Dual Unitary Quantum Circuits

• Dual-unitary quantum circuits are exactly solvable models defined by two-site gates that remain unitary under both time and spatial transformations.
• They enable exact computation of dynamical observables such as entanglement growth, operator spreading, and correlation functions via tensor network techniques.
• Extensions like generalized dual-unitary and biunitary circuits reveal rich phenomena including quantum chaos, thermalization transitions, and emergent solitonic behavior.

Dual unitary quantum circuits are exactly solvable models of discrete quantum dynamics in one or more spatial dimensions, characterized by local two-site gates that are unitary both in the time direction and under a specific spacetime duality transformation—meaning they remain unitary when the inputs and outputs associated with spatial and temporal directions are exchanged. This algebraic structure leads to powerful analytical tractability for critical dynamical properties, including operator spreading, entanglement growth, correlation functions, and thermalization. Dual unitarity enables a direct connection between dynamical quantum many-body systems and exactly computable tensor network structures, and dual-unitary circuits have become central paradigms for exploring both generic and highly structured quantum chaos in interacting lattice systems.

1. Definition and Core Properties

A dual-unitary quantum circuit is a discrete-time, brickwork-structured arrangement of two-site gates $U$ acting on a one-dimensional or higher-dimensional lattice. The dual-unitarity condition requires that the gate $U$, acting on $\mathcal{H}_1 \otimes \mathcal{H}_1$, satisfies not just ordinary unitarity

$U U^\dagger = U^\dagger U = \mathbb{I},$

but also “dual unitarity”: the reshaped or “dual” gate $\widetilde{U}$, defined via index rearrangement (e.g., $$\langle k l|\widetilde{U}|i j\rangle = \langle l j|U|k i\rangle$$), is itself unitary,

$$\widetilde{U} \widetilde{U}^\dagger = \widetilde{U}^\dagger \widetilde{U} = \mathbb{I}.$$

This duality property imposes strong constraints on the structure of $U$ and, by extension, on the global circuit dynamics. As a consequence, dual-unitary circuits admit exact computation of many dynamical observables and act as “solvable” models for quantum thermalization, transport, and information scrambling.

2. Solvable Initial States and Analytic Tractability

Exact analytical results in dual-unitary circuits typically require initial states with particular compatibility conditions. A central class are “solvable” matrix product states (MPS), characterized by a two-site shift-invariant representation and a solvability condition ensuring that a key transfer matrix’s leading eigenvector is a simple product state. For a two-site shift-invariant MPS $$|\Psi_0^L\rangle = \sum_{i_1,\dots,i_{2L}} \mathrm{tr}[A^{(i_1)}B^{(i_2)}\cdots] |i_1,\dots,i_{2L}\rangle$$, solvability is realized if there exists a bond-space matrix $S$ such that

$$\sum_{k=1}^d (A^i B^k) S (A^j B^k)^\dagger = \frac{1}{d} \delta_{ij} S.$$

For such initial states, the time evolution of local observables, correlation functions, and entanglement can be computed exactly by tensor network contraction, benefitting from the factorization imposed by the dual-unitarity and solvability conditions (Piroli et al., 2019).

3. Operator Spreading and Light-Cone Structure

Under dual-unitary evolution, the support of initially local operators grows strictly ballistically at the “speed of light” set by the circuit geometry. This is reflected in a sharply defined light-cone structure for two-point spatiotemporal correlation functions: outside the causal light cone, correlations vanish exactly, while along the light-cone edge they may display model-dependent behaviors, from persistent oscillations (integrable cases) to exponential decay (ergodic cases) (Piroli et al., 2019). Explicitly, the dynamical two-point function for local operators $\sigma^\alpha$ and $\sigma^\beta$ takes the form

$$\mathcal{C}^{(\alpha,\beta)}(j,r,t) = \delta_{r \bmod 2, 1} \, \delta_{j-t \bmod 2, 1} \, \widetilde{\mathcal{C}}^{(\alpha,\beta)}(r,t),$$

where $\widetilde{\mathcal{C}}$ vanishes outside the light cone, and its value along the edge is set by eigenmaps determined by the circuit’s local gate structure.

Operator dynamics can often be analyzed via transfer matrices associated to the gate and initial state; in cases where the transfer matrix is non-diagonalizable (i.e., at an exceptional point), the long-time decay of correlations is polynomially modified, not purely exponential (Hu et al., 12 Jun 2024).

4. Entanglement Spreading and Membrane Picture

Dual-unitary circuits are notable for universal and exactly computable entanglement growth. For solvable initial states, the Rényi entropies of a connected region $A$ of length $\ell$ grow linearly at $t < \ell/2$, saturating to a maximal value at late times:

$$S_A^{(n)}(t) = \min(2t, \ell) \log d + \frac{1}{1-n} \log \left[\mathrm{tr}(\mathbb{O}_{\ell-2t}^{(n)})\right],$$

with $\mathbb{O}_x^{(n)}$ a block determined by the initial state (Piroli et al., 2019).

An alternative and general perspective on entanglement dynamics is the “entanglement membrane” (or minimal cut) approach, where the entropy is given by minimizing an action over a membrane in spacetime characterized by a “line tension” function. In dual-unitary and generalized dual-unitary (DU2) circuits, the line tension is

$$\mathcal{E}_n(v) = |v| + \frac{1 - |v|}{2} \cdot \frac{\log n_\Lambda}{\log d},$$

where $v$ is the membrane’s slope and $n_\Lambda$ parameterizes the isometry rank (Foligno et al., 2023). In pure dual-unitary circuits $n_\Lambda = d^2$ and entanglement growth is maximal; under more general (DU2) conditions, entanglement spreads more slowly even though operator spreading (the “butterfly velocity”) remains maximal.

5. Thermalization, Ergodicity, and Prethermalization

Dual-unitary circuits allow precise classification of their ergodic properties. Ergodic, mixing circuits exhibit thermalization of any finite subsystem to the infinite-temperature state after time proportional to subsystem size, independent of the presence of conserved quantities (Piroli et al., 2019, Claeys et al., 2020). Non-ergodic cases can be constructed by introducing block-diagonal structure into the local gates, yielding circuits with persistent nonthermal steady-states describable by a generalized Gibbs ensemble. Perturbations can induce a controlled crossover: non-ergodic circuits exhibit long-lived prethermalization plateaux in local observables before ultimately thermalizing on timescales set by the square of the perturbation strength (Claeys et al., 2020).

6. Conservation Laws, Solitons, and Many-body Scars

Conserved densities and solitonic operators play a critical role in classifying non-ergodic behavior. In dual-unitary circuits, any width-$w$ conserved density corresponds one-to-one to a $w$-site soliton: an operator that is simply translated (possibly up to a phase) along the causal light cone by the dynamics (Holden-Dye et al., 2023). Explicit constructions exist both as products of elementary solitons (often related to simple local symmetries) and as nontrivial fermionic operators (constructed via Jordan–Wigner transformations).

Importantly, certain dual-unitary circuits can host “quantum many-body scars”—zero- or low-entropy eigenstates coexisting with a broadly thermalizing spectrum—by embedding invariant subspaces (using projector “embedding” constructions) (Logarić et al., 2023). These scarred states fail to thermalize in an otherwise maximally chaotic setting.

7. Structural Stability, Generalizations, and Geometric Constructions

The solvability of dual-unitary circuits is robust to moderate perturbations of the gate set. Perturbed circuits, analyzed via efficient “path-sum” or “skeleton diagram” expansions, retain much of the analytic control, and reveal how deviations from dual unitarity broaden the light-cone and modify the decay of correlations (Kos et al., 2020).

Dual-unitary principles have been extended in several directions:

• Generalized dual-unitary circuits (DU2, DU3, etc.) relax the dual unitarity condition to isometry constraints on blocks of gates, producing richer entanglement and correlation behaviors, including submaximal entanglement velocities and more intricate spatial-temporal spreading (Foligno et al., 2023, Yu et al., 2023).
• Circuits based on biunitary (and triunitary) connections, often realized via geometric arrangements on lattices such as the Kagome lattice, naturally generalize dual-unitary solvability, admit multiple “arrows of time,” and generate models with quantized or tunable entanglement velocities (Rampp et al., 12 Nov 2024).
• Circuits in random geometries, in which spatial and/or temporal structure is randomized, retain exact solvability in their correlation functions’ higher moments and extend the class of exactly solvable models (Kasim et al., 2022).

These extensions allow investigation of nontrivial non-ergodic dynamics—including prethermalization, localization, dualities, and quantum scars—and are also connected to combinatorial and algebraic structures such as noncrossing partitions, free independence, and random matrix theory (Chen et al., 25 Sep 2024).

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Table: Representative Features of Dual-Unitary and Generalized Circuits

Structure	Key Property	Notable Feature
Dual-unitary (DU)	Gate and dual are both unitary	Maximal butterfly and entanglement velocities
Generalized DU (DU2)	Isometric only on subspace, per multi-gate block	Submaximal entanglement velocity
Biunitary/Triunitary	Multi-axial unitarity via geometric arrangement	Multiple light-cone directions, quantized $v_E$
Perturbed DU	Small deviations from exact dual unitarity	Robust solvability; broadened light-cone

8. Applications and Broader Significance

Dual-unitary circuits provide exact benchmarks and minimal models for understanding quantum chaos, operator spreading, thermalization, entanglement dynamics, and measurement-induced transitions in many-body quantum systems. They underpin schemes for measurement-based quantum computation robust to symmetry-respecting perturbations (Stephen et al., 2022), models of discrete holography and spacetime geometry (Masanes, 2023), and provide tractable platforms for investigating the emergence of free probability behavior and deep scrambling (Chen et al., 25 Sep 2024).

Their analytic tractability extends to quantum circuits with projective measurements (enabling exact computation of entanglement in “hybrid” models) (Claeys et al., 2022) and settings with inhomogeneous impurities, where discrepancies between effective descriptions (such as the quasiparticle and membrane approaches) can be precisely diagnosed (Fraenkel et al., 4 Oct 2024).

The development of dual-unitary and generalized dual-unitary circuits continues to have major impacts across quantum information, statistical mechanics, and the theory of quantum simulation, offering a bridge between exactly solvable models and generic chaotic quantum dynamics.