Transitions in the quantum computational power

We construct two spin models on lattices (both two and three dimensional) to study the capability of quantum computational power as a function of temperature and the system parameter. There exists a finite region in the phase diagram such that the thermal equilibrium states are capable of providing a universal fault-tolerant resource for measurement-based quantum computation. Moreover, in such a region the thermal resource states on the three-dimensional lattices can enable topological protection for quantum computation. The two models behave similarly in terms of quantum computational power. However, they have different properties in terms of the usual phase transitions. The first model has a first-order phase transition only at zero temperature whereas there is no transition at all in the second model. Interestingly, the transition in the quantum computational power does not coincide with the phase transition in the first model.

Figure 1

Illustration of bottom-up approach. Panels (a) and (b) illustrate the building block of one unit, which consists of one center spin-3/2 and three outer three virtual bond qubits. Two virtual bond qubits, each from a neighboring unit, form a physical bond spin-3/2 particle, as shown by circles enclosing them in (c). A two-dimensional or three-dimensional structure can be constructed.

Figure 2

Ground-state and the first excited-state energies in each unit for the two models (top: model 1; bottom: model 2). The difference in the two energies is also the gap of the corresponding two- or three-dimensional models (for any finite system as well as in the thermodynamic limit). For the first model, the lowest two energy levels are degenerate for $\delta \le -2$. The ground-state energy exhibits discontinuity in its first derivative with respect to $\delta$, implying a first-order quantum phase transition in the model. However, for the second model, the ground-state energy is analytic for all range of $d_z$, implying nonexistence of phase transition.

Figure 3

Lattices of cluster states and microscopic construction of qubits. Panel (a) is the 2D square-lattice cluster state, and (c) is the unit cell of the 3D bcc (body-center-cubic lattice) cluster state. Both could be constructed from trivalent lattices (with degree-3 nodes) using the modification as shown in (b) and (d), respectively, which converts a degree-4 node to combinations of degree-3 nodes; see also Ref. [27]. Note that with such a construction, (a) will be turned to a brick-wall structure equivalent to the honeycomb lattice shown in Fig. 1. See also Fig. 6 and the Appendix for the conversion. In (b) and (d), each logical qubit (oval) is composed of spins from two units. The spins inside each oval will eventually be converted to one logical qubit. The circle (which includes two virtual qubits, i.e., a single bond particle) between the two ovals is used to entangle neighboring two effective qubits (within the ovals) via measurement of a bond particle. Essentially, the product of two GHZ states from two such units can be converted to a single GHZ via local measurement on the bond particle inside the oval as well as one of the center particle.

Figure 4

2D Phase diagrams in terms of computational power: (a) Upper: the XXZ model; (b) bottom: the anisotropic model. In the region below the solid black curve the equilibrium thermal states provide universal resource for MBQC. For reference, the energies for the ground state and the first excited is shown in white curves (solid and dashed, respective).

Figure 5

3D phase diagrams in terms of computational power. (a) $XXZ$ model; (b) anisotropic model. In the region below the solid black curve the equilibrium thermal states provide universal resource for MBQC. For reference, the energies for the ground state and the first excited is shown in white curves (solid and dashed, respective). Note that the transition temperatures are about three to four times higher than in two dimensions at the Heisenberg point.

Figure 6

Generation of a cluster state on a square lattice: (a) original honeycomb (i.e., brick wall) lattice; (b) a block of spins that give rise to one logical cluster-state qubit; (c) a center logical block and its neighbor four blocks on a square lattice.