Resource Quality of a Symmetry-Protected Topologically Ordered Phase for Quantum Computation

We investigate entanglement naturally present in the 1D topologically ordered phase protected with the on-site symmetry group of an octahedron as a potential resource for teleportation-based quantum computation. We show that, as long as certain characteristic lengths are finite, all its ground states have the capability to implement any unit-fidelity one-qubit gate operation asymptotically as a key computational building block. This feature is intrinsic to the entire phase, in that perfect gate fidelity coincides with perfect string order parameters under a state-insensitive renormalization procedure. Our approach may pave the way toward a novel program to classify quantum many-body systems based on their operational use for quantum information processing.

Figure 1

Schematic of renormalization procedure to manifest the quality of resource states. (a) To perform $z$ buffering, we choose a computational site, and measure $m$ surrounding sites in the Pauli basis. Here $m=2$. (b) If our measurement fails to produce the all-$|z⟩$ outcome, the computational site is measured in the Pauli basis, and we try again on another region. Since all of our measurement outcomes simply induce Pauli operations, the state of the protected space is (up to byproducts) unchanged. (c) If our measurement succeeds, the resource quality of our computational site is improved, at least when ${\zeta }_z$ is finite (Theorem 1).

Figure 2

(a),(b) The gate fidelity for a protected space $\pi /2$ rotation about the $z$ axis, with resource states parametrized by $\phi$, $\theta =\pi /2$ in (a), and by $\theta$, $\phi =\pi /4$ in (b). The renormalized gate fidelity tends toward unity everywhere except at the regions of divergent ${\zeta }_z$, in agreement with Theorem 1. (c),(d) The renormalized order parameter ${\tilde{\mathcal{O}}}_{D_4}^{(z)}$ for the same set of parameters as in (a) and (b), respectively. The RG limit of ${\tilde{\mathcal{O}}}_{D_4}^{(z)}$ is $\frac{1}{2}$ everywhere that the RG limit of the gate fidelity is 1, in agreement with Theorem 2. (e),(f) The $z$-correlation length and the renormalized correlation length, ${\zeta }_z$ and $\tilde{\xi }$, shown for the same set of parameters as in (a) and (b), respectively. While both diverge at the poles of our parameter space, where our toy model is long range correlated, the divergence of ${\zeta }_z$ at $\phi =\pi /2$ is more surprising, and leads to a transition in the resource quality of our state there, as seen in (a).