Spin-$s$ $U(1)$-eigenstate preparation

TL;DR

This work develops a deterministic quantum-state preparation method for general $U(1)$-eigenstates of spin-$s$ chains by leveraging Gray codes for bounded integer compositions to order the $n$-qudit basis by fixed digit sum. It introduces a two-qudit Gray gate and a complete algorithm that initializes a product state and applies a sequence of Gray gates to build the target superposition with amplitudes $a_{\vec{m}}$, maintaining the $S^z$ eigenvalue constraint. The authors demonstrate exact state preparation for nontrivial cases, including the AKLT ground state (spin-$1$), spin-$s$ Dicke states, and spin-$s$ Bethe states of integrable XXX Hamiltonians, highlighting both the method’s broad applicability and potential for VQE-style variational approaches. They also discuss extensions to other $U(1)$-invariant models and open boundary conditions, and acknowledge the high circuit cost for Bethe-state preparation, pointing to future work exploiting integrable structure for efficiency gains.

Abstract

We formulate a deterministic algorithm for preparing a general $U(1)$-eigenstate of a spin-$s$ chain of length $n$. These states consist of linear combinations of computational basis states $|\vec{m}\rangle$ of $n$ qudits, each with $(2s+1)$ levels and $s= 1/2, 1, 3/2, \ldots$, whose ditstrings $\vec{m}$ have a fixed digit sum. Exploiting a Gray code for bounded integer compositions, whose consecutive ditstrings obey the Gray property, the quantum state is prepared by applying corresponding ``Gray gates.'' We use this algorithm to prepare exact eigenstates of integrable spin-$s$ XXX Hamiltonians.

Figures (3)

• Figure 1: The convex polytope for bounded compositions of $k=3$ into $n=3$ parts with $0 \le x_i \le 2$. The hexagon is the intersection of the simplex $x_1+x_2+x_3=3$ and the cube $0 \le x_i \le 2$.
• Figure 2: Circuit diagram for the Gray gate $G_{i,j}^{m_i,m_j}(\theta,\phi)$\ref{['Ggate']}. In its symbolic form shown on the right, we use sharp ($\sharp$) and flat ($\flat$) to designate qudits $i$ and $j$, respectively.
• Figure 3: The quantum circuit \ref{['genalgorithm']} for preparing a state with $(n,k,s) = (3, 3, 1)$, see \ref{['state-example']} and Table \ref{['table:GrayExample']}. The Gray gates are defined in Fig. \ref{['fig:Ggate']}, and the circles denote controls.