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Efficient Quantum Simulation of Variable-Coefficient Transport with Continuous Source Injection

Mohammad Mehedi Hasan Akash, Turag Dev, Nhat-Quang Nguyen, Yanzhu Chen, Huixuan Wu, Kourosh Shoele

arXiv:2608.27712v1quant-phmath.NA

TL;DR

Existing quantum transport architectures face challenges incorporating persistent source injection without register-expanding embeddings. This paper presents a fixed-register hybrid algorithm that combines skew-symmetric advection, Fourier-basis diffusion, and classical source updates, with second-order temporal accuracy, fourth-order Richardson-extrapolated accuracy, and a principal limitation in classical readout and re-preparation.

  • Problem

    Persistent source injection in existing architectures would demand register-expanding embeddings, and a fixed-register treatment combining variable-coefficient advection, diffusion, and source injection had not been demonstrated.

  • Method

    The hybrid algorithm uses skew-symmetric unitary advection, ancilla-assisted Fourier-space diffusion, and classical arithmetic for source injection within an n_q + 1-qubit architecture.

  • Results

    The numerical evidence supports second-order temporal accuracy, fourth-order convergence with Richardson extrapolation, third-order one-step augmented-map defect convergence under the stated refinement, and dissipation-linked ancilla loss across a fifty-fold viscosity sweep.

  • Takeaways & Limitations

    The fixed-width kernel allocates quantum resources according to operator character, providing a hybrid building block whose advection, diffusion, and source components use distinct implementations.

  • Takeaways & Limitations

    Classical readout and re-preparation between steps remain the principal limitation, with coherent multistep evolution identified as an immediate next test.

Abstract

from arXiv · show

Quantum time-marching algorithms for transport PDEs often represent variable coefficients and forcing through register-expanding dilations, block-encoding oracles, or repeated postselection. We present an alternative algorithm for a forced variable-coefficient advection-diffusion equation in flow-inspired skew-symmetric form that incorporates spatially varying velocity, viscous dissipation, and persistent source injection with a peak logical requirement of $n_q+1$ qubits. A centered skew-symmetric discretization makes the advection operator strictly skew-Hermitian for arbitrary velocity profiles, enabling an ancilla-free unitary realization using a Gray-code Trotter sequence of controlled-$R_y$ rotations. Diffusion is applied in the Fourier basis through a uniformly controlled rotation on one postselected ancilla, which is measured, reset, and reused between the two diffusion half-steps, while the source is incorporated classically through second-order Strang splitting. Statevector simulations for $N=16$ and $32$ recover second-order temporal convergence against high-accuracy classical solutions, while Richardson extrapolation gives fourth-order accuracy and reduces kernel calls by factors of four to fourteen. Independent tests through $N=256$ confirm second-order spatial consistency. We further show that the per-step ancilla failure probability is proportional to the instantaneous viscous dissipation rate, making postselection cost self-regulating over a fifty-fold viscosity range. Stable evolution is demonstrated for $5\times10^4$ time steps without observable secular error growth, while Gray-code advection accounts for $71$--$95\%$ of transpiled controlled-NOT gates. The fixed-width kernel provides a qubit-efficient building block for near-term hardware studies, although classical readout and state re-preparation remain the main obstacles to coherent multistep evolution.

I. INTRODUCTION

The paper targets forced variable-coefficient transport, where existing quantum time-marching constructions do not jointly provide spatially varying advection, viscous diffusion, persistent forcing, and fixed-register operation. It introduces a skew-symmetric hybrid architecture that treats advection unitarily, diffusion with one reusable ancilla, and source injection classically.

  • Amplitude encoding stores N = 2^nq field values in nq qubits, motivating quantum time marching for spatially resolved flow simulations.
  • Existing constructions address pure advection, uniform diffusion, or related PDE settings, while persistent source injection would require register-expanding embeddings.
  • The paper combines spatially varying skew-symmetric advection, viscous diffusion, and persistent source injection within an nq + 1 logical-qubit architecture.
  • The centered advection discretization is skew-Hermitian for real velocity profiles, so its exponential is unitary and preserves the discrete 2-norm.
  • Diffusion acts mode-by-mode in Fourier space through an ancilla-assisted controlled rotation, while source injection is a classical update and advection uses Gray-code Trotterization.

B. Strang Splitting and Classical Source Integration

The algorithm places source injection outside the quantum circuit and advances the forced equation with symmetric Strang splitting. Classical source half-steps surround a homogeneous quantum advection-diffusion step, preserving deterministic source handling.

  • The full update applies a classical source half-step, a homogeneous D–A–D quantum propagator, and a second classical source half-step.
  • Symmetric source and diffusion half-steps with one unsplit advection step give global second-order temporal accuracy, O(∆t^2).
  • Source injection is deterministic, requiring no ancilla, post-selection, or amplitude amplification, with P_src succ = 1 by construction.
  • After the first source half-step, the vector is normalized and Fourier-transformed before quantum processing, then the stored norm is restored after readout.

C. Sparse Fourier State Preparation

Sparse Fourier state preparation classically constructs the current source-corrected state and loads retained Fourier modes directly into the quantum register. This removes a redundant initial basis transformation before diffusion.

  • Direct Fourier-basis preparation eliminates the initial QFT because the homogeneous step begins with Fourier-diagonal diffusion.
  • The Fourier-state convention uses the discrete transform, complex conjugation, and 1/√N normalization to match the quantum circuit basis.
  • Binary-tree preparation retains modes with |Ŝ_k| > δ and loads them through controlled-Ry rotations and phase gates computed classically.

D. Homogeneous Quantum Step: Diffusion-Advection-Diffusion

The homogeneous quantum step uses diffusion–advection–diffusion splitting: Fourier-space diffusion is non-unitary and postselected, whereas physical-space advection acts directly on the data register through Gray-code Trotterization.

  • Diffusion in Fourier Space: Diffusion is implemented by a mode-dependent UCRY rotation on one ancilla, followed by measurement and post-selection of the |0⟩ ancilla outcome.
  • Diffusion in Fourier Space: For small ν and ∆t, diffusion success remains close to unity, while unsuccessful attempts can be handled by restart-until-success from a classical checkpoint.
  • Advection via Gray-Code Trotter Decomposition: Gray-code ordering makes each nearest-neighbor advection coupling a single-target controlled rotation, and the even-odd-even sequence has local error O(∆t^3).

2. Advection via Gray-Code Trotter Decomposition

The centered advection operator is strictly skew-Hermitian, so its evolution is exactly unitary and can be implemented without dilation or postselection. Gray-code ordering converts nearest-neighbor couplings into single-target controlled rotations within a symmetric second-order Trotter composition.

  • The centered-difference advection operator couples nearest-neighbor grid points and has a skew-Hermitian structure that makes its exponential exactly unitary.
  • Gray-code ordering relabels adjacent grid points so successive labels differ by one bit, avoiding multi-bit transitions in the advection circuit.For nq = 2, Gray-code ordering replaces the binary transition 01 → 10 with the one-bit transitions 00 → 01 → 11 → 10.
  • Each relabeled nearest-neighbor coupling becomes a single-target transition controlled by the remaining nq −1 qubits, while its basis-gate cost depends on hardware resources.
  • The advection update partitions even and odd edges and applies a symmetric even-odd-even Suzuki decomposition with local splitting error O(∆t^3).The edge rotation angles are determined by the centered-difference off-diagonal coefficients, with layer-dependent τ values.
  • The complete homogeneous update combines diffusion and advection through a symmetric D–A–D composition, with the successful branch evaluated from a Fourier-prepared input state.Because the input is prepared directly in the Fourier basis, no initial QFT is required.

E. Accuracy decomposition and commutator-based error estimate

The accuracy analysis separates spatial and temporal errors by comparing against matched classical discretizations and derives a conservative commutator-based global estimate. The analysis also accounts for Fourier truncation and the grid-dependent diffusion norm when auditing temporal convergence.

  • Matched RK45 and DOP853 solutions use the same spatial discretization, providing temporal-integration references rather than continuum benchmarks.Separating spatial discretization from temporal integration errors is required for consistent error assessment.
  • The Fourier Laplacian is spectrally accurate for smooth periodic fields, while the centered advection stencil determines the algebraic spatial order.
  • The augmented state incorporates the source component, and the even-odd generators satisfy Ke + Ko = Kadv within the semidiscrete evolution.
  • Applying the Baker–Campbell–Hausdorff expansion to the nested symmetric compositions yields a global semidiscrete error estimate controlled by M and the generator norms Λh.Λh is the sum of the norms of the source, diffusion, and even and odd advection generators.
  • Fourier truncation contributes a normalized one-step error bounded by 2ηn, while the diffusion norm grows as N^2 and motivates keeping ∆tN^2 constant under grid refinement.A fixed ∆t can leave the small-step regime on finer grids; conditional UCRY diffusion remains exact, so post-selection changes cost rather than algorithmic bias.

F. Qubit Budget and Circuit Cost

The dynamic-circuit kernel uses nq data qubits and one diffusion ancilla, with the ancilla measured, reset, and reused between diffusion half-steps. Although the logical register is fixed-width, end-to-end cost remains dominated by state preparation, repeated attempts, and phase-sensitive readout.

  • The homogeneous quantum kernel requires nq data qubits plus one diffusion ancilla, giving a peak logical requirement of nq + 1 qubits.The same ancilla is measured, reset, and reused between the two diffusion half-steps.
  • The classical source update and binary-tree preparation add no logical qubits, but the peak qubit count does not determine gate or sampling complexity.
  • 3N/2 (nq −1)-controlled Ry rotations, two N-mode UCRY blocks, and three Fourier transforms occur in one successful kernel call.
  • 3N/2 is a hardware-independent logical rotation count, whereas an unconditional O(Nnq) CX claim is unjustified under the fixed nq + 1-qubit budget.The multicontrolled-rotation cost depends on connectivity, clean work ancillas, and relative-phase synthesis.
  • The end-to-end hybrid step requires classical FFTs, source-corrected state preparation, an expected 1/Pn kernel attempts, and phase-sensitive amplitude reconstruction.Computational-basis sampling returns |um|^2 but cannot provide the signs or phases needed for the next classical source addition.
  • Table I compares relative L2 errors against a matched split-step reference and DOP853, alongside diffusion-ancilla post-selection statistics and source-step success over 2000 applications.The matched split-step reference isolates advection-circuit Trotter error, while DOP853 measures total temporal error.

III. RESULTS

Statevector tests validate the quantum time-marching pipeline against high-accuracy classical references, showing second-order temporal and spatial accuracy. Richardson extrapolation reaches fourth-order temporal convergence and substantially reduces ideal-kernel cost, while Fourier truncation remains an additional error source.

  • Validation against classical references: The N = 16 and N = 32 statevector simulations use Gray-code advection and classical Strang source updates against matched split-step and DOP853 references.The matched split-step reference isolates Gray-code Trotter error because it shares the discrete operators and Strang composition, while evaluating advection exactly.
  • State preparation accuracy: 4.1 × 10−16 and 6.4 × 10−16 maximum Fourier Hermitian-symmetry residuals support accurate real-valued initial-state encoding for nq = 4 and nq = 5.The imaginary parts of the zero-frequency and Nyquist coefficients are below 10−17.
  • Temporal convergence: Approximately 2.000 observed temporal order at both resolutions confirms second-order accuracy through the complete quantum pipeline.The tested composition combines Strang source splitting, D-A-D factorization, and even-odd advection Trotterization.
  • Temporal convergence: 16.0-fold successive error reduction identifies fourth-order convergence for the Richardson-extrapolated solution.The time-symmetric construction removes the leading ∆t^2 term and exposes ∆t^4 behavior.
  • Kernel cost: 300 ideal kernel calls reach error 1.65×10−6 with Richardson extrapolation, versus 800 calls reaching 3.5 × 10−5 without extrapolation.Projected ideal-kernel reductions are approximately fourfold at target error 10−4 and thirteenfold at 10−6, excluding preparation, retries, and reconstruction.
  • Spatial convergence: Spatial refinement through N = 256 approaches second-order consistency while separating O(h^2) spatial error from O(∆t^2) accumulated splitting error.The manufactured-solution check covers the complete forced PDE, not only the semi-discrete system.
  • State preparation accuracy: Threshold-based sparse Fourier truncation adds an independent error source whose retained support can approach the full register at stringent tolerances.At the 10−10 threshold, the recorded N = 64 trajectory reaches K_n = N.

D. Energy budget and the ancilla–dissipation identity

The energy and post-selection tests connect ancilla loss to viscous dissipation while confirming the discrete energy budget. Across a fifty-fold viscosity sweep, retries remain inexpensive and accuracy improves with viscosity, although sampling and forcing-bandwidth effects remain bounded by the tested settings.

  • Ancilla–dissipation identity: 4 × 10−9 from unity correlation between measured ancilla loss and step-centered dissipation estimates verifies the ancilla–dissipation identity across 1000 steps.The comparison reproduces the full non-monotonic time dependence, and a remainder bound controls magnitude agreement.
  • Energy budget: 1.2 × 10−4 energy-budget residual confirms dE/dt = ⟨uf⟩−ε within the tested split-step evolution.Advection contributes identically zero to the discrete budget because the transport circuit is exactly unitary.
  • Post-selection statistics: A fifty-fold viscosity sweep from ν = 0.001 to 0.05 reduces cumulative success probability from 0.773 to 0.268, a factor of approximately 2.9.Per-step success probability recovers toward unity as viscous damping empties high-wavenumber modes.
  • Post-selection statistics: Below 0.14% per-step retry overhead is observed throughout the viscosity sweep under the reciprocal-mean Repeat-Until-Success proxy.Each retry restarts from the same classically stored input and introduces no additional algorithmic approximation in the model.
  • Finite-shot behavior: Sampling error decreases from 4.3 × 10−2 at 10^3 shots to 7.0×10−3 at 5×10^4 shots, consistent with N_s^−1/2 scaling.The four-harmonic source produces equal or smaller total variation distances at every tested shot count.
  • Forcing complexity: No additional sampling penalty appears in the tested two-mode versus four-harmonic comparison, but arbitrary forcing bandwidth is not established.The four-harmonic rerun uses eight active conjugate Fourier modes and reports mean per-step success probabilities near 0.99924.

G. Circuit resources

The resource audit separates compact logical width from substantial gate, copy, and hybrid-workflow costs. Gray-code advection dominates the homogeneous kernel’s transpiled CX count, while end-to-end cost remains unresolved by kernel-only diagnostics.

  • Transpiled kernel cost: 71%–95% of the homogeneous kernel’s CX gates come from the Gray-code advection block across tested register sizes.The diagnostic excludes state preparation, source arithmetic, reconstruction, measurement/reset, and routing.
  • Transpiled kernel cost: The transpilation results indicate kernel-level scaling rather than end-to-end hardware cost or feasibility.Complete assessment requires dynamic Fourier-input preparation, ancilla reuse and reset, and target-device connectivity.
  • Hardware-independent resources: The compact register grows from five to thirteen logical qubits over the displayed range, while loaded velocity data, rotations, and measurement copies grow with N.At nq = 12, one homogeneous kernel contains 6144 logical advection rotations and 8192 logical diffusion-mode rotations.
  • Hardware-independent resources: At nq = 12, generic phase-sensitive reconstruction at one-percent accuracy has a 4.096 × 10^7-copy proxy.These figures support a few-step kernel target but not an end-to-end exponential speedup for the present readout-and-repreparation workflow.
  • Resource allocation: The fixed-width kernel uses nq + 1 qubits by assigning unitary advection, post-selected diffusion, and classical source updating to separate physical roles.The source update succeeds with probability one, while diffusion uses one reused post-selected ancilla.
  • Post-selection scaling: The per-step failure probability tracks relative viscous dissipation to first order in ∆t, with correlation coefficient differing from unity by less than 4 × 10^-9.A fifty-fold viscosity increase reduces cumulative success probability by less than a factor of three.

C. Cost structure and the hybrid boundary

The paper’s hybrid boundary is defined by classical feedback between quantum kernel applications: the ideal statevector pipeline is accurate, but preparation, reconstruction, and recursive feedback remain central unresolved costs and correctness issues.

  • Cost boundary: The classical driver reads out and re-prepares the field at every time step, preventing the present workflow from retaining exponential memory compression end to end.Generic amplitude reconstruction requires state copies growing at least linearly with N at fixed accuracy.
  • Correctness boundary: The correctness target must connect the normalized quantum state and stored norm to the physical field after every classical feedback step.The local error decomposes into preparation, synthesis, and feedback terms, with additional accumulated channel errors.
  • Correctness boundary: Ideal restart recovery changes only expected cost, while conditional on successful UCRY outcomes diffusion contributes no additional bias.This separates post-selection retry overhead from algorithmic approximation in the idealized pipeline.
  • Correctness boundary: A high-probability recursive bound for feedback error is missing because one-step amplitude error becomes input to the next source update.The paper proposes replacing conservative amplification with a discrete semigroup or logarithmic analysis.
  • Future extensions: Fully coherent multistep evolution would remove intermediate readout and re-preparation but requires coherent source injection, norm tracking, and observable estimation.The proposed direction adds an ancilla and a second post-selection channel beyond the current architecture.
  • Future extensions: The framework can extend to multiple dimensions and modified non-periodic boundary stencils, although exact skew-Hermiticity weakens locally near boundaries.The multidimensional Gray-code construction applies edge decompositions independently along coordinate directions.
  • Cost boundary: 71%–95% of CX gates arise from advection in the homogeneous-kernel diagnostic, but preparation and phase-sensitive reconstruction dominate the unresolved end-to-end boundary.The transpiled share does not represent total step cost because dynamic preparation and tomography are excluded.
  • Validated scope: The numerical evidence establishes second-order temporal accuracy, fourth-order Richardson behavior, third-order local splitting defect, and second-order spatial consistency through N = 256.The conclusions concern the ideal statevector pipeline rather than end-to-end hardware execution.

Appendix A: Robustness and supplementary accuracy audits

Supplementary audits confirm that the algorithm’s temporal behavior is robust across grid resolution and source complexity, while resource projections expose the tradeoff between Richardson accuracy and kernel cost. Fourier preparation must track the retained support of the evolving source-corrected state, and noisy probability-space extrapolation can amplify shot noise.

  • Grid-resolution robustness: The nq = 5 validation reproduces the nq = 4 error structure against DOP853 and matched split-step references.The companion study uses T = 1.0 and ∆t = 10^-3.
  • Source-complexity robustness: For the four-harmonic source, temporal order, extrapolation behavior, and diffusion success probability remain quantitatively similar to the single-sine case.The four-harmonic source has eight Fourier modes, while its measured success probability at nq = 5 is 0.99923 versus 0.99937 for the single-sine case.
  • Richardson cost projection: At nq = 4, Richardson extrapolation reduces ideal kernel counts by 4.38 at L2 error 10^-4 and 13.86 at 10^-6.At nq = 5, the corresponding reductions are 4.20 and 13.33; these are ideal homogeneous-kernel comparisons.
  • Fourier preparation: The relevant preparation complexity is the retained support Kn of the complete evolving source-corrected state, not the prescribed source’s Fourier-mode count.Advection and source injection can redistribute spectral amplitude and substantially increase Kn during evolution.
  • Fourier preparation: Aggressive truncation reduces preparation count O(Kn nq), but stringent accuracy can drive Kn toward N and remove the sparsity advantage.End-to-end resource estimates therefore retain Kn explicitly.

2. Ancilla–dissipation remainder refinement

The refinement study verifies the quadratic remainder in the ancilla–dissipation identity, while long-horizon tests show accurate evolution without observable secular error growth. The circuit audit distinguishes Gray-code encoding from second-order Trotter ordering and cautions that transpilation counts are not end-to-end hardware costs.

  • Remainder refinement: The ancilla–dissipation remainder has median observed order 2.00006, with maxn |rn|/∆t^2 approaching 0.6103.This verifies the quadratic remainder over a factor-eight time-step refinement.
  • Long-horizon validation: After 5 × 10^4 quantum steps to T = 50, the quantum and DOP853 fields agree to relative L2 error 1.8 × 10^-5.The energy grows after the initial viscous transient as the quasi-mean mode charges toward its forced steady state.
  • Advection circuit structure: Gray-code relabeling simplifies nearest-neighbor edge encoding, whereas the even-odd-even sequence supplies second-order temporal accuracy.These constructions serve distinct roles in the advection circuit.
  • Resource-accounting boundary: The transpilation audit excludes step-dependent state preparation, source arithmetic, reconstruction, and device connectivity, so its counts are not end-to-end hardware costs.The diagnostic circuit also uses two diffusion ancillas and a leading QFT, unlike the production Fourier-input construction.
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