1D QT Ideal Solver Framework

• 1d-qt-ideal-solver is a framework that integrates quantum simulation and tensor-network algorithms for one-dimensional systems with polynomial scaling.
• It employs numerical techniques such as split-operator factorization, FFT-based spectral differentiation, and complex absorbing potentials to achieve high-precision results.
• The framework also leverages tensor-programmable quantum circuits for efficiently solving 1D PDEs and tridiagonal QUBO/QUDO challenges.

The 1d-qt-ideal-solver framework refers to a class of specialized quantum and quantum-inspired algorithms, as well as high-fidelity classical simulation tools, for one-dimensional systems. Architectures bearing this name solve problems ranging from one-dimensional quantum dynamics to exact optimization over tridiagonal quadratic forms, utilizing a spectrum of numerical and quantum circuit—theoretic methods. Key implementations are found in quantum tunneling solvers for the time-dependent Schrödinger equation (Herho et al., 27 Dec 2025), tensor-network-based exact optimizers for tridiagonal QUBO/QUDO (&&&1&&&), and circuit-based solvers for 1D PDEs via tensor-programmable quantum circuits (Siegl et al., 6 Feb 2025). Broadly, these tools exploit the numerically tractable structure of 1D systems—locality, low entanglement, and/or sparse interactions—to achieve polynomial time and resource scaling for physically and computationally important tasks.

1. Quantum Tunneling Simulation: Algorithmic and Numerical Foundations

The 1d-qt-ideal-solver library (Herho et al., 27 Dec 2025) implements numerically exact real-time propagation of quantum wavepackets through parametric 1D potentials, with an emphasis on ideal coherence and minimal algorithmic artifacts. The solver targets the time-dependent Schrödinger equation: $$i\,\frac{\partial\psi(x,t)}{\partial t} = \biggl[-\frac{1}{2}\,\frac{\partial^2}{\partial x^2} + V(x,t)\biggr]\psi(x,t)$$ in atomic units ($\hbar = m = 1$), posed on $x \in [x_{\min}, x_{\max}]$.

Key numerical methods include:

• Split-Operator Trotter–Suzuki Factorization: Real-time evolution is approximated by second-order Strang splitting,

$$e^{-i\hat H\,\Delta t}\approx e^{-\,\frac{i}{2}\hat V\Delta t}\; e^{-\,i\hat T\Delta t}\; e^{-\,\frac{i}{2}\hat V\Delta t} + \mathcal O(\Delta t^3)$$

• FFT-Based Spectral Differentiation: The kinetic operator is diagonalized in $k$-space via FFTs, enabling spectral accuracy for smooth $\psi(x)$.
• Complex Absorbing Potentials (CAPs): A quartic-cosine mask is multiplied at boundaries to eliminate nonphysical reflections without distorting interior dynamics: $$\mathcal M(x) = \begin{cases} 1 - s[1-\cos^4(\frac{\pi\,\xi_L}{2})],\;\;\;\; x\in[x_{\min},x_{\min}+\Delta x_b]\ 1, \;\;\;\; x\in[x_{\min}+\Delta x_b, x_{\max}-\Delta x_b]\ 1 - s[1-\cos^4(\frac{\pi\,\xi_R}{2})],\;\;\;\; x\in[x_{\max}-\Delta x_b,x_{\max}] \end{cases}$$ This strategy achieves machine-precision conservation of norm and energy over typical femtosecond integration windows.

Just-in-time compilation via Numba accelerates all per-grid operations, yielding up to 50$\times$ improvement over native Python for $N=2048$ and $>10^4$ time steps.

2. Test Cases, Benchmarks, and Analysis Metrics

The solver is validated on canonical scenarios:

1. Rectangular Barrier Tunneling: Models field emission across oxide layers using potentials $V_{\rm rect}(x)$ and initial Gaussian wavepackets.
2. Gaussian Barrier Transmission: Emulates scanning tunneling microscopy tip-sample interactions.

Key observables:

• Transmission/Reflection/Absorption: Post-propagation spatial integrals yield $T$, $R$, and $A$ with closure $T+R+A=1\pm10^{-3}$.
• Energy Drift: Numerical precision is established with $|\Delta E/E|<10^{-5}$.
• Statistical Measures: Jensen–Shannon divergence of probability densities, Shannon entropy, and nonparametric hypothesis tests (KS, Mann–Whitney) assess the substantive difference between barrier profiles.
• Phase-space Coherence: Phase isotropy and phase coherence index (PCI $<10^{-3}$) confirm full decoherence spatially and temporally averaged.

For high-energy (over-barrier) cases, the rectangular barrier gives $T=0.826735$, $R=0.171434$, $A=0.001109$; Gaussian barriers yield $T=0.786226$, $R=0.209228$, $A=0.001595$. Jensen–Shannon divergence is $0.0171$ bits (practically negligible). These benchmarks are directly computable with built-in routines for all library users.

3. Tensor-Network Algorithms for 1D QUBO/QUDO (Quantum-Inspired Ideal Solvers)

In the 1d-qt-ideal-solver context, Mata Ali et al. introduce a tensor-network algorithm that exactly minimizes one-dimensional tridiagonal QUBO/QUDO instances with linear time complexity (Ali et al., 2023). The model cost function is

$$C(x) = \sum_{i=1}^n a_i x_i + \sum_{i=1}^{n-1} b_i x_i x_{i+1}$$

with binary ($x_i\in\{0,1\}$) or $D$-ary variables.

Components:

• Quantum State Mapping: The cost minimization is reframed as finding the $x^*$ that maximizes $A_x = e^{-\tau C(x)}$ in the state $$|\psi(\tau)\rangle = \sum_x e^{-\tau C(x)}|x\rangle$$ for large $\tau$.
• MPS × MPO Representation: The imaginary-time propagator $e^{-\tau H}$ is encoded as a matrix product operator of bond dimension 2, permitting sequential contraction with the site-MPS.
• Greedy Marginalization: Bits are selected sequentially by computing marginal probabilities in $O(n)$ time via environment recycling.

The algorithm is provably polynomial ($T(n) = O(n)$ for binary chains), exact in the $\tau\to\infty$ limit, robust to degeneracy, and compatible with chain/QUDO/tree topologies. For full-graph QUBO, the bond dimension requirement increases exponentially, restricting exact solvability to 1D or tree-like graphs.

4. Tensor-Programmable Quantum Circuits for 1D Differential Equations

Siegl et al. extend the "ideal solver" paradigm to quantum circuits, supporting both linear and nonlinear 1D PDEs in amplitude encoding via tensor-programmable quantum circuits (Siegl et al., 6 Feb 2025).

Key methodologies:

• MPO Representation of Differential Operators: General 1D finite-difference stencils (e.g., $d^2/dx^2$, advection, local nonlinearities) are encoded as MPOs with small bond dimension.
• Isometric Embedding and Unitary Extension: The (generally non-unitary) differential operator MPO is lifted to an isometric, then padded as a unitary circuit by Gram–Schmidt; this unitary acts on both system and auxiliary registers.
• Mid-circuit Measurement and Norm Correction: Post-selective measurement on ancillas implements the non-unitary operation up to a success probability $\alpha_{\text{succ}}$, with state-dependent norm correction factor

$$f_{\mathcal O, j} = \frac{\sin^2(\phi/2) + \alpha_{\text{succ}} \cos^2(\phi/2)}{\sin\phi}$$

allowing probabilistic solution of non-unitary dynamics.

Resource requirements per Euler/Fourier step are

• system qubits $n$,
• $\log_2 Z$ MPO-auxiliary qubits ($Z$ = MPO bond dim.),
• 1 global ancilla; total two-qubit gate count $O(nZ^2)$.

Compared to Trotterization, the MPO method achieves lower depth and gate count when operator locality is strong ($Z \ll n$), and is extensible to a range of 1D PDEs, including boundary-dissipative phenomena.

5. Scope, Limitations, and Usage Context

The "idealized" descriptor—in both software and quantum circuit contexts—refers to deliberate exclusion of dissipation, environmental coupling, many-body interactions, spin, and thermal noise from the dynamical model. This ensures unitary evolution and clean numerical/final-state interpretations at the expense of direct applicability to experiment or device modeling.

Major limitations include:

• Validity is restricted to isolated, coherent, single-particle (or low local-dimension) settings.
• Many-body, entanglement-growth, or general graph connectivity preclude efficient exact solvers.
• Inclusion of open-system or stochastic effects requires extension beyond current package capabilities.

Within these boundaries, the 1d-qt-ideal-solver toolset excels for pedagogical demonstration, comparative analysis of barrier transmission, and as a reference implementation for quantum-inspired or hybrid algorithms.

6. Practical Workflow and Performance Table

For practical users, the standard workflow and empirical benchmarks are as follows:

Stage	Methodology	Performance Example
Grid setup	Uniform grid, $N=2048$	$\Delta x \approx 0.03$ nm
CAP configuration	Quartic cosine mask	$\Delta x_b = 3$ nm, $s=0.9$
Propagation	Split-op, Numba-FFT	$\leq$ 135 s per $12,000$ steps
Barrier types	Rectangular, Gaussian	$T=0.83/0.79$ at $E_k > V_0$
Analysis	Entropy, JS divergence	$H\sim 3.1$, $D_{\rm JS}=0.017$
Memory/complexity	FFT: $O(N)$, QUBO: $O(n)$	On typical laptop (i7-8550U)

This reflects validated physical accuracy, efficient computation, and interpretability, provided the user works within the stated physical idealizations.

7. Extensions and Theoretical Frontiers

A plausible implication is that 1d-qt-ideal-solver techniques, both quantum and classical, offer a scalable foundation for exploratory algorithm development in low-dimensional quantum simulation and combinatorial optimization. Extension to time-dependent potentials, non-unitary (Lindblad) dynamics, and modest growth in system size is possible by leveraging the modularity of tensor-network encodings and circuit modularization. Notably, for general QUBO on arbitrary graphs or for many-body quantum dynamics, one must abandon polynomial scaling and turn to approximate schemes (e.g., MPS with truncated bond dimension, PEPS, or hybrid variational quantum approaches).

In summary, the 1d-qt-ideal-solver unifies high-fidelity numerical simulation, quantum-inspired tensor algorithms, and programmable quantum circuits for idealized one-dimensional quantum and optimization problems, with rigorous validation on benchmark test cases and clear delineation of applicability and limits (Herho et al., 27 Dec 2025, Ali et al., 2023, Siegl et al., 6 Feb 2025).