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The three-input TOFFOLI gate is the workhorse of circuit synthesis for classical logic operations on quantum data, e.g., reversible arithmetic circuits. In physical implementations, however, TOFFOLI gates are decomposed into six CNOT gates…

Quantum Physics · Physics 2011-11-03 Vivek V. Shende , Igor L. Markov

We present quantum circuits for comparison and increment operations that achieve an asymptotically optimal gate count of $\Theta(n)$ and depth of $\Theta(\log n)$ over the Clifford+Toffoli gate set, while using a provably minimal number of…

Quantum Physics · Physics 2026-03-16 Vivien Vandaele

Quantum Error Correction (QEC) is the cornerstone of practical Fault-Tolerant Quantum Computing (FTQC), but incurs enormous resource overheads. Circuits must decompose into Clifford+T gates, and the non-transversal T gates demand costly…

Quantum Physics · Physics 2025-07-15 Archisman Ghosh , Avimita Chatterjee , Swaroop Ghosh

The decomposition for controlled-$ZX$ gate in [Phys. Rev. A, 87, 062318 (2013)] has a shallow circuit depth $8n-20$ with no ancilla. Here we modify this decomposition to decompose $n$-qubit Toffoli gate with only $2n-3$ additional…

Quantum Physics · Physics 2023-05-25 Jian Leng , Fan Yang , Xiang-Bin Wang

Clifford circuit optimization is an important step in the quantum compilation pipeline. Major compilers employ heuristic approaches. While they are fast, their results are often suboptimal. Minimization of noisy gates, like 2-qubit CNOT…

Quantum Physics · Physics 2025-04-02 Irfansha Shaik , Jaco van de Pol

We consider the problem of the variational quantum circuit synthesis into a gate set consisting of the CNOT gate and arbitrary single-qubit (1q) gates with the primary target being the minimization of the CNOT count. First we note that…

Quantum Physics · Physics 2023-05-10 Nikita A. Nemkov , Evgeniy O. Kiktenko , Ilia A. Luchnikov , Aleksey K. Fedorov

We consider quantum gates for trapped ions using state-selective displacement of the ions. We generalize earlier work in order to treat arbitrary separations between the traps. This requires the impact of anharmonicity arising from the…

Quantum Physics · Physics 2016-09-08 Marek Sasura , Andrew Steane

An $(n+1)$-bit Toffoli gate is mainly utilized to construct other quantum gates and operators, such as Fredkin gates, arithmetical adders, and logical comparators, where $n \geq 2$. Several researchers introduced different methods to…

Quantum Physics · Physics 2025-11-03 Shanyan Chen , Ali Al-Bayaty , Xiaoyu Song , Marek Perkowski

In fault-tolerant quantum circuit synthesis, T gates supplied via magic states dominate space-time cost, while Clifford gates incur negligible overhead. Conventional flows minimize AND count in an {XOR, AND, NOT} basis as a proxy for T,…

Quantum Physics · Physics 2026-05-18 Hanyu Wang , Mingfei Yu , Xinrui Wu , Jason Cong

While implementing a quantum algorithm it is crucial to reduce the quantum resources, in order to obtain the desired computational advantage. For most fault-tolerant quantum error-correcting codes the cost of implementing the non-Clifford…

Quantum Physics · Physics 2023-02-10 Vlad Gheorghiu , Michele Mosca , Priyanka Mukhopadhyay

The synthesis of quantum operators involves decomposing general quantum gates into the gate set supported by a given quantum device. Multi-controlled gates are essential components in this process. In this work, we present an improved…

Quantum Physics · Physics 2025-08-13 Jefferson D. S. Silva , Adenilton J. da Silva

Fault-tolerant quantum computation (FTQC) is essential to implement quantum algorithms in a noise-resilient way, and thus to enjoy advantages of quantum computers even with presence of noise. In FTQC, a quantum circuit is decomposed into…

Quantum Physics · Physics 2025-07-23 Kohdai Kuroiwa , Yuya O. Nakagawa

Quantum computing has garnered significant interest for its potential to solve certain computational problems much faster than the best-known classical algorithms. A fully functional and scalable quantum computer could transform various…

Quantum Physics · Physics 2025-09-08 M. AbuGhanem

The quantum Fourier transform (QFT) is a ubiquitous quantum operation that is used in numerous quantum computing applications. The major obstacle to constructing a QFT circuit is that numerous elementary gates are required. Among the…

Quantum Physics · Physics 2024-07-23 Byeongyong Park , Doyeol Ahn

In this paper we show that it is possible to adapt a qudit scheme for creating a controlled-Toffoli created by Ralph et al. [Phys. Rev. A 75 011213] to be applicable to qubits. While this scheme requires more gates than standard schemes for…

Quantum Physics · Physics 2013-04-23 Katherine L. Brown , Anmer Daskin , Sabre Kais , Jonathan P. Dowling

Quantum computers have the potential to break classical cryptographic systems by efficiently solving problems such as the elliptic curve discrete logarithm problem using Shor's algorithm. While resource estimates for factoring-based…

Quantum Physics · Physics 2025-10-28 Quan Gu , Han Ye , Junjie Chen , Xiongfeng Ma

A fundamental problem of fault-tolerant quantum computation with quantum low-density parity-check (qLDPC) codes is the tradeoff between connectivity and universality. It is widely believed that in order to perform native logical…

Quantum Physics · Physics 2026-01-13 Guanyu Zhu , Ryohei Kobayashi , Po-Shen Hsin

Numerical optimization is used to design linear-optical devices that implement a desired quantum gate with perfect fidelity, while maximizing the success rate. For the 2-qubit CS (or CNOT) gate, we provide numerical evidence that the…

Quantum Physics · Physics 2015-05-13 Dmitry B. Uskov , Lev Kaplan , A. Matthew Smith , Sean D. Huver , Jonathan P. Dowling

Clifford gates play a role in the optimisation of Clifford+T circuits. Reducing the count and the depth of Clifford gates, as well as the optimal scheduling of T gates, influence the hardware and the time costs of executing quantum…

Quantum Physics · Physics 2019-06-18 Alexandru Paler , Robert Basmadjian

We present an algorithm for the approximate decomposition of diagonal operators, focusing specifically on decompositions over the Clifford+$T$ basis, that minimize the number of phase-rotation gates in the synthesized approximation circuit.…

Quantum Physics · Physics 2016-06-13 Jonathan Welch , Alex Bocharov , Krysta M. Svore