Measurement-Based Quantum Computation Using the Spin-1 XXZ Model with Uniaxial Anisotropy

TL;DR

This work investigates measurement-based quantum computation (MBQC) using the ground state of a spin-1 XXZ chain with uniaxial anisotropies in the Haldane phase as a resource for single-qubit gates. It provides both numerical and analytic results showing that high-fidelity rotation and universal single-qubit gates (F > 0.99) can be achieved by tuning anisotropy parameters $D$ and $J$, and by partitioning the chain into blocks to realize arbitrary sequences of rotations. A key theoretical contribution is a general fidelity expression for $R_z(\theta)$ in the Haldane phase, $F_{R_z(\theta)}=1-\frac{\sin^2\theta}{2}(1+g_{\rm corr})-\frac{(1-\cos\theta)}{2}g_{\rm fail}$, linking gate performance to post-measurement correlations $g_{\rm corr}$ and failure probability $g_{\rm fail}$; near AFM boundaries, AFM correlations suppress failure states and boost fidelity. The findings suggest practical MBQC resources in anisotropic 1D spin chains and point toward potential cold-atom realizations, with future work needed to extend to two-qubit gates.

Abstract

We demonstrate that the ground state of a spin-1 XXZ chain with uniaxial anisotropies, single-ion anisotropy $D$ and Ising-like anisotropy $J$, within the Haldane phase can serve as a resource state for measurement-based quantum computation implementing single-qubit gates. The gate fidelity of both elementary rotation gates and general single-qubit unitary gates composed of rotations about the $x$-, $y$-, and $z$-axes is evaluated, and is found to exceed 0.99 when $D$ or $J$ is appropriately tuned. Furthermore, we derive an analytic expression for the rotation-gate fidelity under the assumption that the state lies within the $\mathbb Z_2\times\mathbb Z_2$-protected Haldane phase, showing that it is determined by the post-measurement spin-spin correlation function and the failure probability. The observed enhancement of gate fidelity in the spin-1 XXZ chain originates from the strengthening of antiferromagnetic (AFM) correlations near the AFM phase, which effectively suppresses failure states.

Figures (9)

• Figure 1: Scematic illustration of the AKLT state. The pair of black dots connected by a line represents a spin-singlet pair of spin-1/2s, and the gray circles represent the spin-1 sites. The AKLT state is constructed by projecting spin-1/2 singlet pairs onto the physical spin-1 space. The two spin-1/2s at the ends enable input and output of single-qubit states in MBQC.
• Figure 2: Graphical representation of the MPS in Eq. (\ref{['eq:AKLT_MPS']}) for the AKLT state. Circles and squares respectively denote MPS tensors $\Lambda$ and $\Gamma^m$ in Eq. (\ref{['eq:AKLTMPS_tensor']}). Horizontal lines which connect $\Lambda$ and $\Gamma^m$ represent multiplying each matrix and vertical lines represent phisycal degrees of spin-$1/2$s and spin-$1$s.
• Figure 3: Schematic representation of measurement basis in Eqs. (\ref{['eq:measurement_basis']}) and (\ref{['eq:measurement_basis_Rz']}). Basis $|x(\theta)\rangle,~|y(\theta)\rangle$ are obtained by rotating basis $|x\rangle,~|y\rangle$ about $z$-axis by an angle $\theta/2$ respectively.
• Figure 4: Gate fidelity of $R_z(\theta)$ for the ground state of the BLBQ model with system size $L=7$ and $\alpha\in [-1,1]$. The gate fidelities of the identity, $T$-, $S$-, and $Z$-gates (corresponding to $\theta=0$, $\pi/4$, $\pi/2$, $\pi$, respectively) are calculated. The ground state is obtained using the density matrix renormalization group (DMRG) method SCHOLLWOCK2011White1992Ostlund1995itensor with a maximum bond dimension of 100. The gate fidelity is evaluated using Eqs. (\ref{['eq:Rz_GF_expand']}) and (\ref{['eq:Rz_GF_eachterm']}).
• Figure 5: (a) The gate fidelity of the identity gate ($\theta = 0$) and (b) the gate fidelity of the $S$-gate ($\theta = \pi/2$) for the ground state of the XXZ model with the system size $L=15$ and $J,D\in[-6,6]$. The ground state is obtained using DMRG with a maximum bond dimension of 100.