Toffoli Gate Research Articles

Toffoli gate and quantum correlations: a geometrical approach

By using a geometrical approach, we investigate the behavior of the quantum Toffoli gate in connection to quantum correlations. Special attention is paid to states with maximally mixed marginals. Finally and in the same vein, we scrutinize the Hadamard gate.

Majority-layered T hybridization using quantum-dot cellular automata

AbstractThe atomistic quantum-dot cellular automata (QCA) based implementations of the reversible circuits have got tremendous exposures in the last few days, due to “room-temperature workability” of the QCA. The researchers are in serious need of a methodology that can realize the area-efficient QCA counterparts of reversible benchmark circuits. In this work, a novel methodology named majority-layered T hybridization is proposed to synthesize the reversible circuits using QCA. Firstly the reversible library consisting of CNTS Gates have been generated to validate the usability of the proposed methodology. Then, an elementary QCA module of 3×3 Toffoli Gate have been proposed and extended in the realization of 4×4, 5×5 and 6×6 Toffoli Gates (multi-control Toffoli Gates). The proper mathematical modelling of the several QCA design metrics like effective area, delay and O-cost has been established. The QCA counterpart of 3×3 Toffoli Gate reports 18.61% less effective area and 8.33% less O-cost compared to th...

Generation of an arbitrary concatenated Greenberger–Horne–Zeilinger state with single photons

The concatenated Greenberger–Horne–Zeilinger (C-GHZ) state is a new kind of logic-qubit entangled state, which may have extensive applications in future quantum communication. In this letter, we propose a protocol for constructing an arbitrary C-GHZ state with single photons. We exploit the cross-Kerr nonlinearity for this purpose. This protocol has some advantages over previous protocols. First, it only requires two kinds of cross-Kerr nonlinearities to generate single phase shifts ±θ. Second, it is not necessary to use sophisticated m-photon Toffoli gates. Third, this protocol is deterministic and can be used to generate an arbitrary C-GHZ state. This protocol may be useful in future quantum information processing based on the C-GHZ state.

Design of reversible sequential circuits using electro-optic effect of lithium-niobate-based Mach–Zehnder interferometers

In recent years, it has been shown that reversible logic can play an important role in power optimization for computer design. The various reversible logic gates such as Feynman, Fredkin, Peres, and Toffoli gates have been discussed by researchers, but very little work has been done on reversible sequential circuits. Design of reversible sequential circuits using lithium-niobate-based Mach–Zehnder interferometers is proposed. Here, flip-flops are designed with the help of basic reversible logic gates such as Feynman, Fredkin, and Peres gates. Theoretical descriptions along with mathematical formulation of the devices are provided. The devices are also analyzed through finite difference-beam propagation method and MATLAB® simulation.

Designing High-Fidelity Single-Shot Three-Qubit Gates: A Machine-Learning Approach

Three-qubit quantum gates are key ingredients for quantum error correction and quantum information processing. We generate quantum-control procedures to design three types of three-qubit gates, namely Toffoli, Controlled-Not-Not and Fredkin gates. The design procedures are applicable to a system comprising three nearest-neighbor-coupled superconducting artificial atoms. For each three-qubit gate, the numerical simulation of the proposed scheme achieves 99.9% fidelity, which is an accepted threshold fidelity for fault-tolerant quantum computing. We test our procedure in the presence of decoherence-induced noise as well as show its robustness against random external noise generated by the control electronics. The three-qubit gates are designed via the machine learning algorithm called Subspace-Selective Self-Adaptive Differential Evolution (SuSSADE).

Implementing All-Optical New Reversible Gate using SOA-MZI Architecture

Reversible circuits have gained a lot of attention lately because of its remarkable property of dissipating lesser power as compared to irreversible circuits due to the fact that they do not lose information. Because of continuing revolution in VLSI technology transistor count and thus power dissipation has increased to such an extent that it has put a limit on packing density and performance of the circuits. To overcome these limits reversible logic has came into picture and various reversible logic gates such as Feynman gate, Fredkin gate, Toffoli gate, Peres gate etc has already been designed in optical domain. Semiconductor optical amplifier (SOA) based mach-Zehnder interferometer (MZI) plays a promising role in this field of ultra fast all-optical information processing because of its advantages like high speed, low power, easy of fabrication and fast switching time. Thus various circuits designed by these basic gates has been analysed in terms of cost parameters and it was seen that optical cost and delay of digital circuits is more than desired, as a result performance of system is rendered. Thus new gates are designed by various researchers that focus on improving cost and thus performance of the system. Studying all these modified gates and the digital circuits implemented in literature, an idea of New All-optical universal reversible gate has been proposed which will further improve the performance of digital circuits. Applications of this gate include the design of full adder, half adder, multiplexer and various other combinational circuits and also 13 standard Boolean expressions can be implemented with improved cost. Simulation of this new gate is done using VPI Photonics software.

Optimal and asymptotically optimal NCT reversible circuits by the gate types

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.

Automatic synthesis of quaternary quantum circuits

Quaternary encoded binary circuits are more compact than their binary counterpart. Although quaternary reversible circuits are realizable, design of such circuits is still in its infancy. This work proposes a new, enhanced method of quaternary Galois field sum of products (QGFSOP) synthesis for quaternary quantum circuits. To reduce QGFSOP product terms, the algorithm makes use of 11 newly defined quaternary Galois field (QGF) expansions (for a total of 21 QGF expansions). This algorithm achieves QGFSOP minimization with the assistance of a pseudo-Kronecker Galois field decision diagram (QGFDD). This is a novel approach for QGFSOP synthesis. Finally, QGFSOP expressions are translated into quantum cost optimized quaternary quantum circuits using: (1) newly developed quaternary quantum gate realizations of controlled Feynman and Toffoli gate that are optimized in terms of quantum cost, (2) use of composite literals consisting of 1 digit and M---S gates. Performance evaluation against existing works in the literature determined that our proposed method achieves an average QGFSOP expression product term savings of 32.66 %. Also, the synthesized QGFSOP circuits were evaluated in terms of quantum cost.

Quantum networks for implementing locally and conclusively a nonlocal qudit Toffoli gate: Designing and optimizing

Local implementation of a nonlocal d -dimensional quantum Toffoli gate is considered. We construct respectively two quantum networks for realizing this nonlocal quantum gate conclusively. The first one is based on Eisert’s insight but follow a different pathway which results in a nonlocal d -dimensional Toffoli gate design. This scheme involves a local d -dimensional three-body interaction, which does not appear naturally in physical system. The other one can improve significantly the local implementation of this nonlocal gate if the target node harnesses a qutrit as a catalyser. In its simplest form this quantum network that does the improved scheme is built entirely of local single-body and two-body elementary gates, and with only 3 two-body elementary gates at the target node. The latter is simpler but conclusive, and more efficient but with less resource, which will make it more feasible with the current experimental technology and more suitable for large-scale quantum network.

Coherent modification of entanglement: benefits due to extended Hilbert space

A quantum computing system is typically represented by a set of non-interacting (local) two-state systems—qubits. Many physical systems can naturally have more accessible states, both local and non-local. We show that the resulting non-local network of states connecting qubits can be efficiently addressed via continuous time quantum random walks, leading to substantial speed-up of multiqubit entanglement manipulations. We discuss a three-qubit Toffoli gate and a system of superconducting qubits as an illustration.

Design and performance analysis of SOA–MZI based reversible toffoli and irreversible AND logic gates in a single photonic circuit

With the increasing flow of information, data rate is increasing very rapidly. This has resulted in increased amount of heat dissipation from the processing nodes. Generally electronic processors dissipate heat if the data rate is very high. Therefore electronic processors should be replaced by optical processors with optical gates as their foundation. Another reason for heat dissipation is the loss of information bits at the processing nodes. This can be reduced by introducing reversible computing. A combination of optical reversible and irreversible gates at the processing nodes can recover data and minimize power dissipation and consumption. In this article semiconductor optical amplifier–Mach–Zehnder interferometer based all optical reversible Toffoli gate and AND gate in a single photonic circuit has been designed. The performance of the gates has been analyzed through extinction ratio and quality factor. Different types of gates (reversible, irreversible) on a single photonic chip will slowly but surely lead towards all optical processing of data with added advantage of recovery of input data.

Comment on “Linear-depth quantum circuits forn-qubit Toffoli gates with no ancilla”

Multiqubit quantum gates are natural quantum operations derived from various decompositions of quantum evolutions. Recently, linear-depth quantum circuits were proposed for an arbitrary $n$-qubit Toffoli gate with ancillary qubits [M. Saeedi and M. Pedram, Phys. Rev. A 87, 062318 (2013)]. Their circuits are based on an equivalent representation of the not gate regardless of the global phase. However, we find that this equivalence cannot be extended to the controlled-not gate or the $n$-qubit Toffoli gate. By reevaluating their circuits, we argue that their linear-depth quantum circuits could realize the joint operations of an $n$-qubit Toffoli gate and an $n$-qubit phase-flip gate.

Hyper-parallel Toffoli gate on three-photon system with two degrees of freedom assisted by single-sided optical microcavities.

Encoding qubits in multiple degrees of freedom (DOFs) of a quantum system allows less-decoherence quantum information processing with much less quantum resources. We present a compact and scalable quantum circuit to determinately implement a hyper-parallel controlled-controlled-phase-flip (hyper-C2PF) gate in a three-photon system in both the polarization and spatial DOFs. In contrast with the one with many qubits encoding on one DOF only, our hyper-C2PF gate operating two independent C2PF gates on a three-photon system with less decoherence, and reduces the quantum resources required in quantum information processing by a half. Additional photons, necessary for many approaches, are not required in the present scheme. Our calculation shows that this hyper-C2PF gate is feasible in experiment.

Distributed Photonic Quantum Computations Assisted by Atomic Ensembles

Distributed quantum computation requires that quantum operations may be acted on remote logical qubits. We investigate the possibility of the distributed quantum computation for nonlocal photons assisted by cavity quantum electrodynamics. We first give a compact circuit for the controlled-NOT gate on a remote two-photon system. For the Toffoli gate, we introduce two circuits for a bipartite system with one photonic Einstein-Podolski-Rosen (EPR) entanglement. Two EPR pairs are required for a tripartite system. These Toffoli gates cost a half of entanglements required in the previous teleportation-based quantum computation. These elementary gates may be combined flexibly up to special decompositions of joint system in large-scale quantum applications.

Multifractality in fidelity sequences of optimized Toffoli gates

We analyze the multifractality of the fidelity in an engineered Toffoli gate. Using quantum control methods, we define several optimization problems whose global solutions realize the gate in a chain of three qubits with XY Heisenberg interaction. We perturb the system by introducing imperfections in the form of $1/f$ noise to the interqubit couplings. Multifractal analysis shows that the degree of multifractality in the gate fidelity increases when optimized control pulses used to render the fidelity are less sensitive to variations in the interqubits coupling strengths.

Quantum gate learning in qubit networks: Toffoli gate without time-dependent control

AbstractWe put forward a strategy to encode a quantum operation into the unmodulated dynamics of a quantum network without the need for external control pulses, measurements or active feedback. Our optimisation scheme, inspired by supervised machine learning, consists in engineering the pairwise couplings between the network qubits so that the target quantum operation is encoded in the natural reduced dynamics of a network section. The efficacy of the proposed scheme is demonstrated by the finding of uncontrolled four-qubit networks that implement either the Toffoli gate, the Fredkin gate or remote logic operations. The proposed Toffoli gate is stable against imperfections, has a high fidelity for fault-tolerant quantum computation and is fast, being based on the non-equilibrium dynamics.

Conditional operation using three degrees of freedom of a laser beam for application in quantum information

We propose and experimentally demonstrate a conditional operation by using three degrees of freedom of a classical laser beam that simulates the operations of the quantum Toffoli gate. A tripartite non-separable mode, a classical analog of the entangled Greenberger–Horne–Zeilinger (GHZ) state, can be produced in a very simple way. In addition, we present a discussion showing the possibility to perform a quantum Toffoli gate by using the quantum counterpart of our experiment, i.e., by attenuating the laser beam to a photon counting level.

Design and Analysis of n:2n Reversible Decoder

ABSTRACTEvery irreversible circuit has a heat dissipation problem. Heat dissipation of any irreversible circuit depends on number of losses of information and the heat generates due to irreversibility is minimum of Joules per bit loss. In order to avoid heat dissipation, it is needed to build reversible circuits for reduction of loss of information. This paper proposes to design a reversible decoder. The proposed design of reversible decoders has a common pattern for different values of n. The proposed design is a nesting structure where higher order decoder uses lower order decoder's components. Basic quantum gates such as Toffoli gate, CNOT, and Pauli X gates are used in this design.

Experimental replication of single-qubit quantum phase gates

We experimentally demonstrate the underlying physical mechanism of the recently proposed protocol for superreplication of quantum phase gates [W. D\"ur, P. Sekatski, and M. Skotiniotis, Phys. Rev. Lett. 114, 120503 (2015)], which allows to produce up to $N^2$ high-fidelity replicas from N input copies in the limit of large N. Our implementation of 1->2 replication of the single-qubit phase gates is based on linear optics and qubits encoded into states of single photons. We employ the quantum Toffoli gate to imprint information about the structure of an input two-qubit state onto an auxiliary qubit, apply the replicated operation to the auxiliary qubit, and then disentangle the auxiliary qubit from the other qubits by a suitable quantum measurement. We characterize the replication protocol by full quantum process tomography and observe good agreement of the experimental results with theory.

Universal quantum gates for photon-atom hybrid systems assisted by bad cavities.

We present two deterministic schemes for constructing a CNOT gate and a Toffoli gate on photon-atom and photon-atom-atom hybrid quantum systems assisted by bad cavities, respectively. They are achieved by cavity-assisted photon scattering and work in the intermediate coupling region with bad cavities, which relaxes the difficulty of their implementation in experiment. Also, bad cavities are feasible for fast quantum operations and reading out information. Compared with previous works, our schemes do not need any auxiliary qubits and measurements. Moreover, the schematic setups for these gates are simple, especially that for our Toffoli gate as only a quarter wave packet is used to interact the photon with each of the atoms every time. These atom-cavity systems can be used as the quantum nodes in long-distance quantum communication as their relatively long coherence time is suitable for multi-time operations between the photon and the system. Our calculations show that the average fidelities and efficiencies of our two universal hybrid quantum gates are high with current experimental technology.