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Design, Automation, and Test in Europe

Extending the Design Space of Dynamic Quantum Circuits for Toffoli based Network

Abhoy Kole, Arighna Deb, Kamalika Datta, Rolf Drechsler

Published Apr 1, 2023• 7 citations
Publisher

Abstract

Recent advances in fault tolerant quantum systems allow to perform non-unitary operations like mid-circuit measurement, active reset and classically controlled gate operations in addition to the existing unitary gate operations. Real quantum devices that support these non-unitary operations enable us to execute a new class of quantum circuits, known as Dynamic Quantum Circuits (DQC). This helps to enhance the scalability, thereby allowing execution of quantum circuits comprising of many qubits by using at least two qubits. Recently DQC realizations of multi-qubit Quantum Phase Estimation (QPE) and Bernstein-Vazirani (BV) algorithms have been demonstrated in two separate experiments. However the dynamic transformation of complex quantum circuits consisting of Toffoli gate operations have not been explored yet. This motivates us to: (a) explore the dynamic realization of Toffoli gates by extending the design space of DQC for Toffoli networks, and (b) propose a general dynamic transformation algorithm for the first time to the best of our knowledge. More precisely, we introduce two dynamic transformation schemes (dynamic-1 and dynamic-2) for Toffoli gates, that differ with respect to the required number of classically controlled gate operations. For evaluation, we consider the Deutsch-Jozsa (DJ) algorithm composed of one or more Toffoli gates. Experimental results demonstrate that dynamic DJ circuits based on dynamic-2 Toffoli realization scheme provides better computational accuracy over the dynamic-1 scheme. Further, the proposed dynamic transformation scheme is generic and can also be applied to non-Toffoli quantum circuits, e.g. BV algorithm.

DOI
10.23919/date56975.2023.10137250

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Recent advances in fault tolerant quantum systems allow to perform non-unitary operations like mid-circuit measurement, active reset and classically controlled gate operations in addition to the existing unitary gate operations. Real quantum devices that support these non-unitary operations enable us to execute a new class of quantum circuits, known as Dynamic Quantum Circuits (DQC). This helps to enhance the scalability, thereby allowing execution of quantum circuits comprising of many qubits by using at least two qubits. Recently DQC realizations of multi-qubit Quantum Phase Estimation (QPE) and Bernstein-Vazirani (BV) algorithms have been demonstrated in two separate experiments. However the dynamic transformation of complex quantum circuits consisting of Toffoli gate operations have not been explored yet. This motivates us to: (a) explore the dynamic realization of Toffoli gates by extending the design space of DQC for Toffoli networks, and (b) propose a general dynamic transformation algorithm for the first time to the best of our knowledge. More precisely, we introduce two dynamic transformation schemes (dynamic-1 and dynamic-2) for Toffoli gates, that differ with respect to the required number of classically controlled gate operations. For evaluation, we consider the Deutsch-Jozsa (DJ) algorithm composed of one or more Toffoli gates. Experimental results demonstrate that dynamic DJ circuits based on dynamic-2 Toffoli realization scheme provides better computational accuracy over the dynamic-1 scheme. Further, the proposed dynamic transformation scheme is generic and can also be applied to non-Toffoli quantum circuits, e.g. BV algorithm.
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.
The Toffoli gate is a fundamental building block for quantum arithmetic and reversible logic, yet its efficient realization remains a major challenge in both near-term and fault-tolerant quantum architectures. Recent advances in dynamic quantum circuit capabilities, including mid-circuit measurement and classical feedforward, provide new opportunities for reducing the resource overhead of non-Clifford operations. In this work, we propose a set of dynamic decomposition strategies for multi-controlled Toffoli gates that exploit adaptive circuit execution and ancilla-assisted constructions. Our methods systematically reduce entangling-gate count, T-count, and T-depth compared with conventional static decompositions, while preserving fault-tolerance guarantees. Through analytical cost models and experimental evaluation, we demonstrate that relative-phase primitives and measurement-conditioned corrections enable scalable implementations with improved depth and resource efficiency.
The execution of large-scale quantum algorithms is currently constrained by the limited number of available qubits, qubit connectivity restrictions, and the inherent noise in quantum processors. To address these limitations, a design methodology known as Dynamic Quantum Circuits (DQC) has emerged. DQC leverages non-unitary operations-such as active reset, midcircuit measurement, and classically controlled gate operationsto reduce qubit requirements during circuit design. Recently, DQC-based implementations have been explored for various algorithms, including Shor’s Prime Factorization, Quantum Phase Estimation (QPE), and Bernstein-Vazirani (BV), as well as for key operations like state preparation, Toffoli networks, and nonlocal gates. While DQC offers a significant reduction in qubit usage, it introduces a trade-off in the form of increased circuit depth. Therefore, assessing the reliability of such circuits becomes crucial in the context of current quantum hardware architectures. In this paper, we analyze the reliability of DQC-based quantum circuit realizations as a function of qubit count and circuit depth. We present empirical results for two algorithms and evaluate how architectural parameters impact their reliability.
Dynamic quantum circuits (DQCs) provide a hardware-efficient route to quantum computing by reducing physical-qubit overhead and compressing circuit topology through mid-circuit measurements, qubit reset and reuse, and classical feed-forward control. Here, we demonstrate the advantages of DQCs on a single hybrid superconducting qubit-cavity processor by implementing a hierarchy of algorithms with increasing complexity. This hybrid architecture consists of a high-dimensional cavity qudit serving as the computational register and a dispersively coupled superconducting transmon ancilla that is repeatedly measured, reset, and reused to enable dynamic control. Using this device, we implement a 10-bit Bernstein-Vazirani algorithm with an average success probability of 82%, surpassing state-of-the-art dynamic and static implementations in both scale and performance; an 8-bit quantum phase-estimation protocol with estimation errors below 10-3; and the first dynamic-circuit implementation of Shor's algorithm on a superconducting platform, factoring 15 over all coprime bases with squared statistical overlap values above 99.8%. These results provide concrete benchmarks for future DQC implementations and highlight the versatile advantages of DQCs with the hybrid qubit-qudit architecture, establishing it as a promising route toward scalable, programmable quantum computation.
Over the last two decades there has been immense progress in the field of quantum computing. Although today we have demonstrable quantum computers with more than 1000 qubits, researchers are still trying to show how these machines can be utilized to get substantial benefit for certain applications. We are also in the Noisy Intermediate Scale Quantum (NISQ) era that imposes certain restrictions in utilizing the entire physical qubit space. Implementation of various large-scale quantum algorithms suffers from limited number of available qubits and also the qubit coupling restrictions of the target quantum processor. To this end an advanced class of quantum circuits called Dynamic Quantum Circuits (DQC) has been proposed, which can work with very few additional qubits using various non-unitary operations (viz., active reset, mid-circuit measurement and classically controlled gate operations). This paper particularly presents various design automation challenges that exists in the current NISQ era and shows how DQC can be exploited to overcome some of the challenges.
Qubit reuse offers a promising way to reduce the hardware demands of quantum circuits, but current approaches are largely restricted to reordering measurements and applying qubit resets. In this work, we present an approach to further optimize quantum circuits by fully utilizing the potential of dynamic quantum circuits-more precisely by moving measurements and introducing dynamic circuit primitives such as classically controlled gates in a way that forges entirely new pathways for qubit reuse. This significantly reduces the number of required qubits for a variety of circuits, creating new opportunities for running complex circuits on near-term devices with limited qubit counts. We show that the proposed approach drastically outperforms existing methods, reducing qubit requirements where previous approaches are unable to do so for popular quantum circuits such as Quantum Phase Estimation (QPE), Quantum Fourier Transform~(QFT), and Variational Quantum Eigensolver (VQE) ansätze, as well as leading to improvements of up to 95% for sparse random circuits.