Quantum Linear Solver Advances

• Quantum linear solvers are quantum algorithms that encode the solution of A⃗x = b⃗ into a quantum state, potentially achieving exponential speedups over classical methods.
• They employ methodologies such as phase estimation, adiabatic evolution, and variational optimization to address sparsity, error tolerance, and condition number challenges.
• Recent advances integrate classical-quantum hybrid techniques and resource-efficient strategies, broadening applications in simulation, machine learning, and scientific computing.

A quantum linear solver is a quantum algorithmic framework designed to find or encode the solution to a linear system of equations $A\vec{x} = \vec{b}$ as a quantum state, typically with query and circuit complexity that fundamentally outperforms the best known classical methods under specific assumptions. The emergence of quantum linear solvers—most notably initiated by the Harrow-Hassidim-Lloyd (HHL) algorithm—has led to numerous generalizations and theoretical advancements, each addressing different aspects of efficiency, robustness, or applicability across scientific and engineering domains. Quantum linear solvers are now categorized into several paradigm classes, with practical and performance trade-offs influenced by error tolerance, sparsity, condition number, and target computational regime.

1. Foundational Algorithms and Historical Context

The HHL algorithm established the possibility of achieving an exponential speedup in solving large, sparse linear systems by preparing a state %%%%1%%%% using quantum phase estimation, controlled rotation, and measurement/postselection. For a sufficiently sparse $N \times N$ Hermitian matrix $A$ with condition number $\kappa$ and sparsity $s$, the run time is $$\mathcal{O}(\log N\cdot s^2\cdot \kappa^2/\epsilon)$$ to achieve accuracy $\epsilon$ in estimation of expectation values, a scaling unattainable for dense or ill-conditioned systems using classical solvers (Cai et al., 2013). Subsequent work extended experimental demonstrations to superconducting processors with fidelity characterization (Zheng et al., 2017), showcased minimal-qubit protocols for single-variable extraction (Doronin et al., 2019), and highlighted quantum resource trade-offs.

The next wave of research introduced new algorithmic templates: adiabatic and discrete-adiabatic QLS solvers (An et al., 2019, Costa et al., 2021), variational quantum linear solvers (VQLS) (Bravo-Prieto et al., 2019), Krylov-based and LCU/Chebyshev polynomial methods (Xu et al., 10 May 2024, Lefterovici et al., 27 Mar 2025), classical-quantum hybrid methods (Chen et al., 2019, Saito et al., 2023), and resource-efficient shadow or neural approaches for NISQ and post-NISQ scenarios (Ghisoni et al., 13 Sep 2024, De et al., 10 Apr 2025).

The field has also seen the development of optimal-scaling algorithms—those whose complexity is linear in $\kappa$ and polylogarithmic in $1/\epsilon$, matching known lower bounds (Costa et al., 2021, Jennings et al., 2023). Systematic surveys categorize these methods by functional principle, oracle usage, and performance guarantees (Morales et al., 4 Nov 2024, Lefterovici et al., 27 Mar 2025).

2. Algorithmic Principles and Methodologies

2.1. HHL Algorithm and Variants

The HHL framework proceeds through:

• Phase Estimation: Encodes eigenvalues $\lambda_j$ of $A$, prepared in the eigenbasis expansion $|b\rangle = \sum_j \beta_j |u_j\rangle$ (where $A|u_j\rangle = \lambda_j|u_j\rangle$).
• Conditional Rotation: Implements the operation $\sum_j \beta_j |u_j\rangle|1\rangle (C/\lambda_j)$, introducing $A^{-1}$ in amplitude.
• Uncomputation and Postselection: Reverse phase estimation disentangles the register, and postselection on the ancilla yields $|x\rangle \propto A^{-1}|b\rangle$ up to normalization.

Generalizations include experimental resource optimizations (e.g. minimal use of entangling gates (Cai et al., 2013)), high-fidelity realization on superconducting circuits (Zheng et al., 2017), and single-component solvers via embedded unitaries under norm constraints (Doronin et al., 2019).

2.2. Adiabatic and QAOA-based Methods

Adiabatic methods interpolate between an initial ${H}_0$ and target ${H}_1$ Hamiltonian, engineering a schedule $f(s)$ such that the system evolves into a state encoding $A^{-1}|b\rangle$. Optimally tuned schedules enable complexity scaling as $$\mathcal{O}(\kappa \cdot \text{polylog}(\kappa/\epsilon))$$ (An et al., 2019), with rigorous discrete adiabatic error bounds and filtering techniques achieving strictly optimal $\mathcal{O}(\kappa \log(1/\epsilon))$ scaling (Costa et al., 2021, Jennings et al., 2023). QAOA can approximate the adiabatic trajectory with a sequence of parameterized unitaries, matching the optimal complexity under suitable angle schedules.

Key formulae include

$$H(s) = (1-f(s)) H_0 + f(s) H_1, \quad \text{where} \;\; f'(s) \propto (\text{gap}(s))^p.$$

Filtering the final state using QSP polynomials or Chebyshev windows exponentially suppresses error components outside the desired subspace.

2.3. Variational Quantum Linear Solvers (VQLS) and NISQ Optimizations

Variational quantum algorithms prepare the solution $|x\rangle \approx V(\alpha)|0\rangle$ by minimizing a cost function such as

$$C = 1 - \frac{|\langle b |A|x\rangle|^2}{\langle x|A^{\dagger}A|x\rangle}$$

or related Rayleigh quotients and local projectors (Bravo-Prieto et al., 2019). Shallow hardware-efficient ansatzes and cost-specific Hadamard tests enable scaling up to $1024\times1024$ systems, with termination criteria $C \gtrsim \epsilon^2/\kappa^2$ providing an operational stopping rule. Resource and trainability improvements arise from dynamic ansatz layering (Patil et al., 2021) and alternate operator bases exploiting sparsity (Gnanasekaran et al., 25 Apr 2024). VQLS is particularly suitable for noisy intermediate-scale quantum (NISQ) devices due to its shallow circuit depth and hybrid classical-quantum optimization loop.

2.4. Krylov Subspace and Polynomial/LCU Methods

Krylov-subspace techniques approximate the solution as a linear combination of time-evolved basis states $|\phi_n\rangle = e^{-iH n\tau}|b\rangle$, with coefficients found by solving a reduced classical system. This reduces redundancy compared to the Fourier/LCU function-based inversion and improves scaling in $\kappa$ and $\epsilon$ (Xu et al., 10 May 2024, Lefterovici et al., 27 Mar 2025).

Other approaches implement $A^{-1}$ as polynomial (Chebyshev, Fourier, QSVT) or LCU approximants, often with amplitude amplification, and perform best in query complexity and circuit depth when benchmarked on realistic instances (Morales et al., 4 Nov 2024, Lefterovici et al., 27 Mar 2025).

2.5. Hybrid and Modulo-Specific Techniques

Hybrid solvers combine classical and quantum modules—such as using a classical Neumann series expansion and quantum random walk sampling with $O(\log N)$ qubits (Chen et al., 2019), or multi-resolution quantum measurement to reduce sampling cost to $O(n\log(1/\epsilon))$ (Saito et al., 2023). For modulo-2 systems, specialized variational circuits (such as a product of single-qubit $R_y(\theta)$ rotations) and binary matrix–vector product implementations provide efficient solution pipelines and show linear scaling in resources (Aboumrad et al., 2023).

3. Performance, Complexity, and Benchmarking

Algorithm Class	Key Complexity	Error/Precision Scaling
HHL-type	$\mathcal{O}(\log N\; s^2\; \kappa^2/\epsilon)$	Linear in $1/\epsilon$
Adiabatic/Discrete AQC	$\mathcal{O}(\kappa\; \log(1/\epsilon))$	Optimal in $\kappa$, log in $1/\epsilon$
Krylov-LCU	$$\mathcal{O}(d\kappa \log(1/\epsilon) \log(\kappa/\epsilon))$$	Nearly optimal
Variational (VQLS/SQLS)	Polylog in $N$, linear/sublinear in $\kappa$	Empirically log or linear in $1/\epsilon$
Hybrid/Random Walk	$O(\log(N))$ time, $O(\log(N))$ qubits per step	Robust to NISQ noise
Modulo-2 VQLS	$\mathcal{O}(N)$ gates for $N$ nonzeros	Linear scaling, no exponentials

Sophisticated benchmarking frameworks have shown that HHL is generally outperformed by polynomial and QSVT-based methods on practical datasets, such as those arising from PDE discretization, linear programming, and even random matrices when evaluated for query complexity and resource overhead, while the precise scaling constants and pre-factors remain an area of active research (Lefterovici et al., 27 Mar 2025). Implementations employing efficient circuit synthesis and block encoding, especially for structured sparse matrices, further reduce practical gate counts (Goldfriend et al., 29 Jul 2025, Gnanasekaran et al., 25 Apr 2024).

4. Applications and Impact

Quantum linear solvers form a foundational computational primitive for quantum simulation, data fitting, differential equations, and quantum machine learning. Notable applications include:

• Partial Differential Equations: Discretized PDEs, such as heat or Vlasov–Ampère equations, can be mapped to high-dimensional sparse linear systems amenable to quantum inversion, with numerically demonstrated resource advantages via optimized block encoding synthesis (Goldfriend et al., 29 Jul 2025, Gnanasekaran et al., 25 Apr 2024).
• Quantum Machine Learning: Fast solution of kernel methods (support vector machines), regression tasks, and PCA in the quantum access model, exploiting the polylogarithmic scaling for high-dimensional vector spaces (Yi et al., 2023).
• Control and Optimization: Linear complementarity problems in rigid body simulations leveraged VQLS and dequantized variants for efficient collision resolution (De et al., 10 Apr 2025).
• Quantum Chemistry and Many-body Physics: Accurate Green’s function estimation for interacting systems, leveraging QSVT or Krylov-based linear solvers (Morales et al., 4 Nov 2024).
• Modular Arithmetic and Cryptography: Binary-valued (mod 2) systems solved efficiently by specialized quantum circuits, with implications for cryptanalysis and integer factorization workflows (Aboumrad et al., 2023).

5. Resource Optimization and NISQ Realization

For near-term quantum hardware, circuit depth, qubit count, and measurement overhead are primary bottlenecks. Variational and hybrid algorithms, particularly those using classical-quantum resource separation and classical shadows for expectation estimation, allow shallow circuits and reduced qubit/measurement loads (Ghisoni et al., 13 Sep 2024). Dynamic ansatz strategies and basis selection (Pauli vs. structured sigma) further optimize gate counts and mitigate noise accumulation (Patil et al., 2021, Gnanasekaran et al., 25 Apr 2024). High-level synthesis environments (e.g., Qmod, Classiq) allow for automated optimization and mapping of linear solver primitives to underlying hardware constraints, with orders-of-magnitude reductions in two-qubit gate counts empirically demonstrated (Goldfriend et al., 29 Jul 2025).

6. Challenges and Future Directions

Open challenges remain in:

• Developing mechanisms that are robust to noise and hardware errors while retaining theoretical scaling advantages.
• Generalizing quantum linear solvers to non-Hermitian, dense, or poorly conditioned systems and integrating error mitigation or adaptive preconditioning.
• Benchmarking across broader classes of real-world problems and incorporating quantum error-correction overheads into resource modeling.
• Extending efficient decomposition strategies (e.g., optimized polynomial approximants or tailored LCU bases) and integrating with state preparation oracles for arbitrary input vectors.
• Enabling hybrid quantum–classical pipelines, where quantum modules solve the dominant computational bottleneck within a larger classical framework.

Recent results point toward a convergence between theoretical optimality (e.g., QSVT-based or adiabatic solvers reaching $\mathcal{O}(\kappa\log(1/\epsilon))$ complexity) and practical hardware feasibility via variational, shadow, and hybrid methods (Morales et al., 4 Nov 2024, Ghisoni et al., 13 Sep 2024, Lefterovici et al., 27 Mar 2025). Scientific and engineering domains with structured sparse systems, high-dimensional simulation tasks, and large-scale machine learning remain primary targets for quantum linear solver deployment, provided that efficient oracular access and suitable preconditioning mechanisms are in place.