Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2507.04335

Node Replacement based Approximate Quantum Simulation with Decision Diagrams

Yan, Yexin, Stefan Hillmich, Robert Wille, Christian Mayr

Published Jul 6, 2025• 0 citations
arXivHTMLPDF

Abstract

Simulating a quantum circuit with a classical computer requires exponentially growing resources. Decision diagrams exploit the redundancies in quantum circuit representation to efficiently represent and simulate quantum circuits. But for complicated quantum circuits like the quantum supremacy benchmark, there is almost no redundancy to exploit. Therefore, it often makes sense to do a trade-off between simulation accuracy and memory requirement. Previous work on approximate simulation with decision diagrams exploits this trade-off by removing less important nodes. In this work, instead of removing these nodes, we try to find similar nodes to replace them, effectively slowing down the fidelity loss when reducing the memory. In addition, we adopt Locality Sensitive Hashing (LSH) to drastically reduce the computational complexity for searching for replacement nodes. Our new approach achieves a better memory-accuracy trade-off for representing a quantum circuit with decision diagrams with minimal run time overhead. Notably, our approach shows good scaling properties when increasing the circuit size and depth. For the first time, a strong better-than-linear trade-off between memory and fidelity is demonstrated for a decision diagram based quantum simulation when representing the quantum supremacy benchmark circuits at high circuit depths, showing the potential of drastically reducing the resource requirement for approximate simulation of the quantum supremacy benchmarks on a classical computer.

Related papers

Approximating Decision Diagrams for Quantum Circuit Simulation

Stefan Hillmich, Alwin Zulehner, Richard Kueng +2 more

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.

ACM Transactions on Quantum Computing•Apr 25, 2022•14 citations

As Accurate as Needed, as Efficient as Possible: Approximations in DD-based Quantum Circuit Simulation

Stefan Hillmich, Richard Kueng, Igor L. Markov +1 more

Quantum computers promise to solve important problems faster than conventional computers. However, unleashing this power has been challenging. In particular, design automation runs into (1) the probabilistic nature of quantum computation and (2) exponential requirements for computational resources on non-quantum hardware. In quantum circuit simulation, Decision Diagrams (DDs) have previously shown to reduce the required memory in many important cases by exploiting redundancies in the quantum state. In this paper, we show that this reduction can be amplified by exploiting the probabilistic nature of quantum computers to achieve even more compact representations. Specifically, we propose two new DD-based simulation strategies that approximate the quantum states to attain more compact representations, while, at the same time, allowing the user to control the resulting degradation in accuracy. We also analytically prove the effect of multiple approximations on the attained accuracy and empirically show that the resulting simulation scheme enables speed-ups up to several orders of magnitudes.

arXiv:2012.05615•Design, Automation, and Test in Europe•Feb 1, 2021•17 citations

Approximation of Quantum States Using Decision Diagrams

Alwin Zulehner, Stefan Hillmich, Igor L. Markov +1 more

The computational power of quantum computers poses major challenges to new design tools since representing pure quantum states typically requires exponentially large memory. As shown previously, decision diagrams can reduce these memory requirements by exploiting redundancies. In this work, we demonstrate further reductions by allowing for small inaccuracies in the quantum state representation. Such inaccuracies are legitimate since quantum computers themselves experience gate and measurement errors and since quantum algorithms are somewhat resistant to errors (even without error correction). We develop four dedicated schemes that exploit these observations and effectively approximate quantum states represented by decision diagrams. We empirically show that the proposed schemes reduce the size of decision diagrams by up to several orders of magnitude while controlling the fidelity of approximate quantum state representations.

arXiv:2002.04904•Asia and South Pacific Design Automation Conference (ASP-DAC)•Jan 1, 2020•12 citations

Parallelizing Quantum Simulation With Decision Diagrams

Shaowen Li, Yusuke Kimura, Hiroyuki Satō +1 more

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

Stripping Quantum Decision Diagrams of their Identity

Aaron Sander, Ioan-Albert Florea, Lukas Burgholzer +1 more

Classical representations of quantum states and operations as vectors and matrices are plagued by an exponential growth in memory and runtime requirements for increasing system sizes. Based on their use in classical computing, an alternative data structure known as Decision Diagrams (DDs) has been proposed, which, in many cases, provides both a more compact representation and more efficient computation. In the classical realm, decades of research have been conducted on DDs and numerous variations tailored for specific applications exist. However, DDs for quantum computing are just in their infancy and there is still room for tailoring them to this new technology. In particular, existing representations of DDs require extending all operations in a quantum circuit to the full system size through extension by nodes representing identity matrices. In this work, we make an important step forward for quantum DDs by stripping these identity structures from quantum operations. This significantly reduces the number of nodes required to represent them as well as eases the pressure on key building blocks of their implementation. As a result, we obtain a structure that is more natural for quantum computing and significantly speeds up computations—with a runtime improvement of up to 70 × compared to the state-of-the-art.

arXiv:2406.11959•IEEE International Conference on Quantum Software (QSW)•Jul 7, 2024•2 citations

Scaling Quantum Circuit Simulation: Optimizations for the FTDD Framework

Vicente López-Oliva, Badía Contelles Jose M., Maribel Castillo Catalán

Classical simulation remains essential for the development of quantum algorithms; however, traditional matrix-based methods struggle with scalability. While the Fast Tensor Decision Diagram (FTDD) framework offers a hybrid approach by combining the strengths of Tensor Networks and Decision Diagrams (DDs) for memory-efficient simulation, its core operations are often bottlenecked by memory management structures. Our primary motivation is to address the high computational and spatial cost associated with the hash tables, as well as to simplify the complexity of the simulation workflow. This work introduces systematic, low-level optimizations to the framework. Our main contribution is the enhancement of the memory management layer for the internal tables. We implemented a node reuse mechanism and designed a controlled strategy for hash table growth. These improvements minimize the memory footprint and reduce the costly hashing overhead inherent in DD-based simulators. We performed a thorough experimental analysis using a diverse set of quantum circuits, which were characterized by well-established metrics. Experimental results demonstrate that our enhancements improve the tool's performance and scalability. Notably, these advancements enable the simulation of complex circuits, such as Quantum Fourier Transform, with hundreds of qubits, on a standard server with 2-Tb of memory.

•Apr 27, 2026•0 citations
Simulating a quantum circuit with a classical computer requires exponentially growing resources. Decision diagrams exploit the redundancies in quantum circuit representation to efficiently represent and simulate quantum circuits. But for complicated quantum circuits like the quantum supremacy benchmark, there is almost no redundancy to exploit. Therefore, it often makes sense to do a trade-off between simulation accuracy and memory requirement. Previous work on approximate simulation with decision diagrams exploits this trade-off by removing less important nodes. In this work, instead of removing these nodes, we try to find similar nodes to replace them, effectively slowing down the fidelity loss when reducing the memory. In addition, we adopt Locality Sensitive Hashing (LSH) to drastically reduce the computational complexity for searching for replacement nodes. Our new approach achieves a better memory-accuracy trade-off for representing a quantum circuit with decision diagrams with minimal run time overhead. Notably, our approach shows good scaling properties when increasing the circuit size and depth. For the first time, a strong better-than-linear trade-off between memory and fidelity is demonstrated for a decision diagram based quantum simulation when representing the quantum supremacy benchmark circuits at high circuit depths, showing the potential of drastically reducing the resource requirement for approximate simulation of the quantum supremacy benchmarks on a classical computer.
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 computers promise to solve important problems faster than conventional computers. However, unleashing this power has been challenging. In particular, design automation runs into (1) the probabilistic nature of quantum computation and (2) exponential requirements for computational resources on non-quantum hardware. In quantum circuit simulation, Decision Diagrams (DDs) have previously shown to reduce the required memory in many important cases by exploiting redundancies in the quantum state. In this paper, we show that this reduction can be amplified by exploiting the probabilistic nature of quantum computers to achieve even more compact representations. Specifically, we propose two new DD-based simulation strategies that approximate the quantum states to attain more compact representations, while, at the same time, allowing the user to control the resulting degradation in accuracy. We also analytically prove the effect of multiple approximations on the attained accuracy and empirically show that the resulting simulation scheme enables speed-ups up to several orders of magnitudes.
The computational power of quantum computers poses major challenges to new design tools since representing pure quantum states typically requires exponentially large memory. As shown previously, decision diagrams can reduce these memory requirements by exploiting redundancies. In this work, we demonstrate further reductions by allowing for small inaccuracies in the quantum state representation. Such inaccuracies are legitimate since quantum computers themselves experience gate and measurement errors and since quantum algorithms are somewhat resistant to errors (even without error correction). We develop four dedicated schemes that exploit these observations and effectively approximate quantum states represented by decision diagrams. We empirically show that the proposed schemes reduce the size of decision diagrams by up to several orders of magnitude while controlling the fidelity of approximate quantum state representations.
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.
Classical representations of quantum states and operations as vectors and matrices are plagued by an exponential growth in memory and runtime requirements for increasing system sizes. Based on their use in classical computing, an alternative data structure known as Decision Diagrams (DDs) has been proposed, which, in many cases, provides both a more compact representation and more efficient computation. In the classical realm, decades of research have been conducted on DDs and numerous variations tailored for specific applications exist. However, DDs for quantum computing are just in their infancy and there is still room for tailoring them to this new technology. In particular, existing representations of DDs require extending all operations in a quantum circuit to the full system size through extension by nodes representing identity matrices. In this work, we make an important step forward for quantum DDs by stripping these identity structures from quantum operations. This significantly reduces the number of nodes required to represent them as well as eases the pressure on key building blocks of their implementation. As a result, we obtain a structure that is more natural for quantum computing and significantly speeds up computations—with a runtime improvement of up to 70 × compared to the state-of-the-art.
Classical simulation remains essential for the development of quantum algorithms; however, traditional matrix-based methods struggle with scalability. While the Fast Tensor Decision Diagram (FTDD) framework offers a hybrid approach by combining the strengths of Tensor Networks and Decision Diagrams (DDs) for memory-efficient simulation, its core operations are often bottlenecked by memory management structures. Our primary motivation is to address the high computational and spatial cost associated with the hash tables, as well as to simplify the complexity of the simulation workflow. This work introduces systematic, low-level optimizations to the framework. Our main contribution is the enhancement of the memory management layer for the internal tables. We implemented a node reuse mechanism and designed a controlled strategy for hash table growth. These improvements minimize the memory footprint and reduce the costly hashing overhead inherent in DD-based simulators. We performed a thorough experimental analysis using a diverse set of quantum circuits, which were characterized by well-established metrics. Experimental results demonstrate that our enhancements improve the tool's performance and scalability. Notably, these advancements enable the simulation of complex circuits, such as Quantum Fourier Transform, with hundreds of qubits, on a standard server with 2-Tb of memory.