Protected qubits and Chern-Simons theories in Josephson junction arrays

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.

Figure 1

Energy spectrum of the $5\times 5$ and $4\times 4$ systems in the units of $J_z$ coupling as a function of $J_x∕J_z$. We show energies of the lowest 40 states for $5\times 5$ (upper pane) and lowest 20 states for the $4\times 4$ system (lower pane). One clearly sees that at large anisotropy a well-defined low energy band is formed which contains $2^5$ states for $5\times 5$ system and $2^4$ states for $4\times 4$ one. In order to verify that low energy states are in one-to-one correspondence with $P_i$ eigenvalues for large $J_x∕J_z$ we have calculated the second lowest eigenstate in $P_i=1$ sector (first one is the ground state). As shown in the lower pane by the dashed line, this state indeed has a large gap for $J_x∕J_z\gtrsim 1.2$.

Figure 2

Ground state splitting by random field in $z$ direction for $5\times 5$ and $4\times 4$ systems. The random field acted on each spin and was randomly distributed in the interval (−0.05,0.05). Note that the effect of the random field in $z$ direction becomes larger for $J_x⪡J_z$ as expected. Because near $J_x=J_z$ isotropic point the gap for $5\times 5$ system is significantly smaller than the gap for the $4\times 4$ system, this relatively large disorder has almost the same effect on these systems at $J_x\sim J_z$. We have verified numerically that decrease of the disorder by a factor of 2 leads to dramatically smaller effects for $5\times 5$ system confirming the scaling $E_1-E_0\propto {\delta }^n$. For $J_x∕J_z>1.2$ the difference $E_1-E_0$ is difficult to resolve numerically.

Figure 3

Schematics of the array equivalent to the spin model with the interaction (5) in the vertical direction. (a) The main element of the array, the superconducting rhombus frustrated by magnetic flux ${\mathrm{\Phi }}_0∕2$. Josephson energy of each rhombus is minimal for $\theta =0$ and $\theta =-\pi ∕2$. Significant charging energy induces the transitions $\theta =0\leftrightarrow \theta =-\pi ∕2$ between these energy minima. (b) The array geometry. The superconducting boundary conditions chosen here ensure that $P_i{P}_j=1$ thereby eliminating all low lying states in the appropriate regime. (c) The requirement that continuous phase does not fluctuate much while the discrete variables have large fluctuations is easier to satisfy in very big arrays $(L>20)$ if one replaces the vertical links by the rhombi with junctions with ${\widetilde{ϵ}}_J$, ${\widetilde{ϵ}}_C$ frustrated by the flux ${\mathrm{\Phi }}_0(\delta +1∕2)$.

Figure 4

Phase variation along the horizontal axis after a flip of a single rhombus (solid curve) and after consecutive flips of two rhombi (thick curve) located at a distance twice the core size of each rhombi, $\sqrt{E_J∕{ϵ}_J}$. The horizontal axis shows the distance, $X=\sqrt{{ϵ}_J∕E_J}x$, measured in the units of the vortex core size.

Figure 5

Schematics of the array. (a) The elementary Josephson circuit emulating spin 1/2 consists of a four junction loop penetrated by magnetic flux ${\mathrm{\Phi }}_0∕2$. (b) Implementation of the spin chain by Josephson junction loops; here elementary rhombi are connected by weak links, the appropriate parameters for these links can be obtained if each link in fact consists of two elementary Josephson junctions as shown in (c). (d). Full array implementing the spin model (1). The natural boundary conditions for this array are periodic, i.e., the point $A$ should be connected to $A^{\prime }$, $B$ to $B^{\prime }$, etc.

Figure 6

(a) An oriented link $\left(ij\right)$ (dashed line) and the four oriented links adjacent to it (thick lines) which enter in the set $\mathcal{N}\left(ij\right)$. (b) For a given site $i$, the loop $\mathcal{L}\left(i\right)$ defined in text (thick lines).

Figure 7

The electrical field operator, ${\mathcal{E}}_{i_0{i}_1}$ moves one flux quantum from the plaquette $r$ to $r^{\prime }$. In order to represent this process in the gauge where all horizontal links $A_{ij}$ are 0, we use the gauge transformation on the string of sites $i_1\dots i_n$ that produce the electric field operators on vertical links (shown by light arrow) and the phase factors associated with the shaded area. In the flux representation the effect of vertical electric field is to remove the flux from plaquette $r$ and to add one flux to plaquette $r^{\prime }$. Note that the phase factors associated with all plaquettes except $r$ and $r^{\prime }$ are doubled.