Digital Quantum Simulation of the Kitaev Quantum Spin Liquid

TL;DR

This work presents a concrete protocol for digital quantum simulation of the Kitaev quantum spin liquid (KQSL) on a honeycomb lattice, enabling preparation of the ground state and controlled manipulation of vison and Majorana fermion excitations. The core idea is to decompose state preparation and excitation into a fermionic rotation within a fixed vison sector, $U_1$, and a vison-sector change, $U_2$, yielding an overall unitary that maps between eigenstates with polynomial resource scaling. Demonstrated experimentally on IBM hardware for eight and twelve qubits, with spin-correlations, vison/Wilson-loop measurements, and energy comparisons supporting faithful state control; numerics extend to system sizes up to 450 qubits, illustrating scalability in the ideal limit. The work provides a scalable blueprint for exploring topological order and Majorana physics on near-term devices and offers a pathway to broader digital simulations of QSLs and related gauge-theoretic models, while outlining practical error-mitigation strategies for larger implementations.

Abstract

The ground state of the Kitaev quantum spin liquid on a honeycomb lattice is an intriguing many-body state characterized by its topological order and massive entanglement. One of the significant issues is to prepare and manipulate the ground state as well as excited states in a quantum simulator. Here, we provide a protocol to manipulate the Kitaev quantum spin liquid via digital quantum simulation. A series of unitary gates for the protocol is explicitly constructed, showing its circuit depth is an order of O(N) with the number of qubits, N. We demonstrate the efficiency of our protocol on the IBM Heron r2 processor for N = 8 and 12. We further validate our theoretical framework through numerical simulations, confirming high-fidelity quantum state control for system sizes up to N = 450, and discuss the possible implications of these results.

Figures (13)

• Figure 1: (a) Quantum circuit representation of four processes: ground state preparation, vison manipulation, Majorana fermion control, and Majorana fermion readout. (b) The geometry of Kitaev honeycomb model on the torus, ($L_1=L_2=2$ and $M=0$). $M$ is the twisting parameter of torus. Dashed arrows connect the identical sites on the torus. Purple (orange) loop indicates non-contractible loop $W_{X}$ ($W_{Y}$) on the torus.
• Figure 2: (a) Schematic diagram of the quantum circuit for GS preparation. The dashed line divides the quantum circuit into two parts: preparation of the state $|(\Phi_{4},f_{4,\text{gs}}^{\text{dim}})\rangle$ ($S$ part), and creation of KQSL ground state through $U_{1}$ operator. (b) Graphical representation of the mapping implemented by $U_{1}$ operator. The Hamiltonian $H_{\text{dim}}$\ref{['Eq:H_dim']} only allows the Kitaev interaction in $x$-direction links. (c) Schematic diagram of the quantum circuit for vison manipulation. (d) Graphical representation of the mapping implemented by $U_{2}U_{1}^{\prime}$ operator. The vison pair is annihilated from full-vison sector.
• Figure 3: Schematic diagram of the quantum circuit for Majorana fermion control. The $U_{1}^{\prime\prime}$ operator annihilates the lowest mode fermion and creates the second lowest mode fermion.
• Figure 4: Ground state preparation. (a) Geometry of KQSL on the torus. Dashed arrow connects identical sites. Purple (orange) loop indicates non-contractible loop $W_{X}=-\sigma_{1}^{z}\sigma_{2}^{z}\sigma_{3}^{z}\sigma_{4}^{z}$ ($W_{Y}=-\sigma_{1}^{y}\sigma_{2}^{y}\sigma_{7}^{y}\sigma_{8}^{y}$) on torus. (b) The spin correlation obtained from the data set (4096 shots in total). The spin correlation ($\langle\sigma^{\alpha}_{i}\sigma^{\alpha}_{j}\rangle$) is measured for 12 links connecting the pair of nearest neighbors on the torus, with specific $\alpha$. The dashed line indicates the spin correlation value obtained from theory. The color (gray, red, and blue) indicates the $\alpha$=$x$, $y$, and $z$, respectively. (c) The measured spin correlation function as a function of distance between two sites. The spin correlation function is obtained for all 28 pairs on the torus. (d) The measured expectation value of four $Z_{2}$ flux operator (green) and two Wilson loop operators (orange). The dashed line indicates the value obtained from theory. (e) The (quasi) probability distribution over 256 bit strings obtained from data set (4096 shots in total). Blue (red) color indicates that theory predicts its nonzero (zero) probability. (Inset) The probability distribution over 48 bit strings. The solid line indicates the theoretically predicted probability distribution.
• Figure 5: Vison manipulation. (a) Graphical representation of the mapping implemented by $U_{2}U_{1}^{\prime}$ operator. The vison pair ($W_1$ and $W_2$) is annihilated in this process. (b) The spin correlation obtained from the data set (4096 shots in total). The spin correlation ($\langle\sigma^{\alpha}_{i}\sigma^{\alpha}_{j}\rangle$) is measured for 12 links connecting the pair of nearest neighbors on the torus, with specific $\alpha$. The dashed line indicates the spin correlation value obtained from theory. The theory predicts zero spin correlation for $\langle{}X_{3}X_{4}\rangle$, $\langle{}X_{5}X_{6}\rangle$, $\langle{}Y_{1}Y_{4}\rangle$, and $\langle{}Y_{5}Y_{8}\rangle$. The color (gray, red, and blue) indicates the $\alpha$=$x$, $y$, and $z$, respectively. (c) The measured spin correlation function as a function of distance between two sites. The spin correlation function is obtained for all 28 pairs on the torus. (d) The measured expectation value of four $Z_{2}$ flux operator (green) and two Wilson loop operators (orange). (e) The (quasi) probability distribution over 256 bit strings obtained from data set (4096 shots in total). Blue (red) color indicates that theory predicts its nonzero (zero) probability. (Inset) The probability distribution over 24 bit strings. The solid line indicates the theoretically predicted probability distribution.