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Do Qubit States have to be non-degenerate two-level systems?

Zhuoran Bao, Daniel F. V. James

TL;DR

The paper asks whether quantum computation requires a non-degenerate two-level system and proposes using multiply-degenerate atomic sublevels as qubits, exemplified by atomic S and P manifolds. By exploiting spherical symmetry and linearly polarized driving, the degenerate manifold decomposes into independent rank-2 subspaces, each acting as a qubit, with Rabi oscillations enabling single-qubit gates. A degenerate Hadamard gate is realized in the zero-field limit; with a weak static field, the time evolution is expanded in powers of the Zeeman energy, and the average fidelity is given to second order in $(\mu_B B_0)/(\hbar \Omega)$. For two degenerate atoms, a two-qubit interaction is analyzed, and a sequence of local unitaries together with the two-atom evolution can implement a CZ gate under a timing condition $\Omega'^2 \hbar^2 = \tan^2(\Omega' t)$; decoherence from time-varying magnetic fields is discussed. The results show that degeneracy lifting is not strictly required for quantum computation, though it increases noise, and suggest a link between symmetry and qubit-like behavior that warrants further study.

Abstract

A qubit, or quantum bit, is conventionally defined as "a physical system for storing information that is capable of existing in either of two quantum states or in a superposition of both". In this paper, we examine the simple question of whether two distinct levels, each consisting of multiply degenerate sub-states, could serve as a practical quantum bit. We explore this idea using a well-characterized atomic system of the kind employed in several quantum computing implementations. We approximate the atom as a two-level system without degeneracy lifting in the magnetic quantum number while using the angular momentum addition rules to select the desired state transition. We find that, in the continuous presence of the field, the atom still undergoes Rabi oscillations, which are suitable for quantum gate construction. In addition, we compute the average fidelity in quantum gate performance for a single degenerate atom and postulate the required form of two-atom interaction to construct a controlled Z gate.

Do Qubit States have to be non-degenerate two-level systems?

TL;DR

The paper asks whether quantum computation requires a non-degenerate two-level system and proposes using multiply-degenerate atomic sublevels as qubits, exemplified by atomic S and P manifolds. By exploiting spherical symmetry and linearly polarized driving, the degenerate manifold decomposes into independent rank-2 subspaces, each acting as a qubit, with Rabi oscillations enabling single-qubit gates. A degenerate Hadamard gate is realized in the zero-field limit; with a weak static field, the time evolution is expanded in powers of the Zeeman energy, and the average fidelity is given to second order in . For two degenerate atoms, a two-qubit interaction is analyzed, and a sequence of local unitaries together with the two-atom evolution can implement a CZ gate under a timing condition ; decoherence from time-varying magnetic fields is discussed. The results show that degeneracy lifting is not strictly required for quantum computation, though it increases noise, and suggest a link between symmetry and qubit-like behavior that warrants further study.

Abstract

A qubit, or quantum bit, is conventionally defined as "a physical system for storing information that is capable of existing in either of two quantum states or in a superposition of both". In this paper, we examine the simple question of whether two distinct levels, each consisting of multiply degenerate sub-states, could serve as a practical quantum bit. We explore this idea using a well-characterized atomic system of the kind employed in several quantum computing implementations. We approximate the atom as a two-level system without degeneracy lifting in the magnetic quantum number while using the angular momentum addition rules to select the desired state transition. We find that, in the continuous presence of the field, the atom still undergoes Rabi oscillations, which are suitable for quantum gate construction. In addition, we compute the average fidelity in quantum gate performance for a single degenerate atom and postulate the required form of two-atom interaction to construct a controlled Z gate.

Paper Structure

This paper contains 5 sections, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Rabi Oscillation and Single Qubit Gate
  3. The Fidelity of the Degenerate Hadamard Gate with the presence of a weak static Magnetic Field
  4. Controlled Z Gate between two Qubits
  5. Conclusion

Figures (2)

  • Figure 1: All Possible transitions between the degenerate states in energy levels 0 and 1. Blue solid lines represent the transition mediated by $\pi_0$, linearly polarized photons, while the dashed lines represent the transition mediated by $\sigma_{\pm}$, circularly polarized photons.
  • Figure 2: By using linearly polarized light and defining the polarization direction of the electromagnetic field as the z-direction of the atom, only one Rabi frequency remains.