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Disentangling strategies and entanglement transitions in unitary circuit games with matchgates

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Disentangling strategies and entanglement transitions in unitary circuit games with matchgates

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AbstractIn unitary circuit games, two competing parties, an "entangler" and a "disentangler", can induce an entanglement phase transition in a quantum many-body system. The transition occurs at a certain rate at which the disentangler acts. We analyze such games within the context of matchgate dynamics, which equivalently corresponds to evolutions of non-interacting fermions. We first investigate general entanglement properties of fermionic Gaussian states (FGS). We introduce a representation of FGS using a minimal matchgate circuit capable of preparing the state and derive an algorithm based on a generalized Yang-Baxter relation for updating this representation as unitary operations are applied. This representation enables us to define a natural disentangling procedure that reduces the number of gates in the circuit, thereby decreasing the entanglement contained in the system. We then explore different strategies to disentangle the systems and study the unitary circuit game in two different scenarios: with braiding gates, i.e., the intersection of Clifford gates and matchgates, and with generic matchgates. For each model, we observe qualitatively different entanglement transitions, which we characterize both numerically and analytically.Popular summaryQuantum systems consisting of many particles can organize into distinct phases, in the same way that water can be ice, liquid, or steam depending on temperature. In quantum matter, a key quantity that distinguishes phases is entanglement: the degree to which the individual particles are correlated with one another. In a highly entangled phase, knowing the state of one part of the system reveals information about distant parts; in a weakly entangled phase, correlations remain mostly local. A natural question is: which feature determines whether entanglement builds up or is suppressed in a system? One way to probe this is through "unitary circuit games." A quantum circuit is a sequence of elementary operations, so-called gates, applied to a row of quantum bits (qubits), much like a sequence of logical steps in a computation. In a circuit game, two opposing players apply gates to a chain of qubits. The entangler tries to build up quantum correlations across the system; the disentangler tries to destroy them. Depending on how often each player acts, the system settles into either a highly entangled or a weakly entangled phase, with a sharp transition between the two. Importantly, an earlier work showed that the outcome depends not just on the players' rates of play, but on which set of gates they are allowed to use. In this work, we study such games in the setting of matchgate circuits, which describe the physics of non-interacting fermions. A central contribution is the right standard form (RSF): a way of representing any state that can be generated by these circuits, using a representation that is provably optimal in the number of gates. Building on this, we define a "gate disentangler", which is a strategy that reduces the number of gates in the circuit rather than directly minimizing a standard measure of entanglement. This turns out to be more effective. Using it, we find a sharp phase transition between entangled and disentangled phases, demonstrating that successful disentanglement is not simply a matter of how often you act, but depends critically on the structure of the gates available and on choosing the right quantity to optimize.► BibTeX data@article{MorralYepes2026disentangling, doi = {10.22331/q-2026-04-28-2087}, url = {https://doi.org/10.22331/q-2026-04-28-2087}, title = {Disentangling strategies and entanglement transitions in unitary circuit games with matchgates}, author = {Morral-Yepes, Ra{\'{u}}l and Langer, Marc and Gammon-Smith, Adam and Kraus, Barbara and Pollmann, Frank}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2087}, month = apr, year = {2026} }► References [1] Matthew P.A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay. ``Random quantum circuits''. 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[6] Kaito Kobayashi, Benjamin Sappler, and Frank Pollmann, "Holographic Representation of One-Dimensional Many-Body Quantum States via Isometric Tensor Networks", arXiv:2512.11967, (2025). [7] Michael A. Rampp, Suhail A. Rather, and Pieter W. Claeys, "Solvable Quantum Circuits from Spacetime Lattices", arXiv:2512.15871, (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-04-28 10:09:49). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-04-28 10:09:48: Could not fetch cited-by data for 10.22331/q-2026-04-28-2087 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractIn unitary circuit games, two competing parties, an "entangler" and a "disentangler", can induce an entanglement phase transition in a quantum many-body system. The transition occurs at a certain rate at which the disentangler acts. We analyze such games within the context of matchgate dynamics, which equivalently corresponds to evolutions of non-interacting fermions. We first investigate general entanglement properties of fermionic Gaussian states (FGS). We introduce a representation of FGS using a minimal matchgate circuit capable of preparing the state and derive an algorithm based on a generalized Yang-Baxter relation for updating this representation as unitary operations are applied. This representation enables us to define a natural disentangling procedure that reduces the number of gates in the circuit, thereby decreasing the entanglement contained in the system. We then explore different strategies to disentangle the systems and study the unitary circuit game in two different scenarios: with braiding gates, i.e., the intersection of Clifford gates and matchgates, and with generic matchgates. For each model, we observe qualitatively different entanglement transitions, which we characterize both numerically and analytically.Popular summaryQuantum systems consisting of many particles can organize into distinct phases, in the same way that water can be ice, liquid, or steam depending on temperature. In quantum matter, a key quantity that distinguishes phases is entanglement: the degree to which the individual particles are correlated with one another. In a highly entangled phase, knowing the state of one part of the system reveals information about distant parts; in a weakly entangled phase, correlations remain mostly local. A natural question is: which feature determines whether entanglement builds up or is suppressed in a system? One way to probe this is through "unitary circuit games." A quantum circuit is a sequence of elementary operations, so-called gates, applied to a row of quantum bits (qubits), much like a sequence of logical steps in a computation. In a circuit game, two opposing players apply gates to a chain of qubits. The entangler tries to build up quantum correlations across the system; the disentangler tries to destroy them. Depending on how often each player acts, the system settles into either a highly entangled or a weakly entangled phase, with a sharp transition between the two. Importantly, an earlier work showed that the outcome depends not just on the players' rates of play, but on which set of gates they are allowed to use. In this work, we study such games in the setting of matchgate circuits, which describe the physics of non-interacting fermions. A central contribution is the right standard form (RSF): a way of representing any state that can be generated by these circuits, using a representation that is provably optimal in the number of gates. Building on this, we define a "gate disentangler", which is a strategy that reduces the number of gates in the circuit rather than directly minimizing a standard measure of entanglement. This turns out to be more effective. Using it, we find a sharp phase transition between entangled and disentangled phases, demonstrating that successful disentanglement is not simply a matter of how often you act, but depends critically on the structure of the gates available and on choosing the right quantity to optimize.► BibTeX data@article{MorralYepes2026disentangling, doi = {10.22331/q-2026-04-28-2087}, url = {https://doi.org/10.22331/q-2026-04-28-2087}, title = {Disentangling strategies and entanglement transitions in unitary circuit games with matchgates}, author = {Morral-Yepes, Ra{\'{u}}l and Langer, Marc and Gammon-Smith, Adam and Kraus, Barbara and Pollmann, Frank}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2087}, month = apr, year = {2026} }► References [1] Matthew P.A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay. ``Random quantum circuits''. 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Disentangling strategies and entanglement transitions in unitary circuit games with matchgates

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