Coefficient-Decoupled Matrix Product Operators as an Interface to Linear-Combination-of-Unitaries Circuits

Abstract

We introduce a coefficient-decoupled matrix product operator (MPO) representation for Pauli-sum operators that separates reusable symbolic operator support from a tunable coefficient bridge across a fixed bipartition. This representation provides a direct interface to linear-combination-of-unitaries (LCU) circuits: the symbolic left/right dictionaries define a static \textsc{Select} oracle that is compiled once, while coefficient updates modify only a dynamic \textsc{Prep} oracle. As a proof of concept, we construct compact state-adapted Pauli pools by sampling Pauli strings from a pretrained matrix product state (MPS), pruning them to a fixed symbolic pool, optimizing only their coefficients, and transferring the resulting weights directly to the LCU interface. The resulting workflow provides a reusable classical-to-quantum compilation strategy in which the symbolic operator structure is compiled once, and subsequent updates are confined to a low-dimensional coefficient object.

Figures (5)

• Figure 1: Coefficient-decoupled MPO and its LCU interpretation. Top: Coefficient-decoupled factorization across a fixed cut, $\hat{O}=\sum_{(\alpha,\beta)\in \mathcal{E}} C_{\alpha\beta}\bigl(\hat{P}^{\mathsf L}_\alpha\otimes \hat{P}^{\mathsf R}_\beta\bigr)$, where the left/right graphs $\mathcal{G}_{\mathsf L}$ and $\mathcal{G}_{\mathsf R}$ encode the unique Pauli-fragment patterns $\{\hat{P}^{\mathsf L}_\alpha\}$ and $\{\hat{P}^{\mathsf R}_\beta\}$, and the typically sparse bridge matrix $\mathbf{C}=[C_{\alpha\beta}]$ stores the coupling coefficients. Bottom: LCU block encoding: Prep loads the coefficient distribution over an index register $\ket{\alpha}_{\mathsf L}\ket{\beta}_{\mathsf R}$, Select applies the indexed unitaries $\hat{U}_{\alpha\beta}=\hat{P}^{\mathsf L}_\alpha\otimes \hat{P}^{\mathsf R}_\beta$, and Prep$^\dagger$ uncomputes, yielding the $\ket{0}^{\otimes a}$-block proportional to $(\hat{O}/\lambda)\ket{\psi}$, with $\lambda=\sum_{(\alpha,\beta)\in \mathcal{E}} |C_{\alpha\beta}|$. Here $a$ denotes the number of ancilla qubits in the index register. For simplicity, sign and phase handling are omitted; negative or complex coefficients can be incorporated in the standard LCU manner.
• Figure 2: Examples of coefficient-decoupled MPO constructions with an explicit bridge. (a) The $\mathrm{H}_2$ electronic Hamiltonian in the sto-3g basis, mapped to qubits and written as a Pauli sum $\hat{H}=\sum_i c_i(R)\hat{P}_i$. As the bond length $R$ varies, the coefficients change while the Pauli support, and hence the symbolic graph structure, remain fixed. The graph separates into a diagonal $I/Z$ sector and an off-diagonal sector containing strings with at least one $X$ or $Y$, reflecting symmetry-resolved structure that can persist in larger molecules with additional symmetries singh2011tensorsingh2012tensor. (b) The $\mathrm{H}_8$ chain Hamiltonian in the sto-3g basis with 8 orbitals (16 qubits), shown using a randomly selected subset of 50 Pauli terms for visualization. In both panels, the left and right subgraphs encode deduplicated Pauli-string fragments on either side of the cut, while the bridge stores the coefficients coupling the two symbolic graphs.
• Figure 3: Coefficient-decoupled MPOs and their LCU oracle interface. (a) Coefficient-decoupled generator. Across a bipartition $\mathsf{L}\cup\mathsf{R}$, the generator $\hat{G}(\vec{\theta})=\sum_{(\alpha,\beta)\in\mathcal{E}}s_{\alpha\beta}(\vec{\theta})\bigl(\hat{P}^{\mathsf L}_\alpha\otimes \hat{P}^{\mathsf R}_\beta\bigr)$ is specified by fixed left/right Pauli-fragment dictionaries and a tunable bridge $\mathbf S_g(\vec{\theta})$. (b) Effective environment contractions. Contracting $\bra{\phi_0}\hat{G}^\dagger(\vec{\theta})\hat{H}\hat{G}(\vec{\theta})\ket{\phi_0}$ yields left and right environments and an effective problem at the cut, such as $H_{\rm eff}$ and $N_{\rm eff}$, acting only on the bridge indices. (c) Static-Select/dynamic-Prep interface. The bridge coefficients define a dynamic Prep oracle on the index register $(\alpha,\beta)$, while a static Select oracle applies $\hat{U}_{\alpha\beta}=\hat{P}^{\mathsf L}_\alpha\otimes \hat{P}^{\mathsf R}_\beta$. The resulting circuit block encodes $\hat{G}(\vec{\theta})/\Lambda(\vec{\theta})$, where $\Lambda(\vec{\theta})=\sum_{(\alpha,\beta)\in\mathcal{E}}|s_{\alpha\beta}(\vec{\theta})|$. Signs and phases may be absorbed into the selected unitaries in the standard LCU manner.
• Figure 4: Polar map of Pauli strings sampled from pretrained MPSs for $\mathrm{C}_2\mathrm{H}_4$, classified by excitation order: (a) $\chi=8$ and (b) $\chi=32$. The radial coordinate is the normalized Pauli weight $r(P)=w(P)/w_{\max}$, where $w(P)=|\{i:P_i\neq I\}|$. The angle $\theta(P)\in[0,2\pi)$ is assigned through a bitmask-based hash of the operator support and is used only to distribute different support patterns visually around the circle; here $b(P)=\sum_i \mathbf{1}[P_i\in\mathcal{A}]\,2^{i-1}$, with $\mathcal{A}=\{X,Y\}$ for excitation strings and $\mathcal{A}=\{Z\}$ for diagonal strings. Dot area scales as $\sqrt{\Pi(P)}$, so more probable strings appear larger, and the hollow circle at the origin denotes $I^{\otimes N}$. In both panels, the sampled distribution is dominated by diagonal strings, while the off-diagonal weight is concentrated mainly in the double-excitation sector; higher-excitation strings are comparatively rare.
• Figure 5: Optimized energy of the molecule $C_2H_4$ in the sto-3g basis with $10$ orbitals ($20$ qubits) with the ansatz $\ket{\psi(\vec{\alpha})}=\hat{G}(\vec{\alpha})\ket{\Phi_0}$ as a function of the number $N_{\rm samples}$ of Pauli strings sampled from a pretrained MPS, for reference bond dimensions $\chi=8$ and $\chi=32$ corresponding to Figure \ref{['fig:pauli_polar']}. For each value of $N_{\rm samples}$, a curated pool is constructed by retaining strings containing at least one $X$ or $Y$, together with only a small retained subset of purely diagonal $I/Z$-type strings. The dashed horizontal lines show the corresponding DMRG benchmark energies. Increasing $N_{\rm samples}$ systematically lowers the variational energy, but the result saturates above the DMRG benchmark, indicating convergence within the sampled operator span rather than to the full DMRG manifold. The $\chi=32$ reference captures richer correlations and therefore requires larger values of $\mathcal{N}_{\rm samples}$ to approach its DMRG energy.