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arXiv:2606.24789 · Lecture notes in computer science

Faster algorithm for achieving minimal-size quantum decision diagrams

Juul Sanders, Sebastiaan Brand, Arend-Jan Quist, Tim Coopmans

quant-phcs.DSSubmitted Jun 23, 2026• Announced Jun 24, 2026• 0 citations
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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.

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Efficient methods for the representation and simulation of quantum states and quantum operations are crucial for the optimization of quantum circuits. Decision diagrams (DDs), a well-studied data structure originally used to represent Boolean functions, have proven capable of capturing relevant aspects of quantum systems, but their limits are not well understood. In this work, we investigate and bridge the gap between existing DD-based structures and the stabilizer formalism, an important tool for simulating quantum circuits in the tractable regime. We first show that although DDs were suggested to succinctly represent important quantum states, they actually require exponential space for certain stabilizer states. To remedy this, we introduce a more powerful decision diagram variant, called Local Invertible Map-DD (LIMDD). We prove that the set of quantum states represented by poly-sized LIMDDs strictly contains the union of stabilizer states and other decision diagram variants. Finally, there exist circuits which LIMDDs can efficiently simulate, while their output states cannot be succinctly represented by two state-of-the-art simulation paradigms: the stabilizer decomposition techniques for Clifford + T circuits and Matrix-Product States. By uniting two successful approaches, LIMDDs thus pave the way for fundamentally more powerful solutions for simulation and analysis of quantum computing.

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A decision diagram (DD) is a graph-like data structure for homomorphic compression of Boolean and pseudo-Boolean functions. Over the past decades, decision diagrams have been successfully applied to verification, linear algebra, stochastic reasoning, and quantum circuit analysis. Floating-point errors have, however, significantly slowed down practical implementations of real- and complex-valued decision diagrams. In the context of quantum computing, attempts to mitigate this numerical instability have thus far lacked theoretical scaling guarantees and have had only limited success in practice. Here, we focus on the analysis of quantum circuits consisting of Clifford gates and $T$ gates (a common universal gate set). We first hand-craft an algebraic representation for complex numbers, which replace the floating point coefficients in a decision diagram. Then, we prove that the sizes of these algebraic representations are linearly bounded in the number of $T$ gates and qubits, and constant in the number of Clifford gates. Furthermore, we prove that both the runtime and the number of nodes of decision diagrams are upper bounded as $2^t \cdot poly(g, n)$, where $t$ ($g$) is the number of $t$ gates (Clifford gates) and $n$ the number of qubits. Our proofs are based on a $T$-count dependent characterization of the density matrix entries of quantum states produced by circuits with Clifford+$T$ gates, and uncover a connection between a quantum state's stabilizer nullity and its decision diagram width. With an open source implementation, we demonstrate that our exact method resolves the inaccuracies occurring in floating-point-based counterparts and can outperform them due to lower node counts. Our contributions are, to the best of our knowledge, the first scaling guarantees on the runtime of (exact) quantum decision diagram simulation for a universal gate set.

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Quantum-mechanical phenomena are playing an increasing role in information processing as transistor sizes approach the nanometer level, while the securest forms of communication rely on quantum data encoding. When they involve a finite number of basis states, these phenomena can be modeled as quantum circuits, the quantum analogue of conventional or classical logic circuits. Simulation of quantum circuits can therefore be used as a tool to evaluate issues in the design of quantum information processors. Unfortunately, simulating such phenomena efficiently is exceedingly difficult. The matrices representing quantum operators (gates) and vectors modeling quantum states grow exponentially with the number of quantum bits. The information represented by quantum states and operators often exhibits structure that can be exploited when simulating certain classes of quantum circuits. We study the development of simulation methods that run on classical computers and take advantage of such repetitions and redundancies. In particular, we define a new data structure for simulating quantum circuits called the quantum information decision diagram (QuIDD). A QuIDD is a compressed graph representation of a vector or matrix and permits computations to be performed directly on the compressed data. We develop a comprehensive set of algorithms for operating on QuIDDs in both the state-vector and density-matrix formats, and evaluate their complexity. These algorithms have been implemented in a general-purpose simulator program for quantum-mechanical applications called QuIDDPro. Through extensive experiments conducted on representative quantum simulation applications, including Grover's search algorithm, error characterization, and reversible circuits, we demonstrate that QuIDDPro is faster than other existing quantum-mechanical simulators such as the National Institute of Standards and Technology's QCSim program, and is far more memory-efficient. Using QuIDDPro, we explore the advantages of quantum computation over classical computation, simulate quantum errors and error correction, and study the impact of numerical precision on the fidelity of simulations. We also develop several novel algorithms for testing quantum circuit equivalence and compare them empirically. The QuIDDPro software is equipped with a user-friendly interface and is distributed with numerous example scripts. It has been used as a laboratory supplement for quantum computing courses at several universities.

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Since people became aware of the power of quantum phenomena in the domain of traditional computation, a great number of complex problems that were once considered intractable in the classical world have been tackled. The downsides of quantum supremacy are its high cost and unpredictability. In order to obtain a stronger control over quantum interactions(by reducing its noise) and reduce its cost, numerous researchers are relying on quantum simulators running on classical computers. The critical obstacle facing classical computers in the task of quantum simulation is its limited memory space. Quantum simulation intrinsically models the state evolution of quantum subsystems, i.e., qubits. Qubits are mathematically constructed in Hilbert space whose size grows exponentially with its number. Consequently, the scalability of the straightforward statevector approach is limited. It has been proven effective to adopt decision diagrams to mitigate the memory cost issue in various fields. In recent years, researchers have adapted decision diagrams into different forms for representing quantum states and performing quantum calculations efficiently. This leads to the study of decision diagram based quantum simulation. However, their advantage of memory efficiency does not let it replace the mainstream statevector and tensor network based approaches. We argue the reason is the lack of effective parallelization strategies in performing calculations on decision diagrams. In this work, we explore several strategies for parallelizing decision diagram operations with the focus on leveraging them for quantum simulations. The target is to find the optimal parallelization strategies and improve the performance of decision diagram based quantum simulation. Based on the experiment results, our proposed strategy achieves a 2-3 times faster simulation of Grover's algorithm and random circuits than the state-of-the-art single-thread DD-based simulator DDSIM.

IEEE Transactions on Quantum Engineering•Jan 1, 2024•7 citations
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(n3)O(n^3)O(n3) to O(n2)O(n^2)O(n2) for an nnn-qubit DD node with a single child node while keeping the O(n3)O(n^3)O(n3) 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.
Efficient methods for the representation and simulation of quantum states and quantum operations are crucial for the optimization of quantum circuits. Decision diagrams (DDs), a well-studied data structure originally used to represent Boolean functions, have proven capable of capturing relevant aspects of quantum systems, but their limits are not well understood. In this work, we investigate and bridge the gap between existing DD-based structures and the stabilizer formalism, an important tool for simulating quantum circuits in the tractable regime. We first show that although DDs were suggested to succinctly represent important quantum states, they actually require exponential space for certain stabilizer states. To remedy this, we introduce a more powerful decision diagram variant, called Local Invertible Map-DD (LIMDD). We prove that the set of quantum states represented by poly-sized LIMDDs strictly contains the union of stabilizer states and other decision diagram variants. Finally, there exist circuits which LIMDDs can efficiently simulate, while their output states cannot be succinctly represented by two state-of-the-art simulation paradigms: the stabilizer decomposition techniques for Clifford + T circuits and Matrix-Product States. By uniting two successful approaches, LIMDDs thus pave the way for fundamentally more powerful solutions for simulation and analysis of quantum computing.
Tensor networks serve as a powerful tool for efficiently representing and manipulating high-dimensional data in applications such as quantum physics, machine learning, and data compression. Tensor Decision Diagrams (TDDs) offer an efficient framework for tensor representation by leveraging decision diagram techniques. However, the current implementation of TDDs and other decision diagrams fail to exploit tensor isomorphisms, limiting their compression potential. This paper introduces Local Invertible Map Tensor Decision Diagrams (LimTDDs), an extension of TDDs that incorporates local invertible maps (LIMs) to achieve more compact representations. Unlike LIMDD, which uses Pauli operators for quantum states, LimTDD employs the XPXPXP-stabilizer group, enabling broader applicability across tensor-based tasks. We present efficient algorithms for normalization, slicing, addition, and contraction, critical for tensor network applications. Theoretical analysis demonstrates that LimTDDs achieve greater compactness than TDDs and, in best-case scenarios and for quantum state representations, offer exponential compression advantages over both TDDs and LIMDDs. Experimental results in quantum circuit tensor computation and simulation confirm LimTDD's superior efficiency. Open-source code is available at https://github.com/Veriqc/LimTDD.
Exact quantum decision diagrams with scaling guarantees for Clifford+TTT circuits and beyond
A decision diagram (DD) is a graph-like data structure for homomorphic compression of Boolean and pseudo-Boolean functions. Over the past decades, decision diagrams have been successfully applied to verification, linear algebra, stochastic reasoning, and quantum circuit analysis. Floating-point errors have, however, significantly slowed down practical implementations of real- and complex-valued decision diagrams. In the context of quantum computing, attempts to mitigate this numerical instability have thus far lacked theoretical scaling guarantees and have had only limited success in practice. Here, we focus on the analysis of quantum circuits consisting of Clifford gates and TTT gates (a common universal gate set). We first hand-craft an algebraic representation for complex numbers, which replace the floating point coefficients in a decision diagram. Then, we prove that the sizes of these algebraic representations are linearly bounded in the number of TTT gates and qubits, and constant in the number of Clifford gates. Furthermore, we prove that both the runtime and the number of nodes of decision diagrams are upper bounded as 2t⋅poly(g,n)2^t \cdot poly(g, n)2t⋅poly(g,n), where ttt (ggg) is the number of ttt gates (Clifford gates) and nnn the number of qubits. Our proofs are based on a TTT-count dependent characterization of the density matrix entries of quantum states produced by circuits with Clifford+TTT gates, and uncover a connection between a quantum state's stabilizer nullity and its decision diagram width. With an open source implementation, we demonstrate that our exact method resolves the inaccuracies occurring in floating-point-based counterparts and can outperform them due to lower node counts. Our contributions are, to the best of our knowledge, the first scaling guarantees on the runtime of (exact) quantum decision diagram simulation for a universal gate set.
Quantum computers promise to solve important problems faster than conventional computers ever could. Underneath is a fundamentally different computational primitive that introduces new challenges for the development of software tools that aid designers of corresponding quantum algorithms. The different computational primitives render classical simulation of quantum circuits particularly challenging. While the logic simulation of conventional circuits is comparatively simple with linear complexity with respect to the number of gates, quantum circuit simulation has to deal with the exponential memory requirements to represent quantum states on non-quantum hardware with respect to the number of qubits. Decision Diagrams (DDs) address this challenge through exploitation of redundancies in matrices and vectors to provide significantly more compact representations in many cases. Moreover, the probabilistic nature of quantum computations enables another angle to tackle the complexity: Quantum algorithms are resistant to some degree against small inaccuracies in the quantum state as these only lead to small changes in the outcome probabilities. We propose to exploit this resistance against (small) errors to gain even more compact decision diagrams. In this work, we investigate the potential of approximation in quantum circuit simulation in detail. To this end, we first present four dedicated schemes that exploit the error resistance and efficiently approximate quantum states represented by decision diagrams. Subsequently, we propose two simulation strategies that utilize those approximations schemes in order to improve the efficiency of DD-based quantum circuit simulation, while, at the same time, allowing the user to control the resulting degradation in accuracy. We empirically show that the proposed approximation schemes reduce the size of decision diagrams substantially and also analytically prove the effect of multiple approximations on the attained accuracy. Eventually, this enables speed-ups of the resulting approximate quantum circuit simulation of up to several orders of magnitudes—again, while controlling the fidelity of the result.
Quantum-mechanical phenomena are playing an increasing role in information processing as transistor sizes approach the nanometer level, while the securest forms of communication rely on quantum data encoding. When they involve a finite number of basis states, these phenomena can be modeled as quantum circuits, the quantum analogue of conventional or classical logic circuits. Simulation of quantum circuits can therefore be used as a tool to evaluate issues in the design of quantum information processors. Unfortunately, simulating such phenomena efficiently is exceedingly difficult. The matrices representing quantum operators (gates) and vectors modeling quantum states grow exponentially with the number of quantum bits. The information represented by quantum states and operators often exhibits structure that can be exploited when simulating certain classes of quantum circuits. We study the development of simulation methods that run on classical computers and take advantage of such repetitions and redundancies. In particular, we define a new data structure for simulating quantum circuits called the quantum information decision diagram (QuIDD). A QuIDD is a compressed graph representation of a vector or matrix and permits computations to be performed directly on the compressed data. We develop a comprehensive set of algorithms for operating on QuIDDs in both the state-vector and density-matrix formats, and evaluate their complexity. These algorithms have been implemented in a general-purpose simulator program for quantum-mechanical applications called QuIDDPro. Through extensive experiments conducted on representative quantum simulation applications, including Grover's search algorithm, error characterization, and reversible circuits, we demonstrate that QuIDDPro is faster than other existing quantum-mechanical simulators such as the National Institute of Standards and Technology's QCSim program, and is far more memory-efficient. Using QuIDDPro, we explore the advantages of quantum computation over classical computation, simulate quantum errors and error correction, and study the impact of numerical precision on the fidelity of simulations. We also develop several novel algorithms for testing quantum circuit equivalence and compare them empirically. The QuIDDPro software is equipped with a user-friendly interface and is distributed with numerous example scripts. It has been used as a laboratory supplement for quantum computing courses at several universities.
Since people became aware of the power of quantum phenomena in the domain of traditional computation, a great number of complex problems that were once considered intractable in the classical world have been tackled. The downsides of quantum supremacy are its high cost and unpredictability. In order to obtain a stronger control over quantum interactions(by reducing its noise) and reduce its cost, numerous researchers are relying on quantum simulators running on classical computers. The critical obstacle facing classical computers in the task of quantum simulation is its limited memory space. Quantum simulation intrinsically models the state evolution of quantum subsystems, i.e., qubits. Qubits are mathematically constructed in Hilbert space whose size grows exponentially with its number. Consequently, the scalability of the straightforward statevector approach is limited. It has been proven effective to adopt decision diagrams to mitigate the memory cost issue in various fields. In recent years, researchers have adapted decision diagrams into different forms for representing quantum states and performing quantum calculations efficiently. This leads to the study of decision diagram based quantum simulation. However, their advantage of memory efficiency does not let it replace the mainstream statevector and tensor network based approaches. We argue the reason is the lack of effective parallelization strategies in performing calculations on decision diagrams. In this work, we explore several strategies for parallelizing decision diagram operations with the focus on leveraging them for quantum simulations. The target is to find the optimal parallelization strategies and improve the performance of decision diagram based quantum simulation. Based on the experiment results, our proposed strategy achieves a 2-3 times faster simulation of Grover's algorithm and random circuits than the state-of-the-art single-thread DD-based simulator DDSIM.