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Journal of Applied Physics

Single spin Toffoli–Fredkin logic gate

Amit Ranjan Trivedi, Supriyo Bandyopadhyay

Published May 15, 2008• 9 citations
Publisher

Abstract

The Toffoli–Fredkin (TF) gate is a universal reversible logic gate capable of performing logic operations without dissipating energy. Here, we show that a linear array of three quantum dots, each hosting a single electron, can realize the TF gate, if we encode logic bits in the spin polarization of the electrons and allow nearest neighbor exchange coupling. The dynamics of the TF gate is realized by selectively driving spin resonances in the coupled spin system with an ac magnetic field. The conditions for gate operation are established, and an estimate of the switching speed and gate error are provided.

DOI
10.1063/1.2937200

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The Toffoli–Fredkin (TF) gate is a universal reversible logic gate capable of performing logic operations without dissipating energy. Here, we show that a linear array of three quantum dots, each hosting a single electron, can realize the TF gate, if we encode logic bits in the spin polarization of the electrons and allow nearest neighbor exchange coupling. The dynamics of the TF gate is realized by selectively driving spin resonances in the coupled spin system with an ac magnetic field. The conditions for gate operation are established, and an estimate of the switching speed and gate error are provided.
Atomic-scale logic and the minimization of heating (dissipation) are both very high on the agenda for future computation hardware. An approach to achieve these would be to replace networks of transistors directly by classical reversible logic gates built from the coherent dynamics of a few interacting atoms. As superpositions are unnecessary before and after each such gate (inputs and outputs are bits), the dephasing time only needs to exceed a single gate operation time, while fault tolerance should be achieved with low overhead, by classical coding. Such gates could thus be a spin-off of quantum technology much before full-scale quantum computation. Thus motivated, we propose methods to realize the 3-bit Toffoli and Fredkin gates universal for classical reversible logic using a single time-independent 3-qubit Hamiltonian with realistic nearest neighbour two-body interactions. We also exemplify how these gates can be composed to make a larger circuit. We show that trapped ions may soon be scalable simulators for such architectures, and investigate the prospects with dopants in silicon.
Rising energy demand from data-center and AI applications has renewed interest in reversible computation, where logic need not dissipate heat at every step if information is uncomputed. Implementations have so far been classical: adiabatic CMOS recovers part of the switching energy but still moves thousands of kBTk_BTkB​T per logic node at room temperature. Here we propose classical reversible logic implemented by coherent spin dynamics in a spin quantum-dot array, with inputs and outputs in classical basis states and no algorithmic use of superposition. The same spin stores, transports, and computes, with unitary rotation replacing irreversible switching. The universal building block is an iToffoli gate driven by DC voltage pulses and exchange between hole spins in Ge/SiGe quantum dots. Simulations with realistic model parameters reproduce the Toffoli truth table and yield a testable error landscape. Because shuttling transports the bit without measurement, logic and data movement remain unitary until readout. Millivolt pulses on femtofarad gates with superconducting lines dissipate only dielectric loss and have no thermodynamic floor. With the loss parameters assumed for a demonstrated device the gate energy is ∼102 kBTln⁡2\sim 10^{2}\,k_BT\ln 2∼102kB​Tln2 at 4 K; at a design point within reach of existing devices it can fall below the Landauer scale kBTln⁡2k_BT\ln 2kB​Tln2, five to seven orders of magnitude less than a CMOS Toffoli. The same semiconductor hardware therefore serves both purposes, supporting quantum algorithms when superposition is used and classical reversible logic otherwise.
The three-qubit Toffoli gate plays an important role in quantum error correction and complex quantum algorithms such as Shor's factoring algorithm, motivating the search for efficient implementations of this gate. Here we introduce a Toffoli gate suitable for exchange-coupled electron spin qubits in silicon quantum dot arrays. Our protocol is a natural extension of a previously demonstrated resonantly driven cnot gate for silicon spin qubits. It is based on a single exchange pulse combined with a resonant microwave drive, with an operation time on the order of 100 ns and fidelity exceeding 99%. We analyze the impact of calibration errors and 1/f1/f1/f noise on the gate fidelity and compare the gate performance to Toffoli gates synthesized from two-qubit gates. Our approach is readily generalized to other controlled three-qubit gates such as the Deutsch and Fredkin gates.
The possibility of performing single-electron computing without dissipation in an array of tunnel-coupled quantum dots is studied theoretically, taking the spin gate NOT (inverter) as an example. It is shown that the logical operation can be implemented at the stage of unitary evolution of the electron subsystem, although complete switching of the inverter cannot be achieved in a reasonable time at realistic values of model parameters. The optimal input magnetic field is found as a function of the interdot tunneling energy and intradot Coulomb repulsion energy.
This work tries to model Single, Double and Triple electrons, spins based gates using quantum dot. These gates form universal gates in quantum computation and are building block of all logic circuits based on quantum gates. These spins are based on electrons which now days can be easily confined in quantum dots using a combination of hetero-structure and metal electrodes [5]. The modelling has taken analytical approach (One and Two Electron System [15], [10], [14]) wherever possible and numerical approach (Three Electron System). The spin manipulations in above three system, show us how to implement some of the important gates in quantum computing [NOT, CNOT,TOFFOLI etc.] in spin based quantum dot system. We think the TOFFOLI implementation is a novel architecture for a electron spin based system. We will also comment on the physical aspect of the simulation work wherever possible . We will also workout the aspect of quantum noise of the system which would be based on Davies Master(DM) Equation [8] based on Markovian approximation and Weak Coupling to the surrounding