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Explicit Quantum Circuits for Simulating Linear Differential Equations via Dilation

Seonggeun Park

TL;DR

This work addresses simulating linear ODEs with non-Hermitian generators by embedding the dynamics into a unitary dilation and then realizing this dilation on quantum hardware. It develops a concrete SBP-based discretization $F_h$ that becomes exactly skew-Hermitian in the $\ell^2$ sense, enabling exact moment conditions and a provable global error bound of $O(M^{-3/2})$ (plus boundary effects). The authors then construct an end-to-end quantum circuit framework using LCU, QSVT, and QFT-adder techniques to prepare the ancillary state $|r_h\rangle$, block-encode the dilated Hamiltonian, and perform the evaluation step with postselection and amplitude amplification. They provide explicit circuit primitives, analyze resource requirements (scaling between $O(\log M)$ and $O((\log M)^2)$ for different stages), and discuss practical implications for dissipative PDEs and open quantum systems, while acknowledging limitations and avenues for refinement and experimental validation.

Abstract

Quantum simulation has primarily focused on unitary dynamics, while many physical and engineering systems can be modeled by linear ordinary differential equations whose generators include non-Hermitian terms. Recent studies have shown that such equations, which give rise to nonunitary dynamics, can be embedded into a larger unitary framework via dilation techniques. However, their concrete realization on quantum circuits remains underexplored. In this paper we present a concrete pipeline that connects the dilation formalism with explicit quantum circuit constructions. On the analytical side, building on the recent dilation framework, we introduce a discretization of the continuous dilation operator that is tailored for quantum implementation. This construction ensures an exactly skew-Hermitian ancillary generator, which allows the moment conditions to be satisfied without imposing artificial constraints. We prove that the resulting scheme achieves a global error bound of order $O(M^{-3/2})$, up to exponentially small boundary effects. This error can be suppressed by refining the discretization, where $M$ denotes the discretization parameter. On the algorithmic side, we demonstrate that the dilation triple $(F_h, |r_h\rangle, \langle l_h|)$ can be efficiently implemented on quantum circuits. Using linear combinations of unitaries, QFT-adder operators, and quantum singular value transformation, the framework requires resources ranging from $O(\log M)$ to $O((\log M)^2)$, depending on the stage of the pipeline.

Explicit Quantum Circuits for Simulating Linear Differential Equations via Dilation

TL;DR

This work addresses simulating linear ODEs with non-Hermitian generators by embedding the dynamics into a unitary dilation and then realizing this dilation on quantum hardware. It develops a concrete SBP-based discretization that becomes exactly skew-Hermitian in the sense, enabling exact moment conditions and a provable global error bound of (plus boundary effects). The authors then construct an end-to-end quantum circuit framework using LCU, QSVT, and QFT-adder techniques to prepare the ancillary state , block-encode the dilated Hamiltonian, and perform the evaluation step with postselection and amplitude amplification. They provide explicit circuit primitives, analyze resource requirements (scaling between and for different stages), and discuss practical implications for dissipative PDEs and open quantum systems, while acknowledging limitations and avenues for refinement and experimental validation.

Abstract

Quantum simulation has primarily focused on unitary dynamics, while many physical and engineering systems can be modeled by linear ordinary differential equations whose generators include non-Hermitian terms. Recent studies have shown that such equations, which give rise to nonunitary dynamics, can be embedded into a larger unitary framework via dilation techniques. However, their concrete realization on quantum circuits remains underexplored. In this paper we present a concrete pipeline that connects the dilation formalism with explicit quantum circuit constructions. On the analytical side, building on the recent dilation framework, we introduce a discretization of the continuous dilation operator that is tailored for quantum implementation. This construction ensures an exactly skew-Hermitian ancillary generator, which allows the moment conditions to be satisfied without imposing artificial constraints. We prove that the resulting scheme achieves a global error bound of order , up to exponentially small boundary effects. This error can be suppressed by refining the discretization, where denotes the discretization parameter. On the algorithmic side, we demonstrate that the dilation triple can be efficiently implemented on quantum circuits. Using linear combinations of unitaries, QFT-adder operators, and quantum singular value transformation, the framework requires resources ranging from to , depending on the stage of the pipeline.

Paper Structure

This paper contains 12 sections, 10 theorems, 103 equations, 6 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical Framework
  3. Dilation framework and moment conditions
  4. Discretization of the triple (F,|r),(l|
  5. Error Analysis and Global Bound
  6. Sources of error and the main theorem
  7. Proof of Theorem \ref{['thm:main']}
  8. Quantum Circuit Implementation
  9. Preparation of the ancillary state |r_h
  10. Block-encoding of theta Fh and IH + i theta Fh K
  11. Evaluation step
  12. Summary and Discussion

Key Result

Theorem 1

Let $F$ be a linear operator acting on the ancillary Hilbert space $\mathcal{H}_A$, and let $|r) \in \mathbb{X}$ together with a linear functional $(l|$ on $\mathbb{X}$, where $\mathcal{H}_A \subset \mathbb{X}$. Let $H(s)$ and $K(s)$ be Hermitian operators on the system Hilbert space $\mathcal{H}_S$ then the dilated evolution reproduces the exact solution of the target dynamics: where $A(s) = -iH

Figures (6)

  • Figure 1: Decomposition of $\mathrm{Prep}_{\mathrm{init}}$ for $a=3$. Since all amplitudes are real and nonnegative, $\mathrm{Prep}_{\mathrm{init}}$ uses only $R_Y$ rotations. Gate counts: $2^a-2=6$ CNOT gates and $2^a-1=7$$R_Y$ gates.
  • Figure 2: Block encoding of $\hat{H}_{\mathrm{init}}$ using LCU.$\mathrm{Prep}_\mathrm{init}$ makes $\sum_j\sqrt{\omega_j}|j\rangle$, while $\mathrm{Select}_\mathrm{init}$ applies controlled $\{I,-Z_0,-Z_1,\dots,-Z_{m-1}\}$ up to global phase. The construction $(\mathrm{Prep}_\mathrm{init}^\dagger\!\otimes I)\,\mathrm{Select}_\mathrm{init}\,(\mathrm{Prep}_\mathrm{init}\otimes I)$ realizes a $(1,a,0)$ block-encoding of $\hat{H}_{\mathrm{init}}$.
  • Figure 3: QSVT sequence for the monomial $f(x)=x^\beta$. Here $\Pi=|0\rangle\!\langle 0|^{\otimes a}$. In the diagram, the $C_{\Pi}\mathrm{NOT}$ gate denotes the controlled operation $X\otimes\Pi + I\otimes(I-\Pi)$. The interleaving phases $\{\phi_j\}$ implement the degree-$\beta$ monomial transform.
  • Figure 4: Block encoding of $R$ via a QFT adder and block encoding of $\theta F_h$. (a) The $(m\!+\!1)$-qubit QFT layer $F_{2^{m+1}}$ together with the parallel phase gates $\bigotimes_{k=0}^{m} R_Z(-\pi/2^{k})$ realizes the cyclic shift. This structure gives a $(1,1,0)$ block encoding of $R=\sum_{i=0}^{M-1}|i\rangle\!\langle i+1|$. (b) The LCU structure produces $\tfrac{1}{2}(U_DU_R-U_R^\dagger U_D)$ on the computational block, yielding an $(\alpha_{\theta F}, a_{\theta F}, 0)$ block encoding of $\theta F_h$ with $\alpha_{\theta F}=\alpha_D/2=\tfrac{\theta}{2}(2M+1)$ and $a_{\theta F}=2a+3=O(\log\log M)$.
  • Figure 5: Block encoding of $I\!\otimes\!H + i\,\theta F_h\!\otimes\!K$. An $R_Y$ gate prepares amplitudes proportional to $\sqrt{\alpha_H}$ and $\sqrt{\alpha_{\theta F}\alpha_K}$; the branch unitary applies $I\!\otimes\!U_H$ on $|0\rangle$ and $i\,U_{\theta F}\!\otimes\!U_K$ on $|1\rangle$. The overall unitary $U_{\mathrm{tot}}$ is a $(\alpha_H+\alpha_{\theta F}\alpha_K,\ 1+a_H+a_{\theta F}+a_K,\ \epsilon_{\mathrm{tot}})$ block encoding of $I\!\otimes\!H+i\,\theta F_h\!\otimes\!K$.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Theorem 1: Moment conditions for exact dilation, li2025linear
  • Remark 2: On the notation $(l|,|r)$
  • Lemma 3: li2025linear
  • proof : Sketch of proof
  • Lemma 4: Bound for $C_{M,\theta}$
  • proof : Proof sketch
  • Theorem 5: Global error at mid-indices
  • Lemma 6: Solution of the exact dilation
  • proof
  • Lemma 7: Second-order interior error li2025linear
  • ...and 8 more