Faster algorithm for achieving minimal-size quantum decision diagrams

Abstract: The decision diagram (DD) data structure enables fast linear-algebra calculations by bringing vectors into a normal form and subsequently merging equivalent ones, yielding a minimally-sized DD modulo the equivalence relation. A fruitful application area is quantum-circuit simulation, where the vectors represent quantum states. The Local Invertible Map Decision Diagram (LIMDD) type, merges LIM-equivalent (typically Pauli-gate equivalent) vectors, can efficiently simulate Clifford circuits as well as some high-T-count circuits, and has theoretically been proven exponentially faster for simulation than other well-developed data structures, including other common DD variants. However, these exponential advantages have not fully materialized yet in existing implementations, for which the normal-form procedure, which is a highly complex algorithm, is either absent or only partially implemented. We here present a novel normal-form algorithm for Pauli-LIMDDs, achieving a worst-case speedup from $O(n^3)$ to $O(n^2)$ for an $n$-qubit DD node with a single child node while keeping the $O(n^3)$ run time in case of two distinct children nodes. We implement the algorithm as part of QolDDer, our Pauli-LIMDD simulator for quantum circuits, written from scratch in C/C++. The implementation realizes the theoretically-proven advantages of Pauli-LIMDDs on Clifford circuits, is significantly faster than the existing LIMDD simulators on such circuits, and on a public quantum-circuit data set often outperforms them by an order of magnitude. In the future, we envision that our work will enable further application and development of LIMDD variants, not only for quantum design tasks, but also for analysis of linear-algebra-based systems in general.

• The paper introduces an O(n²) algorithm for canonicalizing Pauli-LIMDDs, reducing complexity for single-child nodes from O(n³).
• It leverages group-theoretic properties and stabilizer subgroup intersections to enforce strict canonicity, ensuring exponential performance gains in simulating Clifford circuits.
• The implementation in QolDDer and the reusable PauliLim library validate the method, demonstrating improved memory and computational efficiency over previous approaches.

Faster Algorithm for Achieving Minimal-Size Quantum Decision Diagrams

Introduction and Motivation

This paper addresses the bottleneck in practical simulation of quantum circuits using quantum decision diagrams (QDDs), focusing specifically on Local-Invertible Map Decision Diagrams (LIMDDs) and their core canonicalization step—normal-form computation. Previous work established the theoretical exponential advantage of LIMDDs (especially Pauli-LIMDDs) over alternatives such as QMDDs and Algebraic Decision Diagrams, particularly for efficiently simulating Clifford and certain Clifford+T circuits. However, previous implementations failed to achieve these gains in practice due to the complexity and computational overhead of the canonicity-enforcing algorithms, predominantly governed by $O(n^3)$ procedures for nodes with $n$ qubits.

The paper introduces a new normal-form (high determinism) algorithm for Pauli-LIMDDs that reduces the worst-case complexity from $O(n^3)$ to $O(n^2)$ for single-child nodes, which are central for exploiting the succinctness of decision diagram encodings. This advance is operationalized in QolDDer, a C/C++ Pauli-LIMDD simulator, and the algorithm is made available as the standalone library PauliLim.

Technical Contributions

Background on Quantum Decision Diagrams

Quantum states and operators are, in general, exponentially sized objects in the number of qubits $n$. QDDs, notably QMDDs and LIMDDs, represent these exponentially sized vectors and matrices in a compressed, canonical DAG structure, enabling polynomial-time equivalence checking and more efficient simulation provided canonicity is enforced. Whereas QMDDs recognize equivalence up to scalar multiples, LIMDDs generalize mergeability under equivalence by local invertible maps—specifically, tensor products of invertible $2\times2$ matrices (for Pauli-LIMDDs, restricted to complex multiples of Pauli words).

Critical to the DD's efficacy is strict canonicity: two subfunctions must collapse to the same node if and only if they represent the same function up to the allowed equivalence relation. For LIMDDs, canonicity requires more sophisticated procedures than in QMDDs, particularly for handling nontrivial equivalences (e.g., under stabilizer group actions for Pauli-LIMDDs).

Previous Limitations

Prior implementations (e.g., MQT-LIMDD, LimTDD) were either incomplete—failing to strictly enforce canonicity—or computationally too expensive to realize the theoretical speedups. This typically arises from the high determinism step: computing the minimally-labeled equivalent node modulo the Pauli group, which ensures that all equivalent states are mapped to a unique canonical representative.

The New Algorithm

The presented algorithm reformulates the high determinism procedure by leveraging group-theoretic properties of stabilizer states and efficiently intersecting stabilizer subgroups, building on the Zassenhaus algorithm and Gaussian elimination in the binary Pauli word domain.

• Single-child node case ($v_0 = v_1$): The key insight is that the minimal label in the equivalence class can be found by minimizing over products of stabilizers and applying the minimal-vector algorithm, whose complexity is $O(n^2)$ rather than $O(n^3)$.
• Two-distinct-children case: The overall framework matches prior $O(n^3)$ complexity but implements optimizations in intermediate steps (reuse of matrix computations), reducing constant factors.
• Stabilizer group generators: The algorithms recursively compute stabilizer sets in reduced row-echelon form, enabling merge and normalization rules to be efficiently enforced at each node.
• Canonical form/completeness: The correctness proofs ensure uniqueness and completeness with respect to the LIMDD equivalence relation, overcoming subtle bugs present in earlier implementations.

Implementation: QolDDer and PauliLim

QolDDer is a from-scratch implementation, with modularity and precise canonicalization at its core. It handles floating-point numerical imprecision with careful scalar comparison logic and can be configured to tune tolerance parameters. The PauliLim algorithm is separately available to support reuse and benchmarking.

The canonicalization steps are tightly integrated within the insertion and merge logic, with careful attention to algorithmic edge cases and canonicalization rules—particularly zero-edge handling, total order enforcement, low factoring, and high determinism.

Empirical Results

Benchmarks and Baselines

Empirical evaluation uses (1) random Clifford circuit benchmarks (RANDCLIFF), which are known to exhibit exponential state complexity in non-canonical QDD representations, and (2) the broad-spectrum MQT-BENCH suite.

Comparisons include:

• QolDDer (proposed)
• MQT-DDSIM (QMDD, canonic)
• MQT-LIMDD (partial canonicity)
• LimTDD (generalized LIMs, non-canonical)

Key Findings

• Canonicity essential for exponential speedup: On random Clifford circuits, only fully canonical Pauli-LIMDDs yield exponential speedup (both in time and memory) compared to QMDDs and partial/non-canonical LIMDDs. This is observed as an order-of-magnitude speedup, validating the theoretical argument that succinctness and practical simulation speedups require fully enforced canonicity.
• $n$0 efficiency realized: Empirical data demonstrates sublinear or near-linear scaling (on logarithmic plots) in canonicalized QolDDer runs, compared to much worse scaling in prior implementations.
• Not universally advantageous: On circuits where LIM-equivalence rarely holds (e.g., certain structured circuits in MQT-BENCH), overheads may outweigh gains, and in some cases, QMDDs are faster.
• Robustness and correctness: QolDDer demonstrates correct simulation outcomes, as checked against brute-force and cross-tool validation. In contrast, partial canonicalization leads to node-count blow-up or failed simulation on certain inputs.

Implications and Future Directions

Practical Impacts

• Quantum simulation acceleration: The presented algorithm enables scalable simulation of quantum circuits, particularly for highly entangled stabilizer and Clifford+T states. This provides a foundation for classical verification, optimization, and benchmarking of near-term noisy quantum algorithms.
• Tooling and reusable methods: The PauliLim library can be readily integrated into other simulation or verification platforms, or for benchmarking alternative decision diagram approaches.

Theoretical Implications

• Decision diagram completeness: The results explicitly tie the realization of theoretical exponential succinctness to strict canonicity. The formal connection between group-theoretic properties (stabilizer group actions) and algorithmic DD operations strengthens the theoretical understanding of quantum data structures.
• Generality: The approach is extendable to larger LIM groups, as considered in LimTDD, with the challenge being to generalize the group-theoretic minimization algorithm.

Future Work

• Beyond Pauli group: Extending normal-form algorithms to richer local-invertible map groups (e.g., XP groups) shall enable LIMDDs to succinctly simulate beyond the Clifford group, covering broader quantum circuit classes.
• Applications in equivalence checking and synthesis: The approach is directly relevant for quantum circuit equivalence checking, which is essential for program verification and circuit compilation.
• Classical tasks: The DD canonicalization framework may be adapted to general linear algebra tasks, such as efficient probabilistic inference, Markov chain analysis, or other domains where mergeability under local group actions yields meaningful compression.

Conclusion

The paper delivers a theoretically sound, practically efficient algorithm for LIMDD canonicalization, overcoming prior empirical performance barriers. For relevant quantum circuit classes, this translates theoretical exponential advantages into realized practical speedups in both memory and computation time. The approach provides a robust basis for scalable classical simulation and analysis of quantum systems and opens further research directions in both quantum and classical decision diagram data structures (2606.24789).