Enhancing the reachability of variational quantum algorithms via input-state design

TL;DR

This work tackles the expressivity-trainability trade-off in variational quantum algorithms by introducing input-state design, which uses a low-depth encoder to prepare a superposed input state $ig| extPsi_0(m{ abla})ig angle=\sum_{j=1}^m \,igl(oldsymbol{ abla}igr)_j ig| extpsi_j angle$ that expands the reachable set of a fixed ansatz $U(oldsymbol{ heta})$. A rigorous bound shows that for orthogonal candidates, the maximal fidelity to the target $ig| extPsi_ ext{tar}ig angle$ satisfies $ ext{max}_{m{ abla}} F = extstyle\sum_{j=1}^m F_j$, with optimal encoder amplitudes aligned to the overlaps $raket{ extpsi_j}{ extPsi_ ext{tar}}$. The authors provide a concrete six-step protocol to construct the encoder from a small, representative basis, and then jointly optimize $(oldsymbol{ heta},oldsymbol{ abla})$ to achieve higher ground-state fidelities without increasing circuit depth. Numerically, the method yields consistent fidelity and energy improvements across 1D and 2D Transverse-Field Ising models, the cluster-Ising model, and the Fermi-Hubbard model, reducing the required circuit depth and classical optimization effort. Overall, input-state design emerges as a broadly applicable, practical augmentation to circuit design for VQAs, enabling more expressive yet trainable near-term quantum computations.

Abstract

Variational quantum algorithms (VQAs) face an inherent trade-off between expressivity and trainability: deeper circuits can represent richer states but suffer from noise accumulation and barren plateaus, while shallow circuits remain trainable and implementable but lack expressive power. Here, we propose a general framework to address this challenge by enhancing the VQA performance with a specially designed input state constructed using a linear combination technique. This approach systematically modified the set of states reachable by the original circuit, enhancing accuracy while preserving efficiency. We provide a rigorous proof that such framework increases the expressive capacity of any given VQA ansatz, and demonstrate its broad applicability across different ansatz families. As applications, we apply the method to ground-state preparation of the transverse-field Ising, cluster-Ising, and Fermi-Hubbard models, achieving consistently higher accuracy under the same gate budget compared with standard VQAs. These results highlight input-state design as a powerful complement to circuit design in realizing VQAs that are both expressive and trainable.

Figures (8)

• Figure 1: (a) Hardware-efficient ansatz (HEA). Each layer consists of alternating single-qubit rotations $R_y$ and $R_z$ followed by a chain of $\mathrm{CZ}$ gates. The dashed box indicates one circuit layer, which is repeated $p$ times. (b) General Hamiltonian variational ansatz (HVA). Each layer contains a product of unitaries $\prod_{k=1}^{q} e^{-i \theta_k H_k}$, where $\{H_k\}$ are problem-specific Hamiltonian terms. (c-e) Examples of HVA design for three different models. (c) For the transverse-field Ising model. An initial layer of Hadamard gates $H$ prepares $\lvert+\rangle^{\otimes n}$. $U_{ZZ}(\theta) = e^{-i(\theta/2)\,\sigma_i^z\sigma_j^z}$ represents the two-qubit gate for ZZ interaction, while $R_x(\theta)=e^{-i\theta\,\sigma_i^x}$ represents the single-qubit $X$-rotation. (d) For the cluster-Ising model.$U_{ZXZ}(\theta)=e^{-i(\theta/2)\,\sigma_i^z\sigma_j^x\sigma_k^z}$ is a three-qubit gate, and $U_{XX}(\theta)=e^{-i(\theta/2)\,\sigma_i^x\sigma_j^x}$ is a two-qubit gate. (e) For the Fermi-Hubbard model. The upper (lower) register encodes spin-$\uparrow$ (spin-$\downarrow$). On-site interactions between the two spins at site $i$ are implemented as $U_{ZZ}(\theta)$. Hopping terms on odd and even bonds are realized by $U_{XY}(\theta)=e^{-i(\theta/2)\,(\sigma_i^x\sigma_{i+1}^x+\sigma_i^y\sigma_{i+1}^y)}$.
• Figure 2: Reachable sets modified through input-state design. For a fixed unitary $U(\bm{\theta})$, a simple input state $\ket{\Psi_0}$ induces a reachable set (red-shaded) that excludes the target $\ket{\Psi_{\mathrm{tar}}}$, causing optimization to converge to a suboptimal state $\ket{\Psi'(\bm{\theta})}$ (blue path). By contrast, a designed input state $\ket{\Psi_0(\bm{\gamma})}$, prepared by the encoder $V(\bm{\gamma})$, produces a different reachable set (green-shaded) that contains $\ket{\Psi_{\mathrm{tar}}}$, enabling the same $U(\bm{\theta})$ to reach the target (red path).
• Figure 3: Workflow of the Input-state Design Algorithm. (1) The circuit $U(\bm{\theta})$ is first pre-trained on the initial state $\ket{0}^{\otimes n}$ to obtain the optimized parameters $\tilde{\bm{\theta}}_{\mathrm{opt}}$. (2) With $U(\tilde{\bm{\theta}}_{\mathrm{opt}})$ held fixed, $M$ computational basis states $\{\ket{j}\}$ are randomly sampled. (3) For each $\ket{j}$, $E_j$ is evaluated under the pre-trained circuit. (4) According to a selection rule, $m$ promising candidates (including $\ket{0}^{\otimes n}$) out of the $M$ basis states are chosen to form the set $\mathcal{A}_m$. (5) An encoder $V(\bm{\gamma})$ is then constructed to prepare a superposition of these $m$ states. (6) The encoder $V(\bm{\gamma})$ and the original ansatz $U(\bm{\theta})$ are jointly optimized to minimize the energy, with the ansatz parameters initialized at $\bm{\theta}=\tilde{\bm{\theta}}_{\mathrm{opt}}$.
• Figure 4: Simulation results for the one-dimensional transverse-field Ising model using conventional HEA and our input-state design strategy (enhanced HEA). (a)-(b) Performance comparison of variational circuits with different layer depths, showing (a) ground energy and (b) fidelity. The enhanced circuit (orange) reaches an average fidelity of $0.99$ with only 8 layers, while the conventional HEA circuit (blue) requires $12$ layers to achieve the same accuracy. (c)-(d) Training trajectories for the 8-layer enhanced HEA are compared with a 12-layer conventional HEA baseline. The gray dashed line marks the iteration at which the encoder is introduced. After this point, the energy quickly approaches the exact ground energy and the fidelity continues rising. (e) Infidelity (i.e., $1-F$) as a function of the number of computational-basis states selected for the encoder, based on a 5-layer HEA circuit. As $m$ increases, the infidelity decreases approximately exponentially, showing that as few as $m = 6$ carefully chosen states are sufficient for high-fidelity state preparation, enabling expressive yet resource-efficient initialization.
• Figure 5: Simulation results for the 12-qubit 2D Ising model at $h = 0.5$, $1$, and $1.5$. For each field strength $h$, the upper panel shows the ground energy as a function of circuit depth $p$, and the lower panel reports the corresponding fidelity to the exact ground state. The blue curves correspond to the conventional HVA, and the orange curves correspond to the input-state design (enhanced HVA). Each marker represents the mean over $100$ random initializations. Across all three values of $h$, the input-state design consistently achieves lower variational energies and higher fidelities under the same depth, and it reaches the $0.99$ fidelity threshold with fewer layers than the baseline.

Theorems & Definitions (2)

• Theorem 1
• proof