Unitary Circuit Games

• Unitary circuit games are research constructions where unitary operations act as strategic moves on quantum systems, controlling entanglement and energy payoffs.
• They encompass multiple frameworks—from zero-sum shared-register models and entangler–disentangler dynamics to cooperative quantization and higher-order game semantics—that yield concrete measures like half-chain entropy and quantum Shapley values.
• The studies reveal phase transitions between area- and volume-law entanglement, demonstrating that disentangling power is a representation-dependent resource with implications for scalable quantum computation.

Searching arXiv for recent and directly relevant papers on unitary circuit games and related formulations. arXiv search query: "unitary circuit games entanglement transitions matchgates game semantics higher-order unitary quantum computation" Unitary circuit games are research constructions in which unitary transformations function as strategic moves on quantum states, circuits, or higher-order computational objects. The literature includes zero-sum games on a shared many-body register with energy as payoff (Erbanni et al., 2023), one-dimensional entangler–disentangler dynamics in which local gates compete to produce volume-law or area-law steady states (Morral-Yepes et al., 2023, Morral-Yepes et al., 7 Jul 2025), cooperative-game quantization schemes based on Quantum Register Algebra and quantum Shapley values (Eryganov et al., 2023), and a game-semantics model of higher-order unitary quantum computation formulated as a symmetric monoidal closed category (Abramsky et al., 2024). Across these strands, the common element is the treatment of unitary operations as moves constrained by locality, entangling capacity, or semantic type, while the objectives range from energy optimization to entanglement suppression, coalition payoff allocation, and denotational representation.

1. Research scope and principal formulations

Current work on unitary circuit games does not define a single canonical formalism. Instead, the literature develops several distinct but related frameworks, each centered on unitary control under game-like interaction.

Formulation	System	Central quantity
Shared-register zero-sum game	$N$-qubit register	Energy $E=\langle \psi_f|H|\psi_f\rangle$
Entangler–disentangler circuit game	1D chain of $L$ qubits	Half-chain entropy $S_{L/2}$
Cooperative-game quantization	Two- and three-player circuits	Quantum Shapley value $\widetilde\Phi_i$
Higher-order game semantics	Games and strategies in $\G$ or $\UG$	Induced linear map $[\sigma]$

The zero-sum strand studies sequential play on the same register, with one player maximizing and the other minimizing the expectation value of a fixed Hamiltonian (Erbanni et al., 2023). The circuit-dynamical strand studies repeated random application of entangling and disentangling local gates, yielding nonequilibrium steady states with entanglement transitions (Morral-Yepes et al., 2023, Morral-Yepes et al., 7 Jul 2025). The cooperative strand adapts Eisert–Wilkens–Lewenstein-type quantization to coalition games and uses the expected Shapley value with respect to quantum measurement probabilities (Eryganov et al., 2023). The semantic strand treats unitary computation itself as a category of games and strategies expressive enough to realize all finite-dimensional unitaries at base types (Abramsky et al., 2024).

A plausible implication is that the phrase “unitary circuit games” should be read contextually. In some papers it denotes a competitive dynamical model on a lattice, in others a cooperative payoff mechanism, and in others a denotational semantics for unitary higher-order computation.

2. Shared-register zero-sum games and entangling advantage

A formal shared-register game is defined on $N$ qubits with Hilbert space $\mathcal H=(\mathbb C^2)^{\otimes N}$ and Hamiltonian

$E=\langle \psi_f|H|\psi_f\rangle$0

Player $E=\langle \psi_f|H|\psi_f\rangle$1 begins in a known pure product state $E=\langle \psi_f|H|\psi_f\rangle$2, applies $E=\langle \psi_f|H|\psi_f\rangle$3 on a subset of at most $E=\langle \psi_f|H|\psi_f\rangle$4 qubits, and player $E=\langle \psi_f|H|\psi_f\rangle$5, knowing $E=\langle \psi_f|H|\psi_f\rangle$6, applies $E=\langle \psi_f|H|\psi_f\rangle$7 on up to $E=\langle \psi_f|H|\psi_f\rangle$8 qubits. The final state is

$E=\langle \psi_f|H|\psi_f\rangle$9

the energy is measured as $L$0, and the payoffs are $L$1 and $L$2 (Erbanni et al., 2023).

Entangling capability is quantified operationally. One says that $L$3 has a $L$4-qubit entangling advantage if $L$5, in particular $L$6. Any pure state $L$7 prepared by $L$8 has a Schmidt decomposition across $L$9's $S_{L/2}$0-qubit subsystem,

$S_{L/2}$1

where $S_{L/2}$2. The nonzero $S_{L/2}$3 determine how mixed $S_{L/2}$4 is. A simple entanglement-power measure for a unitary $S_{L/2}$5 on $S_{L/2}$6 qubits is

$S_{L/2}$7

with the bound $S_{L/2}$8 (Erbanni et al., 2023).

The origin of the second-mover advantage is explicit when $S_{L/2}$9. In that case, $\widetilde\Phi_i$0 can completely undo $\widetilde\Phi_i$1's action and then passify his register, so that for any $\widetilde\Phi_i$2 on his full support there exists $\widetilde\Phi_i$3 such that

$\widetilde\Phi_i$4

hence

$\widetilde\Phi_i$5

If $\widetilde\Phi_i$6 and $\widetilde\Phi_i$7 acted first, $\widetilde\Phi_i$8 cannot prevent $\widetilde\Phi_i$9 from reaching the global ground state. The analysis yields the saddle-point property

$\G$0

whenever $\G$1, so the second player wins on average (Erbanni et al., 2023).

A larger entangling capacity weakens this advantage. Let $\G$2. By choosing $\G$3 so that the reduced state on $\G$4 has maximal rank $\G$5 when $\G$6, player $\G$7 can enforce

$\G$8

Then $\G$9's best passification can only occupy the $\UG$0 lowest energy levels of

$\UG$1

and

$\UG$2

For $\UG$3, the per-qubit energy admits the closed form

$\UG$4

with two extreme cases singled out in closed form: $\UG$5 The case $\UG$6 corresponds to a completely mixed marginal on $\UG$7, so $\UG$8 can only reach zero energy per qubit (Erbanni et al., 2023).

Several special cases clarify the structure. Absolutely maximally entangled states exist for qubits only at $\UG$9. Whenever $[\sigma]$0, equivalently $[\sigma]$1, $[\sigma]$2 can prepare an AME so that $[\sigma]$3's marginal is full-rank maximally mixed, implying $[\sigma]$4. If no AME exists, such as $[\sigma]$5 or $[\sigma]$6, one can still sample Haar-random $[\sigma]$7; by Page’s theorem the average marginal entropy satisfies $[\sigma]$8, with $[\sigma]$9, and a concentration bound shows that with overwhelming probability $N$0 lies within $N$1 of its mean. For mixed initial product states $N$2, the relevant quantity is ergotropy,

$N$3

and if $N$4 can entangle sufficiently many subsystems before passification then

$N$5

so a first mover with the larger entangler can extract strictly more work than any sequence of local operations (Erbanni et al., 2023).

A worked example at total size $N$6 with $N$7 and $N$8 illustrates the mechanism. Player $N$9 prepares an AME on qubits $\mathcal H=(\mathbb C^2)^{\otimes N}$0 and a Bell pair on $\mathcal H=(\mathbb C^2)^{\otimes N}$1, so that every two-qubit marginal available to $\mathcal H=(\mathbb C^2)^{\otimes N}$2 has rank $\mathcal H=(\mathbb C^2)^{\otimes N}$3 with eigenvalues $\mathcal H=(\mathbb C^2)^{\otimes N}$4. $\mathcal H=(\mathbb C^2)^{\otimes N}$5's optimal two-plus-one-qubit unitaries then yield $\mathcal H=(\mathbb C^2)^{\otimes N}$6, hence $\mathcal H=(\mathbb C^2)^{\otimes N}$7 (Erbanni et al., 2023).

3. One-dimensional entangler–disentangler dynamics

A second major use of the term concerns repeated local competition on a chain of qubits. The system is an open chain of $\mathcal H=(\mathbb C^2)^{\otimes N}$8 qubits with Hilbert space $\mathcal H=(\mathbb C^2)^{\otimes N}$9. At each elementary update, a bond $E=\langle \psi_f|H|\psi_f\rangle$00 is chosen uniformly at random. With probability

$E=\langle \psi_f|H|\psi_f\rangle$01

the disentangler acts, and with probability

$E=\langle \psi_f|H|\psi_f\rangle$02

the entangler acts. A full time step consists of $E=\langle \psi_f|H|\psi_f\rangle$03 such updates. The entangler places a random two-qubit unitary drawn from an allowed ensemble, while the disentangler, using limited local knowledge of the state on the chosen bond, picks a unitary that minimizes the bipartite entanglement entropy across that bond (Morral-Yepes et al., 2023).

The entanglement measure is the bipartite von Neumann entropy

$E=\langle \psi_f|H|\psi_f\rangle$04

with $E=\langle \psi_f|H|\psi_f\rangle$05. A two-qubit gate on bond $E=\langle \psi_f|H|\psi_f\rangle$06 can change $E=\langle \psi_f|H|\psi_f\rangle$07 by at most one unit in the qubit case, and the disentangler selects an allowed local unitary $E=\langle \psi_f|H|\psi_f\rangle$08 so that the post-update entropy across $E=\langle \psi_f|H|\psi_f\rangle$09 is minimal, subject to local consistency with neighboring bipartitions (Morral-Yepes et al., 2023).

Three variants were studied.

Variant	Critical point	Reported critical behavior
Classical discrete height model	$E=\langle \psi_f|H|\psi_f\rangle$10	$E=\langle \psi_f|H|\psi_f\rangle$11, $E=\langle \psi_f|H|\psi_f\rangle$12, $E=\langle \psi_f|H|\psi_f\rangle$13, $E=\langle \psi_f|H|\psi_f\rangle$14
Clifford circuit model	$E=\langle \psi_f|H|\psi_f\rangle$15	$E=\langle \psi_f|H|\psi_f\rangle$16, $E=\langle \psi_f|H|\psi_f\rangle$17, $E=\langle \psi_f|H|\psi_f\rangle$18, $E=\langle \psi_f|H|\psi_f\rangle$19, $E=\langle \psi_f|H|\psi_f\rangle$20
General $E=\langle \psi_f|H|\psi_f\rangle$21 Haar-random circuit	No finite $E=\langle \psi_f|H|\psi_f\rangle$22	Volume law for all $E=\langle \psi_f|H|\psi_f\rangle$23

In the classical discrete-height or RSOS model, quantum entropies are replaced by a nonnegative height field $E=\langle \psi_f|H|\psi_f\rangle$24 with constraints

$E=\langle \psi_f|H|\psi_f\rangle$25

The entangler update is

$E=\langle \psi_f|H|\psi_f\rangle$26

and the disentangler update is

$E=\langle \psi_f|H|\psi_f\rangle$27

Numerically, the model has a continuous transition at $E=\langle \psi_f|H|\psi_f\rangle$28, separating a volume-law phase from an area-law phase. At criticality, $E=\langle \psi_f|H|\psi_f\rangle$29 and the spatial fluctuations $E=\langle \psi_f|H|\psi_f\rangle$30. Dynamically, $E=\langle \psi_f|H|\psi_f\rangle$31 with $E=\langle \psi_f|H|\psi_f\rangle$32 and $E=\langle \psi_f|H|\psi_f\rangle$33. Through a mapping to the stochastic Fredkin chain, one analytically obtains $E=\langle \psi_f|H|\psi_f\rangle$34, consistent with $E=\langle \psi_f|H|\psi_f\rangle$35 (Morral-Yepes et al., 2023).

In the Clifford circuit model, the entangler places a random two-qubit Clifford gate and the disentangler uses local stabilizer information on qubits $E=\langle \psi_f|H|\psi_f\rangle$36 to search a minimal subset of 19 Clifford gates, shown sufficient in the appendix, that maximally reduce the entropy across $E=\langle \psi_f|H|\psi_f\rangle$37. The resulting transition occurs at

$E=\langle \psi_f|H|\psi_f\rangle$38

The reported scaling mirrors the classical model: $E=\langle \psi_f|H|\psi_f\rangle$39, $E=\langle \psi_f|H|\psi_f\rangle$40, growth $E=\langle \psi_f|H|\psi_f\rangle$41, $E=\langle \psi_f|H|\psi_f\rangle$42, and $E=\langle \psi_f|H|\psi_f\rangle$43 at criticality. Deep in the volume-law phase the dynamics recover KPZ scaling with $E=\langle \psi_f|H|\psi_f\rangle$44, while the area-law regime has $E=\langle \psi_f|H|\psi_f\rangle$45 (Morral-Yepes et al., 2023).

The Haar-random $E=\langle \psi_f|H|\psi_f\rangle$46 model behaves differently. The entangler draws each two-qubit gate from Haar measure on $E=\langle \psi_f|H|\psi_f\rangle$47, and the disentangler performs a local continuous optimization over 9 parameters to minimize the von Neumann entropy. The reported numerics show no finite $E=\langle \psi_f|H|\psi_f\rangle$48: for any $E=\langle \psi_f|H|\psi_f\rangle$49, the late-time state is maximally entangled, with

$E=\langle \psi_f|H|\psi_f\rangle$50

as $E=\langle \psi_f|H|\psi_f\rangle$51. The heuristic argument is that Haar-random dynamics rapidly build high-Schmidt-rank multipartite entanglement regions of length $E=\langle \psi_f|H|\psi_f\rangle$52, whose local disentangling time scales as $E=\langle \psi_f|H|\psi_f\rangle$53, while the entangler can grow them linearly in $E=\langle \psi_f|H|\psi_f\rangle$54. The equilibration time diverges exponentially in $E=\langle \psi_f|H|\psi_f\rangle$55 as $E=\langle \psi_f|H|\psi_f\rangle$56 (Morral-Yepes et al., 2023).

This contrast is central to the subject. In restricted models, local unitary disentanglers can drive genuine area-law/volume-law transitions. In fully Haar-random local dynamics, the entangler wins for any nonzero entangling rate. The literature therefore distinguishes sharply between the effectiveness of local unitary “cooling” and the more drastic entanglement suppression associated with non-unitary measurement dynamics (Morral-Yepes et al., 2023).

4. Matchgate realizations and free-fermion entanglement transitions

A later development studies unitary circuit games within matchgate dynamics, equivalently evolutions of non-interacting fermions. The system is an open chain of $E=\langle \psi_f|H|\psi_f\rangle$57 qubits carrying a fermionic Gaussian state. At each time step one bond $E=\langle \psi_f|H|\psi_f\rangle$58 is chosen uniformly, and with probability $E=\langle \psi_f|H|\psi_f\rangle$59 an entangler applies a random two-qubit unitary from an allowed ensemble, while with probability $E=\langle \psi_f|H|\psi_f\rangle$60 a disentangler applies a two-qubit unitary chosen to reduce entanglement according to a specified strategy. Repetition for many time steps yields a steady state whose phase is diagnosed by the half-chain entropy $E=\langle \psi_f|H|\psi_f\rangle$61 (Morral-Yepes et al., 7 Jul 2025).

A nearest-neighbor matchgate on qubits $E=\langle \psi_f|H|\psi_f\rangle$62 has the form

$E=\langle \psi_f|H|\psi_f\rangle$63

and is equivalent via Jordan–Wigner to evolution by a quadratic fermion Hamiltonian. Any pure fermionic Gaussian state is specified by its $E=\langle \psi_f|H|\psi_f\rangle$64 covariance matrix

$E=\langle \psi_f|H|\psi_f\rangle$65

For a bipartition $E=\langle \psi_f|H|\psi_f\rangle$66, the Rényi-$E=\langle \psi_f|H|\psi_f\rangle$67 entropies are determined by the Williamson eigenvalues $E=\langle \psi_f|H|\psi_f\rangle$68 of the reduced $E=\langle \psi_f|H|\psi_f\rangle$69,

$E=\langle \psi_f|H|\psi_f\rangle$70

The circuit representation called Right Standard Form organizes any pure fermionic Gaussian state into at most $E=\langle \psi_f|H|\psi_f\rangle$71 nearest-neighbor matchgates arranged in diagonals. The RSF is optimal in gate count, and generalized Yang–Baxter moves, left–right moves on basis states, and local gate fusion allow any new gate to be absorbed into the RSF structure (Morral-Yepes et al., 7 Jul 2025).

This representation yields a specific disentangling procedure. Given an RSF state and a chosen bond $E=\langle \psi_f|H|\psi_f\rangle$72, one inserts an auxiliary identity gate on bond $E=\langle \psi_f|H|\psi_f\rangle$73 and runs the absorption algorithm. The identity is absorbed by an existing gate, which becomes the target to remove. Reversing the absorption steps produces a two-qubit unitary $E=\langle \psi_f|H|\psi_f\rangle$74 such that

$E=\langle \psi_f|H|\psi_f\rangle$75

where $E=\langle \psi_f|H|\psi_f\rangle$76 has one fewer gate. The disentangling move is then $E=\langle \psi_f|H|\psi_f\rangle$77 on bond $E=\langle \psi_f|H|\psi_f\rangle$78. Repeated application is guaranteed to reduce the total gate count to zero, i.e. to a product state (Morral-Yepes et al., 7 Jul 2025).

The phase structure depends strongly on the allowed gate set and the disentangling rule. For braiding gates, which are the intersection of Clifford gates and matchgates, the local entropy minimizer works perfectly. The critical point is

$E=\langle \psi_f|H|\psi_f\rangle$79

so any finite $E=\langle \psi_f|H|\psi_f\rangle$80 yields an area law. As $E=\langle \psi_f|H|\psi_f\rangle$81, the correlation length diverges as $E=\langle \psi_f|H|\psi_f\rangle$82 with numerically $E=\langle \psi_f|H|\psi_f\rangle$83. At $E=\langle \psi_f|H|\psi_f\rangle$84, entanglement growth is diffusive, $E=\langle \psi_f|H|\psi_f\rangle$85, and the equilibration time scales as $E=\langle \psi_f|H|\psi_f\rangle$86, while the disentangling time scales as $E=\langle \psi_f|H|\psi_f\rangle$87 (Morral-Yepes et al., 7 Jul 2025).

For generic matchgates, the local von Neumann-entropy minimizer does not reveal a clear transition up to $E=\langle \psi_f|H|\psi_f\rangle$88; the numerics are reported as inconclusive. By contrast, the gate disentangler based on RSF gives sharp results. For Rényi-0 entropy, an exact Bell-pair mapping with $E=\langle \psi_f|H|\psi_f\rangle$89 updates yields

$E=\langle \psi_f|H|\psi_f\rangle$90

For $E=\langle \psi_f|H|\psi_f\rangle$91, the steady state has volume law $E=\langle \psi_f|H|\psi_f\rangle$92; for $E=\langle \psi_f|H|\psi_f\rangle$93, it has area law $E=\langle \psi_f|H|\psi_f\rangle$94; and at $E=\langle \psi_f|H|\psi_f\rangle$95 it shows submaximal volume law $E=\langle \psi_f|H|\psi_f\rangle$96. Growth is ballistic in the volume phase with velocity $E=\langle \psi_f|H|\psi_f\rangle$97 and diffusive at criticality (Morral-Yepes et al., 7 Jul 2025).

For the von Neumann entropy with the same gate disentangler, the critical point remains

$E=\langle \psi_f|H|\psi_f\rangle$98

Below criticality,

$E=\langle \psi_f|H|\psi_f\rangle$99

At $L$00, the leading scaling is $L$01, with a fitted subleading term $L$02 and $L$03, while for $L$04 the system is again area law with $L$05 (Morral-Yepes et al., 7 Jul 2025).

These results sharpen the broader entangler–disentangler picture. They show that the existence and location of an entanglement transition are not determined by the competition rate $L$06 alone, but also by algebraic structure of the gate set and by whether the disentangler uses a merely local entropy minimizer or an optimal gate-count compression strategy. A plausible implication is that “disentangling power” in unitary circuit games is best understood as a representation-dependent resource, not simply as a local variational optimization criterion.

5. Cooperative quantization, QRA, and quantum Shapley payoffs

A distinct line of work studies cooperative games by quantizing their circuit representation. The framework is inspired by the Eisert–Wilkens–Lewenstein protocol, modified to represent cooperation between players and extended to $L$07-qubit states. In the two-qubit case, the entangling gate is

$L$08

with matrix form given explicitly in the source, and the initial state is

$L$09

corresponding to “no one cooperates.” Acting on $L$10, this produces

$L$11

which interpolates from product to Bell-type entanglement, with full Bell-type entanglement at $L$12 (Eryganov et al., 2023).

Each player $L$13 chooses a local unitary

$L$14

The classical binary actions are embedded by identifying “Cooperate” with $L$15 and “Defect” with $L$16. The two-player circuit prepares $L$17, applies $L$18, applies local strategy unitaries $L$19, optionally applies $L$20, and finally measures in the computational basis. Algebraically,

$L$21

in the formulation that includes the final disentangler (Eryganov et al., 2023).

The computational formalism is Quantum Register Algebra, built from a real geometric algebra $L$22 with signature $L$23, a pseudo-scalar $L$24, and a Witt basis

$L$25

satisfying

$L$26

With the idempotent

$L$27

basis states are identified as

$L$28

Gates are even multivectors, serial composition is Clifford multiplication, and tensor products require a sign-bookkeeping rule. In particular,

$L$29

while more complex gates such as CNOT, SWAP, and Hadamard have closed-form expressions in the $L$30 language (Eryganov et al., 2023).

Payoffs are assigned through a quantum Shapley value. For final density matrix $L$31, the value for player $L$32 is

$L$33

where $L$34 projects onto computational basis states whose support of “1”s is exactly $L$35. This replaces classical coalition probabilities by quantum measurement probabilities while retaining classical Shapley weights (Eryganov et al., 2023).

The framework also emphasizes symbolic verification. QRA descriptions can be entered into the GAALOPWeb symbolic engine, and the ordering inside

$L$36

is proved commutative by checking

$L$37

for three orderings. The paper describes this as an automated proof of circuit equivalence and presents the construction as a blueprint for scaling to $L$38-player cooperative quantum games (Eryganov et al., 2023).

6. Higher-order game semantics for unitary computation

A more abstract strand replaces concrete players and qubit chains by a category of games whose morphisms represent unitary computation. The basic category $L$39 has objects

$L$40

where $L$41 is a finite set of moves, $L$42 labels each move as Opponent or Player and Question or Answer, $L$43 is a symmetric coherence relation on answer moves, and $L$44 is the set of maximal plays satisfying alternation, well-balanced question–answer structure, existence of maximal $L$45- and $L$46-cliques of answers, and an extension property for partial plays (Abramsky et al., 2024).

From each game one obtains finite-dimensional Hilbert spaces

$L$47

A morphism or strategy $L$48 is a suitable subset of plays in $L$49 satisfying determinacy, receptivity, monotonicity, and $L$50-preservation. Extensional equivalence is equality of the induced linear map

$L$51

Composition is parallel composition plus hiding, or equivalently a trace-operator of a GoI-style construction, and the model proves

$L$52

guaranteeing strict associativity (Abramsky et al., 2024).

The category is symmetric monoidal closed. Tensor product $L$53 is a disjoint sum of moves with Opponent switching between components, yielding

$L$54

The internal hom $L$55 is formed by reversing polarity in $L$56 and imposing the constraint that a maximal play visits $L$57 first and then $L$58. Additives are also defined: $L$59, where Opponent chooses the component, and $L$60, where Player chooses it. The coherence diagrams for units, associators, symmetries, and distributivities commute up to the stated strategy-equivalences (Abramsky et al., 2024).

Base quantum types are encoded directly. The quantum bit is the game

$L$61

so

$L$62

More generally,

$L$63

A strategy $L$64 is called unitary iff $L$65 is a unitary matrix and $L$66 can be written as a finite $L$67-linear combination of reversible partial-injection strategies. These morphisms form a subcategory $L$68 (Abramsky et al., 2024).

Concrete gates appear as strategies. The Hadamard gate is written as

$L$69

with copycat implementing the identity and a NOT-strategy implementing the swap matrix. CNOT on two qubits is represented by a single deterministic reversible strategy whose action on maximal $L$70-cliques is exactly the CNOT truth table. Sequential composition models back-to-back gates, tensor models side-by-side gates on disjoint wires, and the closed structure interprets higher-order circuit constructors such as morphisms

$L$71

The universality theorem states that every unitary operator

$L$72

is realized by some morphism

$L$73

in $L$74 (Abramsky et al., 2024).

This semantic line broadens the meaning of unitary circuit games from explicit multi-agent competition to a structured theory of interactive computation. It shows that “game” can refer not only to strategic conflict or cooperation but also to the combinatorial organization of higher-order unitary processes themselves.