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Physical Review B

Theoretical investigation of a protected quantum bit in a small Josephson junction array with tetrahedral symmetry

R. A. Usmanov, L. B. Ioffe

Published Jun 21, 2004• 5 citations
Publisher

Abstract

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 $1∕2$. 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.

DOI
10.1103/physrevb.69.214513

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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.
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.
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.
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 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.
We propose a design for a quantum bit with four superconducting islands in the topology of a symmetric tetrahedron, uniformly frustrated with one-half flux quantum per loop and one-half Cooper pair per island. This structure emulates a noise-resistant spin-1∕21∕21∕2 system in a vanishing magnetic field. The tetrahedral quantum bit combines a number of advances such as a doubly degenerate ground state minimizing decoherence via phonon radiation, a weak quadratic sensitivity to electric and magnetic noise, relieved constraints on the junction fabrication, a large freedom in manipulation, and attractive measurement schemes. The simultaneous appearance of a degenerate ground state and a weak noise sensitivity are consequences of the tetrahedral symmetry, while enhanced quantum fluctuations derive from the special magnetic frustration. We determine the spectral properties of the tetrahedral structure within a semiclassical analysis and confirm the results numerically. We show how proper tuning of the charge frustration selects a doubly degenerate ground state and discuss the qubit's manipulation through capacitive and inductive coupling to external bias sources. The complete readout of the spin components ${\ensuremath{\sigma}}_{i}$, i=x,y,zi=x,y,zi=x,y,z, is achieved through coupling of the internal qubit currents to external junctions driven close to criticality during the measurement.
Quantum states are usually fragile, making quantum computation less stable than classical computation. Quantum correction codes can protect quantum states but need many physical qubits to encode a single logical qubit. Alternatively, protecting quantum states at the hardware level has been recently developed to maintain the coherence of the quantum information by using symmetry. However, it generally has to pay the expense of increasing the complexity of the quantum devices. In this work, we propose to approach the protection of quantum states at the hardware level without increasing the complexity of the devices. The interplay between the spin-orbit coupling and the Zeeman splitting in the semiconductor allows us to tune the Josephson coupling in terms of the spin degree of freedom of Cooper pairs, the hallmark of the superconducting spintronics. This leads to the implementation of the parity-protected 0-$\ensuremath{\pi}$ superconducting qubit with only one highly transparent superconductor-semiconductor Josephson junction, which makes our proposal immune from the various fabrication imperfections.