Toffoli Gate Research Articles

Experimental quantum teleportation of a Toffoli gate across three spatially distributed parties in a photonic quantum network

Quantum computers may offer significant computational advantages over classical counterparts, revolutionizing the technology landscape in the near future. When networked together, the advantage of quantum computing may be further amplified, and there may emerge innovative applications. Among various active explorations on distributed quantum computing, multiqubit quantum gates between distant networked quantum computers are of particular interest because they offer operational advantages of efficiency and fidelity. Here we report the first experimental demonstration of quantum teleportation for a Toffoli gate, which is a commonly used multiqubit quantum gate, across three spatially distributed parties within a photonic quantum network. Employing the Hofmann method, we estimate the fidelity of the teleported Toffoli gate to be at least 0.706 ± 0.131. This successful demonstration of the quantum teleportation of a Toffoli gate constitutes a critical step toward the ultimate realization of distributed quantum computation.

Quantum circuit syntheses of Boolean matrix multiplication using only CNOT gates

Abstract Boolean matrices and operations on Boolean matrices have become increasingly popular in quantum computing and quantum information because they can be used to represent quantum states and the corresponding operations. In this paper, we first give a quantum circuit direct synthesis of general Boolean matrix multiplication using Toffoli gates. We assign a qubit to each element of the matrix and add Toffoli gates at the corresponding positions where the matrix elements are multiplied according to the arithmetic of matrix multiplication. For $n\times p$ dimensional matrix $A$ multiplied by $p\times m$ dimensional matrix $B$, $np + pm + nm$ qubits and at most $pmn$ Toffoli gates are used in total. Then, an improved heuristic algorithm based on matrix Hamming distance selection of CNOT gate is proposed to synthesize the quantum circuit of a matrix, without using wire switching and SWAP gates, using only CNOT gates. The results show that our method is as good as or better than the classical synthesis. Finally, the quantum circuit of Boolean matrix multiplication with at least one invertible or full-rank matrix using only CNOT gates is given, which is divided into five cases for discussion and analysis. For $n\times n$ dimensional invertible matrix $A$ multiplied by $n\times m$ dimensional matrix $B$, $nm$ qubits and at most $mn^{2}$ CNOT gates are used in total. For $n\times p$ dimensional column full-rank matrix $A$ multiplied by $p\times m$ dimensional matrix $B$, $nm$ qubits and at most $pmn$ CNOT gates are used in total. The results show that this method uses less quantum resources to synthesize the quantum circuit of Boolean matrix multiplication. The correctness of the quantum circuits was verified by the Aer simulator of the IBM Quantum platform.

A Study of q-Deformed Bosons, and Their Implications to Quantum Optics

In this study, we investigate three types of q-deformed boson oscillators, focusing on their mathematical frameworks and thermodynamic properties. We calculate key thermodynamic quantities, such as internal energy and entropy, as functions of the deformation parameter q. Our results reveal that these oscillators are eigenstates of specific deformed boson annihilation operators. We also analyze their unique characteristics and implications in deformed quantum optics. Furthermore, we examine the impact of q-deformation on qutrit logic gates, including cycle, self-shift, controlled cycle, controlled self-shift, Feynman, ternary Toffoli, and Fredkin gates, highlighting their altered computational properties. This research contributes to a deeper understanding of q-deformed systems and their applications in quantum computing. Overall, it opens new avenues for exploring the interplay between deformation parameters and quantum information processing.

Polarization and Orbital Angular Momentum Encoded Quantum Toffoli Gate Enabled by Diffractive Neural Networks.

Controlled quantum gates play a crucial role in enabling quantum universal operations by facilitating interactions between qubits. Direct implementation of three-qubit gates simplifies the design of quantum circuits, thereby being conducive to performing complex quantum algorithms. Here, we propose and present an experimental demonstration of a quantum Toffoli gate fully exploiting the polarization and orbital angular momentum of a single photon. The Toffoli gate is implemented using the polarized diffractive neural networks scheme, achieving a mean truth table visibility of 97.27±0.20%. We characterize the gate's performance through quantum state tomography on 216 different input states and quantum process tomography, which yields a process fidelity of 94.05±0.02%. Our method offers a novel approach for realizing the Toffoli gate without requiring exponential optical elements while maintaining extensibility to the implementation of other three-qubit gates.

Design of ternary reversible Feynman and Toffoli gates in ternary quantum-dot cellular automata

The use of reversible logic gates leads to a reduction in energy loss in logic circuits by preventing information loss. New computing methods, such as quantum-dot cellular automata (QCA), have been offered by nanotechnology emerging with nanoelectronics to make more comprehensive logic circuits. In nanotechnology-based systems, some bits are erased when the system performs any computation, and this causes heat dissipation and energy loss in systems. Adder circuits are the basis of any arithmetic operation and one of the main parts of many circuits for creating complex hardware; therefore, the use of enhanced adder circuits leads to high performance in logic circuits. In irreversible logic, the energy that is transferred from the power supply to the circuit is converted into heat, and energy loss occurs. Power management plays a vital role in modern computational systems, and using ternary logic instead of previous technologies leads to better performance. The main purpose of our study is to design ternary quantum-dot cellular automata (TQCA) reversible logic gates based on ternary quantum-dot cellular technology. Reversible gates are the basis of creating a reversible circuit. In this paper, the Muthukrishnan-Stroud (M-S) gate, which is the basis of all other reversible ternary gates, is implemented in ternary QCA technology, and then, reversible ternary Feynman and Toffoli (C2NOT) gates are designed. More optimal adder circuits can be realized in three-valued technology using Feynman and Toffoli gates. The area, delay, and cell count of the proposed TQCA designs are compared with those of other related works, and the effect of fault on the designs in the presence of cell omission defect is determined. The occupied areas of the proposed Feynman and Toffoli gate designs are 0.069 μm2 and 0.073 μm2, respectively. Moreover, the fault tolerance levels of these TQCA gates are 77% and 92%, respectively.

Implementation of 4-bit Universal Shift Register using Reversible Logic

The field of digital logic design is increasingly exploring reversible logic due to its potential in minimizing power dissipation, an essential criterion in low-power applications. This paper presents the design and implementation of a 4-bit universal shift register (USR) using reversible logic gates. The design leverages Fredkin gates, multiplexer (MUX), and D flip-flops to achieve functionality with minimal power consumption. Simulation results demonstrate the efficiency and effectiveness of the proposed design. Reversible logic is essential to low-power digital design and quantum computing because it makes it possible to reconstruct input signals from output signals, which reduces heat generation and information loss. The suggested shift register can be used for hold, parallel load, and left and right shift operations. To implement the functions of the shift register, we used a variety of reversible gates, including the Toffoli and Fredkin gates. When the architecture is contrasted with traditional non-reversible shift registers, it shows notable gains in reduced gate count and power efficiency

Implementing arbitrary multimode continuous-variable quantum gates with fixed non-Gaussian states and adaptive linear optics

Non-Gaussian quantum gates are essential components for optical quantum information processing. However, the efficient implementation of practically important multimode higher-order non-Gaussian gates has not been comprehensively studied. We propose a measurement-based method to directly implement general, multimode, and higher-order non-Gaussian gates using only fixed non-Gaussian ancillary states and adaptive linear optics. Compared to existing methods, our method allows for a more resource-efficient and experimentally feasible implementation of multimode gates that are important for various applications in optical quantum technology, such as the two-mode cubic quantum nondemolition gate or the three-mode continuous-variable Toffoli gate, and their higher-order extensions. Our results will expedite the progress toward fault-tolerant universal quantum computing with light. Published by the American Physical Society 2024

Reducing the Runtime of Fault-Tolerant Quantum Simulations in Chemistry through Symmetry-Compressed Double Factorization.

Quantum phase estimation based on qubitization is the state-of-the-art fault-tolerant quantum algorithm for computing ground-state energies in chemical applications. In this context, the 1-norm of the Hamiltonian plays a fundamental role in determining the total number of required iterations and also the overall computational cost. In this work, we introduce the symmetry-compressed double factorization (SCDF) approach, which combines a CDF of the Hamiltonian with the symmetry shift technique, significantly reducing the 1-norm value. The effectiveness of this approach is demonstrated numerically by considering various benchmark systems, including the FeMoco molecule, cytochrome P450, and hydrogen chains of different sizes. To compare the efficiency of SCDF to other methods in absolute terms, we estimate Toffoli gate requirements, which dominate the execution time on fault-tolerant quantum computers. For the systems considered here, SCDF leads to a sizable reduction of the Toffoli gate count in comparison to other variants of DF or even tensor hypercontraction, which is usually regarded as the most efficient approach for qubitization.

A Multiple Controlled Toffoli Driven Adaptive Quantum Neural Network Model for Dynamic Workload Prediction in Cloud Environments.

The key challenges in cloud computing encompass dynamic resource scaling, load balancing, and power consumption. Accurate workload prediction is identified as a crucial strategy to address these challenges. Despite numerous methods proposed to tackle this issue, existing approaches fall short of capturing the high-variance nature of volatile and dynamic cloud workloads. Consequently, this paper introduces a novel model aimed at addressing this limitation. This paper presents a novel Multiple Controlled Toffoli-driven Adaptive Quantum Neural Network (MCT-AQNN) model to establish an empirical solution to complex, elastic as well as challenging workload prediction problems by optimizing the exploration, adaption, and exploitation proficiencies through quantum learning. The computational adaptability of quantum computing is ingrained with machine learning algorithms to derive more precise correlations from dynamic and complex workloads. The furnished input data point and hatched neural weights are refitted in the form of qubits while the controlling effects of Multiple Controlled Toffoli (MCT) gates are operated at the hidden and output layers of Quantum Neural Network (QNN) for enhancing learning capabilities. Complimentarily, a Uniformly Adaptive Quantum Machine Learning (UAQL) algorithm has evolved to functionally and effectually train the QNN. The extensive experiments are conducted and the comparisons are performed with state-of-the-art methods using four real-world benchmark datasets. Experimental results evince that MCT-AQNN has up to 32%-96% higher accuracy than the existing approaches.

Low-cost nonlocal Toffoli gate in a linear quantum network topology

Although a Toffoli gate can be equivalently implemented using several single-qubit and two-qubit gates, it will consume much resource, two of which are nonlocal controlled-NOT (CNOT) gates acting on two non-adjacent nodes, especially in distributed quantum computation (DQC). We, for the first time, employ an ancillary qubit to construct a nonlocal Toffoli gate for DQC in linear network topology. The ancillary-qubit-based scheme needs fewer qubits, quantum gates, and entanglement states than that based on quantum teleportation scheme and the entanglement swapping scheme. We also analyze the performance of the three proposed schemes under different application scenarios, and present their pros and cons. Our work will help to implement DQC in noisy intermediate-scale quantum (NISQ) era.

Neural-network-designed three-qubit gates robust against charge noise and crosstalk in silicon

Spin qubits in semiconductor quantum dots are a promising platform for quantum computing, however, scaling to large systems is hampered by crosstalk and charge noise. Crosstalk here refers to the unwanted off-resonant rotation of idle qubits during the resonant rotation of the target qubit. For a three-qubit system with crosstalk and charge noise, it is difficult to analytically create gate protocols that produce three-qubit gates, such as the Toffoli gate, directly in a single shot instead of through the composition of two-qubit gates. Therefore, we numerically optimize a physics-informed neural network to produce theoretically robust shaped pulses that generate a Toffoli-equivalent gate. Additionally, robust π2 X and Controlled-Z gates are also presented in this work to create a universal set of gates robust against charge noise. The robust pulses maintain an infidelity of 10−3 for average quasistatic fluctuations in the voltage of up to a few mV instead of tenths of mV for non-robust pulses.

A novel quantum Dempster's rule of combination for pattern classification

Dempster's rule of combination (DRC) is widely used for uncertainty reasoning in intelligent information system, which is generalized to complex domain recently. However, as the increase of identification framework elements, the computational complexity of Dempster's rule of combination increases exponentially. To address this issue, we propose a novel quantum Dempster's rule of combination (QDRC) by means of Toffoli gate. The QDRC combination process is completely implemented using quantum circuits. To validate the efficiency of the proposed QDRC, we conduct simulation experiments on IBM and IonQ quantum cloud platforms. Besides, we implement the QDRC algorithm to the real quantum computer of IonQ Harmony. The experiment results show that the result of QDRC is getting more and more accurate as the number of measurements increases. In addition, we propose a quantum multisource information fusion method on the basis of QDRC. Finally, we demonstrate its effectiveness to solve pattern classification problem under several diverse real UCI datasets.

Optimal implementation of quantum gates with two controls

We give a detailed proof of a well-known theorem in quantum computing. The theorem characterizes the number of two-qubit gates that is necessary for implementing three-qubit quantum gates with two controls. For example, the theorem implies that five 2-qubit gates are necessary for implementing the Toffoli gate. No detailed proof was available earlier.

Concrete quantum cryptanalysis of binary elliptic curves via addition chain

Thus far, several papers reported concrete resource estimates of Shor’s quantum algorithm for solving the elliptic curve discrete logarithm problem. In this paper, we study quantum FLT-based inversion algorithms over binary elliptic curves. There are two major algorithms proposed by Banegas et al. and Putranto et al., where the former and latter algorithms achieve fewer numbers of qubits and smaller depths of circuits, respectively. We propose two quantum FLT-based inversion algorithms that essentially outperform previous FLT-based algorithms and compare the performance for NIST curves of the degree n. Specifically, for all n, our first algorithm achieves fewer qubits than Putranto et al.’s one without sacrificing the number of Toffoli gates and the depth of circuits, while our second algorithm achieves smaller depths of circuits without sacrificing the number of qubits and Toffoli gates. For example, when n=571\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 571$$\\end{document}, the number of qubits of our first algorithm is 74 % of that of Putranto et al.’s one, while the depth of our second algorithm is 83 % of that of Banegas et al.’s one. The improvements stem from the fact that FLT-based inversions can be performed with arbitrary sequences of addition chains for n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n - 1$$\\end{document} although both Banegas et al. and Putranto et al. follow fixed sequences that were introduced by Itoh and Tsujii’s classical FLT-based inversion. In particular, we analyze how several properties of addition chains, which do not affect the computational resources of classical FLT-based inversions, affect the computational resources of quantum FLT-based inversions and find appropriate sequences.

Full-magnetic implementation of a classical Toffoli gate

The Toffoli gate is the essential ingredient for reversible computing, an energy-efficient classical computational paradigm that evades the energy dissipation resulting from Landauer's principle. In this paper, we analyze different setups to realize a magnetic implementation of the Toffoli gate using three interacting classical spins, each one embodying one of the three bits needed for the Toffoli gate. In our scheme, different control-spin configurations produce an effective field capable of conditionally flipping the target spin. We study what the experimental requirements are for the realization of our scheme, focusing on the degree of local control, the ability to dynamically switch the spin-spin interactions, and the required single-spin anisotropies to make the classical spin stable, showing that these are compatible with current technology. Published by the American Physical Society 2024

Optimized design and investigation of novel reversible toffoli and peres gates using QCA techniques

The QCA is a revolutionary dominating transistor-less computational nanotechnology based on quantum dots. As such, it might be used in the next generation of quantum computational nano-electronics devices. New molecular materials like QCA are being developed for use in nanoscale devices and cables. The major objective of this research is to optimize the design and investigate numerous properties of a QCA reversible logic circuit design. These locations include, limited Toffoli Gate (TG) and Peres Gate (PG). To address the key problems associated with the physical integration of digital circuits is power consumption and power dissipation which leads to synchronization issues, this research proposes a novel reversible logic gates (RLG-TG) a single layer coplanar approach. RLG designs with minimal design area, latency, and quantum cell count (QCA) are given and implemented using a Bijection functional method. Using the QD-E (Energy) tool, the first-order energy dissipation of the proposed shape and the impact of output bias cell temperature are also investigated. The proposed circuit designs were tested using the CVSE parameters, which had high clock signal saturation energies of 9.8e-22 d (Jules), recovery times for damping factors of 1e-15 s, and relative dielectric constants of 12.90 for GaAs and AlGaAs. The number of quantum cells used by the described new RLG-TG and RLG-PG designs is decreased by 38.23 % and 21.14 %, respectively, when compared to the optimal RLG designs employed in the state-of-the-art RLG designs. In this investigation of the proposed four-bit EPG and OPG circuits occupies 18.91 % and 38.27 % less design area, requires 46.15 % and 46.25 % less number of cells, and both designs has been 66.66 % improvement in delay.

Reducing T-Count in Quantum Circuits Using Alternate Forms of the Relative Phase Toffoli Gate

Reducing T-Count in Quantum Circuits Using Alternate Forms of the Relative Phase Toffoli Gate

Efficient Nano-Scale Design of TIEO Based Reversible Logic Toffoli Gate Priority Encoder in Quantum-Dot Cellular Automata

The goal of this research is to create a QCA-based reversible priority encoder. It is one of the most crucial parts of the encoding and decoding process. This novel nanotechnology, known as quantum-dot cellular automata (QCA), shows promise as a foundation for the development of reversible circuits. This paper proposes a simple, reversible, 4-2 line priority encoder design for the QCA platform. Using the simple and cheap Toffoli gate architecture, a reversible encoder circuit may be built. To analyze the proposed designs’ structural soundness, the simulation program QCA Designer is used.

Decomposing (n+1)$(n+1)$‐Qubit Toffoli Gate with Shallow Circuit Depth and No Ancilla

AbstractCircuit depth optimization is crucial for implementing quantum algorithms in practical systems. The decomposition of ‐qubit Toffoli gate is continually optimized since this gate is a fundamental component for many quantum algorithms. Without any ancilla, ‐qubit Toffoli gate can be decomposed by standard gates with circuit depth . Here, the decomposition is further optimized and its circuit depth is reduced to . The decomposition is equivalent to ‐qubit Toffoli gate up to an overall phase and also does not need any ancilla.

Design‐for‐testability for reversible logic circuits based on bit‐swapping

AbstractThe emerging technology of reversible circuits offers a potential solution to the synthesis of ultra low‐power quantum computing systems. A reversible circuit can be envisaged as a cascade of reversible gates only, such as Toffoli gate, which has two components: k control bits and a target bit (k‐CNOT), k ≥ 1. While analysing testability issues in a reversible circuit, the missing‐gate fault model is often used for modelling physical defects in k‐CNOT gates. A new design‐for‐testability (DFT) technique is proposed for reversible circuits that deploys bit‐swapping using Fredkin reversible gates. It is shown that in an (n × n) circuit implemented with k‐CNOT gates, addition of only two extra inputs along with a few Fredkin gates yields easy testability in the circuit. The modified design admits a universal test set of maximum size 2n + 1 that detects all detectable missing gate faults in the original circuit, where n is the number of input/output lines in the circuit. The DFT overhead in terms of quantum cost is much less compared to that of previous approaches. The method is more advantageous for large circuits.