Journal articles: 'Toffoli gate'

1

Song, G., and A. Klappenecker. "Optimal realizations of simplified Toffoli gates." Quantum Information and Computation 4, no. 5 (September 2004): 361–72. http://dx.doi.org/10.26421/qic4.5-2.

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A simplified Toffoli gate coincides with the Toffoli gate except that the result is allowed to differ on one computational basis state by a phase factor. We prove that the simplified Toffoli gate implementation by Margolus is optimal, in the sense that it attains a lower bound of {\em three} controlled-not gates, and subject to that, a sharp lower bound of {\em four} single-qubit gates. We also discuss optimal implementations of other simplified Toffoli gates, and explain why the phase factor $-1$ invariably occurs in such implementations.

2

FANG, BAO-LONG, ZHEN YANG, and LIU YE. "SCHEME FOR IMPLEMENTING AN N-QUBIT CONTROLLED NOT GATE WITH SUPERCONDUCTING QUANTUM INTERFERENCE DEVICES IN CAVITY QED." International Journal of Quantum Information 08, no. 08 (December 2010): 1337–45. http://dx.doi.org/10.1142/s0219749910006307.

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We present a scheme for implementing a Toffoli gate. The superconducting quantum interference devices are coupled to a resonant cavity with nonidentical SQUID–cavity coupling constants. So only one interaction between SQUID and cavity is required, and a Toffoli gate can be obtained. The method can be generalized to the N-qubit case easily and the scheme is insensitive to systematic coupling error.

3

Guan, Zhi Jin, Wei Ping Ding, and Xue Yun Cheng. "Cascade Network in Reversible Logic Gate Based on Series Connection." Applied Mechanics and Materials 241-244 (December 2012): 3075–79. http://dx.doi.org/10.4028/www.scientific.net/amm.241-244.3075.

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This paper analyzes and proves that the relationship between the output results of the homotypic Toffoli gate which is series cascade and the number of the gates which are series cascade. In order to guarantee the convergence of the process of the series cascade, we gave the counting results of the series cascade network for Toffoli gates, and proved that in the input vector (0, 1, …, 2n-1), the number of the bit vectors with Hamming weight H(w)≥n-1 is equal to the bit number of bit vectors plus 1, and obtained the conclusion that there are (n+1)! kinds of transformation for Toffoli gate series cascade network. Simultaneously we provide the series cascade network algorithm of the Toffoli gates. The reversible network cascade system designed by the above algorithm verified the validity of this algorithm.

4

Trivedi, Amit Ranjan, and S. Bandyopadhyay. "Single spin Toffoli–Fredkin logic gate." Journal of Applied Physics 103, no. 10 (May 15, 2008): 104311. http://dx.doi.org/10.1063/1.2937200.

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Huang, He-Liang, Wan-Su Bao, Tan Li, Feng-Guang Li, Xiang-Qun Fu, Shuo Zhang, Hai-Long Zhang, and Xiang Wang. "Deterministic linear optical quantum Toffoli gate." Physics Letters A 381, no. 33 (September 2017): 2673–76. http://dx.doi.org/10.1016/j.physleta.2017.06.034.

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Sarkar, Angik, and T. K. Bhattacharyya. "Universal Toffoli gate in ballistic nanowires." Applied Physics Letters 90, no. 17 (April 23, 2007): 173101. http://dx.doi.org/10.1063/1.2731521.

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7

Backens, Miriam, Aleks Kissinger, Hector Miller-Bakewell, John van de Wetering, and Sal Wolffs. "Completeness of the ZH-calculus." Compositionality 5 (July 12, 2023): 5. http://dx.doi.org/10.32408/compositionality-5-5.

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There are various gate sets used for describing quantum computation. A particularly popular one consists of Clifford gates and arbitrary single-qubit phase gates. Computations in this gate set can be elegantly described by the ZX-calculus, a graphical language for a class of string diagrams describing linear maps between qubits. The ZX-calculus has proven useful in a variety of areas of quantum information, but is less suitable for reasoning about operations outside its natural gate set such as multi-linear Boolean operations like the Toffoli gate. In this paper we study the ZH-calculus, an alternative graphical language of string diagrams that does allow straightforward encoding of Toffoli gates and other more complicated Boolean logic circuits. We find a set of simple rewrite rules for this calculus and show it is complete with respect to matrices over Z[12], which correspond to the approximately universal Toffoli+Hadamard gateset. Furthermore, we construct an extended version of the ZH-calculus that is complete with respect to matrices over any ring R where 1+1 is not a zero-divisor.

8

Liu, Wen, Yangzhi Li, Zhirao Wang, and Yugang Li. "A New Quantum Private Protocol for Set Intersection Cardinality Based on a Quantum Homomorphic Encryption Scheme for Toffoli Gate." Entropy 25, no. 3 (March 16, 2023): 516. http://dx.doi.org/10.3390/e25030516.

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Set Intersection Cardinality (SI-CA) computes the intersection cardinality of two parties’ sets, which has many important and practical applications such as data mining and data analysis. However, in the face of big data sets, it is difficult for two parties to execute the SI-CA protocol repeatedly. In order to reduce the execution pressure, a Private Set Intersection Cardinality (PSI-CA) protocol based on a quantum homomorphic encryption scheme for the Toffoli gate is proposed. Two parties encode their private sets into two quantum sequences and encrypt their sequences by way of a quantum homomorphic encryption scheme. After receiving the encrypted results, the semi-honest third party (TP) can determine the equality of two quantum sequences with the Toffoli gate and decrypted keys. The simulation of the quantum homomorphic encryption scheme for the Toffoli gate on two quantum bits is given by the IBM Quantum Experience platform. The simulation results show that the scheme can also realize the corresponding function on two quantum sequences.

9

Lei, Peng, Yang Zhang, Jiong Cheng, and Wen-Zhao Zhang. "Quantum Toffoli gate in hybrid optomechanical system." Results in Physics 35 (April 2022): 105338. http://dx.doi.org/10.1016/j.rinp.2022.105338.

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Samanta, Debajyoti. "Implementation of polarization-encoded quantum Toffoli gate." Journal of Optics 48, no. 1 (December 6, 2018): 70–75. http://dx.doi.org/10.1007/s12596-018-0496-4.

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Shao, Xiao-Qiang, Tai-Yu Zheng, and Shou Zhang. "Robust Toffoli gate originating from Stark shifts." Journal of the Optical Society of America B 29, no. 6 (May 9, 2012): 1203. http://dx.doi.org/10.1364/josab.29.001203.

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12

Mondal, Joyati, Bappaditya Mondal, Dipak Kumar Kole, Hafizur Rahaman, and Debesh Kumar Das. "Boolean Difference Technique for Detecting All Missing Gate and Stuck-at Faults in Reversible Circuits." Journal of Circuits, Systems and Computers 28, no. 12 (November 2019): 1950212. http://dx.doi.org/10.1142/s0218126619502128.

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Quantum reversible circuit is a new emerging technology attracting the researchers. A reversible circuit is composed of reversible gates. One example of reversible gate is Toffoli gate. A Toffoli gate (also known as [Formula: see text]-CNOT) has two components — the control and the target. Initially, stuck-at fault and other fault models were used for modeling defects in quantum reversible circuits. Later, a new fault model known as missing gate fault model was introduced, which is more effective in capturing the errors in quantum reversible circuit. Boolean Difference is already a known technique to detect stuck-at faults in conventional CMOS circuit. In this paper, Boolean Difference method is applied to derive the test set for detecting each stuck-at fault and missing gate fault in a reversible circuit. Then an optimization algorithm is used to derive an optimal test set, which will detect all possible faults in a circuit. The method is valid also for other fault models.

13

Nemkov, Nikita A., Evgeniy O. Kiktenko, Ilia A. Luchnikov, and Aleksey K. Fedorov. "Efficient variational synthesis of quantum circuits with coherent multi-start optimization." Quantum 7 (May 4, 2023): 993. http://dx.doi.org/10.22331/q-2023-05-04-993.

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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 along with the discrete architecture search suffering from the combinatorial explosion of complexity, optimization over 1q gates can also be a crucial roadblock due to the omnipresence of local minimums (well known in the context of variational quantum algorithms but apparently underappreciated in the context of the variational compiling). Taking the issue seriously, we make an extensive search over the initial conditions an essential part of our approach. Another key idea we propose is to use parametrized two-qubit (2q) controlled phase gates, which can interpolate between the identity gate and the CNOT gate, and allow a continuous relaxation of the discrete architecture search, which can be executed jointly with the optimization over 1q gates. This coherent optimization of the architecture together with 1q gates appears to work surprisingly well in practice, sometimes even outperforming optimization over 1q gates alone (for fixed optimal architectures). As illustrative examples and applications we derive 8 CNOT and T depth 3 decomposition of the 3q Toffoli gate on the nearest-neighbor topology, rediscover known best decompositions of the 4q Toffoli gate on all 4q topologies including a 1 CNOT gate improvement on the star-shaped topology, and propose decomposition of the 5q Toffoli gate on the nearest-neighbor topology with 48 CNOT gates. We also benchmark the performance of our approach on a number of 5q quantum circuits from the ibm_qx_mapping database showing that it is highly competitive with the existing software. The algorithm developed in this work is available as a Python package CPFlow.

14

Haah, Jeongwan, and Matthew B. Hastings. "Codes and Protocols for Distilling T, controlled-S, and Toffoli Gates." Quantum 2 (June 7, 2018): 71. http://dx.doi.org/10.22331/q-2018-06-07-71.

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We present several different codes and protocols to distill T, controlled-S, and Toffoli (or CCZ) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversal T. We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiency γ→1. We also present a Reed-Muller based construction of these codes which obtains a worse γ but performs well at small sizes. Additionally, we present protocols based on checking the stabilizers of CCZ magic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of. Several examples, including a Reed-Muller code for T-to-Toffoli distillation, punctured Reed-Muller codes for T-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a 512 T-gate to 10 Toffoli gate code with distance 8 as well as triorthogonal codes with parameters [[887,137,5]],[[912,112,6]],[[937,87,7]] with very low prefactors in front of the leading order error terms in those codes.

15

van Hoof, Iggy. "Space-efficient quantum multiplication polynomials for binary finite fields with sub-quadratoc Toffoli gate count." Quantum Information and Computation 20, no. 9&10 (August 2020): 721–35. http://dx.doi.org/10.26421/qic20.9-10-1.

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Multiplication is an essential step in a lot of calculations. In this paper we look at multiplication of 2 binary polynomials of degree at most n-1, modulo an irreducible polynomial of degree n with 2n input and n output qubits, without ancillary qubits, assuming no errors. With straightforward schoolbook methods this would result in a quadratic number of Toffoli gates and a linear number of CNOT gates. This paper introduces a new algorithm that uses the same space, but by utilizing space-efficient variants of Karatsuba multiplication methods it requires only O(n^{\log_2(3)}) Toffoli gates at the cost of a higher CNOT gate count: theoretically up to O(n^2) but in examples the CNOT gate count looks a lot better.

16

Stojkovic, Suzana, Milena Stankovic, and Claudio Moraga. "Complexity reduction of Toffoli networks based on FDD." Facta universitatis - series: Electronics and Energetics 28, no. 2 (2015): 251–62. http://dx.doi.org/10.2298/fuee1502251s.

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Synthesis of switching functions by Toffoli gates has become a very important research topic in the last years, since Toffoli gates are used in the synthesis of reversible circuits. Early methods based on the truth-table representation of Boolean functions are applicable to functions with a relatively small number of variables. Later on, methods for synthesis by Toffoli gates based on decision diagrams (BDDs, FDDs or OKFDDs) were introduced and applied to the synthesis of both reversible and irreversible functions. This paper presents a method for the reduction of the number of lines and gates in the Toffoli gate realization of Boolean functions based on their Functional Decision Diagram (FDD) representation. Experiments show that, when the proposed reduction is used, the realization of the given function based on FDD will, on the average, be smaller in terms of the number of lines and the number of gates than the realizations based on an OKFDD, an optimal BDD or based on a FDD by using previously defined algorithms.

17

Fedorov, A., L. Steffen, M. Baur, M. P. da Silva, and A. Wallraff. "Implementation of a Toffoli gate with superconducting circuits." Nature 481, no. 7380 (December 14, 2011): 170–72. http://dx.doi.org/10.1038/nature10713.

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18

Dalla Chiara, M. L., A. Ledda, G. Sergioli, and R. Giuntini. "The Toffoli-Hadamard Gate System: an Algebraic Approach." Journal of Philosophical Logic 42, no. 3 (April 6, 2013): 467–81. http://dx.doi.org/10.1007/s10992-013-9271-9.

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19

Monfared, Asma Taheri, and Majid Haghparast. "Quantum Ternary Multiplication Gate (QTMG): Toward Quantum Ternary Multiplier and a New Realization for Ternary Toffoli Gate." Journal of Circuits, Systems and Computers 29, no. 05 (July 3, 2019): 2050071. http://dx.doi.org/10.1142/s0218126620500711.

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The designs using ternary logic exploit its logarithmic reduction in the number of qudits compared with the binary circuits. In this paper, we propose quantum ternary multiplication gate. We term it as QTMG. Then we present the symbol and the realization of QTMG. Researchers will be able to use this gate as well as its symbol and realizations in their future studies. We also present a new realization of ternary Toffoli gate in specific state. Moreover, in this paper, we propose 1-qutrit multiplier circuit. The symbol and the realization of the proposed 1-qutrit multiplier circuit are also provided. Afterward, we proposed ternary partial products generation circuit (TPPG) and summation network circuit in order to design quantum ternary 2-qutrit multiplier circuit. Overall, the proposed design of QTMG in this paper is suggested for the first time. In addition, the proposed realization of ternary Toffoli gate, TPPG, summation network and 2-qutrit multiplier circuits are compared with the existing designs and the improvements are reported. The proposed gate and circuits are realized using macro-level ternary gates which are built on the top of the ion-trap realizable 1-qutrit gates and 2-qutrit Muthukrishnan–Stroud gates.

20

Sun, Qian, and Liu Ye. "Implementing Toffoli gate via weak cross-Kerr nonlinearity and classical feedback." Modern Physics Letters B 29, no. 09 (April 10, 2015): 1550032. http://dx.doi.org/10.1142/s0217984915500323.

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We design an efficient optical circuit to realize a Toffoli gate via weak cross-Kerr nonlinearity and classical feedback. To obtain a high-fidelity gate, we probe different path detectors to minimize the influence from the environmental noise. Compared to previous proposals, our scheme does not need external auxiliary photons and is easy to implement in experiment.

21

Adamatzky, Andrew. "Fredkin and Toffoli Gates Implemented in Oregonator Model of Belousov–Zhabotinsky Medium." International Journal of Bifurcation and Chaos 27, no. 03 (March 2017): 1750041. http://dx.doi.org/10.1142/s0218127417500419.

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A thin-layer Belousov–Zhabotinsky (BZ) medium is a powerful computing device capable for implementing logical circuits, memory, image processors, robot controllers, and neuromorphic architectures. We design the reversible logical gates — Fredkin gate and Toffoli gate — in a BZ medium network of excitable channels with subexcitable junctions. Local control of the BZ medium excitability is an important feature of the gates’ design. An excitable thin-layer BZ medium responds to a localized perturbation with omnidirectional target or spiral excitation waves. A subexcitable BZ medium responds to an asymmetric perturbation by producing traveling localized excitation wave-fragments similar to dissipative solitons. We employ interactions between excitation wave-fragments to perform the computation. We interpret the wave-fragments as values of Boolean variables. The presence of a wave-fragment at a given site of a circuit represents the logical truth, absence of the wave-fragment — logically false. Fredkin gate consists of ten excitable channels intersecting at 11 junctions, eight of which are subexcitable. Toffoli gate consists of six excitable channels intersecting at six junctions, four of which are subexcitable. The designs of the gates are verified using numerical integration of two-variable Oregonator equations.

22

Park, Dong-Young. "Realization of Multiple-Control Toffoli gate based on Mutiple-Valued Quantum Logic." Journal of Korea Navigation Institute 16, no. 1 (February 29, 2012): 62–69. http://dx.doi.org/10.12673/jkoni.2012.16.1.062.

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23

Barber, Ben, Neil I. Gillespie, and J. M. Taylor. "Post-selection-free preparation of high-quality physical qubits." Quantum 7 (May 4, 2023): 994. http://dx.doi.org/10.22331/q-2023-05-04-994.

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Rapidly improving gate fidelities for coherent operations mean that errors in state preparation and measurement (SPAM) may become a dominant source of error for fault-tolerant operation of quantum computers. This is particularly acute in superconducting systems, where tradeoffs in measurement fidelity and qubit lifetimes have limited overall performance. Fortunately, the essentially classical nature of preparation and measurement enables a wide variety of techniques for improving quality using auxiliary qubits combined with classical control and post-selection. In practice, however, post-selection greatly complicates the scheduling of processes such as syndrome extraction. Here we present a family of quantum circuits that prepare high-quality |0⟩ states without post-selection, instead using CNOT and Toffoli gates to non-linearly permute the computational basis. We find meaningful performance enhancements when two-qubit gate fidelities errors go below 0.2%, and even better performance when native Toffoli gates are available.

24

Shende, V. V., and I. L. Markov. "On the CNOT -cost of TOFFOLI gates." Quantum Information and Computation 9, no. 5&6 (May 2009): 461–86. http://dx.doi.org/10.26421/qic8.5-6-8.

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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 and several one-qubit gates. Though this decomposition has been known for at least 10 years, we provide here the first demonstration of its \CNOT-optimality. We study three-qubit circuits which contain less than six \CNOT\ gates and implement a block-diagonal operator, then show that they implicitly describe the cosine-sine decomposition of a related operator. Leveraging the canonical nature of such decompositions to limit one-qubit gates appearing in respective circuits, we prove that the $n$-qubit analogue of the \TOFFOLI\ requires at least $2n$ \CNOT\ gates. Additionally, our results offer a complete classification of three-qubit diagonal operators by their \CNOT -cost, which holds even if ancilla qubits are available.

25

Gazazyan, E. A., G. G. Grigoryan, V. O. Chaltykyan, and D. Schraft. "Implementation of all-optical Toffoli gate in Λ-systems." Journal of Contemporary Physics (Armenian Academy of Sciences) 47, no. 5 (September 2012): 216–21. http://dx.doi.org/10.3103/s1068337212050040.

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26

Dong, Li, Sen-Lin Wang, Cen Cui, Xue Geng, Qing-Yang Li, Hai-Kuan Dong, Xiao-Ming Xiu, and Ya-Jun Gao. "Polarization Toffoli gate assisted by multiple degrees of freedom." Optics Letters 43, no. 19 (September 20, 2018): 4635. http://dx.doi.org/10.1364/ol.43.004635.

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27

Jiang-Feng, Du, Shi Ming-Jun, Zhou Xian-Yi, Fan Yang-Mei, Wu Ji-Hui, Ye Bang-Jiao, Weng Hui-Min, and Han Rong-Dian. "Realization of the Three-Qubit Toffoli Gate in Molecules." Chinese Physics Letters 17, no. 12 (December 1, 2000): 859–61. http://dx.doi.org/10.1088/0256-307x/17/12/001.

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28

LIU, CHUAN-LONG, YAN-WEI WANG, and YI-ZHUANG ZHENG. "IMPLEMENTATION OF NON-LOCAL TOFFOLI GATE VIA CAVITY QUANTUM ELECTRODYNAMICS." International Journal of Quantum Information 07, no. 03 (April 2009): 669–80. http://dx.doi.org/10.1142/s0219749909003329.

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A scheme for realizing the non-local Toffoli gate among three spatially separated nodes through cavity quantum electrodynamics (C-QED) is presented. The scheme can obtain high fidelity in the current C-QED system. With entangled qubits as quantum channels, the operation is resistive to actual environment noise.

29

Nikolaeva, Anstasiia S., Evgeniy O. Kiktenko, and Aleksey K. Fedorov. "Generalized Toffoli Gate Decomposition Using Ququints: Towards Realizing Grover’s Algorithm with Qudits." Entropy 25, no. 2 (February 20, 2023): 387. http://dx.doi.org/10.3390/e25020387.

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Qubits, which are the quantum counterparts of classical bits, are used as basic information units for quantum information processing, whereas underlying physical information carriers, e.g., (artificial) atoms or ions, admit encoding of more complex multilevel states—qudits. Recently, significant attention has been paid to the idea of using qudit encoding as a way for further scaling quantum processors. In this work, we present an efficient decomposition of the generalized Toffoli gate on five-level quantum systems—so-called ququints—that use ququints’ space as the space of two qubits with a joint ancillary state. The basic two-qubit operation we use is a version of the controlled-phase gate. The proposed N-qubit Toffoli gate decomposition has O(N) asymptotic depth and does not use ancillary qubits. We then apply our results for Grover’s algorithm, where we indicate on the sizable advantage of using the qudit-based approach with the proposed decomposition in comparison to the standard qubit case. We expect that our results are applicable for quantum processors based on various physical platforms, such as trapped ions, neutral atoms, protonic systems, superconducting circuits, and others.

30

Moraga, Claudio, and Fatima Hadjam. "The Fredkin gate in reversible and quantum environments." Facta universitatis - series: Electronics and Energetics 36, no. 2 (2023): 253–66. http://dx.doi.org/10.2298/fuee2302253m.

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Reversible Computing circuits are characterized by low power consumption and their proximity to circuits for quantum computing. The Fredkin gate was one of the earliest proposed controlled reversible circuits, which however, was soon superseded by the Toffoli gate, the NOT, and CNOT gates, which constituting a flexible functionally complete set could also realize the Fredkin gate as a building block. In quantum computing circuits, the Fredkin gate (under the name controlled SWAP) plays an important role regarding the superposition of states. The present paper studies extensions of the Fredkin gate in terms of mixed polarity in the reversible domain and an application in quantum computing.

31

Shi, Y.-Y. "Both Toffoli and Controlled-NOT need little help to universal quantum computing." Quantum Information and Computation 3, no. 1 (January 2003): 84–92. http://dx.doi.org/10.26421/qic3.1-7.

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What additional gates are needed for a set of classical universal gates to do universal quantum computation? We prove that any single-qubit real gate suffices, except those that preserve the computational basis. The Gottesman-Knill Theorem implies that any quantum circuit involving only the Controlled-NOT and Hadamard gates can be efficiently simulated by a classical circuit. In contrast, we prove that Controlled-NOT plus any single-qubit real gate that does not preserve the computational basis and is not Hadamard (or its like) are universal for quantum computing. Previously only a generic gate, namely a rotation by an angle incommensurate with \pi, is known to be sufficient in both problems, if only one single-qubit gate is added.

32

Khesin, A., and K. Ren. "Extending the graph formalism to higher-order gates." Quantum Information and Computation 23, no. 13&14 (November 2023): 1128–41. http://dx.doi.org/10.26421/qic23.13-14-5.

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We present an algorithm for efficiently simulating a quantum circuit in the graph formalism. In the graph formalism, we present states as a linear combination of graphs with Clifford operations on their vertices. We show how a $\calC_3$ gate such as the Toffoli gate or $\frac\pi8$ gate acting on a stabilizer state splits it into two stabilizer states. We also describe conditions for merging two stabilizer states into one. We discuss applications of our algorithm to circuit identities and finding low stabilizer rank presentations of magic states.

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Dalcumune, Edinelço, Luis Antonio Brasil Kowada, André da Cunha Ribeiro, Celina Miraglia Herrera de Figueiredo, and Franklin de Lima Marquezino. "A reversible circuit synthesis algorithm with progressive increase of controls in generalized Toffoli gates." JUCS - Journal of Universal Computer Science 27, no. 6 (June 28, 2021): 544–63. http://dx.doi.org/10.3897/jucs.69617.

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We present a new algorithm for synthesis of reversible circuits for arbitrary n-bit bijective functions. This algorithm uses generalized Toffoli gates, which include positive and negative controls. Our algorithm is divided into two parts. First, we use partially controlled gen- eralized Toffoli gates, progressively increasing the number of controls. Second, exploring the properties of the representation of permutations in disjoint cycles, we apply generalized Toffoli gates with controls on all lines except for the target line. Therefore, new in the method is the fact that the obtained circuits use first low cost gates and consider increasing costs towards the end of the synthesis. In addition, we employ two bidirectional synthesis strategies to improve the gate count, which is the metric used to compare the results obtained by our algorithm with the results presented in the literature. Accordingly, our experimental results consider all 3-bit bijective functions and twenty widely used benchmark functions. The results obtained by our synthesis algorithm are competitive when compared with the best results known in the literature, considering as a complexity metric just the number of gates, as done by alternative best heuristics found in the literature. For example, for all 3-bit bijective functions using generalized Toffoli gates library, we obtained the best so far average count of 5.23.

34

Newman, Michael, and Yaoyun Shi. "Limitations on transversal computation through quantum homomorphic encryption." Quantum Information and Computation 18, no. 11&12 (September 2018): 927–48. http://dx.doi.org/10.26421/qic18.11-12-3.

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Transversality is a simple and effective method for implementing quantum computation fault-tolerantly. However, no quantum error-correcting code (QECC) can transversally implement a quantum universal gate set (Eastin and Knill, {\em Phys. Rev. Lett.}, 102, 110502). Since reversible classical computation is often a dominating part of useful quantum computation, whether or not it can be implemented transversally is an important open problem. We show that, other than a small set of non-additive codes that we cannot rule out, no binary QECC can transversally implement a classical reversible universal gate set. In particular, no such QECC can implement the Toffoli gate transversally.}{We prove our result by constructing an information theoretically secure (but inefficient) quantum homomorphic encryption (ITS-QHE) scheme inspired by Ouyang {\em et al.} (arXiv:1508.00938). Homomorphic encryption allows the implementation of certain functions directly on encrypted data, i.e. homomorphically. Our scheme builds on almost any QECC, and implements that code's transversal gate set homomorphically. We observe a restriction imposed by Nayak's bound ({\em FOCS} 1999) on ITS-QHE, implying that any ITS quantum {\em fully} homomorphic scheme (ITS-QFHE) implementing the full set of classical reversible functions must be highly inefficient. While our scheme incurs exponential overhead, any such QECC implementing Toffoli transversally would still violate this lower bound through our scheme.

35

Jordan, Stephen P. "Strong equivalence of reversible circuits is coNP-complete." Quantum Information and Computation 14, no. 15&16 (November 2014): 1302–7. http://dx.doi.org/10.26421/qic14.15-16-3.

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It is well-known that deciding equivalence of logic circuits is a coNP-complete problem. As a corollary, the problem of deciding weak equivalence of reversible circuits, \emph{i.e.} allowing initialized ancilla bits in the input and ignoring ``garbage'' ancilla bits in the output, is also coNP-complete. The complexity of deciding strong equivalence, including the ancilla bits, is less obvious and may depend on gate set. Here we use Barrington's theorem to show that deciding strong equivalence of reversible circuits built from the Fredkin gate is coNP-complete. This implies coNP-completeness of deciding strong equivalence for other commonly used universal reversible gate sets, including any gate set that includes the Toffoli or Fredkin gate.

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Monfared, Asma Taheri, and Majid Haghparast. "Design of New Quantum/Reversible Ternary Subtractor Circuits." Journal of Circuits, Systems and Computers 25, no. 02 (December 23, 2015): 1650014. http://dx.doi.org/10.1142/s0218126616500146.

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Ternary quantum circuits play a significant role in future quantum computing technology because they have many advantages over binary quantum circuits. Subtraction is considered as being one of the key arithmetic operations; hence, subtractors are very essential for the construction of various computational units of quantum computers and other complex computational systems. In this paper, we have proposed the realization of a quantum reversible ternary half-subtractor circuit using a generalized ternary gate, a ternary Toffoli gate, and a ternary C2NOT gate. Based on the realization of the ternary half-subtractor, we proposed the realization of a ternary full-subtractor.

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Freytes, Hector, Roberto Giuntini, and Giuseppe Sergioli. "Holistic Type Extension for Classical Logic via Toffoli Quantum Gate." Entropy 21, no. 7 (June 27, 2019): 636. http://dx.doi.org/10.3390/e21070636.

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A holistic extension of classical propositional logic is introduced via Toffoli quantum gate. This extension is based on the framework of the so-called “quantum computation with mixed states”, where also irreversible transformations are taken into account. Formal aspects of this new logical system are detailed: in particular, the concepts of tautology and contradiction are investigated in this extension. These concepts turn out to receive substantial changes due to the non-separability of some quantum states; as an example, Werner states emerge as particular cases of “holistic" contradiction.

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Baker, Aneirin J., Gerhard B. P. Huber, Niklas J. Glaser, Federico Roy, Ivan Tsitsilin, Stefan Filipp, and Michael J. Hartmann. "Single shot i-Toffoli gate in dispersively coupled superconducting qubits." Applied Physics Letters 120, no. 5 (January 31, 2022): 054002. http://dx.doi.org/10.1063/5.0077443.

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Ohya, Masanori, and Noboru Watanabe. "On the mathematical treatment of the Fredkin-Toffoli-Milburn gate." Physica D: Nonlinear Phenomena 120, no. 1-2 (September 1998): 206–13. http://dx.doi.org/10.1016/s0167-2789(98)00056-6.

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Chen, Li-Bing, and Hong Lu. "Implementing a Nonlocal Toffoli Gate Using Partially Entangled Qubit Pairs." International Journal of Theoretical Physics 50, no. 11 (June 9, 2011): 3442–50. http://dx.doi.org/10.1007/s10773-011-0849-0.

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Peng Yonggang, 彭永刚. "量子Toffoli门的核磁共振物理实现." Laser & Optoelectronics Progress 60, no. 7 (2023): 0727002. http://dx.doi.org/10.3788/lop220821.

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Maslov, Dmitri. "Optimal and asymptotically optimal NCT reversible circuits by the gate types." Quantum Information and Computation 16, no. 13&14 (October 2016): 1096–112. http://dx.doi.org/10.26421/qic16.13-14-2.

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We report optimal and asymptotically optimal reversible circuits composed of NOT, CNOT, and Toffoli (NCT) gates, keeping the count by the subsets of the gate types used. This study fine tunes the circuit complexity figures for the realization of reversible functions via reversible NCT circuits. An important consequence is a result on the limitation of the use of the T-count quantum circuit metric popular in applications.

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Jang, Kyungbae, Wonwoong Kim, Sejin Lim, Yeajun Kang, Yujin Yang, and Hwajeong Seo. "Quantum Binary Field Multiplication with Optimized Toffoli Depth and Extension to Quantum Inversion." Sensors 23, no. 6 (March 15, 2023): 3156. http://dx.doi.org/10.3390/s23063156.

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The Shor’s algorithm can find solutions to the discrete logarithm problem on binary elliptic curves in polynomial time. A major challenge in implementing Shor’s algorithm is the overhead of representing and performing arithmetic on binary elliptic curves using quantum circuits. Multiplication of binary fields is one of the critical operations in the context of elliptic curve arithmetic, and it is especially costly in the quantum setting. Our goal in this paper is to optimize quantum multiplication in the binary field. In the past, efforts to optimize quantum multiplication have centred on reducing the Toffoli gate count or qubits required. However, despite the fact that circuit depth is an important metric for indicating the performance of a quantum circuit, previous studies have lacked sufficient consideration for reducing circuit depth. Our approach to optimizing quantum multiplication differs from previous work in that we aim at reducing the Toffoli depth and full depth. To optimize quantum multiplication, we adopt the Karatsuba multiplication method which is based on the divide-and-conquer approach. In summary, we present an optimized quantum multiplication that has a Toffoli depth of one. Additionally, the full depth of the quantum circuit is also reduced thanks to our Toffoli depth optimization strategy. To demonstrate the effectiveness of our proposed method, we evaluate its performance using various metrics such as the qubit count, quantum gates, and circuit depth, as well as the qubits-depth product. These metrics provide insight into the resource requirements and complexity of the method. Our work achieves the lowest Toffoli depth, full depth, and the best trade-off performance for quantum multiplication. Further, our multiplication is more effective when not used in stand-alone cases. We show this effectiveness by using our multiplication to the Itoh–Tsujii algorithm-based inversion of F(x8+x4+x3+x+1).

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Deibuk, V. G., I. M. Yuriychuk, and I. Lemberski. "Fidelity of noisy multiple-control reversible gates." Semiconductor Physics, Quantum Electronics and Optoelectronics 23, no. 04 (November 19, 2020): 385–92. http://dx.doi.org/10.15407/spqeo23.04.385.

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The effect of frequency noise on correct operation of the multiple-control Toffoli, Fredkin, and Peres gates has been discussed. In the framework of the Ising model, the energy spectrum of a chain of atoms with nuclear spins one-half in a spinless semiconductor matrix has been obtained, and allowed transitions corresponding to the operation algorithm of these gates have been determined. The fidelities of the obtained transitions were studied depending on the number of control qubits and parameters of the radio-frequency control pulses. It has been shown that correct operation of the Toffoli and Fredkin gates does not depend on the number of control qubits, while the Peres gate fidelity decreases significantly with the increasing number of control signals. The calculated ratios of the Larmor frequency to the exchange interaction constant correspond with the results of other studies.

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Dey, Shuvra, and Sourangshu Mukhopadhyay. "All-optical high frequency clock pulse generator using the feedback mechanism in Toffoli gate with Kerr material." Journal of Nonlinear Optical Physics & Materials 25, no. 01 (March 2016): 1650012. http://dx.doi.org/10.1142/s0218863516500120.

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Optics is found as a potential information carrier in information processing because of its superfast speed of operation. For such operations, optical Kerr materials are highly sensitive for use as optical switches. In this paper, the authors for thr first time propose a new scheme of generation of an all-optical pulse train generator using the feedback mechanism in an all-optical Toffoli gate with Kerr material.

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Gado, Mariam, and Ahmed Younes. "Optimization of Reversible Circuits Using Toffoli Decompositions with Negative Controls." Symmetry 13, no. 6 (June 7, 2021): 1025. http://dx.doi.org/10.3390/sym13061025.

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The synthesis and optimization of quantum circuits are essential for the construction of quantum computers. This paper proposes two methods to reduce the quantum cost of 3-bit reversible circuits. The first method utilizes basic building blocks of gate pairs using different Toffoli decompositions. These gate pairs are used to reconstruct the quantum circuits where further optimization rules will be applied to synthesize the optimized circuit. The second method suggests using a new universal library, which provides better quantum cost when compared with previous work in both cost015 and cost115 metrics; this proposed new universal library “Negative NCT” uses gates that operate on the target qubit only when the control qubit’s state is zero. A combination of the proposed basic building blocks of pairs of gates and the proposed Negative NCT library is used in this work for synthesis and optimization, where the Negative NCT library showed better quantum cost after optimization compared with the NCT library despite having the same circuit size. The reversible circuits over three bits form a permutation group of size 40,320 (23!), which is a subset of the symmetric group, where the NCT library is considered as the generators of the permutation group.

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Song, Li-Cong, Yan Xia, and Jie Song. "Noise resistance of Toffoli gate in an array of coupled cavities." Journal of Modern Optics 61, no. 16 (June 23, 2014): 1290–97. http://dx.doi.org/10.1080/09500340.2014.930194.

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Shao, Xiao-Qiang, Hong-Fu Wang, Li Chen, Shou Zhang, and Kyu-Hwang Yeon. "One-step implementation of the Toffoli gate via quantum Zeno dynamics." Physics Letters A 374, no. 1 (December 2009): 28–33. http://dx.doi.org/10.1016/j.physleta.2009.10.020.

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Freytes, Hector, and Giuseppe Sergioli. "Fuzzy Approach for Toffoli Gate in Quantum Computation With Mixed States." Reports on Mathematical Physics 74, no. 2 (October 2014): 159–80. http://dx.doi.org/10.1016/s0034-4877(15)60014-3.

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Cheng, Chua Shin, Ashutosh Kumar Singh, and Lenin Gopal. "Efficient Three Variables Reversible Logic Synthesis Using Mixed-polarity Toffoli Gate." Procedia Computer Science 70 (2015): 362–68. http://dx.doi.org/10.1016/j.procs.2015.10.035.

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Journal articles on the topic 'Toffoli gate'

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