Classical and Quantum Algorithms for Topological Invariants of Torus Bundles

TL;DR

This work demonstrates that Witten-Reshetikhin-Turaev invariants of torus bundles admit efficient classical and quantum computation by embedding the skein algebra into the symmetric subalgebra of the non-commutative torus, yielding a fixed $N^2$-dimensional representation. The classical dynamic-programming approach updates an $N^2$-sized coefficient table in $\Theta((m+\ell)N^2)$ time, while a quantum algorithm uses only $O(\log N + m)$ qubits to approximate the amplitude via linear combinations of unitaries and the Hadamard test. Exact FG-coefficient counting is #P-complete, yet a quantum routine provides additive approximation for many configurations, establishing a clear complexity separation. The work also clarifies a place-free representation, enabling uniform quantum circuit design and a direct computation of WRT traces rather than knot-based invariants. Overall, it highlights exponential space savings and notable computational distinctions between trace evaluation and exact coefficient extraction in topological quantum computation.

Abstract

Computing topological invariants of 3-manifolds is generally intractable, yet specialized algebraic structures can enable efficient algorithms. For Witten-Reshetikhin-Turaev (WRT) invariants of torus bundles, we exploit the non-commutative torus structure to embed the skein algebra of the closed torus into its symmetric subalgebra at roots of unity. This yields a fixed $N^2$-dimensional representation that supports polynomial-time classical computation with $O(N^2)$ space, and a quantum algorithm using only $O(\log N)$ qubits -- an exponential space advantage. We further prove that extracting individual expansion coefficients is #P-complete, yet there is a quantum algorithm that can efficiently approximate these coefficients for a non-negligible fraction of configurations.

Figures (3)

• Figure 1: Computational landscape enabled by the NCT embedding. The structural foundation (top) enables two distinct computational problems. Left (WRT Trace): The classical dynamic programming algorithm achieves $\Theta((m{+}\ell)N^2)$ time but requires $\Theta(N^2)$ space. The quantum LCU approach maintains polynomial time $O((\ell{+}m) \log^2 N)$ using $O(\log N + m)$ total qubits compared to $\Theta(N^2)$ classical memory—an exponential space advantage. Right (Coefficient Counting): The classical enumeration is #P-complete, requiring exponential $\Omega(2^m)$ time and thus intractable for exact counting. The quantum Hadamard test achieves polynomial-time additive approximation with $O(m \log^2 N)$ circuit depth, demonstrating proven computational advantage via coherent superposition over all $2^m$ terms.
• Figure 2: Classical vs quantum approaches to WRT computation. Classical (left): The Frohman–Gelca two-term rule generates $2^m$ terms conceptually, but these map to an $N^2$ coefficient table enabling polynomial-time dynamic programming with $\Theta(N^2)$ space. Quantum (right): By embedding the skein algebra into the symmetric subalgebra $\mathcal{W}_t^{\iota}$ of the non-commutative torus, we implement each multiplication as a coherent sum via LCU, using $O(\log N)$ data qubits plus $m+1$ ancillas—an exponential space reduction from the $\Theta(N^2)$ classical memory requirement.
• Figure 3: Classical vs Quantum approaches to FG-Coefficient Counting. Top (Classical): Binary tree shows exponential explosion. Each node represents a partial sum, and each edge represents adding $+a_i$ or $-a_i$ (edge labels). Starting from sum 0, each choice doubles the number of paths: level 1 has $2^1{=}2$ paths, level 2 has $2^2{=}4$ paths, reaching $2^m$ total paths at level $m$. Each path from root to leaf corresponds to one sign assignment $\varepsilon \in \{\pm 1\}^m$. The classical algorithm must enumerate all $2^m$ paths to count which ones yield sum $z$ (green nodes)—this is #SIGNED-SUM counting (a signed variant of #SUBSET-SUM), requiring exponential $\Omega(2^m)$ time. Bottom (Quantum): Hadamard test evaluates all $2^m$ assignments simultaneously via superposition. Sampling estimates the normalized coefficient $\alpha = c(z)/2^m$ to additive precision $\epsilon$ in polynomial time $O(m \log^2 N / \epsilon^{2})$ (Theorem \ref{['thm:fgcc-quantum']}) for common targets.

Theorems & Definitions (67)

• Definition 1: WRT invariant for torus bundles
• Theorem : Classical Dynamic Programming Algorithm
• Theorem : Quantum Algorithm
• Theorem : Quantum Advantage for Coefficient Counting
• Theorem : Structural Decomposition
• Definition 2: Witten-Reshetikhin-Turaev TQFT
• Definition 3: Mapping torus
• Definition 4: Kauffman bracket skein algebra
• Remark 1: Parameter conventions
• Definition 5: Primitive pairs