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Physical Review A

Ballistic electron waveguide adder

Michael G. Snyder, L. E. Reichl

Published Nov 30, 2004• 16 citations
Publisher

Abstract

We analyze a quantum network, constructed with ballistic electron waveguides, which can add one-bit binary numbers. We consider two different adder networks, one constructed with Hadamard gates and the other constructed with $\sqrt{\mathrm{NOT}}$ gates. Reflection of electron probability at the gates causes a loss in fidelity of the overall calculation. A fidelity requirement for the implementation of individual rotation gates corresponding to a probability $p>1∕2$ of completing the addition successfully is found.

DOI
10.1103/physreva.70.052330

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We analyze a quantum network, constructed with ballistic electron waveguides, which can add one-bit binary numbers. We consider two different adder networks, one constructed with Hadamard gates and the other constructed with NOT\sqrt{\mathrm{NOT}}NOT​ gates. Reflection of electron probability at the gates causes a loss in fidelity of the overall calculation. A fidelity requirement for the implementation of individual rotation gates corresponding to a probability p>1∕2p>1∕2p>1∕2 of completing the addition successfully is found.
We present a study of ballistic electron waveguides as used to construct simple quantum computational networks.We begin with an introduction to ballistic electron waveguides, some basic theory of quantum computers, and the possible role of electron waveguides in quantum computation.Following that, our study of various waveguide networks and the physics of the specific components of those networks.We continue with a discussion of shot noise in network elements and its possible role in helping to build good computational gates.Finally we conclude with a summary of research and possible future projects.
We show that a network of ballistic electron waveguides can generate entangled Bell-like states from separable states when it is resonant. The network we study is grouped into individual qudits (d=4)(d=4)(d=4) made up of pairs of waveguides. Rotation gates in the network produce coherent superpositions of qudit states. A Coulomb gate entangles the qudits. We construct a unitary matrix which characterizes the network dynamics and allows a more systematic study of that dynamics.
The quantum adder is an essential attribute of a quantum computer, just as classical adder is needed for operation of a digital computer. We model the quantum full adder as a realistic complex algorithm on a large number of qubits in an Ising spin quantum computer. Our results are an important step toward effective modeling of the quantum modular adder which is needed for Shor's and other quantum algorithms. Our full adder has the following features. (i) The near-resonant transitions with small detunings are completely suppressed, which allows us to decrease errors by several orders of magnitude and to model a 1000-qubit full adder. (We add a 1000-bit number using 2001 spins.) (ii) We construct the full adder gates directly as sequences of radio-frequency pulses, rather than breaking them down into generalized logical gates, such as Control-Not and one qubit gates. This substantially reduces the number of pulses needed to implement the full adder. (The maximum number of pulses required to add one bit (F-gate) is 15.) (iii) Full adder is realized in a homogeneous spin chain. (iv) The phase error is minimized: the F-gates generate approximately the same phase for different states of the superposition. (v) Modeling of the full adder is performed using quantum maps instead of differential equations. This allows us to reduce the calculation time to a reasonable value.
Quantum addition based on the quantum Fourier transform can be an integral part of a quantum circuit and proved to be more efficient than the existing classical ripple carry adder. Our study includes identifying the quantum resource required in a quantum adder in any arbitrary dimension and its relationship with the performance indicator in the presence of local noise acting on the circuit and when a limited number of controlled rotation operations is permitted, a procedure known as banding. We analytically prove an upper bound on the number of the controlled rotation gates required to accomplish the quantum addition up to an arbitrary defect in the fidelity between the desired and imperfect output. When the environment interacts with individual qudits, we establish a connection between quantum coherence and fidelity of the output. Interestingly, we demonstrate that when banding is employed in the presence of noise, approximate circuits of constant depth outperform circuits with a higher number of controlled rotations, establishing a complementary relationship between the approximate quantum adder and the strength of the noise. We exhibit that utilizing magnetic fields to prepare an initial state that evolves according to a one-dimensional spin chain for a specific amount of time can be a potential technique to implement quantum addition circuits in many-body systems.
We present a small network for the testing of the entanglement of two ballistic electron waveguide qubits. The network produces different output conditional on the presence or absence of entanglement. The structure of the network allows for the determination of successful entanglement operations through the measurement of the output of a single qubit. We also present a simple model of a dynamic Coulomb-like interaction and use it to describe some characteristics of a proposed scheme for the entanglement of qubits in ballistic electron waveguides.