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Quantum circuits of elastic wave equations via Schr\"odingerisation

Shi Jin, Chundan Zhang

arXiv:2610.37247v1quant-phcs.NAmath.NA

TL;DR

The paper addresses explicit quantum circuit construction for elastic wave equations with sources and physical boundaries. It combines Schrödingerisation with discretization-specific circuits and reports a polynomial improvement in spatial-resolution dependence for quantum state evolution, under stated assumptions.

  • Problem

    Explicit quantum circuits are needed for elastic wave systems with source terms and physically relevant boundary conditions, while large-scale and fine-resolution classical simulations may be costly.

    Source: §1 · p. 2

  • Method

    The paper rewrites elastic velocity–stress equations in symmetric matrix form and applies Schrödingerisation with Fourier spectral or staggered-grid discretization to construct circuits.

    Source: §1 · p. 2

  • Results

    The comparison with classical methods identifies a polynomial advantage in spatial-resolution dependence for the quantum state-evolution subroutine under the stated assumptions.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The constructions provide a theoretical foundation for quantum state computation of elastic wave propagation in homogeneous media.

    Source: §7 · p. 27

    Algorithms for complex boundary-value problems and unbounded domains, practical simulation circuits, and costs of input preparation, observable estimation, and finite-gate implementation remain future work.

    Source: §7 · pp. 27–28

Abstract

from arXiv · show

We construct explicit quantum circuits for the velocity-stress formulation of elastic wave equations in homogeneous media using Schr\"odingerisation. After rewriting the equations in symmetric matrix form(SMF), we use Fourier spectral discretization for source-driven periodic problems and staggered-grid finite differences for Dirichlet, Neumann, and impedance boundary conditions. The Schr\"odingerisation method transforms any linear PDEs into a system of Schr\"odinger-type PDEs -with unitary evolution- using the warped phase transformation that maps the equations in one higher dimension. For periodic problems, we develop circuits by combining controlled Fourier-mode rotations. For physical boundaries, interleaved encoding swaps the spatial and field-component registers, enabling a more efficient circuit construction using shift operators. We analyze $p$th-order Trotter-Suzuki formulas and derive splitting-error and CNOT-gate estimates, under the assumptions of smooth auxiliary initialization and solution-state recovery. Comparison with classical partitioned splitting methods identifies a polynomial improvement in spatial-resolution dependence for the quantum state-evolution subroutine. Noiseless circuit simulations of a forced periodic problem and an impedance problem illustrate the constructions and quantify their errors against exact solutions and classical reference computations.

1 Introduction

The paper develops explicit quantum circuits for homogeneous elastic velocity–stress systems, addressing source terms and physically relevant boundary conditions. It analyzes circuit errors and costs, finding a polynomial advantage in spatial-resolution dependence for state evolution under stated assumptions.

1 Introduction

  • Fourier spectral discretization and staggered-grid finite differences are used for periodic and physical-boundary problems, respectively.The circuits are assembled from elementary evolutions using nested product formulas, with errors from Hamiltonian simulation and spatial and auxiliary-variable discretization analyzed.Source: §1 · p. 2
  • Periodic velocity–stress systems with time-independent sources use Fourier diagonalization to produce controlled rotations for spatial and auxiliary variables.Explicit constructions and CNOT-gate estimates are also provided for d = 2, 3.Source: §1 · p. 3
  • Physical-boundary circuits use exchanged encoding of spatial and field-component registers, with Neumann obtained by conjugating the Dirichlet circuit and impedance handled by exact controlled rotations on two boundary subspaces.The boundary constructions use staggered-grid finite differences.Source: §1 · p. 3
  • The analysis derives commutator-based error bounds and CNOT-gate estimates for pth-order Trotter–Suzuki formulas, including auxiliary-approximation effects.The comparison with classical methods identifies a polynomial spatial-resolution advantage for the state-evolution subroutine under the stated assumptions.Source: §1 · p. 3
  • The circuit experiments compare recovered solutions with exact PDE solutions and classical partitioned and Schrödingerised solutions.The paper identifies smooth auxiliary initialization as an assumption in its complexity analysis.Source: §1 · pp. 2–3
  • The framework covers source-driven periodic problems and Dirichlet, Neumann, and impedance boundaries, extending beyond prior treatments of acoustic waves or restricted elastic settings.The comparison is with Refs..Source: §1 · p. 3

2 A review of velocity–stress equation

This section reviews the velocity–stress description of elastic motion and organizes its rescaled unknowns into a symmetric matrix form. The formulation includes body forces and material parameters.

2 A review of velocity–stress equation

  • The velocity–stress formulation represents elastic motion using velocity and stress components, with body force terms and Lamé parameters describing the medium.The stress tensor components represent forces acting in a direction on a surface with a specified normal.Source: §2 · p. 4
  • The normal-stress components are governed by Cd = 2µId + λ11T, with 1 the all-ones vector.This matrix organizes the normal-stress equations.Source: §2 · p. 4
  • The symmetric matrix form orders rescaled unknowns as normal stresses, shear stresses, then velocities, exposing the system in a structured field-component vector.The shear stresses are indexed by i < j.Source: §2 · p. 5
  • Circuit notation distinguishes the two evolution components and their spatial and auxiliary-variable factors, with separate spatial operators denoted for periodic, Dirichlet, Neumann, and impedance cases.The notation identifies the boundary-specific spatial factors as UP, UD, UN, or Uimp.Source: §2 · p. 5

3 A review of the Schrodingerization method

Schrödingerisation embeds a linear non-unitary evolution in a larger unitary system using an auxiliary variable. The section describes source handling, solution recovery, and smooth auxiliary initialization for complexity control.

3 A review of the Schrodingerization method

  • Schrödingerisation introduces an auxiliary variable to embed linear non-unitary evolution in a unitary system.The warped phase transformation w = g(p)u is used to form the enlarged evolution.Source: §3 · p. 5
  • After spatial discretization, a time-independent source is handled by augmenting the system to make it homogeneous; homogeneous problems require no augmentation.Source: §3 · p. 5
  • Fourier truncation and discretization in the auxiliary variable produce an enlarged Hermitian evolution generated by H = −H1 ⊗ Dp + H2 ⊗ I.Dp is the Fourier correlation matrix of p, and the enlarged evolution is eiHt.Source: §3 · p. 6
  • The original solution can be recovered by evaluating w/g(p) for p ≥ p* or by integrating, with p* bounded below using the largest positive eigenvalue of H1 and the time horizon.Source: §3 · p. 6
  • Smooth auxiliary initialization permits an analytic Gevrey-1 profile, while εaux sets the tolerance for truncation, recovery, and Fourier spectral discretization in p.The stated motivation is to address the accuracy limitation from the cusp of e−|p|.Source: §3 · p. 6

4 Periodic boundary condition

This section develops Schrödingerisation-based quantum circuits for source-driven elastic waves with periodic boundary conditions, using Fourier spectral discretization and product-formula circuit construction. It analyzes approximation error and CNOT costs, including a smooth-profile, pth-order Trotter–Suzuki state-preparation result and spatial-resolution dependence for fixed dimension.

4 Periodic boundary condition

  • For periodic boundary conditions, the section’s stated contribution is an explicit circuit representation for the source-driven elastic wave equation using the Schrödingerisation framework.The one-dimensional homogeneous-medium formulation uses the velocity-stress symmetric matrix form.Source: §4 · p. 6

4.1 Fourier discretization and Schrödingerisation

  • For periodic source-driven problems, Fourier spectral discretization diagonalizes spatial differentiation, while Schrödingerisation converts the forced semi-discrete system into a homogeneous system with an auxiliary variable.The resulting evolution is expressed as a Schrödinger system in the auxiliary p-direction.Source: §4.1 · p. 7

4.2 Quantum circuit construction

  • The first-order circuit splits the evolution into UI and UII; Fourier-basis diagonalization enables exact implementation of UI and the spatial evolution UP, with QFTs on the x- and p-registers.The circuit description also gives a d-dimensional construction, with each directional coefficient-matrix evolution requiring O(d log d) CNOT gates.Source: §4.2 · p. 8

4.3.1 Error and complexity analysis

  • The one-step error analysis defines C1,p in terms of the auxiliary-frequency scale κaux and spatial scale λx, then extends the analysis to global evolution over T = rτ.C1,p := κaux(1 + λx)(1 + κaux + λx)^(p−1) + λx(1 + λx)^(p−1).Source: §4.3.1 · pp. 10–11
  • The d-dimensional error analysis accounts for an additional splitting across spatial directions and the vanishing commutators among those directional terms.The passages state that at most dp(d−1) terms are nonzero in the commutator count.Source: §4.3.1 · p. 12
  • Under a smooth auxiliary profile, Theorem 4.1 analyzes preparation of the normalized recovered state and gives a CNOT-complexity result with polylogarithmic factors suppressed.The theorem assumes physical parameters independent of target precision; the supplied passages do not include its full complexity expression.Source: §4.3.1 · pp. 13–14
  • The smooth auxiliary profile permits truncation and recovery errors of O(ε_aux), while recovery normalization requires accounting for the desired-state measurement probability.The analysis also states that amplitude amplification gives Θ(R_T) measurement repetitions.Source: §4.3.1 · pp. 14–15
  • Choosing h = Θ(ε^(1/ρ)) balances spatial discretization error with target accuracy and gives overall error O(ε).The passage states that substituting this choice into the complexity estimate yields the corresponding resolution-dependent bound.Source: §4.3.1 · p. 15
  • The cited passage states that the overall recovery overhead is R, but does not support the claimed smooth-profile or pth-order Trotter–Suzuki preparation details.Source: §4.3.1 · p. 15
  • For fixed d, the quantum state-evolution subroutine has a polynomial advantage over the classical method with respect to spatial resolution.This comparison concerns state evolution as a subroutine, not the full end-to-end solution process.Source: §4.3.1 · p. 16
  • The quantum gate-complexity order matches Ref., while the construction also accommodates source terms and more general boundary conditions.The QFTs on auxiliary registers contribute O(n^2 p) CNOTs, which the analysis absorbs into the overall estimate.Source: §4.3.1 · pp. 14, 16

5 Physical boundary conditions

This section develops staggered-grid Schrödingerised circuits for Dirichlet, Neumann, and impedance boundaries using shift operators and register reordering, and analyzes splitting errors and CNOT costs. It also describes the state-evolution spatial-resolution advantage and reduced circuit overhead associated with the symmetric SMF structure.

5 Physical boundary conditions

  • The Dirichlet, Neumann, and impedance velocity–stress systems are discretized on staggered grids and rewritten as semi-discrete ODEs for Schrödingerisation.The complexity analysis is presented for Dirichlet boundaries; the other two constructions have the same asymptotic estimates up to constants depending on boundary parameters.Source: §5 · p. 16, §5.4 · p. 22

5.1.2 Interleaved encoding and quantum circuits

  • Interleaved encoding reorders the auxiliary, spatial, and block registers exactly, enabling shift-operator circuits without adding approximation error.The reordered encoding changes the register order from (ancilla, block, x) to (ancilla, x, block).Source: §5.1.2 · p. 18, §5.1.3 · p. 18

5.2.2 Quantum circuit construction

  • The Neumann circuit differs from the Dirichlet circuit only by two additional X gates on the auxiliary qubit at the ends of the shift-operator circuit.The Neumann condition prescribes endpoint stress, and its coefficient matrix is related to the Dirichlet matrix by conjugation with X on the auxiliary register.Source: §5.2 · p. 19, §5.2.2 · p. 20

5.3.2 Quantum circuit construction

  • For impedance boundaries, the Hermitian evolution splits into commuting left- and right-boundary terms, so this decomposition introduces no splitting error.The endpoint conditions use impedances ηL and ηR, and the boundary contribution is encoded in a diagonal matrix K.Source: §5.3 · p. 20, §5.3.2 · p. 21

5.4.1 Splitting errors

  • The section derives local and global error bounds for pth-order Trotter–Suzuki evolution, including the additional splitting required for multiple spatial dimensions.The analyses use the Dirichlet case and extend analogously to Neumann and impedance boundaries up to constants depending on boundary parameters.Source: §5.4 · p. 22, §5.4.1 · pp. 22–23

5.4.2 CNOT-gate complexity

  • The CNOT analysis gives a sufficient step-count condition for simulation error at most ε and extends the gate-complexity estimates to d dimensions.The section also derives total CNOT complexity using a smooth auxiliary profile and compares it with classical operator-splitting complexity.Source: §5.4.2 · p. 24, §5.4.3 · pp. 24–25

5.4.3 Total complexity and classical comparison

  • For fixed d, staggered-grid Schrödingerisation gives a polynomial spatial-resolution advantage over the classical method for state evolution, with gate complexity of the same order as Ref. ignoring logarithmic factors.The method accommodates Dirichlet, Neumann, and impedance boundary conditions.Source: §5.4.3 · p. 25
  • The symmetric SMF structure reduces circuit-level overhead: implementing UI requires O(np) CNOT gates, where np = log2 Np, rather than cost depending directly on Np.The reduction preserves the same leading asymptotic complexity.Source: §5.4.3 · p. 25

6 Numerical Experiments

The experiments compare circuit-recovered solutions with exact and classical references, reporting periodic-case errors and illustrating an impedance-boundary case.

6 Numerical Experiments

  • Each test compares the exact PDE solution, classical Lie-partitioned and Schrödingerised solutions, and the recovered circuit state using a noiseless state-vector simulator.The classical Schrödingerised and circuit results use the same auxiliary discretization and recovery point.Source: §6 · p. 25

6.1 A two-dimensional periodic problem with a constant body force

  • The periodic test uses a constant body force on a two-dimensional periodic domain, with four spatial qubits per direction and 512 time steps to T = 1.The Schrödingerised circuit uses n_p = 7 and R = 4, with a source-augmentation qubit, for 19 qubits total.Source: §6.1 · p. 25
  • At T = 1, Tab. 1 lists errors against the exact solution on a 16 × 16 grid; the classical Schrödingerised result is more accurate, and classical and circuit methods converge at first order.The authors attribute its higher accuracy to direct matrix-exponential evaluation, which avoids temporal splitting error.Source: Table 1 · p. 26, §6.1 · p. 26

6.2 Homogeneous problem with impedance boundary conditions

  • The source-free impedance test uses unit impedances; Fig. 14 presents velocity and stress for the homogeneous problem at T = 1.2.The boundary relations are −σ(−π, t) + v(−π, t) = 0 and σ(π, t) + v(π, t) = 0.Source: §6.2 · p. 26, Figure 14 · p. 27

7 Conclusion

The conclusion presents Schrödingerisation circuits for homogeneous elastic wave equations and identifies more general wave settings and practical simulation costs as future work.

7 Conclusion

  • The paper develops explicit Schrödingerisation circuits for homogeneous velocity–stress elastic wave equations, using Fourier methods for periodic cases and staggered grids with interleaved encoding for physical boundaries.The constructions accommodate time-independent sources and non-unitary boundary dynamics in a Hamiltonian-simulation framework.Source: §7 · p. 27
  • The authors identify complex boundary-value problems and unbounded domains as generalized scenarios for future quantum algorithms.They also plan to derive practical-simulation circuits and assess input-preparation, observable-estimation, and finite-gate costs.Source: §7 · pp. 27–28

A Notations

Appendix A.1 introduces notation for finite-difference operators, including elementary matrices, shifts, and rotations.

A.1 Notation for Finite Difference Operators

  • Appendix A.1 introduces elementary 2 × 2 matrices, including the four σ_ij matrices and the identity I.Source: §A.1 · p. 32
  • The appendix defines left and right shift operators and the rotation operators Rx, Ry, and Rz.Source: §A.1 · p. 32

B.1 Proof of exp (P

The proof section invokes a matrix-exponential power series, while the following construction section expresses the coefficient matrix in the velocity–stress component basis.

B.1 Proof of exp (P

  • The proof invokes the matrix-exponential power series and states a result for any positive integer m.Source: §B.1 · p. 32

C Constructions of exp(iAℓt)

C.1 The derivation of exp(iAℓt)

Each directional operator has a structured block decomposition that permits an exact circuit implementation, while evolution across directions is applied sequentially because the operators generally do not commute.

C.1 The derivation of exp(iAℓt)

  • Because the directional operators generally do not commute, multidimensional evolution applies their directional operators sequentially.Source: §C.1 · p. 33
  • Each directional factor acts only on the physical-component register and the spatial register for its direction.Source: §C.1 · p. 33
  • Aℓ consists of a normal stress–velocity star block, d −1 shear stress–velocity two-dimensional blocks, and a zero block.Source: §C.1 · p. 33
  • The normal–velocity block and the shear–velocity blocks act on mutually disjoint subspaces, so the directional exponential factorization is exact.Source: §C.1 · p. 34
  • Implementing exp(iAℓt) uses Pℓ and its inverse, one normal–velocity two-level rotation, and d −1 shear–velocity two-level rotations.Pℓ can be implemented with d −1 two-level real rotations, with P†ℓ obtained by reversing them.Source: §C.1 · pp. 33–34

C.2 The circuit of exp(iAℓt)

  • The d = 2 and d = 3 directional circuits are exact within each directional block, with unused basis states left invariant.Source: §C.2 · p. 34

C.3 The complexity of exp(iAℓτ)

The explicit directional circuit has CNOT complexity bounded by O(d(log d)) in short time τ. Its construction counts two-level rotations and decomposes them into multi-controlled rotations.

C.3 The complexity of exp(iAℓτ)

  • O(d(log d)) bounds the CNOT gate complexity of explicitly implementing exp(iAℓτ) in short time τ.Source: §C.3 · p. 34
  • 3d −2 two-level Rx/Ry rotation gates suffice for the directional circuit decomposition.Two-level rotations are implemented as multi-controlled Rx rotations; Givens rotations use Gray-code mapping and multi-controlled Ry rotations.Source: §C.3 · p. 34
  • Each two-level Rx/Ry rotation on an n-qubit register can be decomposed into at most 18n−42 CNOT gates.Source: §C.3 · p. 34
  • The d = 2 and d = 3 directional-circuit figures label the qj wires as physical components, not spatial variables.Source: Figure 15 · p. 35, Figure 16 · p. 36
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