Toffoli Gate Research Articles

All-optical frequency-encoded Toffoli gate

Abstract The construction of an all-optical frequency-encoded Toffoli gate employing a reflecting semiconductor optical amplifier (RSOA) is proposed in this article. By establishing fields such as quantum computing, optical quantum computing, quantum-dot cellular automata, and superconducting flux logic family, quantum gates have been proved to perform reliably in the present day. A nonzero-mass electron, on the other hand, moves far slower than a quantum particle with zero rest mass, such as a photon. Photons can also be utilized to store data while being sent. These photon qualities have motivated researchers to create quantum gates in the all-optical domain based on them. The RSOA-based implementation of the Toffoli gate gives a significant improvement in the case of high speed, low power, and fast switching time. MATLAB Simulink (R2018a) software is used to simulate the devised design. The theoretical prediction is satisfied by the simulation results.

Early fault-tolerant simulations of the Hubbard model

Simulation of the Hubbard model is a leading candidate for the first useful applications of a fault-tolerant quantum computer. A recent study of quantum algorithms for early simulations of the Hubbard model [Kivlichan et al 2019 Quantum 4 296] found that the lowest resource costs were achieved by split-operator Trotterization combined with the fast-fermionic Fourier transform (FFFT) on an L × L lattice with length L = 2 k . On lattices with length L ≠ 2 k , Givens rotations can be used instead of the FFFT but lead to considerably higher resource costs. We present a new analytic approach to bounding the simulation error due to Trotterization that provides much tighter bounds for the split-operator FFFT method, leading to 16× improvement in error bounds. Furthermore, we introduce plaquette Trotterization that works on any size lattice and apply our improved error bound analysis to show competitive resource costs. We consider a phase estimation task and show plaquette Trotterization reduces the number of non-Clifford gates by a factor 5.5× to 9× (depending on the parameter regime) over the best previous estimates for 8 × 8 and 16 × 16 lattices and a much larger factor for other lattice sizes not of the form L = 2 k . In conclusion, we find there is a potentially useful application for fault-tolerant quantum computers using around one million Toffoli gates.

Phase-state duality in reversible circuit design

The reversible implementation of classical functions accounts for the bulk of most known quantum algorithms. As a result, a number of reversible circuit constructions over the Clifford+$T$ gate set have been developed in recent years which use both the state and phase spaces, or $X$ and $Z$ bases, to reduce circuit costs beyond what is possible at the strictly classical level. We study and generalize two particular classes of these constructions: relative phase circuits, including Giles and Selinger's multiply-controlled $iX$ gates and Maslov's $4$ qubit Toffoli gate, and measurement-assisted circuits, including Jones' Toffoli gate and Gidney's temporary logical-AND. In doing so, we introduce general methods for implementing classical functions up to phase and for measurement-assisted termination of temporary values. We then apply these techniques to find novel $T$-count efficient constructions of some classical functions in space-constrained regimes, notably multiply-controlled Toffoli gates and temporary products.

Toffoli gate with photonic qubits based on weak cross-Kerr nonlinearities

Toffoli gate with photonic qubits based on weak cross-Kerr nonlinearities

Multiple-qubit Rydberg quantum logic gate via dressed-state scheme

We present a scheme to realize multiple-qubit quantum state transfer and quantum logic gate by combining the advantages of Vitanov-style pulses and dressed-state-based shortcut to adiabaticity (STA) in Rydberg atoms. The robustness of the scheme to spontaneous emission can be achieved by reducing the population of Rydberg excited states through STA technology. Meanwhile, the control errors can be minimized through using the well-designed pulses. Moreover, the dressed-state method applied in the scheme makes the quantum state transfer more smoothly turned on or off with high fidelity and also faster than traditional shortcut to adiabaticity methods. By using Rydberg antiblockade (RAB) effect (Su et al., 2020 [1]), the multiple-qubit Toffoli gate can be constructed under a general selection conditions of the parameters.

Generators and Relations for the Group On(Z[1/2

We give a finite presentation by generators and relations for the group O_n(Z[1/2]) of n-dimensional orthogonal matrices with entries in Z[1/2]. We then obtain a similar presentation for the group of n-dimensional orthogonal matrices of the form M/sqrt(2)^k, where k is a nonnegative integer and M is an integer matrix. Both groups arise in the study of quantum circuits. In particular, when the dimension is a power of 2, the elements of the latter group are precisely the unitary matrices that can be represented by a quantum circuit over the universal gate set consisting of the Toffoli gate, the Hadamard gate, and the computational ancilla.

The ZX&-calculus: A complete graphical calculus for classical circuits using spiders

We give a complete presentation for the fragment, ZX&, of the ZX-calculus generated by the Z and X spiders (corresponding to copying and addition) along with the not gate and the and gate. To prove completeness, we freely add a unit and counit to the category TOF generated by the Toffoli gate and ancillary bits, showing that this yields the full subcategory of finite ordinals and functions with objects powers of two; and then perform a two way translation between this category and ZX&. A translation to some extension of TOF, as opposed to some fragment of the ZX-calculus, is a natural choice because of the multiplicative nature of the Toffoli gate. To this end, we show that freely adding counits to the semi-Frobenius algebras of a discrete inverse category is the same as constructing the Cartesian completion. In particular, for a discrete inverse category, the category of classical channels, the Cartesian completion and adding counits all produce the same category. Therefore, applying these constructions to TOF produces the full subcategory of finite ordinals and partial maps with objects powers of two. By glueing together the free counit completion and the free unit completion, this yields "qubit multirelations."

A general protocol for distributed quantum gates

In distributed quantum computation, quantum remote-controlled gates are used frequently and applied on separate nodes or subsystems of a network. One of the universal and well-known controlled gates is the n-qubit controlled-NOT gate, especially Toffoli gate for the case of three qubits, which are frequently used to synthesize quantum circuits. In this paper, we considered a more general case, an n-qubit controlled-U gate, and present a general protocol for implementing these gates remotely with minimum required resources. Then, the proposed method is applied to implement a Toffoli gate in bipartite and tripartite systems. In this method, we considered cases in which a group of qubits belongs to one subsystem of the network. Then, we improved its consumption resources.

A reversible circuit synthesis algorithm with progressive increase of controls in generalized Toffoli gates

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.

Editorial

Welcome to the sixth issue in 2021. I am very pleased to announce the journals’ Scopus CiteScore of 2.0 for 2020 indicating another scientifically successful year. On behalf of the J.UCS team, I would like to thank all authors for their sound research contributions, the reviewers for their very helpful suggestions and the consortium members for their financial support. Your commitment and dedicated work have strongly contributed to the long-lasting success of our journal. In this regular issue, I am very pleased to introduce five accepted papers from six different countries and 17 involved authors. Edinelço Dalcumune, Luis Antonio Brasil Kowada, André da Cunha Ribeiro, Celina Miraglia Herrera de Figueiredo and Franklin de Lima Marquezino from Brazil present in their article a new algorithm for synthesis of reversible circuits for arbitrary n-bit bijective functions using generalized Toffoli gates, which include positive and negative controls. Murat Firat, Derya Yiltas-Kaplan and Ruya Samli introduce their work on a machine learning method - including Artificial Neural Network (ANN), Linear Regression (LR) and Gradient Boosting (GB) - for determining optimal seat capacity that can supply the highest load factor for the flight operation between any two countries. In a collaborative research between Switzerland, China and the Netherlands Fabian Honegger, Yuan Feng and Matthias Rauterberg have investigated in their research effects of visual, auditory, vibration and draught stimuli on the sense of presence. Julio Moreno, David G. Rosado, Luis E. Sánchez, Manuel A. Serrano and Eduardo Fernández-Medina from Spain discuss in their research a security reference architecture for cyber-physical systems. Adem Tuncer from Turkey introduces a new approach based on a Artificial Bee Colony Algorithm for solving the 15-puzzle problem.

High-fidelity method for a single-step N -bit Toffoli gate in trapped ions

Conditional multiqubit gates are a key component for elaborate quantum algorithms. In a recent work, Rasmussen et al. [Phys. Rev. A 101, 022308 (2020)] proposed an efficient single-step method for a prototypical multiqubit gate, a Toffoli gate, based on a combination of Ising interactions between control qubits and an appropriate driving field on a target qubit. Trapped ions are a natural platform to implement this method, since Ising interactions mediated by phonons have been demonstrated in increasingly large ion crystals. However, the simultaneous application of these interactions and the driving field required for the gate results in undesired entanglement between the qubits and the motion of the ions, reducing the gate fidelity. In this work, we propose a solution based on adiabatic switching of these phonon-mediated Ising interactions. We study the effects of imperfect ground-state cooling and use spin-echo techniques to undo unwanted phase accumulation in the achievable fidelities. For gates coupling to all axial modes of a linear crystal, we calculate high-fidelity $(>99%)$ $N$-qubit rotations with $N=3--7$ ions cooled to their ground state of motion and a gate time below 1 ms. Finally, we study the effect of laser intensity fluctuations and find that the proposed gate requires intensity stabilization with subpercentage noise levels. The high fidelities obtained also for large crystals could make the gate competitive with gate-decomposed, multistep variants of the $N$-qubit Toffoli gate, at the expense of requiring ground-state cooling of the ion crystal.

Quantum Logic Gates Based on DNAtronics, RNAtronics, and Proteintronics

Quantum computers (QCs) adopt an n‐state quantum mechanical system to manipulate the superposition state. However, molecular transistors are not used to build up the quantum logic gate. Herein, it is demonstrated that DNA, RNA, and protein are promising media for QCs, and one uses residue pairs, including nucleotide base pairs and amino acid pairs, via proton‐coupled electron transfer to fabricate a quantum logic gate. In the residue pair, the proton transfer between donor and acceptor states fulfills a qubit. Both the DNA‐CG (3‐qubit) nucleobase pair and nucleotide base pair obey the Toffoli gate. AT (2‐qubit) nucleotide base pair behaves as a SWAP gate and a CNOT gate. Furthermore, the AU and RNA‐CG nucleotide base pairs follow the CNOT gate and Toffoli gate, respectively. In addition, a pair of amino acids achieves 1‐qubit and satisfies the Pauli‐X, ‐Y, and ‐Z gates. The generators of universal quantum logic gates are obtained.

Error rates and resource overheads of repetition cat qubits

We estimate and analyze the error rates and the resource overheads of the repetition cat qubit approach to universal and fault-tolerant quantum computation. The cat qubits stabilized by two-photon dissipation exhibit an extremely biased noise where the bit-flip error rate is exponentially suppressed with the mean number of photons. In a recent work, we suggested that the remaining phase-flip error channel could be suppressed using a 1D repetition code. Indeed, using only bias-preserving gates on the cat-qubits, it is possible to build a universal set of fault-tolerant logical gates at the level of the repetition cat qubit. In this paper, we perform Monte-Carlo simulations of all the circuits implementing the protected logical gates, using a circuit-level error model. Furthermore, we analyze two different approaches to implement a fault-tolerant Toffoli gate on repetition cat qubits. These numerical simulations indicate that very low logical error rates could be achieved with a reasonable resource overhead, and with parameters that are within the reach of near-term circuit QED experiments.

Implementation of all‐optical Toffoli gate by 2D Si–air photonic crystal

Implementation of all‐optical Toffoli gate by 2D Si–air photonic crystal

Constructing three-qubit unitary gates in terms of Schmidt rank and CNOT gates

It is known that every two-qubit unitary operation has Schmidt rank one, two or four, and the construction of three-qubit unitary gates in terms of Schmidt rank remains an open problem. We explicitly construct the gates of Schmidt rank from one to seven. It turns out that the three-qubit Toffoli and Fredkin gate, respectively, have Schmidt rank two and four. As an application, we implement the gates using quantum circuits of CNOT gates and local Hadamard and flip gates. In particular, the collective use of three CNOT gates can generate a three-qubit unitary gate of Schmidt rank seven in terms of the known Strassen tensor from multiplicative complexity.

Reversible Gate Mapping into QCA Explicit Cells Packed with Single Layer

MOS based circuits and computing devices are facing challenge related to energy dissipation, short channel effect and device density. The state of the art technology is driven towards Quantum cellular automata (QCA) computing with an emphasis on a high-speed and smaller area. QCA is a new computing technology that is made of quantum cell restraining two electrons and two dots. In this paper, a novel design of Fredkin, Toffoli, and Feynman gate layout is designed using the molecular QCA concept. Presented cell layout of newly circuit validity is verified by QCADesigner software. Further, a large number of articles are reviewed and comparison results are presented to show the optimal results such as cell count, majority gate count, bounded box area, and clock utilize. In this work, QCA and reversible logic combined together and the proposed designs have less bounded box area and delay, which is an essential part of the next-generation computing.

Fast and robust implementation of quantum gates by transitionless quantum driving

As a basic operation in quantum computers, quantum logic gates and particularly controlled NOT (CNOT) gates are important in quantum information processing. In this paper, we propose a one-step scheme to generate a CNOT gate via transitionless quantum driving. In this scheme, an effective Hamiltonian is obtained based on quantum Zeno dynamics to drive the system to evolve into the target state when the parameters are reasonably set. The results of explicit numerical simulations indicate that the proposed scheme is robust against instability related to variation in experimental parameters and decoherence. The scheme can also be extended to generate a Toffoli gate, which is useful for large-scale quantum computers. Moreover, the generation process involves only a single step, greatly simplifying its implementation.

High-Fidelity Hybrid Universal Quantum Controlled Gates on Photons and Quantum-Dot Spins

Both photons and semiconductor quantum-dot (QD) spins are promising candidates for quantum information science and technology. It is of critical significance to realize high-fidelity quantum controlled gates on photon-spin hybrid systems. In this paper, based on the novel balance condition for the interaction between a single input photon and a singly charged QD embedded in an optical single-sided microcavity, we present three schemes for implementing three universal quantum controlled gates, i.e., the two-qubit controlled-NOT (CNOT) gate, the three-qubit Toffoli gate, and the three-qubit Fredkin gate, on composite hybrid quantum systems consisting of flying photons and QD-confined electron spins. By exploiting the balance condition, the noise caused by the unbalanced reflectance of the coupled and uncoupled QD-cavity systems can be efficiently suppressed, so that the fidelity of each quantum gate operation can be raised to unity in principle. The balance condition can be met without the strict requirement of strong coupling, making the high-fidelity quantum gates easier to be demonstrated in experiments. These features can improve the fidelity and feasibility of these schemes, which can be applied to large-scaled quantum computing and quantum information networks.

Realization of a deterministic quantum Toffoli gate with a single photon

Quantum controlled-logic gates, including controlled NOT gate and Toffoli gate, play critical roles in lots of quantum information processing schemes. We design and experimentally demonstrate deterministic Toffoli gate by utilizing orbital-angular-momentum and polarization degrees of freedom of a single photon. In addition, we generate Bell states by using the controlled NOT gate. The effective conversion rate of the Toffoli gate in our experiment is $(95.1\pm3.2)\%$. Furthermore, our experimental setup does not require any auxiliary photons and probabilistic post selections.

Design and implementation of carry Look ahead generator in reversible logic using nano QCA

The idea of reversible logic is another emerging architecture that has developed its ground in research area. This argument implied zero heat dissipation at the device. According to the physical laws of quantum mechanical effect and the law of coulomb the function and heat relations are satisfied. In this paper, a carry-looking generator is built using reversible logic and its QCA have been reported. To achieve the proposed design use was made of Toffoli and BJN gate. Comparing simulation outcomes to theoretical values verifies the design. The proposed model is tested and simulated using version 2.0.3 of the QCA Designer method.