Symmetry-protected topological phases with uniform computational power in one dimension

We investigate the usefulness of ground states of quantum spin chains with symmetry-protected topological order (SPTO) for measurement-based quantum computation. We show that, in spatial dimension 1, if an SPTO phase protects the identity gate, then, subject to an additional symmetry condition that is satisfied in all cases so far investigated, it can also be used for quantum computation.

Figure 1

Boundary conditions of the resource states $|\mathrm{\Phi }\rangle$ and $|\tilde{\mathrm{\Phi }}\rangle$. $|\tilde{\mathrm{\Phi }}\rangle$ has an additional degree of freedom on its right boundary whose number of states is equal to the bond dimension $D_b$.

Figure 2

Accumulated filter functions $f_0^{N_0}{f}_1^{N_1}$ (normalized): (a) $N_0=N_1=1$, (b) $N_0=N_1=5$, and (c) $N_0=N_1=50$. The parameters for this plot are ${\nu }_{00}={\nu }_{11}=1,{\nu }_{10}=0.8,\alpha =\pi /4$.

Figure 3

Measurement of the logical subsystem of the virtual system. (top) Changeover between unitary evolution and measurement. For unitary evolution of the logical subsystem, the angle $\alpha$ in $\mathcal{B}\left(\alpha \right)$ is chosen, $\sim 1/N_M$, where $N_M:=N_0+N_1$. Unitary evolution turns into measurement at $\alpha \approx 1/\sqrt{N_M}$. The measurement is optimal for $\alpha =\pi /4$. In this plot, $N_M=1600$. (bottom) Onset of the large-$N$ limit in measurement. Each dot in the plot is the estimate of $cos{\varphi }_k$, where $e^{i{\varphi }_k}$ are the eigenvalues of the measured observables $C$. Each such estimate is based on a sequence of $N_M$ individual weak measurements. In this example, ${\varphi }_k=\pi k/4,k\in {\mathbb{Z}}_8$. The parameters for both plots are ${\nu }_{00}={\nu }_{11}=1,{\nu }_{10}=0.9$, and the state of the logical subsystem before measurement is completely mixed.

Figure 4

Computation by interaction of the logical system with the state $\nu$. The state $\nu$ represents the “computational essence” of the fixed point state ${\rho }_{\text{fix}}$ of the junk subsystem. $U$ is the unitary that transforms between the wire basis and the measurement basis for the physical degree of freedom. The interaction between the two systems is via the gate Eq. (42).