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Protected quantum bits and Josephson junction arrays

R. A. Usmanov

Published Jan 1, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.7282/t3xk8f8b

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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 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.
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 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 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.
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.
Over the past two decades the performance of superconducting quantum bits (qubits) has been improved tremendously: the coherence time of individual qubits has been increased by five orders of magnitude, from a few nanoseconds to >100 μs. The much-improved coherence, lithographic scalability, and compatibility with microwave control place superconducting qubits in the leading position in the race amongst several quantum systems being considered for quantum information processing. Despite this very impressive progress, experimental realization of quantum error corrections and demonstration of logical qubits remains a considerable challenge.One of possible solutions to this problem is the development of so-called protected qubits whose errors would be suppressed by special symmetries of the underlying Hamiltonian. The realization of such qubits requires elements not found in the conventional superconducting circuit toolbox, such as Josephson elements with cos(ϕ/2) and cos(2ϕ) dependences of the Josephson energy E_J on the phase difference ϕ, circuits with a very large kinetic inductance, and junctions with unusually low E_J.This thesis focuses on design, fabrication and characterization of circuits based on low-E_J Josephson junctions (JJs) and superinductors (SIs). By analyzing limitations on the junction performance imposed by the thermally activated phase slips, we observed a dramatic reduction of the critical currents in the regime E_J≤T which was accompanied by an increase of the zero-bias resistance R_0. The first part of this work provides practical considerations for the use of such junctions in quantum circuits. With the aim of improving elements with a very high kinetic inductance, in the second part we developed superinductors based on the granular Aluminum (grAl) films, in which Josephson junctions are realized between nanoscale grains. The circuits based on such SIs demonstrate low microwave losses at ultra-low temperatures. Superinductors are an essential element of a novel qubit that we have developed – the so-called bifluxon. The qubit consists of a Cooper-pair box (CPB) with low-E_J Josephson junctions shunted by a superinductor, thus forming a superconducting loop. When the loop is threaded by the magnetic flux Φ=Φ_0⁄2 where Φ_0 is the flux quantum, the qubit offers exponential suppression of energy decay from charge and flux noises, and dephasing from flux noise. In the last part of this work, we observed an increase of the energy relaxation time by two orders of magnitude, up to 100μs, by turning on protection in the bifluxon qubit.