Non-invertible Nielsen Circuits

• Non-invertible Nielsen circuits are a framework that generalizes quantum circuit complexity by incorporating non-invertible fusion operations between superselection sectors.
• The approach hybridizes continuous unitary evolution within sectors with a discrete shortest-path search over fusion gates, effectively combining geodesic optimization with graph algorithms.
• This framework finds practical application in rational conformal field theories and holography, where fusion gates mimic shock-like defects that alter boundary conditions in AdS₃ gravity.

Non-invertible Nielsen circuits are a generalization of the geometric framework of quantum circuit complexity developed by Nielsen, designed to incorporate intrinsically non-invertible operations. These circuits are motivated by modern advances in the understanding of symmetry and topological defects in quantum field theories, where fusion operations between superselection sectors—typically encoded by modular tensor categories—cannot be described solely by invertible unitary evolution. The resulting hybrid framework amalgamates continuous optimization over unitary gates within each sector with a discrete shortest-path problem over fusion operations, enabling transformations between sectors previously inaccessible in standard circuit complexity. This approach is realized rigorously in rational conformal field theories and finds interpretation in the context of AdS$_3$ gravity, where fusion gates correspond to shock-like defects at the boundary.

1. Standard Nielsen Complexity Framework

Nielsen's geometric approach to quantum circuit complexity formulates the synthesis of a target unitary $U \in \mathrm{SU}(2^n)$ from the identity as evolving under a time-dependent Hamiltonian $H(t)$: $$\dot{U}(t) = -i H(t) U(t), \quad U(0) = I, \quad U(1) = U$$ A right-invariant metric is imposed on the group manifold using a norm $F(H)$ on the generator $H(t)$, satisfying $F(UHU^\dagger)=F(H)$. Common choices include Finsler $p$-norms, e.g., $F(H) = \sqrt{\mathrm{Tr}[H^2]}$ or $F(H) = \|H\|_p$. The circuit cost functional is

$C[U(t)] = \int_0^1 dt\, F(H(t))$

Optimizing this functional is equivalent to solving the geodesic problem on $\mathrm{SU}(2^n)$ equipped with the chosen metric.

2. Fusion Operations as Non-invertible Gates

Quantum field theories and rational conformal field theories (RCFTs) often possess superselection sectors labeled by $a, b, c$, corresponding to simple objects of a unitary modular tensor category (UMTC). Fusion of sector $a$ with defect $b$ produces a direct sum of sectors $c$ according to fusion rules

$a \otimes b \cong \bigoplus_c N_{ab}^c\, c$

This process is inherently non-invertible but admits a canonical realization as a completely positive, trace-preserving (CPTP) quantum channel: $$\Phi_b^{(a)}: \rho_a \mapsto \sum_{c, \mu} K^{(a, b)}_{c, \mu}\, \rho_a\, K^{(a, b)\dagger}_{c, \mu}$$ where each Kraus operator $$K^{(a, b)}_{c, \mu}: \mathcal{H}_a \to \mathcal{H}_c$$ implements fusion into sector $c$ via channel $\mu = 1, \ldots, N_{ab}^c$. Trace preservation is ensured by the pivotal (unitarity) structure of the UMTC: $\sum_{c, \mu} K^\dagger K = I_{\mathcal{H}_a}$

3. Hybrid Complexity Measure: Continuous and Discrete Components

In the generalized framework, admissible circuit moves at each time $t_i$ are either:

• Continuous unitary transformations generated by $H(t)$,
• Instantaneous non-invertible jumps given by fusion channels $\Phi_b^{(a)}$.

A circuit then alternates between unitary flows within a sector and fusion gates that effect a transition

$$U_{\text{final}} = \left(\text{unitary in sector } a_0\right) \circ \Phi_{b_1}^{(a_0)} \circ \left(\text{unitary in sector } a_1\right) \circ \Phi_{b_2}^{(a_1)} \circ \cdots$$

For each unitary segment in sector $a_k$, the cost is the geodesic length for the sector metric $F_{a_k}(H)$: $$C_{\text{inv}}^{(k)} = \int_{t_k}^{t_{k+1}} dt\, F_{a_k}(H^{(k)}(t))$$ Fusion steps are represented by edges in the fusion graph $G_f$; each edge from $a\to c$ via defect $b$ is assigned a minimal positive weight $w_{a\to c}^{(b)}$, encoding the cost for the fusion gate.

The full complexity is a minimization over all possible sector paths $P$ in $G_f$: $$\mathrm{Complexity} = \min_P \left[ \sum_i \int_{t_i}^{t_{i+1}} dt\, F_{a_i}(H^{(i)}(t)) + \sum_{(a\to b)\in P} w_{ab} \right]$$ This hybrid structure leads to a sector-changing problem that is inherently discrete—the optimal path is determined by a weighted shortest-path computation on $G_f$.

4. Fusion Graphs and Case: The Ising Model

In the $c=\frac{1}{2}$ Ising RCFT, three sectors exist: $1$, $\sigma$, $\epsilon$, with fusion rules

$$\begin{align*} \sigma \times \sigma &= 1 \oplus \epsilon \ \epsilon \times \sigma &= \sigma \ \epsilon \times \epsilon &= 1 \ 1 \times X &= X \end{align*}$$

The fusion graph $G_f$ comprises directed edges between these sectors:

• $\sigma \to 1$ (weight $w_{\sigma\to1}$)
• $\sigma \to \epsilon$ (weight $w_{\sigma\to\epsilon}$)
• $\epsilon \to \sigma$ (weight $w_{\epsilon\to\sigma}$)
• $1 \to \sigma$, etc.

If a uniform penalty $w$ is applied to every nontrivial fusion, all nontrivial fusion edges have weight $w$. Within each sector, the continuous complexity is reduced to the standard $\mathrm{SU}(2^{n_a})$ geodesic length. For the $\sigma$ sector, $H(t) \in \mathfrak{su}(2)$ and

$$d_\sigma(I,U) = \min_{H(t)} \int_0^1 dt\, \sqrt{\mathrm{Tr}[H(t)^2]}$$

For a fixed norm geodesic, this equals $\|\log U\|$.

Comparison of paths returning from $\sigma$ to $\sigma$ depends on geodesic lengths and fusion penalties: $$\begin{align*} P_1:~ \sigma \to 1 \to \sigma &\quad \text{cost} = 2\, d_{\sigma\to1} + 2w \ P_2:~ \sigma \to \epsilon \to \sigma &\quad \text{cost} = 2\, d_{\sigma\to\epsilon} + 2w \end{align*}$$ The minimal cost path is selected according to sector-dependent geometries.

5. Optimization Algorithms on Hybrid Circuit Spaces

Minimizing the total circuit complexity involves a two-stage procedure:

• Continuous geodesic optimization: For each superselection sector $a$, solve the geodesic equation (Euler–Lagrange derived from the cost functional) on $\mathrm{SU}(\dim \mathcal{H}_a)$.
• Discrete shortest-path search: Assign weights $w_{ab}$ to sector transitions in $G_f$, and deploy graph algorithms such as Dijkstra or A* to find the path from initial to target sector with minimal accumulated cost.

The complexity of a transformation is the sum of geodesic lengths in the visited sectors plus the fusion weights accrued.

6. Fusion Gates: CPTP Structure and Category Consistency

Fusion gates as non-invertible operations are formalized via their realization as CPTP maps. In RCFT or UMTC context, the chiral defect algebra provides the fusion data:

• Kraus operators $K_{c, \mu}^{(a, b)}$ implement channels from sector $a$ to $c$,
• The completeness relation $\sum_{c, \mu} K^\dagger K = 1$ arises from quantum dimension constraints,
• Different fusion parenthesizations are unitarily related via associator $F$-symbols, ensuring channel consistency and associativity.

Channel weights $p(c | a, b) = N_{ab}^c d_c / (d_a d_b)$ determine probabilistic branching during fusion. Selection cost for output channel $c$ is $$C_{\text{sel}}(c | a, b) = \arccos(\sqrt{p(c|a, b)})$$, capturing the irreversibility of choosing among probabilistic branches.

7. Physical and Holographic Interpretation

Fusion-induced sector transitions have physical interpretations in holography. In the Chern–Simons formulation of AdS$_3$ gravity, boundary fusion gates translate into instantaneous shifts in the boundary stress-tensor data: $$\Delta T_{++} = \frac{c}{12\pi R} \sum_c p_c \Delta h_c\, \delta(x^+ - \tau_0)$$ This is a shock-like defect affecting the holonomy class of the bulk solution, changing the boundary's Virasoro highest-weight label. Averaging over channels respects the ANEC constraint, but selecting an individual channel can violate ANEC, corresponding to the extra irreversibility cost $C_{\text{sel}}$.

8. Implementation Procedure

Algorithmic steps for constructing and optimizing non-invertible Nielsen circuits are:

• Specify UMTC fusion data $\{N_{ab}^c, d_a, F^{abc}_d\}$,
• Build the fusion graph $G$: edges $a\to c$ whenever $\exists b$ with $N_{ab}^c > 0$,
• For each edge $(a, b \to c)$, calculate $p_c$ and determine $C_{\text{gate}}$ and $C_{\text{sel}}$,
• Minimize total cost via discrete shortest-path (Dijkstra/A*) and continuous geodesic optimization.

This framework provides a systematic approach for evaluating circuit complexity in settings where non-invertible sector transitions are essential, fundamentally expanding the class of transformations accessible by quantum circuits beyond the limitations of standard invertible operations (Demulder, 14 Jan 2026, Demulder, 14 Jan 2026).