Toffoli Gate Research Articles

Compressed sensing quantum process tomography for superconducting quantum gates

We apply the method of compressed sensing (CS) quantum process tomography (QPT) to characterize quantum gates based on superconducting Xmon and phase qubits. Using experimental data for a two-qubit controlled-Z gate, we obtain an estimate for the process matrix $\chi$ with reasonably high fidelity compared to full QPT, but using a significantly reduced set of initial states and measurement configurations. We show that the CS method still works when the amount of used data is so small that the standard QPT would have an underdetermined system of equations. We also apply the CS method to the analysis of the three-qubit Toffoli gate with numerically added noise, and similarly show that the method works well for a substantially reduced set of data. For the CS calculations we use two different bases in which the process matrix $\chi$ is approximately sparse, and show that the resulting estimates of the process matrices match each ther with reasonably high fidelity. For both two-qubit and three-qubit gates, we characterize the quantum process by not only its process matrix and fidelity, but also by the corresponding standard deviation, defined via variation of the state fidelity for different initial states.

Hardness of classically simulating quantum circuits with unbounded Toffoli and fan-out gates

We study the classical simulatability of constant-depth polynomial-size quantum circuits followed by only one single-qubit measurement, where the circuits consist of universal gates on at most two qubits and additional gates on an unbounded number of qubits. First, we consider unbounded Toffoli gates as additional gates and deal with the weak simulation, i.e., sampling the output probability distribution. We show that there exists a constant-depth quantum circuit with only one unbounded Toffoli gate that is not weakly simulatable, unless $\bqp \subseteq \postbpp \cap \am$. Then, we consider unbounded fan-out gates as additional gates and deal with the strong simulation, i.e., computing the output probability. We show that there exists a constant-depth quantum circuit with only two unbounded fan-out gates that is not strongly simulatable, unless $\p = \pp$. These results are in contrast to the fact that any constant-depth quantum circuit without additional gates on an unbounded number of qubits is strongly and weakly simulatable.

Hardness of classically simulating quantum circuits with unbounded Toffoli and fan-out gates

We study the classical simulatability of constant-depth polynomial-size quantum circuits followed by only one single-qubit measurement, where the circuits consist of universal gates on at most two ...

Fuzzy Approach for Toffoli Gate in Quantum Computation With Mixed States

In the framework of quantum computation with mixed states, a fuzzy representation based on continuous t -norms for Toffoli gate is introduced. In this representation, the incidence of nonseparability is specially investigated.

Enhanced architectures for room-temperature reversible logic gates in graphene

We show that reversible two- and three-input logic gates, such as the universal Toffoli gate, can be implemented with three tilted gate electrodes patterned on a monolayer graphene flake. These reversible gates are based on the unique properties of ballistic charge carriers in graphene, which induce bandgaps in transmission for properly chosen potential barriers. The enhanced architectures for reversible logic gate implementation proposed in this paper offer a remarkable design simplification compared to standard approaches based on field-effect transistor circuits, as well as potential high-frequency operation.

Implementing a one-bit reversible full adder using quantum-dot cellular automata

Quantum-dot cellular automata (QCA) technique is one of the emerging and promising nanotechnologies. It has considerable advantages versus CMOS technology in various aspects such as extremely low power dissipation, high operating frequency and small size. In this paper, designing of a one-bit full adder is investigated using a QCA implementation of Toffoli and Fredkin gates. Then, a full adder design with reversible QCA1 gates is proposed regarding to overhead and power savings. Our proposed full adder design is more preferable when considering both circuit area and speed. The proposed design uses only two QCA1 gates and maximizes the circuit density and focuses on a layout of the circuit which is minimal in using QCA cells.

On implementing nondestructive triplet Toffoli gate with entanglement swapping operations via the GHZ state analysis

We investigate an novel implementation of a Toffoli gate using multiple independent auxiliary photons prepared beforehand in single-qubit states. This gate can be performed nondestructively with entanglement swapping via the Greenberger---Horne---Zeilinger state analysis. We evaluate the performance of the proposed Toffoli gate with the fidelity based on different computation bases. The multi-qubit-entanglement gate is no longer theoretical since it can be implemented in principle with single-qubit photons.

Exact Synthesis of Reversible Circuits Using A* Algorithm

With the growing emphasis on low-power design methodologies, and the result that theoretical zero power dissipation is possible only if computations are information lossless, design and synthesis of reversible logic circuits have become very important in recent years. Reversible logic circuits are also important in the context of quantum computing, where the basic operations are reversible in nature. Several synthesis methodologies for reversible circuits have been reported. Some of these methods are termed as exact, where the motivation is to get the minimum-gate realization for a given reversible function. These methods are computationally very intensive, and are able to synthesize only very small functions. There are other methods based on function transformations or higher-level representation of functions like binary decision diagrams or exclusive-or sum-of-products, that are able to handle much larger circuits without any guarantee of optimality or near-optimality. Design of exact synthesis algorithms is interesting in this context, because they set some kind of benchmarks against which other methods can be compared. This paper proposes an exact synthesis approach based on an iterative deepening version of the A* algorithm using the multiple-control Toffoli gate library. Experimental results are presented with comparisons with other exact and some heuristic based synthesis approaches.

Quantum circuits for simple periodic functions

Periodic functions are of special importance in quantum computing, particularly in applications of Shor's algorithm. We explore methods of creating circuits for periodic functions to better understand their properties. We introduce a method for constructing the circuit for a simple monoperiodic function, that is one-to-one within a single period, of a given period $p$. We conjecture that to create a simple periodic function of period $p$, where $p$ is an $n$-bit number, one needs at most $n$ Toffoli gates.

ALL OPTICAL IMPLEMENTATION OF HIGH SPEED AND LOW POWER REVERSIBLE FULL ADDER USING SEMICONDUCTOR OPTICAL AMPLIFIER BASED MACH-ZEHNDER INTERFEROMETER

In the recent years reversible logic design has promising applications in low power computing, optical computing, quantum computing. VLSI design mainly concentrates on low power logic circuit design. In the present scenario researchers have made implementations of reversible logic gates in optical domain for its low energy consumption and high speed. This study is all about designing a reversible Full adder using combination of all optical Toffoli and all optical TNOR and to compare it with the Full adder designed using all optical Toffoli gate in terms of optical cost. All optical TNOR gate can work as a replacement of existing NAND based All optical Toffoli Gate (TG). The gates are designed using Mach-Zehnder Interferometer (MZI) based optical switch. The proposed system is developed with the basic of reversibility to design all optical full Adder implemented with CMOS transistors. The design is efficient in terms of both architecture and in power consumption.

Challenges and advances in Toffoli network optimisation

This study gives a brief overview of the current trends in reversible logic synthesis with emphasis on template matching. The basic building block for reversible circuits considered here is the multiple-control Toffoli gate. Some approaches to synthesis are reviewed and the challenges are explained. Since many practical functions are not reversible, they must be embedded into reversible ones, if they are to be implemented using reversible logic. The complexity of such embeddings is expounded. A two phase synthesis is described where particular attention is devoted to the optimisation phase via template matching. Insights into the properties of the templates, have led to algorithms that aid the generation of templates. Until recently, the application of templates has been guided by different heuristics. A review of an exact template matching algorithm with a discussion of the implications of such an algorithm is given. Exact matching affects both the generation as well as the application of templates. Results from a prototype implementation are encouraging.

Noise resistance of Toffoli gate in an array of coupled cavities

Two schemes of implementing the Toffoli gate are proposed in an array of coupled cavities. In the first one, two classical laser fields with fixed Rabi frequency are used via quantum Zeno dynamics. While in the second scheme, the invariant-based inverse engineering is used to implement the Toffoli gate, which can drive the instantaneous eigenstates to evolve exactly under the Schrödinger equation. The strictly numerical simulations are given and the influences of cavity decay and spontaneous emission on the Toffoli gate operation are analyzed with master equation. The result shows that our schemes for the Toffoli gate are robust against the decoherence because of the short time needed.

Hardware emulation of quantum circuits based on Toffoli gates

This work presents the design of a hardware architecture for the emulation of quantum circuits based on Toffoli gates allowing the emulation of more than 50 qubits. The state of the system is obtained processing each basis state by means of the functions determined by the quantum gates for each qubit; the time required to execute the emulation grows exponentially only with the number of qubits that are used to generate a superposition of states, but not with the total amount of qubits of the system as occurs when the conventional matrix representation is used. Additionally, an array of processing units was designed to decrease the execution time. The synthesis results allow concluding that 9.35 seconds are required to emulate the 8-bit modular exponentiation, which uses 48 qubits, 155,312 quantum gates and requires processing 131,072 basis states. Furthermore these results allow estimating that 256 processing units of 52 qubits can be implemented in the FPGA EP3C120F780I7.

Efficient reversible NOR gates and their mapping in optical computing domain

Reversible logic is a computing paradigm in which there is a one to one mapping between the input and the output vectors. Reversible logic gates are implemented in an optical domain as it provides high speed and low energy computations. In the existing literature there are two types of optical mapping of reversible logic gates: (i) based on a semiconductor optical amplifier (SOA) using a Mach–Zehnder interferometer (MZI) switch; (ii) based on linear optical quantum computation (LOQC) using linear optical quantum logic gates. In reversible computing, the NAND logic based reversible gates and design methodologies based on them are widely popular. The NOR logic based reversible gates and design methodologies based on them are still unexplored. In this work, we propose two NOR logic based n-input and n-output reversible gates one of which can be efficiently mapped in optical computing using the Mach–Zehnder interferometer (MZI) while the other one can be mapped efficiently in optical computing using the linear optical quantum gates. The proposed reversible NOR gates work as a corresponding NOR counterpart of NAND logic based Toffoli gates. The proposed optical reversible NOR logic gates can implement the reversible boolean logic functions with a reduced number of linear optical quantum logic gates or reduced optical cost and propagation delay compared to their implementation using existing optical reversible NAND gates. It is illustrated that an optical reversible gate library having both optical Toffoli gate and the proposed optical reversible NOR gate is superior compared to the library containing only the optical Toffoli gate: (i) in terms of number of linear optical quantum gates when implemented using linear optical quantum computing (LOQC), (ii) in terms of optical cost and delay when implemented using the Mach–Zehnder interferometer.

Implementation of distributed Toffoli gates in separate cavities through virtual excitation processes

We propose a scheme to realize a three-qubit Toffoli gate with three atoms held in separate cavities that are coupled by optical fibers. The gate is achieved by engineering an effective asymmetric X–Y model induced by virtually manipulating the atomic excited states and photons. We study the validity of the scheme as well as the influence of dissipation by numerical simulation, and show that the scheme is robust against decoherence.

The quantum realization of Arnold and Fibonacci image scrambling

The quantum Fourier transform, the quantum wavelet transform, etc., have been shown to be a powerful tool in developing quantum algorithms. However, in classical computing, there is another kind of transforms, image scrambling, which are as useful as Fourier transform, wavelet transform, etc. The main aim of image scrambling, which is generally used as the preprocessing or postprocessing in the confidentiality storage and transmission, and image information hiding, was to transform a meaningful image into a meaningless or disordered image in order to enhance the image security. In classical image processing, Arnold and Fibonacci image scrambling are often used. In order to realize these two image scrambling in quantum computers, this paper proposes the scrambling quantum circuits based on the flexible representation for quantum images. The circuits take advantage of the plain adder and adder modulo $$N$$N to factor the classical transformations into basic unitary operators such as Control-NOT gates and Toffoli gates. Theoretical analysis indicates that the network complexity grows linearly with the size of the number to be operated.

Universal quantum gates on electron-spin qubits with quantum dots inside single-side optical microcavities

We present some compact quantum circuits for a deterministic quantum computing on electron-spin qubits assisted by quantum dots inside single-side optical microcavities, including the CNOT, Toffoli, and Fredkin gates. They are constructed by exploiting the giant optical Faraday rotation induced by a single-electron spin in a quantum dot inside a single-side optical microcavity as a result of cavity quantum electrodynamics. Our universal quantum gates have some advantages. First, all the gates are accomplished with a success probability of 100% in principle. Second, our schemes require no additional electron-spin qubits and they are achieved by some input-output processes of a single photon. Third, our circuits for these gates are simple and economic. Moreover, our devices for these gates work in both the weak coupling and the strong coupling regimes, and they are feasible in experiment.

A Fault Analysis in Reversible Sequential Circuits

In this paper, the researchers propose the design of reversible circuits using reversible gates. Reversible logic is implemented in reversible circuits. Reversible logic is mostly preferred due to less heat dissipation. Conservative logic gates can be designed in any sequential circuits and can be tested using two test vectors. The significance of proposed work lies in the design of reversible sequential circuits and their equivalent circuits for maximum fault coverage. The design of reversible sequential circuits using Toffoli gate and Peres gate is proposed in this literature. The design of Toffoli and Peres equivalent circuits is proposed first time in this literature, in order to achieve maximum fault coverage. The proposed Toffoli and Peres gates surpass the Fredkin gate and MXCQCA gate in terms of area, number of gates and timing. The simulation and coding is performed using cadence tool. Keywords—Reversible logic, Peres gate, Toffoli gate, Feynman gate, latches.

Local Implementations of Non-Local Quantum Gates in Linear Entangled Channel

In this paper, we demonstrate n-party controlled unitary gate implementations locally on arbitrary remote state through linear entangled channel where control parties share entanglement with the adjacent control parties and only one of them shares entanglement with the target party. In such a network, we describe the protocol of simultaneous implementation of controlled-Hermitian gate starting from three party scenarios. We also explicate the implementation of three party controlled-Unitary gates, a generalized form of Toffoli gate and subsequently generalize the protocol for n-party using minimal cost.

A Novel Presentation of Toffoli Gate in Quantum-dot Cellular Automata (QCA)

Quantum dot Cellular Automata (QCA) is one of the emerging nanotechnologies, promising alternative to CMOS technology due to faster speed, smaller size, lower power consumption, higher scale integration and higher switching frequency. The basic element in QCA is majority gate. This paper, present two different design layout of Toffoli gate based on QCA logic gates: majority voter gate (MV), QCA wire and inverter gate. QCADesigner, a common QCA layout design and verification tool is employed to verify and simulate the proposed Toffoli Gate (TG). The simulation result confirmed the correctness of the proposed circuits. The proposed circuit has a promising future in constructing of nano-scale low power exhausting information processing system and can stimulate higher digital applications in QCA. General Terms Nano Technology, Quantum dot Cellular Automata (QCA).