolic/Surrogate Propagation: For variational algorithms, Propaq can symbolically propagate operators to create a surrogate model, allowing for rapid evaluation of expectation values across many parameter settings without re-simulating the circuit.
Implementation Details: The backend uses Rust with rayon for multithreading and pyo3 for Python bindings. It utilizes specialized data structures like Bitset, BasisString (for compile-time optimization of fixed-width bitmasks), and OperatorIndex (to efficiently track non-trivial operations) to minimize memory overhead and maximize cache locality.

Key Contributions

Propaq Package: A performant Python package for Heisenberg propagation that matches or exceeds the speed of state-of-the-art tools (e.g., PauliPropagation.jl, MajoranaPropagation.jl, monoprop, pauli-prop).
Arbitrary Basis Propagation: Propaq is the first library to natively support Heisenberg propagation in user-defined bases, demonstrated by a Weyl-Heisenberg propagation example for qudits (Potts model).
Hybrid Schrödinger-Heisenberg Simulation: The package introduces a hybrid simulation mode that partitions the circuit, offering a space-time tradeoff for specific problems.
Advanced Logging and Analysis: Propaq provides detailed logging tools to track simulation accuracy, runtime, memory usage, and truncation statistics (e.g., discarded $L_1$ coefficient norms), aiding in the tuning of truncation parameters.
Noise and Extrapolation: Native support for noisy simulation and extrapolation methods (Zero-Cutoff Extrapolation and Zero-Noise Extrapolation) to improve accuracy and mitigate truncation errors.
Plugin Architecture: A plugin system allows users to implement custom noise and truncation policies in Rust, C, or AOT-compiled Julia via shared libraries, bypassing Python overhead.

Results

The authors present benchmarks comparing Propaq against pauli-prop, pyrauli, PauliPropagation.jl, MajoranaPropagation.jl, and monoprop:

Runtime: In benchmarks involving Trotterized time evolution of 2D Transverse-Field Ising Models (TFIM) and Hubbard models, Propaq performs similarly to monoprop and significantly outperforms other packages, achieving speedups approaching 100x for large Trotter steps (where term counts reach billions).
Memory Usage: Propaq demonstrates memory usage comparable to or better than state-of-the-art packages, maintaining similar peak memory usage across different backends when using the same coefficient cutoffs.
Scalability: Propaq scales effectively with thread count, showing a 10-20x speedup when moving from 1 to 64 threads on a 2D TFIM simulation.
Accuracy: Examples demonstrate that Propaq can reproduce exact diagonalization results for short times and that extrapolation methods (ZCE and ZNE) can recover values closer to exact results than truncated simulations alone.
Hybrid Efficiency: The hybrid simulation approach is shown to be viable, with the paper noting it as a unique feature among Heisenberg simulators.

Significance

The paper claims that Propaq is a significant tool for quantum information researchers and software developers due to its combination of high performance and flexibility. Its primary significance lies in:

Enabling New Research Directions: By supporting arbitrary bases, it allows researchers to explore optimal bases for specific problems (e.g., qudit systems) which was previously difficult or impossible with existing tools.
Facilitating Hybrid Approaches: The hybrid Schrödinger-Heisenberg capability offers a new avenue for simulating circuits that are too large for pure state evolution but too complex for pure Heisenberg propagation.
Research Infrastructure: The detailed logging, truncation analysis, and plugin architecture provide a robust framework for analyzing simulation accuracy and developing custom propagation engines.
Performance: It establishes a new benchmark for speed in Heisenberg propagation, making large-scale classical simulations of quantum circuits more feasible.

The authors conclude that Propaq is intended to be a community resource to enable fast simulation and future research in (hybrid) Heisenberg propagation, addressing open questions regarding basis selection and circuit partitioning.