Figure 1

Single-qubit operations on a Haldane-phase spin chain. A chain of spin-1 particles terminated on the left by a spin-$\frac{1}{2}$ particle encodes one qubit on its right edge. Single-spin measurements, shown in (a), implement single-qubit operations on the AKLT chain, with measurement in the basis $\{|S_{{\widehat{e}}_i}=0⟩{\}}_{i=1}^3$ for some Cartesian basis $\{{\widehat{e}}_i{\}}_{i=1}^3$ leading to $\pi$ rotation around the outcome axis [9]. Each outcome occurs with probability $1/3$. Fixing a “standard” basis $\{\widehat{x},\widehat{y},\widehat{z}\}$ (called $\{|x⟩,|y⟩,|z⟩\}$ in the spin-1 state space), the first two outcomes of measurement in a basis $\{{\widehat{x}}^{\prime },{\widehat{y}}^{\prime },\widehat{z}\}$ rotated by $\theta$ around the $\widehat{z}$ axis (spin-1 states $\{|\theta ⟩,|\theta +\pi ⟩,|z⟩\}$, for $|\theta ⟩\equiv \frac{1}{2}[(1+e^{-i\theta })|x⟩+(1-e^{-i\theta })|y⟩]$) result in the same rotation $R_z(\theta )$ of the qubit, followed by a corresponding by-product $\pi$ rotation of it around $\widehat{x}$ or $\widehat{y}$ (which can be later corrected). The third outcome is just a by-product $\widehat{z}$ rotation. Induced rotations become noisy for $\beta \ne -\frac{1}{3}$ but can be improved by buffering, depicted in (b). Here the left and right spins are measured first, and the rotation measurement (middle) is attempted only when these are both $\widehat{z}$. Failing this, the middle spin is measured in the standard basis, and the attempt is repeated in the next block. Concatenating block-3 buffering is equivalent to block-9 buffering, as shown in (c).Reuse & Permissions

Figure 2

(a) Fidelity of $\frac{\pi }{2}$ rotation, the worst case, and (b) buffering success probability versus $\beta$ for block lengths $L=1$ (no buffering), 3, and 9 on a chain of length 12. The $L=3$ fidelity decrease for $\beta <-\frac{1}{3}$ can be attributed to a one-off effect in our renormalization map, as described in Fig. 3. In (b) the buffering probability is normalized by the AKLT value $1/3^L$; the $L=9$ factor decays to roughly ${10}^{-5}$ as $\beta \to -1$, though not plotted explicitly due to space constraints. Rotation fidelity is computed by entangling the chain with a fictitious termination spin on the right edge and calculating the overlap of the measured chain with the ideal output state. The unique, entangled ground state of the doubly terminated chain can be quickly found by using sparse matrix methods. This overlap can, in turn, be evaluated by using the expectation of string operators, which for rotation by $\theta$ about the $\widehat{z}$ axis (measurement in the $\{{\widehat{x}}^{\prime },{\widehat{y}}^{\prime },\widehat{z}\}$ basis) are ${\Sigma }_{x^{\prime }}\otimes {\sigma }_x$ and ${\Sigma }_z\otimes {\sigma }_z$. The initial state is an eigenstate of ${\Sigma }_z\otimes {\sigma }_z$ and remains so after the measurement, implying the square of the fidelity of the output state with the ideal state is given by $F^2=\frac{1}{2}(1+⟨{\Sigma }_{x^{\prime }}\otimes {\sigma }_x⟩)$.Reuse & Permissions

Figure 3

Bloch vector of the label space ${\mathcal{H}}_L$ reduced density operator before (dashed line) and after (solid line) a single RG step for $-1\le \beta \le 1$. The state $|{\chi }_{\mathrm{s}}⟩$ of Eq. (2) defines the vertical axis, while the horizontal corresponds to the superposition $|+⟩\equiv (1/\sqrt{2})(|{\chi }_s⟩+|{\overline{\chi }}_s⟩)$, for $|{\overline{\chi }}_s⟩=(1/\sqrt{6})(\sqrt{5}|0⟩+|1⟩)$; the $y$ axis is not shown, as the reduced density operator coefficients are real in this basis. The Heisenberg antiferromagnet ($\beta =0$) is denoted HAF. The norm of the Bloch vector, i.e., the radial distance from the origin, provides the weight of the symmetric three-spin $J=1$ sector in the (pre- and postrenormalized) ground state $|\mathcal{G}(\beta )⟩$, and the projection onto the vertical axis indicates the buffering success probability, given successful $J=1$ projection. Observe that, for $\beta >-\frac{1}{3}$, the RG approximately maps Bloch vectors closer to the AKLT point along the original curve parametrized by $\beta$, meaning the correlation of the reduced state is effectively renormalized to another $\beta$ closer to the AKLT point. Meanwhile, the first iteration takes $\beta <-\frac{1}{3}$ to $\beta >-\frac{1}{3}$. Accordingly, we expect that iteration of our RG map generates a flow toward the AKLT point.Reuse & Permissions