Non-Universality from Conserved Superoperators in Unitary Circuits

TL;DR

This work develops a general commutant-algebra framework to diagnose universality in unitary circuits generated by $k$-local, symmetry-constrained gates. By recasting universality as a connectivity problem in operator space via the dynamical Lie algebra (DLA) and its superoperator commutants, it identifies two fundamental non-universality classes: weak (scar-based) and strong (additional superoperator symmetries). The authors provide analytical tools and numerical methods (including MPS-based and block-diagonalization approaches) to compute DLAs and their centers, with explicit analyses for $U(1)$, $SU(2)$, translation-invariant, and matchgate-like circuits. They also connect these algebraic structures to physical observables, showing how strong non-universality leaves imprints in OTOCs, Rényi entropies, and $k$-design properties, thereby linking symmetry-derived constraints to measurable dynamics. Overall, the paper offers a comprehensive framework to understand when symmetric unitary circuits fail to be universal and to predict observable signatures of such failures.

Abstract

An important result in the theory of quantum control is the "universality" of $2$-local unitary gates, i.e. the fact that any global unitary evolution of a system of $L$ qudits can be implemented by composition of $2$-local unitary gates. Surprisingly, recent results have shown that universality can break down in the presence of symmetries: in general, not all globally symmetric unitaries can be constructed using $k$-local symmetric unitary gates. This also restricts the dynamics that can be implemented by symmetric local Hamiltonians. In this paper, we show that obstructions to universality in such settings can in general be understood in terms of superoperator symmetries associated with unitary evolution by restricted sets of gates. These superoperator symmetries lead to block decompositions of the operator Hilbert space, which dictate the connectivity of operator space, and hence the structure of the dynamical Lie algebra. We demonstrate this explicitly in several examples by systematically deriving the superoperator symmetries from the gate structure using the framework of commutant algebras, which has been used to systematically derive symmetries in other quantum many-body systems. We clearly delineate two different types of non-universality, which stem from different structures of the superoperator symmetries, and discuss its signatures in physical observables. In all, our work establishes a comprehensive framework to explore the universality of unitary circuits and derive physical consequences of its absence.

Figures (6)

• Figure 1: Schematic representation of the setup considered for studying non-universality in Sec \ref{['sec:superoperatoralgebra']}. Given a set of generators $\{h_\alpha\}$ we study the set of unitaries that can be obtained as arbitrary products of $u_\alpha(\theta)=\exp(i\theta h_\alpha)$. The hermitian generators $\{h_\alpha\}$ need not be local as in the picture presented here. The main question we study in this work is if this setup can generate all global unitaries with the same symmetries as $\{u_\alpha(\theta)\}$.
• Figure 2: The operator Hilbert space $\widehat{\mathcal{H}} \mathrel{\mathop:}= \mathrm{End}(\mathcal{H})$ can be interpreted as a ladder Hilbert space through the Liouvillian isomorphosm with $\mathcal{H}\otimes\mathcal{H}$.
• Figure 3: Diagram of the operator and superoperator algebras under consideration. For simplicity, we have omitted most references to the generators ${\mathcal{G}}$. (a) The superoperators in $\widehat{\mathcal{A}}_{\mathcal{G}}$ are generated by the adjoint superoperators ${\mathcal{L}}_{h_\alpha}$, and their action on the generators $\{h_\beta\}$ spans the DLA $\mathfrak{Lie}({\mathcal{G}})$. The algebra generated by $\{h_\beta\}$ is $\mathcal{A}_{\mathcal{G}}$, and its commutant $\mathcal{C}_{\mathcal{G}}$ is the same as $\mathfrak{Lie}({\mathcal{G}})$'s commutant; $\mathcal{C}_{\mathcal{G}}$ is the set of conventional symmetries of the set of gates. (b) By writing an element of $\widehat{\mathcal{A}}_{\mathcal{G}}$ according to the general block-decomposition of equation \ref{['eq:matrixrep']}, we can see that $\mathfrak{Lie}({\mathcal{G}})$ and $\mathcal{A}_{\mathcal{G}}$ are a direct sum of Krylov subspaces. This is because: both algebras are invariant under the action of $\widehat{\mathcal{A}}_{\mathcal{G}}$, and the action of $\widehat{\mathcal{A}}_{\mathcal{G}}$ is irreducible on every Krylov subspace.
• Figure 4: The minimal super-commutant $\widehat{\mathcal{C}}_{\langle\!\langle{\mathcal{G}}\rangle\!\rangle}$ describes the block decomposition associated to the conventional symmetries at operator level $\mathcal{C}_{\mathcal{G}}$. If $\widehat{\mathcal{C}}_{\mathcal{G}}$ contains additional superoperator symmetries, they will be responsible for further splitting of the operator Hilbert space into smaller Krylov subspaces and/or for the formation of additional degeneracies between Krylov subspaces belonging to different symmetry sectors.
• Figure 5: Plots of simulations and predictions for the asymptotic values of physical observables associated to strong universality breaking. (a) Time-averaged OTOC of $Z_j$ for a $\mathbb{Z}_2$-preserving and matchgate Brownian circuit, and for a $\mathbb{Z}_2$-preserving and matchgate Floquet system ($L=6$). (b) Average value of the second Rényi entropies for a universal Brownian circuit and a matchgate Brownian circuit ($L=22$).

Theorems & Definitions (17)

• Lemma D.1: Independence from the particular choice of generators
• proof
• Lemma D.2: Minimal super-commutant and maximal super-bond algebra
• proof
• Definition D.1
• Theorem D.3
• proof
• Lemma D.4: The generators overlap with all symmetry sectors up to central elements
• proof
• Lemma D.5: Conserved superoperators and strong non-universality