Variational Quantum Linear Solvers

• Variational Quantum Linear Solvers (VQLS) are hybrid algorithms that use parameterized quantum circuits and classical optimization to approximate solutions of linear equations.
• They employ shallow, hardware-efficient ansätze and dynamic circuit adaptation to reduce measurement overhead and mitigate noise on NISQ devices.
• VQLS is pivotal in quantum machine learning and simulation, offering high-fidelity results for structured systems with significant resource reductions.

Variational Quantum Linear Solvers (VQLS) are a class of hybrid quantum–classical algorithms designed for the solution of linear systems of equations on near-term quantum devices via variational techniques. They replace deep-circuit subroutines such as quantum phase estimation (as in HHL) with shallow, parameterized ansätze, enabling practical utility on Noisy Intermediate-Scale Quantum (NISQ) hardware. The VQLS framework has been extended beyond real, Hermitian systems to complex, non-symmetric cases, and is now a central primitive for quantum machine learning, scientific computing, and quantum simulation of classical and quantum transport.

1. Core Algorithmic Principles

The VQLS algorithm seeks to prepare a parameterized quantum state |x(θ)⟩ such that the action of the coefficient operator A maps this trial state close to a desired right-hand-side |b⟩, i.e., A|x(θ)⟩ ≈ |b⟩. Mathematically, this is cast for an n-qubit encoding of an N = 2ⁿ-dimensional system with

$$A\ket{x} = \ket{b}, \quad A \in \mathbb{C}^{N \times N},\; \ket{b} \in \mathbb{C}^{N}.$$

A variational quantum circuit V(θ) prepares the ansatz |x(θ)⟩ = V(θ)|0⟩ⁿ, with θ ∈ ℝᴾ as circuit parameters, typically organized in shallow, hardware-efficient layers of single-qubit rotations and entanglers (Bravo-Prieto et al., 2019, Yi et al., 2023). The cost function quantifies the proximity of A|x(θ)⟩ to |b⟩, most commonly through either

• a squared residual norm:

$$C(\theta) = \Vert A|x(θ)⟩ - |b⟩ \Vert^2 = \langle x(θ)|A^\dag A|x(θ)⟩ - 2\Re\langle b|A|x(θ)⟩ + \langle b|b\rangle,$$

• or a normalized global Rayleigh quotient:

$$C_G(\theta) = 1 - \frac{|\langle b|A|x(θ)\rangle|^2}{\langle x(θ)|A^\dag A|x(θ)\rangle},$$

where minimization of the cost, typically to a threshold C < ε²/κ² (with κ the condition number of A), ensures the preparation of an approximate quantum solution (Bravo-Prieto et al., 2019, Yi et al., 2023).

The measurement of the cost is implemented via efficient circuits utilizing Pauli/LinComb-of-Unitary (LCU) decompositions of A (and A†A), with overlaps and inner products estimated by Hadamard-style or swap circuits (Bravo-Prieto et al., 2019). Classical optimization algorithms update θ in a closed parameter loop.

2. Cost Functions, Ansatz Expressibility, and Trainability

The design of the cost function and variational ansatz is critical for algorithmic trainability and performance. In addition to the global cost above, local cost variants are used to mitigate barren plateaus in cost landscapes. These are of the form

$$C_L(\theta) = \frac{\langle x(θ)|H_L|x(θ)\rangle}{\langle x(θ)|A^\dag A|x(θ)\rangle},$$

where

$$H_L = A^\dag U\left(I - \frac{1}{n}\sum_{j=1}^n|0_j\rangle\langle 0_j|\otimes I_{\overline{j}}\right)U^\dag A,$$

with U a state-preparation unitary for |b⟩ (Bravo-Prieto et al., 2019, De et al., 10 Apr 2025).

Ansatz expressibility directly determines success: for typical hardware-efficient R_y/CNOT circuits, depth as low as O(1) suffices for highly structured systems (e.g., Ising-type Hamiltonians), but generic or ill-conditioned A can demand depth O(2ⁿ/n), and even then, convergence may stall at suboptimal cost due to the restricted solution manifold (Patil et al., 2021, Turati et al., 10 Sep 2024).

Dynamic ansatz growth—where circuit depth is increased adaptively when observed improvement plateaus—reduces total resource cost (TRC) and mitigates overparameterization and barren-plateau effects, especially for high-κ or large-n cases. Reductions in TRC of up to 30% are observed for ill-conditioned families (Patil et al., 2021).

3. Measurement, Operator Decomposition, and Complexity

The quantum resource bottleneck of VQLS is dominated by the need to decompose A and A†A into efficient operator bases for cost-evaluation. The standard approach uses a Pauli-string basis,

$$A = \sum_{\ell} \alpha_\ell P_\ell ,\quad P_\ell\in\{I,X,Y,Z\}^{\otimes n},$$

yielding O(N²) terms in the worst case. However, for structured (e.g., banded, block-tridiagonal, or shift-operator based) matrices, alternate bases (e.g., sigma-basis with shift, projector, and swap gates) enable O(log N) scaling in the number of LCU terms (Gnanasekaran et al., 25 Apr 2024, Balducci et al., 6 Dec 2024), substantially reducing number of circuit executions per cost evaluation and the total measurement budget.

For example, heat equation and 1D Poisson systems with tridiagonal A require only O(N) or O(log N) circuit terms using specialized decompositions, compared to O(N²) Pauli-strings (Gnanasekaran et al., 25 Apr 2024, Balducci et al., 6 Dec 2024). This dramatically improves NISQ viability for large sparse problems.

Measurement overhead can be further reduced by Measurement Simplification, which exploits algebraic and circuit simplifications to collapse O(K²) measurement circuits to O(K), yielding up to 180× reductions in classical/quantum runtime and storage (Hahm et al., 2023).

4. Hybrid Optimization and Classical Enhancement

The classical component of VQLS includes optimization of the parameter vector θ using either gradient-free (e.g., SPSA, Powell, COBYLA, Nelder–Mead) or gradient-based (e.g., AMSGrad, BFGS, Conjugate-Gradient) methods (Pellow-Jarman et al., 2021). In the presence of realistic device noise (gate, decoherence, readout), gradient-free optimizers—especially SPSA—are empirically most robust, while methods relying on accurate direct search or conjugate gradients may degrade by 4–5× in termination cost (Pellow-Jarman et al., 2021).

Machine learning-based parameter prediction (PVLS) leverages graph neural networks trained on A and b to provide warm-start initializations for θ, reducing initial cost by 81%, halving the number of quantum-classical iterations, and providing overall runtime speedups exceeding 2.6× (Yang, 4 Dec 2025). This is particularly effective in circumventing barren plateaus in large-scale systems.

Classical preconditioning (e.g., incomplete LU) of the input system (A, b) can further halve the required circuit depth and accelerate convergence by orders of magnitude, by improving the spectral condition of the linear system and thus reducing the complexity of the required ansatz (Hosaka et al., 2023).

5. Applications and Empirical Performance

VQLS has been demonstrated in quantum machine learning (as a scalable replacement for HHL in QSVMs), quantum transport, fluid dynamics, rigid-body simulations, and nonlinear ODE/PDE integration. Canonical examples include:

• Classification with least-squares SVM: VQLS solves the normal equations arising from kernel-based SVMs, replacing HHL in LS-SVM algorithms (Yi et al., 2023). With preprocessing (SVD, normalization), three-qubit VQLS achieves 100% accuracy on Iris benchmarks for κ up to 100, with accuracy degrading only for κ ≫ 10³.
• Nonlinear physics and fluid transport: VQLS is integrated in implicit solvers for the Navier–Stokes and Stokes PDEs. Multi-ansatz tree architectures coupled with domain decomposition enable accurate solution of shock-tube tests and 2D potential flows, with qubit and circuit depth cost remaining logarithmic in system size (Liu et al., 2023, Yao et al., 28 Aug 2025).
• Quantum transport in semiconductors: By introducing hybrid and normalized-residual cost functions for complex, non-symmetric A, VQLS can solve the sequence of NEGF equations for Green's functions in device modeling, with circuit resources again scaling logarithmically with the full grid size (Yang et al., 6 Sep 2025).
• Linear complementarity problems: VQLS is used within Newton solvers for LCPs, showing competitive convergence with classical and neural analogues up through 8-qubit systems (De et al., 10 Apr 2025).
• Binary systems: The Mod2VQLS variant leverages a circuit-level XOR matvec for equations over 𝔽₂, matching the performance of the block Wiedemann algorithm for n ≤ 9 (Aboumrad et al., 2023).

Empirical studies show that for highly structured systems (e.g., Ising-like A, sparse tridiagonal), VQLS achieves solution fidelities >0.99 with O(100)–O(300) classical-quantum iterations on log₂N qubits and shallow circuit depth. For generic or poorly conditioned A, performance is limited by ansatz expressibility and measurement overhead, with convergence rates stalling for n>8 even with 10⁵ evaluations (Turati et al., 10 Sep 2024).

6. Scalability, Limitations, and Prospects

The practical viability of VQLS on NISQ devices centers around three axes:

• Ansatz Expressibility and Trainability: Hardware-efficient circuits suffice for structured A, but adaptive/deep/branched ansätze are necessary for general or ill-conditioned systems. Dynamic ansatz, multi-branch architectures, and learning-based initialization are active areas (Patil et al., 2021, Yao et al., 28 Aug 2025, Yang, 4 Dec 2025).
• Measurement and Operator Decomposition: The exponential complexity of operator decomposition (O(N²) Pauli terms) precludes black-box A. Structure-adapted bases and measurement simplification are crucial for bringing large, sparse systems into feasible scope (Gnanasekaran et al., 25 Apr 2024, Hahm et al., 2023).
• Hybrid Quantum–Classical Synergy: Preconditioning, SVD compression, feature normalization, and classical-quantum subroutine partitioning are all essential. For LS-SVMs and other machine learning integrations, careful identification and reduction of quantum-classical communication bottlenecks are necessary for performance (Yi et al., 2023).
• Resource Requirements: For an N-dimensional structured problem, VQLS routinely operates with n = log₂N qubits, ansatz depth O(1–10), and O(10³)–O(10⁴) circuit executions per cost evaluation per iteration; but total resources balloon for generic/random Hamiltonians (Hosaka et al., 2023, Gnanasekaran et al., 25 Apr 2024, Turati et al., 10 Sep 2024).

Current limitations include trainability for highly ill-conditioned/complex A, measurement cost for general (unstructured) inputs, shot limitations, and noise. Quantum advantage is forecast for cases where classical bottlenecks (e.g., the “curse of dimensionality” in NEGF transport) are mapped to logarithmic quantum scaling, under structured decomposition and optimal ansatz (Yang et al., 6 Sep 2025).

Ongoing research includes the design of adaptive ansatz architectures, scalable measurement grouping, learning-driven warm starts, improved error mitigation, and empirical benchmarking on emerging quantum hardware.

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References (by arXiv ID):

• (Bravo-Prieto et al., 2019) – Foundational VQLS: algorithm definition, scalability, and termination bounds
• (Yi et al., 2023) – VQLS-enhanced QSVM and practical NISQ implementation
• (Patil et al., 2021) – Dynamic ansatz for resource-efficient trainability
• (Pellow-Jarman et al., 2021) – Classical optimizer performance for VQLS under noise
• (Turati et al., 10 Sep 2024) – Empirical scalability and bottlenecks of VQLS
• (Yang, 4 Dec 2025) – ML-based parameter initialization for VQLS (PVLS)
• (Hosaka et al., 2023) – Preconditioning in VQLS
• (Gnanasekaran et al., 25 Apr 2024) – Efficient LCU decomposition for structured sparse matrices
• (Balducci et al., 6 Dec 2024, 2207.14630) – VQLS for tridiagonal (Poisson) and heat equation systems
• (Hahm et al., 2023) – Measurement simplification protocols
• (Liu et al., 2023, Yao et al., 28 Aug 2025) – VQLS for fluid mechanics and multi-ansatz approaches
• (Yang et al., 6 Sep 2025) – VQLS for quantum transport with complex, non-symmetric A
• (De et al., 10 Apr 2025) – VQLS for linear complementarity and hybrid algorithms
• (Aboumrad et al., 2023) – Mod2VQLS for binary systems