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arXiv:cond-mat/0403712 · Physical Review B

Protected qubits and Chern-Simons theories in Josephson junction arrays

Benoît Douçot, M. V. Feigel’man, L. B. Ioffe, Alexei S. Ioselevich

Published Jan 7, 2005• 187 citations
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Abstract

We present general symmetry arguments that show the appearance of doubly degenerate states protected from external perturbations in a wide class of Hamiltonians. We construct the simplest spin Hamiltonian belonging to this class and study its properties both analytically and numerically. We find that this model generally has a number of low energy modes which might destroy the protection in the thermodynamic limit. These modes are qualitatively different from the usual gapless excitations as their number scales as the linear size (instead of volume) of the system. We show that the Hamiltonians with this symmetry can be physically implemented in Josephson junction arrays and that in these arrays one can eliminate the low energy modes with a proper boundary condition. We argue that these arrays provide fault tolerant quantum bits. Further we show that the simplest spin model with this symmetry can be mapped to a very special ${\mathbb{Z}}_{2}$ Chern-Simons model on the square lattice. We argue that appearance of the low energy modes and the protected degeneracy is a natural property of lattice Chern-Simons theories. Finally, we discuss a general formalism for the construction of discrete Chern-Simons theories on a lattice.

DOI
10.1103/physrevb.71.024505

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We present general symmetry arguments that show the appearance of doubly degenerate states protected from external perturbations in a wide class of Hamiltonians. We construct the simplest spin Hamiltonian belonging to this class and study its properties both analytically and numerically. We find that this model generally has a number of low energy modes which might destroy the protection in the thermodynamic limit. These modes are qualitatively different from the usual gapless excitations as their number scales as the linear size (instead of volume) of the system. We show that the Hamiltonians with this symmetry can be physically implemented in Josephson junction arrays and that in these arrays one can eliminate the low energy modes with a proper boundary condition. We argue that these arrays provide fault tolerant quantum bits. Further we show that the simplest spin model with this symmetry can be mapped to a very special Z2{\mathbb{Z}}_{2}Z2​ Chern-Simons model on the square lattice. We argue that appearance of the low energy modes and the protected degeneracy is a natural property of lattice Chern-Simons theories. Finally, we discuss a general formalism for the construction of discrete Chern-Simons theories on a lattice.
In this thesis we consider a Josephson junction device whose symmetry is described by the point group Td. It can be visualized as a tetrahedron that contains two Josephson junctions on each edge. We find the conditions under which the ground state of the system is degenerate or almost degenerate. In this case, the low-energy degrees of freedom can be mapped to the Hilbert space of a quantum spin 1/2. We evaluate effects of different physical perturbations on the degenerate ground state and find that they are small for most perturbations. We argue that this system can be considered as a very promising candidate for a protected quantum bit with built-in error correction. We propose and discuss an experimental method that allows to test validity of some of the theoretical results obtained for the tetrahedral Josephson junction array and other similar symmetric circuits. We have chosen a simpler pyramidal array to demonstrate the main ideas of our method. Even though the noise resistance and theoretical decoherence time of the pyramidal array are worse than those of the more complex tetrahedral systems, it is much easier to realize the pyramid experimentally. The proposed design can be used with any symmetric Josephson junction circuit. We explore a natural generalization of the tetrahedral quantum bit and consider devices whose symmetry can be described by one of the higher-order permutation groups Sn. We study the level structure and the associated built-in protection of some conceptually simple circuits and show that these circuits have many interesting properties. In particular, their ground state can be highly degenerate and stable with respect to perturbations violating the symmetry. Unfortunately, these highly symmetric systems consist of a large number of identical Josephson junctions. This makes them too complicated for experimental realization.
A realistic implemetation of the macroscopic quantum computer requires that individual qubits are realized as quantum systems with degenerate ground states which are well protected from the effects of the environment. I show how this protection can be achieved in the new class of Josephson arrays with non-trivial symmetry or topology. For such "topologically protected" arrays the effect of noise is exponentially small in the array size which allows one in principle to get extremely small error rates and extremely long dephasing times in these systems. In this review I present a simple physical picture of the topological protection and explain why it is related to the presence of the topological order parameter in these systems. I formulate a set of general mathematical requirements on a model that ensures the appearance of the protected degenerate states and show how these conditions can be satisfied in a simple spin model. Finally I present Josephson junction array that is described by the mathematical model which satisfy these conditions and discuss its physical properties and how one can test these predictions experimentally.
We consider a Josephson junction device which has a symmetry of a tetrahedron; it can be visualized as a tetrahedron that contains two Josephson junctions at each edge. We find the conditions for which the ground state of the system is degenerate or almost degenerate; in this case, the low-energy degrees of freedom can be mapped on a quantum spin 1∕21∕21∕2. We evaluate the effect of the physical perturbations and imperfections on the level splitting in this system and find that they are small for most perturbations. We argue that this system can provide a possible physical implementation of a protected quantum bit with a built-in error correction. We propose a way of manipulating the effective quantum spin by means of electrical potentials and an experimental scheme to read the information that it contains.
We introduce a class of Josephson arrays which have nontrivial topology and exhibit a novel state at low temperatures. This state is characterized by long-range order in a two Cooper pair condensate and by a discrete topological order parameter. These arrays have degenerate ground states with this degeneracy ``protected'' from the external perturbations (and noise) by the topological order parameter. We show that in ideal conditions the low order effect of the external perturbations on this degeneracy is exactly zero and that deviations from ideality lead to only exponentially small effects of perturbations. We argue that this system provides a physical implementation of an ideal quantum computer with a built-in error correction and show that even a small array exhibits interesting physical properties such as superconductivity with double charge, 4e,4e,4e, and extremely long decoherence times.
We outline the physical principles underlying the design of finite size superconducting arrays that may serve as protected qubits. These systems are built from fully frustrated rhombi consisting of four Josephson junctions; their low energy spectrum is composed of localized Z2 charge excitations and π phase vortices. Protection from both the phase and flip errors requires a balance between charging and Josephson energies; optimization of the array size results in the hierarchical construction of these arrays. We have developed computational methods, based on a combination of perturbation theory and numerical diagonalization, which allow us to compute the global phase stiffness and the energy of localized charge excitations in various geometries. We present the most promising designs, which are currently under experimental investigation.
We construct the Hamiltonian description of the Chern-Simons theory with Zn{Z}_{n}Zn​ gauge group on a triangular lattice. We show that the Z2{Z}_{2}Z2​ model can be mapped onto free Majorana fermions and compute the excitation spectrum. In the bulk the spectrum turns out to be gapless but acquires a gap if a magnetic term is added to the Hamiltonian. On a lattice edge one gets additional nongauge invariant (matter) gapless degrees of freedom whose number grows linearly with the edge length. Therefore, a small hole in the lattice plays the role of a charged particle characterized by a nontrivial projective representation of the gauge group, while a long edge provides a decoherence mechanism for the fluxes. We discuss briefly the implications for the implementations of protected qubits.