Symmetry-Protected Phases for Measurement-Based Quantum Computation

Ground states of spin lattices can serve as a resource for measurement-based quantum computation. Ideally, the ability to perform quantum gates via measurements on such states would be insensitive to small variations in the Hamiltonian. Here, we describe a class of symmetry-protected topological orders in one-dimensional systems, any one of which ensures the perfect operation of the identity gate. As a result, measurement-based quantum gates can be a robust property of an entire phase in a quantum spin lattice, when protected by an appropriate symmetry.

Figure 1

One generator of the $(Z_2\times Z_2{)}^{\times N}$ symmetry in the 2D cluster state. The other generators can be obtained from this one by a displacement of 1 horizontally and/or an even number vertically. The circles represent qubits in the 2D square lattice.