Hybrid QPE-Ansatz Strategy for Reliable Excited-State Variational Quantum Deflation

Abstract

We introduce a spin $z$-component ($S_{z}$) conserving symmetry-preserving ansatz and a shallow quantum phase estimation (QPE) routine of spin $x$ ($S_x$), and combine them into a spin-filtering variational quantum deflation (sfVQD) scheme for noisy intermediate-scale quantum (NISQ) computing era excited state calculations. The scheme encodes the spin information into a small ancilla register through controlled rotations under $\mathrm{exp} (iθ\hat{S}_{x})$ with only modest circuit overhead. The encoded information is then utilized to suppress spin contamination by screening, avoiding costly explicit evaluation on the total spin $\langle\hat{S}^{2}\rangle$. Because the screening module operates independently of the variational ansatz, it can also be employed with other excited-state calculation schemes based on variational quantum eigensolvers. As a demonstration, we apply sfVQD to LiH and BeH$_2$ with varying geometries to show markedly improved separation of singlet and triplet manifolds over conventional VQD without QPE-derived screening. These results suggest that ancilla-assisted symmetry screening provides a modular and NISQ-compatible route to securing excited state calculations of physically meaningful properties. We discuss how our scheme may naturally be extended to computing other conserved quantities.

Figures (8)

• Figure 1: Circuit representation of $A\left(\theta, \phi\right)$, where $R\left(\theta, \phi\right) = R_{z}\left(\phi+\pi\right)R_{y}\left(\theta+\pi/2\right)$.
• Figure 2: Composite quantum circuit of the SP ansatz that conserves the total number of particles.
• Figure 3: Construction of the SSP ansatz with the SP ansatz gates applied separately on $\alpha$ and $\beta$ spin domains followed by phase gates for providing up to three different phases to three different occupations. Bullets denote the input and the output qubits of the corresponding gates.
• Figure 4: Scheme of screening statevector with controlled-$\hat{R}_{x}$ operators.
• Figure 5: Schematic illustration of the $\hat{S}_x$-based phase estimation. When there is a spin state $\ket{S,m_z}$, shown in this case with $\ket{2,2}$ on the left, it is in general a superposition of $\hat{S}_x$-eigenstates with $\ket{S,m_z}=\sum_{m_x} c_{m_x}\ket{S,m_x}$. Rotation by $U(\theta)=e^{i\hat{S}_x\theta}$ tags each component with a phase $e^{i m_x\theta}$, and standard QPE registers the estimates of $m_x$ on the ancilla. When a state with $S$ is targeted, (1) an initial state is set to $m_z = S$, (2) the state is then propagated through the ansatz circuit, and (3) invalid states with $|m_x| > S$ are marked for filtering out or for penalization.