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arXiv:quant-ph/9503017 · Physical Review Letters

Conditional Quantum Dynamics and Logic Gates

Adriano Barenco, David Deutsch, ARTUR K. EKERT, Richard Jozsa

Published May 15, 1995• 969 citations
arXivHTMLPDF

Abstract

Quantum logic gates provide fundamental examples of conditional quantum dynamics. They could form the building blocks of general quantum information processing systems which have recently been shown to have many interesting nonclassical properties. We describe a simple quantum logic gate, the quantum controlled-NOT, and analyze some of its applications. We discuss two possible physical realizations of the gate, one based on Ramsey atomic interferometry and the other on the selective driving of optical resonances of two subsystems undergoing a dipole-dipole interaction.

DOI
10.1103/physrevlett.74.4083

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Quantum logic gates provide fundamental examples of conditional quantum dynamics. They could form the building blocks of general quantum information processing systems which have recently been shown to have many interesting nonclassical properties. We describe a simple quantum logic gate, the quantum controlled-NOT, and analyze some of its applications. We discuss two possible physical realizations of the gate, one based on Ramsey atomic interferometry and the other on the selective driving of optical resonances of two subsystems undergoing a dipole-dipole interaction.
In this chapter, we shall discuss some practical applications of quantum control and dynamics. Our goal is not to give an exhaustive introduction to areas of application, but to point out the role of control and dynamics so as to further motivate the field as well as the techniques described in the previous chapters. The basics of mathematical modeling of quantum mechanical control systems were described in Chapter 2. Here we focus on more practical aspects in the areas of nuclear magnetic resonance and control of molecular and atomic systems. We also discuss a possible implementation of quantum information processing, namely the one with trapped ions. The dynamics here is used to implement quantum logic gates. We shall show how, changing the physical set-up, it is possible to create different Hamiltonians. The control is obtained by switching among these different Hamiltonians.
In order to realize quantum logical operations, Quantum Computation (QC) requires that its basic tools and concepts obey the laws of physics. One of the fundamental concepts in QC is the conditional quantum dynamics \cite{barenco}, some times called controlled-unitary operation \cite{kumar}, which is established by the conditional "If-Then" sentence. The best know example is the c-not gate, which operates on the computational basis as follows: \textit{If the control qubit is set to % |1>, then apply the single qubit quantum NOT gate on the target. Otherwise, if the control qubit is set to ∣0>|0>∣0>, then the target qubit is unchanged}. This gate represents the paradigm for the conditional quantum dynamics, where the flipping of the target qubit is conditioned to the state of the control qubit. Other gates have been defined in a similar way of conditional evolution; for instance, the control phase gates \cite{paulo1}. However, to the best of our knowledge, such conditional quantum dynamics has not been yet used to define the SWAP gate. Here we propose a possible conditional definition, in the form of If-Then sentence, to construct a SWAP gate in the case of two qubits. This definition suggests a classification in two classes, which depends upon the number of qubits that have to undergo a conditional quantum dynamics.
The promise of tremendous computational power, coupled with the development of robust error-correcting schemes, has fuelled extensive efforts to build a quantum computer. The requirements for realizing such a device are confounding: scalable quantum bits (two-level quantum systems, or qubits) that can be well isolated from the environment, but also initialized, measured and made to undergo controllable interactions to implement a universal set of quantum logic gates. The usual set consists of single qubit rotations and a controlled-NOT (CNOT) gate, which flips the state of a target qubit conditional on the control qubit being in the state 1. Here we report an unambiguous experimental demonstration and comprehensive characterization of quantum CNOT operation in an optical system. We produce all four entangled Bell states as a function of only the input qubits' logical values, for a single operating condition of the gate. The gate is probabilistic (the qubits are destroyed upon failure), but with the addition of linear optical quantum non-demolition measurements, it is equivalent to the CNOT gate required for scalable all-optical quantum computation.
Nonlinear interactions between two photons are required for the construction of optical quantum logic gates, but those interactions are normally very weak due to the small magnitude of the electric field associated with a single photon. We note that exchange interactions can have a large effect even when there is no physical interaction between two particles, and we exploit this property for the construction of optical quantum logic gates. We show that the probability of there being two virtually-excited atoms in a medium can be a factor of 2 larger when two nonresonant photons propagate through the same medium as compared to the case in which they propagate through two separate media, in analogy with photon bunching. As a result, the application of one or more laser pulses will produce a nonlinear phase shift that can be used to construct an XOR quantum logic gate. This provides an example of a quantum control process in which one photon can control the state of another photon even when there is no sequence of physical interactions linking the two photons. From a classical point of view, it is not possible to identify a path for the flow of information or a specific cause for the outcome of the control process.
We propose a new system for implementing quantum logic gates: neutral atoms trapped in a very far-off-resonance optical lattice. Pairs of atoms are made to occupy the same well by varying the polarization of the trapping lasers, and then a near-resonant electric dipole is induced by an auxiliary laser. A controlled-NOT can be implemented by conditioning the target atomic resonance on a resolvable level shift induced by the control atom. Atoms interact only during logical operations, thereby suppressing decoherence.
A system of particles in a double well potential is a widely studied and useful example for understanding quantum mechanics. This simple system has recently been used in theoretical proposals and related experiments as a way to make quantum logic gates for ultracold atoms confined in optical lattices. Such quantum gates are the fundamental building blocks for quantum information processing; in these proposals, a regular array of cold atoms in an optical lattice serves as the quantum register. We explain how this research can be understood in terms of well-known principles for systems of identical particles.