Completely Reducible Circuits

• Completely reducible circuits are systems whose global dynamics decompose into independent, zero-synergy contributions from constituent parts, applicable in both quantum and classical contexts.
• They are rigorously characterized using methodologies from partial information decomposition and algebraic-topological invariants such as knot theory to signal the absence of higher-order synergy.
• These circuits enable exactly solvable dynamics with predictable entanglement propagation and offer insights into operator spreading in chaotic non-integrable regimes.

A completely reducible circuit is a system—classically or quantum-mechanically defined—whose global behavior can be decomposed, with zero synergy, into the independent or lower-order contributions of its constituent parts. This mathematical notion arises in multiple contexts, notably in the PID (Partial Information Decomposition) formalism for classical systems, and in the study of exactly solvable quantum dynamics where tensor network contractions can be reduced without reference to the detailed forms of underlying gates. The paradigm yields sharp criteria for the vanishing of all high-order synergy contributions, providing both operational and structural classification of circuit irreducibility, and admits rigorous characterization via both information-theoretic and algebraic-topological (knot-theoretic) invariants.

1. Definition and Formalism

In the quantum context, a completely reducible circuit is a one-dimensional, non-integrable, exactly solvable circuit constructed from two-site dual-unitary gates arranged in a regular, potentially non-rectangular spacetime lattice. The defining property is that every tensor-network contraction required for key dynamical quantities—such as propagator Rényi-$\alpha$ operator entropies—can be reduced by repeated use of unitarity and dual-unitarity identities alone, ultimately yielding results that depend only on permutation-state overlaps. Explicitly, for any dual-unitary gate $U$ and contraction network $$Z_\alpha(m,n) = \operatorname{tr}\left[\left\{\operatorname{tr}_A(\mathcal{U} \otimes \mathcal{U}^\dagger)\right\}^\alpha\right]$$, a circuit is completely reducible if $Z_\alpha(m,n) = q^{-(\alpha-1) N(m,n)}$, independent of the details of $U$, for all $\alpha$ and all relevant system sizes (Rampp et al., 17 Dec 2025).

In PIN (Partial Information Decomposition) theory applied to classical circuits, complete reducibility requires that the mutual information between $n$ predictors $X_1,...,X_n$ and a target $Y$ can be decomposed such that every synergy-based irreducibility measure vanishes. This is captured by the vanishing of four measures—$\IbS, \IbP, \IbC, \IbD$—each corresponding to decreasingly restrictive partitions, culminating in the “all-parts” measure $\IbD$ (Griffith et al., 2013):

$\IbD(X_1, ..., X_n; Y) = I(X_1 \cdots X_n; Y) - \Icup (A_1, ..., A_n; Y) = 0$

where $A_j$ is the 'Almost': the set of all predictors except $X_j$. All synergy regions in the PID diagram must be identically zero.

2. Construction and Verification

In the quantum setting, the construction proceeds by selecting a base dual-unitary gate $U$ (acting on $\mathbb{C}^q \otimes \mathbb{C}^q$), and arranging multiple copies in a staggered 'brickwork' or more generalized lattice, with careful control to ensure only non-overlapping local interactions. Unlike classical dual-unitary square-lattice brickwork—which enforces a global spacetime self-duality—completely reducible circuits allow broader classes of spacetime tilings, including those supporting multiple distinct directions of information flow (multi-ray circuits).

The global condition for complete reducibility is checked by attempting to contract the relevant folded Rényi-$\alpha$ tensor network using only unitarity and dual-unitarity moves. If the contraction must always yield a product of disconnected permutation overlaps (i.e., no irreducible subdiagrams remain even when defects are introduced), the circuit is completely reducible (Rampp et al., 17 Dec 2025).

For classical circuits, verification consists of explicit calculation of the four irreducibility measures for the chosen function, following the PID-based approach, and confirming their vanishing, notably $\IbD=0$.

3. Entanglement Dynamics and Information Flow

The entanglement propagation in completely reducible quantum circuits is governed by a minimal-membrane picture, characterized by an explicit, exactly computable 'line tension' $\tau(v)$ that quantifies entanglement growth at velocity $v$. This profile is piecewise linear, with kinks corresponding to distinct permitted information velocities $v_i$. For a finite set of rays:

$$\tau(v) = \sum_{i:v_i<v} \frac{n_i}{2N}(v-v_i) + \sum_{i:v_i>v} \frac{n_i}{2N}(v_i-v)$$

for worldlines of multiplicity $n_i$ per block of size $2N$ (Rampp et al., 17 Dec 2025). The entanglement entropy in the scaling limit is $S_\alpha(x, t) \simeq s_{eq} \tau(v)t$, with all Rényi indices sharing the same profile.

This structure yields immediate access to operational quantities such as the butterfly velocity $v_B$ (the maximal signal velocity, $\max_i |v_i|$), the entanglement velocity $v_E = \tau(0)$, and the decay of out-of-time-order correlators (OTOCs), which is exponential in $[\tau(v)-v]t$ for $|v|<v_B$.

In the classical framework, the complete absence of synergy implies that the output is fully determined by the (possibly overlapping) collection of lower-order parts—there is no higher-order information flow requiring multi-variable coordination (Griffith et al., 2013).

4. Knot-Theoretic and Algebraic Structure

Completely reducible quantum circuits admit an algebraic-topological reformulation in terms of knot invariants. Given the tensor network for $Z_2(m, n)$, replacing every $U/U^\dagger$ with SWAP gates transforms the contraction into a link diagram $L_{m, n}$. The contraction using only unitarity and dual-unitarity corresponds to performing only Reidemeister II moves. Complete reducibility is thus equivalent to the link being isotopic to the unlink by these moves alone. The tensor contraction becomes

$Z_2(m, n) = q^{-2(m+n)} \langle L_{m, n} \rangle_K$

where $\langle \cdot \rangle_K$ is the Kauffman bracket with $A^2 + A^{-2} = -q$ (Rampp et al., 17 Dec 2025). The presence of nontrivial linkings (such as crossings of zero-velocity worldlines) signals irreducibility.

This perspective unifies the solvable dynamics of completely reducible circuits with algebraic invariants, drawing a direct parallel with the Yang–Baxter relation (Reidemeister III) in integrable systems. It enables systematic generation and classification of new exactly solvable circuit families based on the topology of underlying lattice wiring diagrams.

5. Characteristic Examples

Both quantum and classical domains exhibit canonical examples that span the reducibility spectrum.

Quantum Circuits:

• Dual-unitary brickwork (two-ray): $N=1$, $v_i = \pm1$, $\tau(v) = 1$ for $|v| < 1$, maximal butterfly and entanglement velocities.
• Kagome (three-ray): $N=2$, $v_i \in \{0, \pm1\}$, $\tau(v) = \frac{1}{2}(1+|v|)$ for $|v| < 1$, $v_E = 1/2$.
• Four-ray (pyramid/rocket): $N=4$, $v_i \in \{\pm 1/3, \pm 1\}$, submaximal $v_E$.
• Five-ray circuits: $N=5$, $v_i \in \{0, \pm 1/3, \pm 1\}$, $\tau(v)$ given by a distinct, piecewise-linear profile.

Classical Circuits (PID):

Circuit	Boolean Function	Irreducibility Vector $[\IbS, \IbP, \IbC, \IbD]$
Xor (n=2)	$Y = X_1 \oplus X_2$	$[1, 1, 1, 1]$
XorUnique	$Y =$ "digit" $= X_1 \oplus X_2;$ "letter" $= X_3$	$[1, 0, 0, 0]$
DoubleXor	$Y =$ (left $= X_1 \oplus X_2$, right $= X_2 \oplus X_3$)	$[2, 1, 0, 0]$
TripleXor	$$Y = (X_1 \oplus X_2, X_1 \oplus X_3, X_2 \oplus X_3)$$	$[3, 2, 1, 0]$
Parity	$Y = X_1 \oplus X_2 \oplus X_3$	$[1, 1, 1, 1]$

In this list, only circuits with $\IbD=0$ are fully reducible in the PID sense (Griffith et al., 2013). The spectrum $\IbS \geq \IbP \geq \IbC \geq \IbD$ quantifies the degree and order of synergy present.

6. Broader Implications and Open Directions

Completely reducible circuits establish a unified analytic framework for non-integrable, exactly solvable dynamics that encompasses and extends conventional dual-unitary models. Despite their solvability, these systems remain non-integrable, displaying hallmark features of quantum chaos such as random-matrix statistics and vanishing Thouless time, but with varying entanglement velocities.

The connection to knot invariants provides a novel algebraic handle on non-integrable but solvable dynamics, analogous in role to the Yang–Baxter structure for integrability. By classifying spacetime lattices whose associated links are unlinked under Reidemeister II moves, completely reducible models can be systematically enumerated and explored.

Furthermore, the explicit relation $d^2\tau/dv^2 = 2 n(v) \geq 0$ suggests that, even in non-reducible, generic chaotic dynamics, the convex entanglement profile $\tau(v)$ may admit a “quasiparticle” interpretation tied to a continuous spectrum of information flow densities. This points toward the potential for new hydrodynamic descriptions of operator spreading and entanglement growth, with implications beyond the exactly solvable regime (Rampp et al., 17 Dec 2025).

The synergy-based irreducibility spectrum from $\IbS$ down to $\IbD$ remains a foundational tool for systematically classifying circuit decomposability in both quantum and classical information-processing models, establishing a rigorous criterion for when a system “forgets” its full many-body structure in favor of independent or lower-order partwise contributions (Griffith et al., 2013).