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

ACM/IEEE Design Automation Conference (DAC)

Efficient Circuits for Quantum Search over 2D Square Lattice Architecture

Shaohan Hu, Dmitri Maslov, Marco Pistoia, Jay Gambetta

Published May 23, 2019• 9 citations
Publisher

Abstract

Quantum computing has increasingly drawn interest and investments from the academic, industrial, and governmental research communities worldwide. Among quantum algorithms, Quantum Search is important for its quadratic speedup over its classical-computing counterpart. A key ingredient in its implementation is the Multi-Control Toffoli (MCT) gate, which creates a Boolean product of control variables and XORs it into the target. On an idealized quantum computer, all-to-all connectivity would eliminate the need to use SWAP gates to communicate information. This is, however, not affordable in the current Noisy Intermediate-Scale Quantum (NISQ) computing era. In this work, we discuss how to efficiently implement MCT gates on 2D Square Lattices (2DSL), suitable for superconducting circuits, by taking advantage of relative-phase Toffoli gates and H-tree layouts to drastically reduce resulting circuits' depths and the amount of SWAPping required.

DOI
10.1145/3316781.3322464

Related papers

Layout Synthesis for Quantum Circuits Considering Toffoli Gate Decomposition

Po-Wei Chen, Sheng-Tan Huang, Shao‐Yun Fang

State-of-the-art studies on quantum layout synthesis have proposed various approaches based on the assumption that the input circuit is only composed of single-qubit and two-qubit gates. This assumption greatly simplifies the layout synthesis problem, and thus they only require ensuring that all controlled-NOT (CNOT) gates satisfy the hardware constraints imposed by a given coupling graph. However, during the design of quantum circuits, multi-controlled Toffoli (MCT) gates are usually used to better characterize the function of the circuits. Directly decomposing them into single and two-qubit gates with a fixed routine ignores the flexibility and optimization opportunity provided by various decomposition (i.e., logic synthesis) possibilities and thus suffers from sub-optimal results. This article proposes a co-optimization approach for quantum logic and layout synthesis. The MCT gates are first decomposed into Toffoli gates, and an efficient qubit mapping checking process is proposed to optimally solve the SWAP-free layout synthesis problem by automatically determining the decomposition result of each Toffoli gate. If a SWAP-free result cannot be found, the proposed algorithm flow is then used to obtain a solution that minimizes the total cost that simultaneously counts the cost caused by the different decomposition methods for Toffoli gates and the cost induced by SWAP gates. Compared with a state-of-the-art method, the proposed approach reduces the number of additional CNOT gates by 16% with around 25X runtime speedup for the cases with SWAP-free solutions, which cannot be obtained without the co-optimization approach.

ACM Transactions on Design Automation of Electronic Systems•Jun 16, 2025•1 citation

Optimal Toffoli-Depth Multi-Controlled Toffoli Decomposition in 2D Qubit Layout

Anik Basu Bhaumik, Suman Dutta, Anupam Chattopadhyay

The multi-controlled Toffoli (MCT) gate is a key primitive in quantum arithmetic, oracle construction, and quantum cryptanalysis. Although recent work has established optimal Toffoli-depth MCT decompositions under all-to-all qubit connectivity, their realization on near-term quantum hardware with restricted qubit connectivity remains largely unexplored. While general-purpose quantum mappers can route arbitrary circuits, they do not explicitly exploit the repeated interaction patterns inherent in MCT decompositions. In our present paper, we study architecture-aware mappings of optimal Toffoli-depth MCT decompositions onto restricted two-dimensional qubit layouts. We begin with a structured geometric placements that preserve the parallelism of state-of-the-art Toffoli and MCT decompositions with no additional depth overhead. We further introduce a motif-based packing framework in which decomposition layers are represented by interaction motifs derived from basic Toffoli gates. By embedding these motifs vertex-disjointly into hardware graphs, we characterize the minimum-size topologies supporting the required qubit resources and derive explicit bounds on the resulting depth overhead under tight qubit budgets. Finally, we compare these bounds with routing-aware placement heuristics and empirically evaluate the effectiveness of embedding different motifs across a range of hardware topologies.

quant-pharXiv:2606.15113•Jun 16, 2026•0 citations

Device-Aware Decomposition and Mapping of Multiple-Control Toffoli Circuits in Quantum Computing Systems

Yibo Jiang, Xueyun Cheng, Pengcheng Zhu +2 more

Abstract With the rapid development of quantum computing, quantum computing devices have entered the Noisy Intermediate-Scale Quantum(NISQ)era. However, current NISQ quantum computing devices still face the challenge of physical limitations. To enable quantum algorithms to be directly executed on quantum computing devices, high-level quantum gates (such as MCT gates) in quantum circuits need to be decomposed and mapped onto quantum computing devices. This paper proposes a quantum device-aware method for decomposing high-level quantum gates. This method selects an optimal sub-topology of the quantum device as the target based on the circuit scale and the goal of reducing error rates. According to the proposed MCT gate decomposition rules, an associated gate-pairs generation algorithm is designed to facilitate subsequent decomposition. The control bits of the MCT gates are partitioned based on the associated gate-pair to better match the selected sub-topology. After eliminating redundant gates through the inverse decomposition of associated gate-pairs, the mapping is performed. Experimental results show that the proposed decomposition method reduces the number of basic quantum gates in quantum circuits, with an average optimization rate of 14.6%. For benchmark circuits decomposed using the method, applying the routing algorithm for mapping requires fewer additional gates, with an average optimization rate of 20.6%. The fidelity of the circuits improved by an average of 14.1% after mapping.

•Sep 13, 2024•0 citations

A T-depth two Toffoli gate for 2D square lattice architectures

Alexandru Paler, Evan E. Dobbs, Joseph S. Friedman

We present a novel Clifford+T decomposition of a Toffoli gate. Our decomposition requires no SWAP gates in order to be implemented on 2D square lattices of qubits. This decomposition enables shallower, more fault-tolerant quantum computations on both NISQ and error-corrected architectures. We present the derivation of the circuit, and illustrate the qubit mapping on a Sycamore-like architecture.

arXiv:2311.12510•IEEE/ACM International Symposium on Nanoscale Architectures (NANOARCH)•Dec 18, 2023•0 citations

A Cost-Effective Layout-Aware Quantum Circuit Synthesis For Triangular, Square, and Heavy-Hex Layouts

Yang, Sonia, Ali Al-Bayaty, Marek A. Perkowski

The quantum layout and the mapping of logical to physical qubits are crucial in quantum circuit synthesis for a real quantum computer. Circuits that include large $n$-bit Toffoli gates ($n \geq 3$), such as those designed from cost-expensive gates and hard-to-decompose Exclusive-or Sum of Products (ESOP) expressions, have complications of effective mappings into contemporary quantum layouts, such as the square grid and heavy-hex layouts. These complications are primarily caused by the limited connectivity among the physical qubits in such layouts, leading to the insertion of many additional SWAP gates. This paper introduces a new quantum circuit synthesis methodology by exploring the advantage of a Positive Davio lattice (PDL) as an intermediate representation to create our proposed triangular layout and layout-aware circuits. From these circuits, we introduce and form the SWAT gate, composed of a SWAP gate followed by a 3-bit Toffoli gate. To illustrate the usefulness of our method for existing industrial quantum layouts, we also introduce cost-effective mappings of the resulting circuits onto square grid and heavy-hex layouts without additional SWAP gates. This is done with the help of the SWAT gate. Our research highlights PDLs as an efficient tool for layout-aware quantum circuit synthesis.

arXiv:2506.20098•Jun 25, 2025•0 citations

Dynamic Realization of Multiple Control Toffoli Gate

Abhoy Kole, Arighna Deb, Kamalika Datta +1 more

Dynamic Quantum Circuits (DQC) is an inevitable solution for today's Noisy Intermediate Scale Quantum (NISQ) systems. This enables realization of an n-qubit (where,$n > 2$) quantum circuit using only 2-qubits with the aid of additional non-unitary operations which is evident from the recent dynamic realizations of algorithms like Quantum Phase Estimation (QPE) and Bernstein- Vazirani (BV) as well as 3-qubit Toffoli operation. In this work, we introduce two different dynamic realization schemes for Multiple Control Toffoli (MCT) gates, for the first time to the best of our knowledge. We compare the respective realizations in terms of resources (e.g., gate, depth and nearest neighbor overhead) and computational accuracy. For this purpose, we apply the proposed dynamic MCT gates in Deutsch-Jozsa (DJ) algorithm, thereby realizing the traditional DJ algorithm as DQCs. Experimental evaluations show that one dynamic scheme for MCT gates leads to DQCs with better computational accuracy, while the other one results in DQCs with better computational resources.

Design, Automation, and Test in Europe•Mar 25, 2024•6 citations
Quantum computing has increasingly drawn interest and investments from the academic, industrial, and governmental research communities worldwide. Among quantum algorithms, Quantum Search is important for its quadratic speedup over its classical-computing counterpart. A key ingredient in its implementation is the Multi-Control Toffoli (MCT) gate, which creates a Boolean product of control variables and XORs it into the target. On an idealized quantum computer, all-to-all connectivity would eliminate the need to use SWAP gates to communicate information. This is, however, not affordable in the current Noisy Intermediate-Scale Quantum (NISQ) computing era. In this work, we discuss how to efficiently implement MCT gates on 2D Square Lattices (2DSL), suitable for superconducting circuits, by taking advantage of relative-phase Toffoli gates and H-tree layouts to drastically reduce resulting circuits' depths and the amount of SWAPping required.
State-of-the-art studies on quantum layout synthesis have proposed various approaches based on the assumption that the input circuit is only composed of single-qubit and two-qubit gates. This assumption greatly simplifies the layout synthesis problem, and thus they only require ensuring that all controlled-NOT (CNOT) gates satisfy the hardware constraints imposed by a given coupling graph. However, during the design of quantum circuits, multi-controlled Toffoli (MCT) gates are usually used to better characterize the function of the circuits. Directly decomposing them into single and two-qubit gates with a fixed routine ignores the flexibility and optimization opportunity provided by various decomposition (i.e., logic synthesis) possibilities and thus suffers from sub-optimal results. This article proposes a co-optimization approach for quantum logic and layout synthesis. The MCT gates are first decomposed into Toffoli gates, and an efficient qubit mapping checking process is proposed to optimally solve the SWAP-free layout synthesis problem by automatically determining the decomposition result of each Toffoli gate. If a SWAP-free result cannot be found, the proposed algorithm flow is then used to obtain a solution that minimizes the total cost that simultaneously counts the cost caused by the different decomposition methods for Toffoli gates and the cost induced by SWAP gates. Compared with a state-of-the-art method, the proposed approach reduces the number of additional CNOT gates by 16% with around 25X runtime speedup for the cases with SWAP-free solutions, which cannot be obtained without the co-optimization approach.
The multi-controlled Toffoli (MCT) gate is a key primitive in quantum arithmetic, oracle construction, and quantum cryptanalysis. Although recent work has established optimal Toffoli-depth MCT decompositions under all-to-all qubit connectivity, their realization on near-term quantum hardware with restricted qubit connectivity remains largely unexplored. While general-purpose quantum mappers can route arbitrary circuits, they do not explicitly exploit the repeated interaction patterns inherent in MCT decompositions. In our present paper, we study architecture-aware mappings of optimal Toffoli-depth MCT decompositions onto restricted two-dimensional qubit layouts. We begin with a structured geometric placements that preserve the parallelism of state-of-the-art Toffoli and MCT decompositions with no additional depth overhead. We further introduce a motif-based packing framework in which decomposition layers are represented by interaction motifs derived from basic Toffoli gates. By embedding these motifs vertex-disjointly into hardware graphs, we characterize the minimum-size topologies supporting the required qubit resources and derive explicit bounds on the resulting depth overhead under tight qubit budgets. Finally, we compare these bounds with routing-aware placement heuristics and empirically evaluate the effectiveness of embedding different motifs across a range of hardware topologies.
Abstract With the rapid development of quantum computing, quantum computing devices have entered the Noisy Intermediate-Scale Quantum(NISQ)era. However, current NISQ quantum computing devices still face the challenge of physical limitations. To enable quantum algorithms to be directly executed on quantum computing devices, high-level quantum gates (such as MCT gates) in quantum circuits need to be decomposed and mapped onto quantum computing devices. This paper proposes a quantum device-aware method for decomposing high-level quantum gates. This method selects an optimal sub-topology of the quantum device as the target based on the circuit scale and the goal of reducing error rates. According to the proposed MCT gate decomposition rules, an associated gate-pairs generation algorithm is designed to facilitate subsequent decomposition. The control bits of the MCT gates are partitioned based on the associated gate-pair to better match the selected sub-topology. After eliminating redundant gates through the inverse decomposition of associated gate-pairs, the mapping is performed. Experimental results show that the proposed decomposition method reduces the number of basic quantum gates in quantum circuits, with an average optimization rate of 14.6%. For benchmark circuits decomposed using the method, applying the routing algorithm for mapping requires fewer additional gates, with an average optimization rate of 20.6%. The fidelity of the circuits improved by an average of 14.1% after mapping.
The quantum layout and the mapping of logical to physical qubits are crucial in quantum circuit synthesis for a real quantum computer. Circuits that include large nnn-bit Toffoli gates (n≥3n \geq 3n≥3), such as those designed from cost-expensive gates and hard-to-decompose Exclusive-or Sum of Products (ESOP) expressions, have complications of effective mappings into contemporary quantum layouts, such as the square grid and heavy-hex layouts. These complications are primarily caused by the limited connectivity among the physical qubits in such layouts, leading to the insertion of many additional SWAP gates. This paper introduces a new quantum circuit synthesis methodology by exploring the advantage of a Positive Davio lattice (PDL) as an intermediate representation to create our proposed triangular layout and layout-aware circuits. From these circuits, we introduce and form the SWAT gate, composed of a SWAP gate followed by a 3-bit Toffoli gate. To illustrate the usefulness of our method for existing industrial quantum layouts, we also introduce cost-effective mappings of the resulting circuits onto square grid and heavy-hex layouts without additional SWAP gates. This is done with the help of the SWAT gate. Our research highlights PDLs as an efficient tool for layout-aware quantum circuit synthesis.
Dynamic Quantum Circuits (DQC) is an inevitable solution for today's Noisy Intermediate Scale Quantum (NISQ) systems. This enables realization of an n-qubit (where,n>2n > 2n>2) quantum circuit using only 2-qubits with the aid of additional non-unitary operations which is evident from the recent dynamic realizations of algorithms like Quantum Phase Estimation (QPE) and Bernstein- Vazirani (BV) as well as 3-qubit Toffoli operation. In this work, we introduce two different dynamic realization schemes for Multiple Control Toffoli (MCT) gates, for the first time to the best of our knowledge. We compare the respective realizations in terms of resources (e.g., gate, depth and nearest neighbor overhead) and computational accuracy. For this purpose, we apply the proposed dynamic MCT gates in Deutsch-Jozsa (DJ) algorithm, thereby realizing the traditional DJ algorithm as DQCs. Experimental evaluations show that one dynamic scheme for MCT gates leads to DQCs with better computational accuracy, while the other one results in DQCs with better computational resources.