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EPJ Quantum Technology

A co-optimization qubit mapping algorithm via two-stage search and bidirectional look

Kang Xu, Peian Chen, Yukun Wang, Dandan Li

Published Aug 21, 2026• 0 citations
Publisher

Abstract

Abstract Current quantum devices typically lack full qubit connectivity, making it difficult to directly execute logical circuits on quantum devices. This limitation necessitates quantum circuit mapping algorithms to insert SWAP gates, dynamically remapping logical qubits to physical qubits, and transforming logical circuits into physical circuits that comply with device connectivity constraints. However, the insertion of SWAP gates increases both the gate count and circuit depth, ultimately reducing the fidelity of quantum algorithms. To achieve a balanced optimization of these two objectives, we propose the TANGO algorithm. The algorithm first formulates an evaluation function that balances the impact of qubit mapping on both mapped and unmapped nodes is introduced to enhance initial mapping quality, thereby reducing the number of SWAP gates inserted during routing. Next, we design an innovative two-stage routing algorithm that prioritizes the number of executable gates as the primary evaluation metric while also considering quantum gate distance, circuit depth, {and a novel bidirectional-look SWAP strategy}, which optimizes SWAP gate selection in conjunction with preceding gates, improving the effectiveness of the mapping algorithm. Finally, by integrating advanced quantum gate optimization techniques, the algorithm's overall performance is further enhanced. Experimental results demonstrate that, compared to state-of-the-art methods, the proposed algorithm achieves multi-objective co-optimization of gate count and circuit depth across various benchmarks and quantum devices, exhibiting significant performance advantages.

DOI
10.1140/epjqt/s40507-026-00553-4

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Abstract Current quantum devices typically lack full qubit connectivity, making it difficult to directly execute logical circuits on quantum devices. This limitation necessitates quantum circuit mapping algorithms to insert SWAP gates, dynamically remapping logical qubits to physical qubits, and transforming logical circuits into physical circuits that comply with device connectivity constraints. However, the insertion of SWAP gates increases both the gate count and circuit depth, ultimately reducing the fidelity of quantum algorithms. To achieve a balanced optimization of these two objectives, we propose the TANGO algorithm. The algorithm first formulates an evaluation function that balances the impact of qubit mapping on both mapped and unmapped nodes is introduced to enhance initial mapping quality, thereby reducing the number of SWAP gates inserted during routing. Next, we design an innovative two-stage routing algorithm that prioritizes the number of executable gates as the primary evaluation metric while also considering quantum gate distance, circuit depth, {and a novel bidirectional-look SWAP strategy}, which optimizes SWAP gate selection in conjunction with preceding gates, improving the effectiveness of the mapping algorithm. Finally, by integrating advanced quantum gate optimization techniques, the algorithm's overall performance is further enhanced. Experimental results demonstrate that, compared to state-of-the-art methods, the proposed algorithm achieves multi-objective co-optimization of gate count and circuit depth across various benchmarks and quantum devices, exhibiting significant performance advantages.
Quantum computing has already demonstrated great computational potential across multiple domains and has received more and more attention. However, due to the connectivity limitations of Noisy Intermediate-Scale Quantum (NISQ) devices, most of the quantum algorithms cannot be directly executed without the help of inserting SWAP gates. Nevertheless, more SWAP gates lead to a longer execution time and, inevitably, lower fidelity of the algorithm. To this end, this paper proposes an optimized qubit mapping algorithm based on a dynamic look-ahead strategy to minimize the number of SWAP gates inserted. Firstly, a heuristic algorithm is proposed based on maximizing physical qubit connectivity to generate the optimal initial qubit mapping, which reduces the need for logical qubit shifts during subsequent SWAP gate insertion. Secondly, in the form of directed acyclic graphs, we identify quantum gates that violate the constraints of physical coupling and insert SWAP gates to remap qubits, thereby overcoming the limitations of qubit interactions. Finally, the optimal SWAP gate insertion strategy is built by comparing the cost of different SWAP gate insertion strategies through a multi-window look-ahead strategy to reduce the number of SWAP gates inserted. The experimental results show that the strategy in this paper decreases the number of SWAP gate insertions and significantly reduces the depth of quantum circuits when performing qubit mapping compared with state-of-the-art methods.
Due to little considerations in the hardware constraints, e.g., limited connections between physical qubits to enable two-qubit gates, most quantum algorithms cannot be directly executed on the Noisy Intermediate-Scale Quantum (NISQ) devices. Dynamically remapping logical qubits to physical qubits in the compiler is needed to enable the two-qubit gates in the algorithm, which introduces additional operations and inevitably reduces the fidelity of the algorithm. Previous solutions in finding such remapping suffer from high complexity, poor initial mapping quality, and limited flexibility and control. To address these drawbacks mentioned above, this paper proposes a SWAP-based Bidirectional heuristic search algorithm (SABRE), which is applicable to NISQ devices with arbitrary connections between qubits. By optimizing every search attempt, globally optimizing the initial mapping using a novel reverse traversal technique, introducing the decay effect to enable the trade-off between the depth and the number of gates of the entire algorithm, SABRE outperforms the best known algorithm with exponential speedup and comparable or better results on various benchmarks.
In the era of noisy intermediate-scale quantum (NISQ) computing, the limited connectivity between qubits is one of the common physical limitations faced by current quantum computing devices. Quantum circuit mapping methods transform quantum circuits into equivalent circuits that satisfy physical connectivity constraints by remapping logical qubits, making them executable. The optimization problem of quantum circuit mapping has NP-hard computational complexity, and existing heuristic mapping algorithms still have significant potential for optimization in terms of the number of quantum gates generated. To reduce the number of SWAP gates inserted during mapping, the solution space of the mapping problem is represented as a tree structure, and the mapping process is equivalent to traversing this tree structure. To effectively and efficiently complete the search process, a beam search framework (BSF) is proposed for solving quantum circuit mapping. By iteratively selecting, expanding, and making decisions, high-quality target circuits are generated. Experimental results show that this method can significantly reduce the number of inserted SWAP gates on medium to large circuits, achieving an average reduction of 44% compared to baseline methods, and is applicable to circuits of various sizes and complexities.
The design of quantum circuits is often idealized, however, executing these circuits on quantum devices necessitates meeting the connectivity constraints imposed on physical devices by inserting SWAP gates. Various methods based on exact and heuristic solutions have been proposed to address the challenge of minimizing the degradation of quantum circuit performance due to factors such as circuit depth and the number of inserted SWAP gates while solving the qubit mapping problem. However, exact methods suffer from long runtime, lack of flexibility, and difficulty in scaling to larger circuits, while heuristic methods struggle to find the global optimum. Recently, solving the qubit mapping problem using machine learning approaches has emerged as a novel and promising solution. This paper studies qubit mapping optimization techniques based on machine learning, optimizing a model that solves the qubit mapping problem using the Deep Q-Network (DQN) algorithm, resulting in a SA-DQN model with superior performance in terms of circuit depth. The optimization efforts include improving the SWAP insertion strategy during the qubit mapping process and optimizing the model's initial mapping. Our SA-DQN model, tested on the IBM QX5 architecture and Rigetti 19Q Acorn architecture with a set of test circuits, achieved an improvement of 7.62% to 8.51% and 8.37% to 9.17% in circuit depth ratio compared to the original DQN model, respectively.
The rapid progress of physical implementation of quantum computers paved the way of realising the design of tools to help users write quantum programs for any given quantum devices. The physical constraints inherent to the current NISQ architectures prevent most quantum algorithms from being directly executed on quantum devices. To enable two-qubit gates in the algorithm, existing works focus on inserting SWAP gates to dynamically remap logical qubits to physical qubits. However, their schemes lack the consideration of the depth of generated quantum circuits. In this work, we propose a depth-aware SWAP insertion scheme for qubit mapping problem in the NISQ era.
Despite rapid advances in quantum computing technologies, the qubit connectivity limitation remains to be a critical challenge. Both near-term NISQ quantum computers and relatively long-term scalable quantum architectures do not offer full connectivity. As a result, quantum circuits may not be directly executed on quantum hardware, and a quantum compiler needs to perform qubit routing to make the circuit compatible with the device layout. During the qubit routing step, the compiler inserts SWAP gates and performs circuit transformations. Given the connectivity topology of the target hardware, there are typically multiple qubit routing candidates. The state-of-the-art compilers use a cost function to evaluate the number of SWAP gates for different routes and then select the one with the minimum number of SWAP gates. After qubit routing, the quantum compiler performs gate optimizations upon the circuit with the newly inserted SWAP gates.In this paper, we observe that the aforementioned qubit routing is not optimal, and qubit routing should not be independent on subsequent gate optimizations. We find that with the consideration of gate optimizations, not all of the SWAP gates have the same basis-gate cost. These insights lead to the development of our qubit routing algorithm, NASSC (Not All Swaps have the Same Cost). NASSC is the first algorithm that considers the subsequent optimizations during the routing step. Our optimization-aware qubit routing leads to better routing decisions and benefits subsequent optimizations. We also propose a new optimization-aware decomposition for the inserted SWAP gates. Our experiments show that the routing overhead compiled with our routing algorithm is reduced by up to 69.30% (21.30% on average) in the number of CNOT gates and up to 43.50% (7.61% on average) in the circuit depth compared with the state-of-the-art scheme, SABRE.