Family of $\left[\right[6k,2k,2\left]\right]$ codes for practical and scalable adiabatic quantum computation

In this work, we introduce a family of $\left[\right[6k,2k,2\left]\right]$ codes designed specifically to be compatible with adiabatic quantum computation. These codes support computationally universal sets of weight 2 logical operators and are particularly well suited for implementing dynamical decoupling error suppression. For Hamiltonians embeddable on a planar graph of fixed degree, our encoding maintains a planar connectivity graph and increases the graph degree by only two.

Figure 1

The Gottesman $\left[\right[2k+2,2k,2\left]\right]$ quantum error detecting code. Notice the hub-and-spoke configuration taken by the single-qubit logical operators. Solid lines indicating $\overline{X}=XX$ operators and dashed lines indicating $\overline{Z}=ZZ$ operators. The stabilizer generators are the $(2k+2)$-body operators, $X^{\otimes 2k+2}$ and $Z^{\otimes 2k+2}$. The high connectivity demanded of physical qubits 1 and $2k+2$ makes this code challenging to implement for $k$ much larger than 2.

Figure 2

Schematic for a linear layout of the $\left[\right[6k,2k,2\left]\right]$ code indicating the physical qubits involved in each of the stabilizer generators (pale lines) and single-qubit logical operators (dark lines). Each logical qubit is composed of a row of three physical qubits. All code operators are weight 2 with the exception of two weight-$4k$ stabilizer generators. Qubits which are initially entangled are enclosed in black ovals.

Figure 3

A two-dimensional (2D) grid of logical qubits remains planar upon encoding in the $\left[\right[6k,2k,2\left]\right]$ code. We have emphasized a single logical qubit and its interactions. Again, all interactions between logical qubits take place through the $(i,0)$ qubits.