Infinite-Level Hierarchy of Solvable Quantum Circuits

Abstract: Dual-unitary circuits have emerged as a paradigm of exactly solvable yet non-integrable quantum dynamics. Recently, a generalization of dual unitarity attempting to extend the phenomenology of exactly solvable circuits has been introduced through a hierarchy of conditions, with dual unitarity as the first level. However, beyond the second level the proposed generalized dual-unitary hierarchy ceases to be solvable in the whole spacetime. We present an infinite hierarchy of solvability conditions remedying this problem. These new conditions can be combined with the generalized dual-unitary hierarchy to obtain circuits for which correlation functions and entanglement dynamics can be analyzed exactly in the whole spacetime. We show that this novel hierarchy possesses non-trivial solutions at every level. Our results demonstrate that dual unitarity can be systematically extended while preserving solvability, opening up investigations of exactly solvable non-integrable systems with more general properties.

• The paper presents an infinite-level hierarchy (FDUₖ) that extends dual-unitarity to cover full spacetime domains for exact solvability.
• It employs analytical constructions and tensor network methods to evaluate correlation functions and entanglement dynamics with precise domain characterizations.
• The work offers a framework for designing quantum circuits and benchmarking quantum devices within non-integrable regimes.

Infinite-Level Hierarchy of Solvable Quantum Circuits: Technical Overview

Context and Motivation

Dual-unitary circuits represent a class of quantum models combining exact solvability with the absence of integrability, allowing analytical treatment of dynamical correlation functions and entanglement in non-integrable regimes. However, previous hierarchies of generalized dual-unitary conditions (notably via DU$_k$ constraints) only provide full solvability up to the second level, with higher-level generalizations suffering from restricted solvability domains in spacetime. This paper introduces a complementary infinite hierarchy—full dual-unitary (FDU$_k$)—expanding the solvability landscape by constructing logically independent complementary conditions for each hierarchy level. The result is a systematic extension of dual-unitarity, enabling exact analysis of correlation and entanglement dynamics throughout the entire spacetime for a broad class of non-integrable quantum circuits.

Hierarchy Construction and Solvability Domains

The hierarchy consists of two logically independent sets of constraints:

• Generalized Dual Unitarity (DU$_k$): Extends the standard (space-time) dual-unitarity to higher levels, but restricts the exact solvability of correlation functions and entanglement to regions near the edges of the causal light-cone.
• Complementary Dual Unitarity (DU$^*_k$): Introduced in this work, these conditions are independent from DU$_k$ and fill in the complementary spacetime regions where DU$_k$ fails to provide solvability.
• Full Dual Unitarity (FDU$_k$): Defined by combining DU$_k$ and DU$^*_k$, ensuring that solvability extends to the entire spacetime.

For each $k$-th level, FDU$_k$0 retains non-trivial solutions, systematically proving that dual unitarity can be extended beyond previously known limits without sacrificing solvability. Explicitly, FDU$_k$1 gates exhibit a complementary domain structure: DU$_k$2 governs solvability above a threshold velocity $_k$3; DU$_k$4 governs below this threshold. The solvability domains are precisely characterized, and analytical constructions for non-trivial FDU$_k$5 gates at all levels are presented.

Exact Determination of Correlation Functions

The dynamical correlation functions are efficiently evaluated using low-dimensional quantum channels. The tensor network representations for these circuits facilitate contraction procedures that are tractable, even for finite regions inside the causal light-cone inaccessible to generic (random or integrable) models. The analysis demonstrates:

• For DU$_k$6 and DU$_k$7 circuits, correlation functions are non-zero only in well-defined regions of spacetime, and tensor contractions yield explicit expressions via sequences of quantum channels.
• In FDU$_k$8 circuits, correlation functions exhibit non-trivial support along rays with velocity $_k$9 and vanish outside the combined solvable region, precisely matching the union of DU$_k$0 and DU$_k$1 domains.
• Consistency conditions between channels ensure the piecewise linear decay of correlations with velocity, highlighting the deviation from generic many-body systems with curved decay exponents.

Entanglement Dynamics: Entanglement Line Tension

The entanglement line tension (ELT), $_k$2, extracted from the scaling limit of operator entanglement dynamics (measured via Rényi entropies), is analytically characterized for all hierarchy levels:

• In FDU$_k$3 circuits, the ELT assumes a piecewise linear form with at most five kinks, corresponding to distinct velocities dictating information flow. Explicit formulas are derived for both the entanglement velocity and the full ELT profile.
• The ELT profile directly encodes discrete propagation directions of information, and the entanglement velocity is accessible via eigenvalues of associated transfer matrices.
• Notably, FDU$_k$4 gates can be constructed with ELT profiles and correlation structures that are manifestly distinct from dual-unitary or random circuits, accentuating the non-integrable yet solvable nature of these models.

Analytical and Numerical Construction of FDU$_k$5 Gates

A principal contribution is the explicit construction of infinite families of FDU$_k$6 gates using spacetime lattice configurations and complex Hadamard matrices. The conditions for full dual-unitarity are non-linear and high-dimensional tensor equations, yet the authors provide:

• Analytical constructions utilizing spacetime lattice transformations and biunitary connections that yield FDU$_k$7 gates at any hierarchy level, including reduced Hilbert space implementations via CHMs.
• Demonstrations that non-trivial FDU$_k$8 solutions exist for all $_k$9, including in systems with non-prime local Hilbert space dimensions, such as Clifford group gates in dimension 4.
• Numerical tangent space analysis proving that the sets of analytical solutions do not exhaust the solution space, identifying additional free parameters and establishing the local structure of the solution manifold.

Practical and Theoretical Implications

The introduction of the FDU$^*_k$0 hierarchy provides a new framework for constructing exactly solvable non-integrable quantum circuits beyond previous limits. The implications are multifaceted:

• Quantum Information: FDU$^*_k$1 circuits allow for the design and benchmarking of quantum devices where dynamical quantities (entropy growth, correlation spreading) can be computed analytically, bypassing the limitations of randomness or integrability.
• Many-Body Dynamics: These models serve as analytically tractable testbeds for quantum chaos and ergodicity in non-integrable regimes, facilitating precise study of spectral statistics and deep thermalization.
• Mathematical Physics: The hierarchy extends the taxonomy of solvability conditions, motivating searches for organizing principles connecting geometric spacetime constructions (lattice tilings) and tensor solvability constraints.
• Quantum Computing: The identification of Clifford group FDU$^*_k$2 gates in non-prime dimensions opens avenues for experimental realization of these circuits on digital quantum platforms.

Speculative future directions include the investigation of quenched dynamics from low-entanglement initial states in FDU$^*_k$3 circuits, analytical studies of chaos indicators (e.g., spectral form factor) in these models, and the search for further reduction in local Hilbert space dimension for full dual-unitary gates. The comprehensive characterization of the solution space and its minimality properties remains open.

Conclusion

By constructing an infinite hierarchy of solvability conditions (FDU$^*_k$4), this work systematically generalizes dual-unitarity to encompass broader classes of exactly solvable non-integrable quantum circuits. Explicit analytical and numerical methods confirm the existence of non-trivial solutions at every level. The techniques for evaluating dynamical correlation functions and entanglement line tension provide robust tools for probing quantum information dynamics in a wide range of non-integrable settings. Theoretical innovations in the hierarchy structure and practical advances in circuit construction offer promising directions for further exploration in both mathematical physics and quantum technology.