-# MQT YAQS — Scalable simulation and characterization for open systems, noisy circuits, and realistic hardware
+# MQT YAQS — Simulation and characterization of quantum systems and their environments
-MQT YAQS (pronounced "yaks" like the animals) is a Python library designed for
-**scalable, computationally efficient** simulation and characterization of open
-quantum dynamics, noisy quantum circuits, and hardware-realistic device models.
-MQT YAQS uses state-of-the-art techniques in these areas such as parallelized
-trajectories, tensor network compression, and problem-size-appropriate backends
-wherever possible (see [Cite This](#cite-this)). It is part of the
+MQT YAQS (pronounced "yaks") is a Python library for simulating quantum systems
+and studying their interaction with the environment. It supports analog
+evolution, noisy quantum circuits, circuit equivalence checking, and
+characterization of environmental memory and noise models. It is part of the
[_Munich Quantum Toolkit (MQT)_](https://mqt.readthedocs.io).
+YAQS primarily uses tensor networks and quantum trajectories, with statevector
+and density matrix backends available for smaller simulations. Tensor network
+simulation cost depends strongly on entanglement and accuracy settings, which
+can make systems far larger than those accessible to statevector or density
+matrix methods practical to simulate. We encourage you to explore larger system
+sizes with our package, as we have put thought and effort into making these
+methods scalable.
+
+YAQS selects simulation methods from your inputs and provides presets that
+balance speed and numerical accuracy. You can get started without tuning every
+numerical setting, while still having control over individual settings when
+needed.
+
-## Getting Started
-
-`mqt.yaqs` is available via [PyPI](https://pypi.org/project/mqt.yaqs/).
-
-```console
-uv pip install mqt.yaqs
-```
-
-### Simulation
-
-Noisy analog Hamiltonian simulation
-([guide](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/analog_simulation.html)):
-
-```python
-from mqt.yaqs import AnalogSimParams, Hamiltonian, NoiseModel, Observable, Simulator, State
-
-sim = Simulator(show_progress=False)
-state = State(length=3, initial="zeros")
-H = Hamiltonian.ising(length=3, J=1.0, g=0.5)
-noise = NoiseModel([{"name": "lowering", "sites": [i], "strength": 0.05} for i in range(3)])
-params = AnalogSimParams(
- observables=[Observable("z", sites=0)],
- elapsed_time=0.5,
- dt=0.1,
- preset="fast",
- num_traj=8,
-)
-print(sim.run(state, H, params, noise).expectation_values[0][-1])
-```
-
-Noisy digital circuit simulation
-([guide](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/circuit_observables.html)):
-
-```python
-from qiskit.circuit import QuantumCircuit
-
-from mqt.yaqs import NoiseModel, Observable, Simulator, State, DigitalSimParams
-
-circuit = QuantumCircuit(3)
-circuit.h(0)
-circuit.cx(0, 1)
-circuit.cx(1, 2)
-noise = NoiseModel([{"name": "lowering", "sites": [i], "strength": 0.05} for i in range(3)])
-params = DigitalSimParams(observables=[Observable("z", sites=0)], preset="fast", num_traj=8)
-result = Simulator(show_progress=False).run(State(3, initial="zeros"), circuit, params, noise)
-print(result.expectation_values[0])
-```
-
-### Characterization
-
-Environmental memory characterization
-([guide](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/characterization.html)):
+## Cite This
-```python
-from mqt.yaqs import AnalogSimParams, Hamiltonian, MemoryCharacterizer
+If you use YAQS in academic work, please cite both the
+[YAQS software](#yaqs-software) and the [MQT Handbook](#mqt-handbook). Also cite
+the papers for the methods you use.
-ham = Hamiltonian.ising(length=3, J=1.0, g=0.5)
-params = AnalogSimParams(dt=0.1)
-result = MemoryCharacterizer(show_progress=False).characterize(
- ham,
- params,
- num_interventions=4,
- cut=2,
- n_pasts=4,
- n_futures=4,
-)
-print(result.summary())
-```
+### YAQS software
-Noise model characterization
-([guide](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/digital_twin.html)):
+We expect to replace this software citation with a dedicated YAQS publication.
+Please use the citation below for now.
-```python
-import numpy as np
-
-from mqt.yaqs import AnalogSimParams, Hamiltonian, NoiseCharacterizer, NoiseModel, Observable, State
-
-n = 2
-ham = Hamiltonian.ising(length=n, J=1.0, g=2.0)
-state = State(n, initial="zeros")
-observables = [Observable("z", sites=s) for s in range(n)]
-params = AnalogSimParams(observables=observables, elapsed_time=0.5, dt=0.1, sample_timesteps=True)
-reference = NoiseModel([{"name": "pauli_z", "sites": [s], "strength": 0.1} for s in range(n)])
-guess = NoiseModel([{"name": "pauli_z", "sites": [s], "strength": 0.3} for s in range(n)])
-result = NoiseCharacterizer(show_progress=False).characterize(
- ham,
- params,
- init_state=state,
- init_guess=guess,
- observables=observables,
- reference_model=reference,
- x_low=np.zeros(n),
- x_up=np.full(n, 0.5),
- max_iter=30,
- popsize=6,
- seed=0,
-)
-print(result.optimal_model)
+```bibtex
+@misc{YAQS,
+ author = {Aaron Sander},
+ title = {{YAQS}: Yet Another Quantum Simulator},
+ year = {2025},
+ howpublished = {\url{https://github.com/munich-quantum-toolkit/yaqs}}
+}
```
-**Documentation:**
-[Quickstart](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/quickstart.html)
-·
-[Analog simulation](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/analog_simulation.html)
-·
-[Circuit observables](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/circuit_observables.html)
-·
-[Environmental memory](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/characterization.html)
-·
-[Noise characterization](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/digital_twin.html)
-· [Full guide](https://mqt.readthedocs.io/projects/yaqs)
-
-## System Requirements
+### MQT Handbook
-MQT YAQS can be installed on all major operating systems with all
-[officially supported Python versions](https://devguide.python.org/versions/).
-Building (and running) is continuously tested under Linux, macOS, and Windows
-using the
-[latest available system versions for GitHub Actions](https://github.com/actions/runner-images).
+[[8]](https://doi.org/10.1109/QSW62656.2024.00013) R. Wille et al. The MQT
+Handbook: A Summary of Design Automation Tools and Software for Quantum
+Computing. _IEEE International Conference on Quantum Software (QSW)_ (2024).
-## Cite This
-
-Please cite the work that best fits your use case.
-
-### Peer-Reviewed Research
+### Research publications
When citing the underlying methods and research, please reference the most
-relevant peer-reviewed publications from the list below:
+relevant publications from the list below:
[[1]](https://www.nature.com/articles/s41467-025-66846-x) A. Sander, M.
Fröhlich, M. Eigel, J. Eisert, P. Gelß, M. Hintermüller, R. M. Milbradt, R.
@@ -262,23 +278,17 @@ Eisert, M. Fröhlich, T. Peham, R. Wille. Computational regimes in
matrix-product-state-based quantum trajectory simulations.
_arXiv:2606.13779 (2026)._
-### The Munich Quantum Toolkit (the project)
+[[5]](https://arxiv.org/abs/2607.01323) M. Fröhlich, A. Sander, M. Eigel, R.
+Wille, M. Hintermüller. Noisy quantum circuit simulation with the tensor jump
+method. _arXiv:2607.01323 (2026)._
-When discussing the overall MQT project or its ecosystem, cite the MQT Handbook:
+[[6]](https://arxiv.org/abs/2608.24668) A. R. Ramos Ramos, M. Fröhlich, A.
+Sander, R. Wille, M. Eigel, P. Gelß, S. Pokutta. Scalable Lindblad Noise
+Learning via Stochastic Tensor-Network Simulation. _arXiv:2608.24668 (2026)._
-```bibtex
-@inproceedings{mqt,
- title = {The {{MQT}} Handbook: {{A}} Summary of Design Automation Tools and Software for Quantum Computing},
- shorttitle = {{The MQT Handbook}},
- author = {Wille, Robert and Berent, Lucas and Forster, Tobias and Kunasaikaran, Jagatheesan and Mato, Kevin and Peham, Tom and Quetschlich, Nils and Rovara, Damian and Sander, Aaron and Schmid, Ludwig and Schoenberger, Daniel and Stade, Yannick and Burgholzer, Lukas},
- year = 2024,
- booktitle = {IEEE International Conference on Quantum Software (QSW)},
- doi = {10.1109/QSW62656.2024.00013},
- eprint = {2405.17543},
- eprinttype = {arxiv},
- addendum = {A live version of this document is available at \url{https://mqt.readthedocs.io}}
-}
-```
+[[7]](https://arxiv.org/abs/2608.16994) M. Fröhlich, R. M. Milbradt, M. Eigel,
+A. Sander, R. Wille, C. B. Mendl. Basis-update and Galerkin time integration in
+canonical matrix-product-state form. _arXiv:2608.16994 (2026)._
---
diff --git a/docs/references.md b/docs/references.md
index 5bfad2532..411b90251 100644
--- a/docs/references.md
+++ b/docs/references.md
@@ -1,19 +1,36 @@
# References
-MQT YAQS implements algorithms from peer-reviewed research. **When you use MQT
-YAQS in academic work, please cite the publications that correspond to the
-features you use:**
+MQT YAQS implements algorithms from research publications and preprints. When
+you use the library in academic work, please cite both the **YAQS software**
+{footcite:p}`YAQS` and the **MQT Handbook** {footcite:p}`mqt`.
+
+We expect to replace the YAQS software citation with a dedicated YAQS
+publication. Please use the current software citation for now.
+
+Also cite the research papers for the methods you use:
- {footcite:p}`sander2025_TJM` for open **analog** system simulation (tensor
jump method),
- {footcite:p}`sander2025_CircuitTDVP` for **digital circuit** simulation,
-- {footcite:p}`sander2025_EquivalenceChecking` for **equivalence checking**, and
+- {footcite:p}`froehlich2026_NoisyCircuitTJM` for **noisy quantum circuit**
+ simulation,
+- {footcite:p}`sander2025_EquivalenceChecking` for **equivalence checking**,
- {footcite:p}`sander2026_computationalregimes` for **trajectory unravellings**
- and their computational trade-offs.
+ and their computational trade-offs,
+- {footcite:p}`ramos2026_NoiseLearning` for **noise characterization**, and
+- {footcite:p}`froehlich2026_BUG` for
+ **basis-update and Galerkin (BUG) time integration**.
Representative BibTeX entries:
```bibtex
+@misc{YAQS,
+ author = {Aaron Sander},
+ title = {{YAQS}: Yet Another Quantum Simulator},
+ year = {2025},
+ howpublished = {\url{https://github.com/munich-quantum-toolkit/yaqs}}
+}
+
@article{sander2025_TJM,
title = {Large-scale stochastic simulation of open quantum systems},
author = {Sander, Aaron and Fr\"{o}hlich, Maximilian and Eigel, Martin and Eisert, Jens and Gel\ss{}, Patrick and Hinterm\"{u}ller, Michael and Milbradt, Richard M. and Wille, Robert and Mendl, Christian B.},
@@ -51,19 +68,39 @@ Representative BibTeX entries:
archiveprefix = {arXiv},
primaryclass = {quant-ph},
}
-```
-MQT YAQS is part of the Munich Quantum Toolkit, which is described in
-{cite:p}`mqt`.
+@misc{froehlich2026_NoisyCircuitTJM,
+ title = {Noisy quantum circuit simulation with the tensor jump method},
+ author = {Maximilian Fr\"{o}hlich and Aaron Sander and Martin Eigel and Robert Wille and Michael Hinterm\"{u}ller},
+ year = {2026},
+ url = {https://arxiv.org/abs/2607.01323},
+ eprint = {2607.01323},
+ archiveprefix = {arXiv},
+ primaryclass = {quant-ph},
+}
+
+@misc{ramos2026_NoiseLearning,
+ title = {Scalable {Lindblad} Noise Learning via Stochastic Tensor-Network Simulation},
+ author = {Alejandro R. Ramos Ramos and Maximilian Fr\"{o}hlich and Aaron Sander and Robert Wille and Martin Eigel and Patrick Gel\ss{} and Sebastian Pokutta},
+ year = {2026},
+ url = {https://arxiv.org/abs/2608.24668},
+ eprint = {2608.24668},
+ archiveprefix = {arXiv},
+ primaryclass = {quant-ph},
+}
+
+@misc{froehlich2026_BUG,
+ title = {Basis-update and {Galerkin} time integration in canonical matrix-product-state form},
+ author = {Maximilian Fr\"{o}hlich and Richard M. Milbradt and Martin Eigel and Aaron Sander and Robert Wille and Christian B. Mendl},
+ year = {2026},
+ url = {https://arxiv.org/abs/2608.16994},
+ eprint = {2608.16994},
+ archiveprefix = {arXiv},
+ primaryclass = {quant-ph},
+}
+```
A full list of references is given below.
```{footbibliography}
-:filter: False
-
-sander2025_TJM
-sander2025_CircuitTDVP
-sander2025_EquivalenceChecking
-sander2026_computationalregimes
-mqt
```
diff --git a/docs/refs.bib b/docs/refs.bib
index 1d949905b..ab66e37f7 100644
--- a/docs/refs.bib
+++ b/docs/refs.bib
@@ -1,3 +1,23 @@
+@misc{froehlich2026_BUG,
+ title = {Basis-update and {Galerkin} time integration in canonical matrix-product-state form},
+ author = {Maximilian Fr\"{o}hlich and Richard M. Milbradt and Martin Eigel and Aaron Sander and Robert Wille and Christian B. Mendl},
+ year = {2026},
+ url = {https://arxiv.org/abs/2608.16994},
+ eprint = {2608.16994},
+ archiveprefix = {arXiv},
+ primaryclass = {quant-ph},
+}
+
+@misc{froehlich2026_NoisyCircuitTJM,
+ title = {Noisy quantum circuit simulation with the tensor jump method},
+ author = {Maximilian Fr\"{o}hlich and Aaron Sander and Martin Eigel and Robert Wille and Michael Hinterm\"{u}ller},
+ year = {2026},
+ url = {https://arxiv.org/abs/2607.01323},
+ eprint = {2607.01323},
+ archiveprefix = {arXiv},
+ primaryclass = {quant-ph},
+}
+
@inproceedings{mqt,
title = {The {{MQT}} Handbook: {{A}} Summary of Design Automation Tools and Software for Quantum Computing},
shorttitle = {The MQT Handbook},
@@ -10,6 +30,16 @@ @inproceedings{mqt
addendum = {A live version of this document is available at \url{https://mqt.readthedocs.io}},
}
+@misc{ramos2026_NoiseLearning,
+ title = {Scalable {Lindblad} Noise Learning via Stochastic Tensor-Network Simulation},
+ author = {Alejandro R. Ramos Ramos and Maximilian Fr\"{o}hlich and Aaron Sander and Robert Wille and Martin Eigel and Patrick Gel\ss{} and Sebastian Pokutta},
+ year = {2026},
+ url = {https://arxiv.org/abs/2608.24668},
+ eprint = {2608.24668},
+ archiveprefix = {arXiv},
+ primaryclass = {quant-ph},
+}
+
@misc{sander2025_CircuitTDVP,
title = {Quantum circuit simulation with a local time-dependent variational principle},
author = {Aaron Sander and Maximilian Fr\"{o}hlich and Mazen Ali and Martin Eigel and Jens Eisert and Michael Hinterm\"{u}ller and Christian B. Mendl and Richard M. Milbradt and Robert Wille},
@@ -64,3 +94,10 @@ @misc{sander2026_computationalregimes
archiveprefix = {arXiv},
primaryclass = {quant-ph},
}
+
+@misc{YAQS,
+ title = {{YAQS}: Yet Another Quantum Simulator},
+ author = {Aaron Sander},
+ year = {2025},
+ howpublished = {\url{https://github.com/munich-quantum-toolkit/yaqs}},
+}
+ 
@@ -32,41 +44,143 @@ wherever possible (see [Cite This](#cite-this)). It is part of the
## Key Features
-- **Analog simulation**: Large-scale open-system and unitary time evolution
- using parallelized quantum trajectories when a noise model is attached [1]
- (trajectory guidance [4]).
-- **Digital circuit simulation**: Noisy circuits at scale, final and mid-circuit
- observables, shot-based readout, and OpenQASM 2 inputs [3]
- (`uv pip install mqt-yaqs[qasm3]` for OpenQASM 3).
-- **Digital–analog simulation**: Compose analog evolution and digital operations
- in one ordered program with automatic state handoff, program-wide observable
- traces, and continuous stochastic trajectories across segment boundaries
- ([guide](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/digital_analog_simulation.html)).
-- **Equivalence checking**: Scalable comparison of quantum circuits [2].
-- **Process characterization**: Quantify non-Markovian memory in multi-time
- quantum processes, how much temporal history a process retains, with exact
- reference checks where needed
- ([guide](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/characterization.html)).
-- **Process tensor surrogates**: Train a causal Transformer surrogate for fast
- prediction of non-Markovian response to local interventions and measurement
- over time
- ([guide](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/characterization.html)).
-- **Noise model characterization**: Fit Markovian Lindblad jump rates from
- observable dynamics
- ([guide](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/digital_twin.html)).
-- **Hardware-oriented modeling**: Realistic noise models including Gaussian and
- other strength distributions, plus hardware dynamics such as
- transmon–resonator systems, and heterogeneous site dimensions
- ([examples](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/realistic_noise_models.html)).
-- **Multiple backends**: Monte Carlo wavefunction and master equation evolution
- are available for analog simulation on smaller systems, alongside the scalable
- MPS trajectory path.
+Analog, digital, and analog–digital simulations support both noiseless and noisy
+dynamics.
+
+| Capability | What users can do |
+| ----------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------- |
+| [Analog simulation](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/analog_simulation.html) | Simulate large quantum systems using tensor networks and Monte Carlo methods. |
+| [Digital simulation](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/circuit_observables.html) | Simulate quantum circuits using tensor networks and sample measurement outcomes. |
+| [Analog–digital simulation](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/digital_analog_simulation.html) | Combine analog evolution and quantum circuits in one simulation. |
+| [Equivalence checking](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/equivalence_checking.html) | Check whether two quantum circuits behave the same, measure differences, and study the effects of noise on compilation. |
+| [Environmental memory](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/characterization.html) | Study a system's environment and understand memory effects from past interactions. |
+| [Surrogate models](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/memory_surrogate.html) | Train models to predict dynamics that depend on past interactions (non-Markovian dynamics; requires PyTorch). |
+| [Noise characterization](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/digital_twin.html) | Learn a system's noise model from its observed dynamics. |
+| [Hardware modeling](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/hamiltonians.html) | Use device Hamiltonians together with realistic noise models. |
If you have any questions, feel free to create a
[discussion](https://github.com/munich-quantum-toolkit/yaqs/discussions) or an
[issue](https://github.com/munich-quantum-toolkit/yaqs/issues) on
[GitHub](https://github.com/munich-quantum-toolkit/yaqs).
+## Getting Started
+
+MQT YAQS requires **Python 3.11 or newer** and runs on Linux, macOS, and
+Windows. Install it from [PyPI](https://pypi.org/project/mqt.yaqs/) in a virtual
+environment using `uv`:
+
+```console
+uv pip install mqt.yaqs
+```
+
+Or use `pip`:
+
+```console
+python -m pip install mqt.yaqs
+```
+
+See the
+[installation guide](https://mqt.readthedocs.io/projects/yaqs/en/latest/installation.html)
+for environment setup and development installation.
+
+Optional extras support
+[OpenQASM 3 input](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/circuit_observables.html)
+(`qasm3`) and
+[surrogate training and prediction](https://mqt.readthedocs.io/projects/yaqs/en/latest/examples/memory_surrogate.html)
+(`torch`). The linked guides include installation steps.
+
+### Noisy analog evolution
+
+Evolve a 50-site Ising chain with local damping and print the final mean ⟨Z₀⟩.
+
+```python
+from mqt.yaqs import AnalogSimParams, Hamiltonian, NoiseModel, Observable, Simulator, State
+
+if __name__ == "__main__":
+ length = 50
+ state = State(length, initial="zeros")
+ hamiltonian = Hamiltonian.ising(length, J=1.0, g=0.5)
+ noise = NoiseModel([{"name": "lowering", "sites": [site], "strength": 0.05} for site in range(length)])
+ params = AnalogSimParams(
+ observables=[Observable("z", sites=0)],
+ elapsed_time=1.0,
+ dt=0.1,
+ )
+ simulator = Simulator()
+ result = simulator.run(state, hamiltonian, params, noise)
+ print(f"Final mean