Quantum Linear System Algorithms

• Quantum linear system (QLS) algorithms are quantum procedures that output a normalized state proportional to the solution of a linear system, leveraging oracles and block-encoding techniques.
• They encompass various families such as HHL, LCU-based solvers, QSVT methods, adiabatic schemes, and variational approaches that offer improved query complexities.
• QLS algorithms are crucial in applications like PDE discretizations, optimization, and quantum machine learning, addressing challenges including condition number barriers and quantum preconditioning.

Quantum linear system (QLS) algorithms are quantum algorithms for preparing a quantum state proportional to the solution of a linear system, rather than outputting the classical solution vector itself. In the standard Hermitian formulation used repeatedly in the recent literature, the task is: given Hermitian $A \in \mathbb{C}^{N\times N}$ with $\|A\|=1$, condition number $\kappa$, sparsity $s$, a vector $b\in\mathbb{C}^N$, and oracle access to $A$ and $b$, output a normalized state $|\tilde x\rangle$ such that $$\big\||\tilde x\rangle-|x^\star\rangle\big\| \le \varepsilon$$, where $x^\star=A^{-1}b$ (Kim et al., 2024). The modern literature spans HHL, LCU-based solvers, QSVT-based solvers, adiabatic-family methods, eigenstate-filtering and resonant-transition constructions, optimization-based schemes, and hybrid variational approaches; a recent survey characterizes this landscape as a progression from the Harrow–Hassidim–Lloyd solver toward post-HHL enhancements and optimal lower bounds with respect to error tolerance and condition number, with applications in differential equations, quantum machine learning, and many-body physics (Morales et al., 2024).

1. Formal problem statement and access models

A central distinction in QLS is between the classical linear system problem $\|A\|=1$0 and the quantum linear system problem, whose output is the normalized solution state

$\|A\|=1$1

In the formulation used by the catalyst framework, the input consists of a Hermitian matrix $\|A\|=1$2 with $\|A\|=1$3, condition number $\|A\|=1$4, sparsity $\|A\|=1$5, a vector $\|A\|=1$6, and oracles for state preparation of $\|A\|=1$7 together with either sparse access or a block-encoding of $\|A\|=1$8; the goal is to output a normalized state $\|A\|=1$9 satisfying $\kappa$0 (Kim et al., 2024). A recurring implication is that QLS algorithms are naturally suited to downstream quantum subroutines, while full classical recovery of all components of $\kappa$1 is a separate task with its own overhead.

Two oracle models dominate the literature. In the sparse-access model, one has black-box access to nonzero locations and values. In the block-encoding model, a unitary $\kappa$2 is an $\kappa$3-block-encoding of $\kappa$4 if

$\kappa$5

Sparse access can be turned into a block encoding with $\kappa$6 using a constant number of oracle calls and polylog overhead (Kim et al., 2024). This equivalence is one reason QSVT- and qubitization-based methods have become central.

Several papers also widen the problem class beyond the basic Hermitian setting. The momentum-accelerated Schrödingerization framework formulates the target as either $\kappa$7 or the normal equations

$\kappa$8

and interprets “generalized linear systems” to include augmented dynamical systems and PDE discretizations such as Helmholtz and biharmonic equations (Hu et al., 20 Sep 2025). This suggests that the QLS label now covers a broader family of linear-algebraic formulations than direct inversion alone.

2. Main algorithmic families and representative complexity scalings

The historical baseline is HHL, followed by LCU-based refinements, QSVT-based inversion, and adiabatic-family solvers. A compact summary of representative scalings reported in recent comparative and survey-style works is given below.

Family	Representative complexity	Source
HHL	$\kappa$9	(Kim et al., 2024)
Ambainis VTAA	$s$0	(Kim et al., 2024)
CKS (LCU + Chebyshev)	$s$1	(Kim et al., 2024)
Costa et al. (discrete adiabatic theorem)	$s$2	(Kim et al., 2024)

Within the narrower class of functional QLS—algorithms that directly implement an approximate matrix inversion function—the comparison paper studies four methods under the same sparse-access input model: HHL, QLS-Fourier, QLS-Chebyshev, and a QSVT-based solver (Lefterovici et al., 27 Mar 2025). In that setting, HHL retains a $s$3 dependence, whereas QLS-Chebyshev and QSVT have $s$4 dependence through polynomial approximation. The same paper reports that HHL underperforms in comparison to the other methods across all data sets, often by orders of magnitude, while the QSVT-based method shows the best performance (Lefterovici et al., 27 Mar 2025). A plausible implication is that “best asymptotic worst-case complexity” and “best instance-level query count” are not interchangeable notions in QLS.

The benchmarking evidence is especially important because several families share similar asymptotic dependence in $s$5 and $s$6 but differ sharply in constants, oracle counts, and amplification overhead. QLS-Fourier uses many time-evolution unitaries $s$7, QLS-Chebyshev implements Chebyshev polynomials through a quantum walk oracle, and QSVT applies a polynomial approximation of $s$8 directly to a block-encoding of $s$9 (Lefterovici et al., 27 Mar 2025). The resulting practical ranking—QSVT first, Chebyshev next, Fourier behind, HHL last—has become one of the clearest empirical syntheses currently available in the functional-QSL literature.

3. Positive-definite structure, condition-number barriers, and preconditioning

A persistent question is whether positive-definite structure permits a quantum analogue of the classical $b\in\mathbb{C}^N$0 improvement familiar from conjugate gradient. In the standard black-box setting, the answer is negative. For PD-QLS with access to $b\in\mathbb{C}^N$1 via state preparation and to $b\in\mathbb{C}^N$2 via either sparse oracle or normalised block-encoding, any solver achieving constant precision has query complexity $b\in\mathbb{C}^N$3; in particular, for $b\in\mathbb{C}^N$4, one needs $b\in\mathbb{C}^N$5 queries (Orsucci et al., 2021). This corrects the common intuition that positive-definite structure alone should automatically reduce the quantum $b\in\mathbb{C}^N$6-dependence.

The same work nevertheless identifies broad structured subclasses where the lower bound can be circumvented. One route assumes a normalised block-encoding of

$b\in\mathbb{C}^N$7

for suitable $b\in\mathbb{C}^N$8, and then approximates $b\in\mathbb{C}^N$9 on the spectrum of $A$0 rather than $A$1 on the spectrum of $A$2. The resulting polynomial degree is $A$3, and in favorable overlap regimes the full PD-QLS solver attains $A$4 dependence (Orsucci et al., 2021). A second route, for Sum-QLS instances $A$5, constructs

$A$6

and uses a pseudo-inverse solver for $A$7, giving gate complexity $A$8, where $A$9 quantifies overlap with the relevant support (Orsucci et al., 2021).

Preconditioning remains central, but its quantum status is subtler than its classical counterpart. The catalyst framework imports the proximal point algorithm and replaces $b$0 by repeated inversion of $b$1, producing the effective condition number

$b$2

For suitable $b$3, one-step PPA yields $b$4, giving a constant-factor speedup for any wrapped QLSP solver while preserving the target solution state (Kim et al., 2024). By contrast, the 3D heterogeneous Poisson analysis proves that block encoding the system matrix and preconditioner separately does not improve the effective condition number that dominates the QLS runtime: if $b$5 and $b$6 are separate block encodings, then the composite effective condition number satisfies $b$7 (Pechan et al., 10 Aug 2025). A common misconception is therefore that classical preconditioning transfers unchanged to block-encoding-based QLS; the recent literature indicates that this is not generally the case.

This tension is historically significant. The early “Preconditioned quantum linear system algorithm” integrated a quantum-compatible sparse approximate inverse into an HHL-style pipeline (Clader et al., 2013). Later work shows that modern block-encoding runtimes are governed by an effective condition number that is not automatically improved by separately encoding a classical preconditioner (Pechan et al., 10 Aug 2025). This suggests that quantum preconditioning must often be designed at the level of the quantum representation itself rather than imported as a purely classical preprocessing step.

4. Reformulations beyond direct matrix inversion

A substantial fraction of current QLS research avoids direct approximation of $b$8 and instead reformulates the problem as filtering, dynamics, or iterative block embeddings. One example transforms $b$9 into the problem of finding the right singular vector with singular value zero of an augmented matrix

$|\tilde x\rangle$0

or, equivalently, the zero-eigenvalue eigenstate of

$|\tilde x\rangle$1

This yields two algorithms: a quantum eigenstate filtering method with query complexity $|\tilde x\rangle$2, and a quantum resonant transition method with query complexity $|\tilde x\rangle$3 (Wang et al., 2022).

Another line embeds classical stationary iterations into a larger block system. The qRLS method encodes the whole relaxation history into a single block linear system

$|\tilde x\rangle$4

whose condition number scales as $|\tilde x\rangle$5, independent of the dimension $|\tilde x\rangle$6 and the condition number $|\tilde x\rangle$7 of the original FEM system (Raisuddin et al., 2023). Solving that block system with a QSP-based QLSA gives overall complexity

$|\tilde x\rangle$8

with $|\tilde x\rangle$9 qubits (Raisuddin et al., 2023). This is a distinct paradigm: the quantum solver is applied not to the original operator $$\big\||\tilde x\rangle-|x^\star\rangle\big\| \le \varepsilon$$0, but to an iteration-encoding surrogate with explicitly controlled conditioning.

Schrödingerization-based methods follow a related dynamical route. The MAG algorithm starts from the momentum accelerated gradient iteration

$$\big\||\tilde x\rangle-|x^\star\rangle\big\| \le \varepsilon$$1

rewrites it as a stable linear ODE on an enlarged state, and then maps that ODE to Hamiltonian evolution through Schrödingerization (Hu et al., 20 Sep 2025). The resulting quantum algorithm has query complexity

$$\big\||\tilde x\rangle-|x^\star\rangle\big\| \le \varepsilon$$2

and the paper emphasizes linear dependence on the condition number up to polylogarithmic factors while avoiding VTAA and RM overheads (Hu et al., 20 Sep 2025). The same paper argues that carefully chosen auxiliary variables are essential because, in the damped-system alternative, auxiliary variables can dominate the error measure and degrade practical convergence.

Optimization-based and variational schemes form a further branch. A 2025 proposal treats $$\big\||\tilde x\rangle-|x^\star\rangle\big\| \le \varepsilon$$3 as minimizing

$$\big\||\tilde x\rangle-|x^\star\rangle\big\| \le \varepsilon$$4

and performs coherent gradient descent in a density-state-like representation $$\big\||\tilde x\rangle-|x^\star\rangle\big\| \le \varepsilon$$5, giving a final complexity

$$\big\||\tilde x\rangle-|x^\star\rangle\big\| \le \varepsilon$$6

with no explicit dependence on $$\big\||\tilde x\rangle-|x^\star\rangle\big\| \le \varepsilon$$7 in the stated bound (Nghiem, 19 Feb 2025). At the NISQ-oriented end of the spectrum, the Shadow Quantum Linear Solver combines a VQLS-style local cost with classical shadows, avoids large controlled unitaries, and uses $$\big\||\tilde x\rangle-|x^\star\rangle\big\| \le \varepsilon$$8 qubits with no ancilla for cost evaluation (Ghisoni et al., 2024). These approaches do not displace fault-tolerant QSVT-style solvers, but they demonstrate that “QLS algorithm” is now a genuinely heterogeneous category.

5. Applications across scientific computing, optimization, and quantum algorithms

The most concrete application domain in the recent literature is PDE discretization. In finite element analysis, qRLS is demonstrated on FEM Poisson problems in 1D and 2D, with the entire iterative process encoded into a well-conditioned block system and solved using a QSP-based QLSA (Raisuddin et al., 2023). In tensor-structured scientific computing, an adiabatic-inspired solver for

$$\big\||\tilde x\rangle-|x^\star\rangle\big\| \le \varepsilon$$9

shows that under $x^\star=A^{-1}b$0 and $x^\star=A^{-1}b$1 the total complexity of the implementation is polylogarithmic in the dimension $x^\star=A^{-1}b$2, matching the per-iteration complexity of classical heuristic tensor methods while providing a full gate-model analysis (Wu et al., 2024).

For PDEs with difficult coefficient structure, the 3D heterogeneous Poisson study is especially instructive. After finite-difference discretization, the matrix $x^\star=A^{-1}b$3 is sparse with at most 7 nonzeros per row/column, and a block encoding of the rescaled operator $x^\star=A^{-1}b$4 is constructed with cost $x^\star=A^{-1}b$5 under structured heterogeneity assumptions (Pechan et al., 10 Aug 2025). A discrete Poincaré argument gives $x^\star=A^{-1}b$6, so the effective condition number scales as $x^\star=A^{-1}b$7, yielding QLS runtime

$x^\star=A^{-1}b$8

and exponential memory savings relative to classical storage of the full 3D system (Pechan et al., 10 Aug 2025). The same paper uses groundwater flow through geologic fracture networks as the motivating example, making it one of the clearest end-to-end feasibility analyses for QLS in real scientific computing.

Optimization is another major outlet. In an inexact infeasible quantum interior point method for linear optimization, the only quantum subroutine is the solution of modified normal equations via a QLSA plus quantum tomography, embedded inside a rigorously analyzed classical IPM loop (Mohammadisiahroudi et al., 2022). Iterative refinement is then used so that the II-QIPM solves only moderate-precision subproblems, avoiding exponential dependence on final accuracy and yielding overall polynomial complexity under the hybrid quantum–classical model (Mohammadisiahroudi et al., 2022). This is representative of a broader pattern: in optimization, QLS often appears not as the entire algorithm but as the Newton-step engine.

The application frontier also includes wave and higher-order PDEs. The MAG–Schrödingerization paper studies 1D and 2D Helmholtz equations and 1D and 2D biharmonic equations, both with zero and non-zero boundary conditions, and reports that the “MAG solution”, the Schrödingerized solution, and the analytical (or high-fidelity numerical) reference solution coincide within numerical precision (Hu et al., 20 Sep 2025). At a broader level, the survey literature continues to identify differential equations, quantum machine learning, and many-body physics as the canonical application triad for QLS algorithms (Morales et al., 2024).

6. Lower bounds, benchmarking, and practical assessment

The modern lower-bound picture is now substantially sharper. For constant sparsity, the query complexity of QLS in the sparse-access model is

$x^\star=A^{-1}b$9

and for constant error the sparsity-dependent lower bound is

$\|A\|=1$00

The rigorous $\|A\|=1$01 bound settles a long-standing folklore expectation about sparsity dependence, while the joint $\|A\|=1$02 dependence remains open (Mori et al., 23 Jan 2026). These results imply that improvements in QLS are now constrained less by asymptotic exponents in $\|A\|=1$03 and more by access models, constants, structure, and problem reformulations.

Benchmarking has correspondingly moved beyond asymptotic scaling. Instance-based comparisons of functional QLS under a common sparse-access model find that HHL is consistently dominated, often by orders of magnitude, whereas the QSVT-based method shows the best performance on random instances, simplex-derived linear systems, and Poisson equations (Lefterovici et al., 27 Mar 2025). This is an important corrective to the common pedagogical centrality of HHL: historically foundational, but not competitive as a practical functional solver according to current query-count estimates.

At the same time, NISQ-oriented work points in a different direction. SQLS does not compete with fault-tolerant QSVT asymptotically, but it replaces Hadamard-test evaluation by classical shadows, removes the ancilla needed by VQLS cost estimation, and gives a number of circuits per cost-function evaluation

$\|A\|=1$04

to be contrasted with the VQLS scaling

$\|A\|=1$05

for a simplified Pauli-sum setting (Ghisoni et al., 2024). The paper’s numerical experiments indicate comparable or slightly better convergence in number of cost evaluations, together with an exponential advantage of SQLS in circuit execution per cost function evaluation (Ghisoni et al., 2024).

Several controversies or recurring misconceptions can therefore be stated precisely. Positive-definite structure does not, in general, imply $\|A\|=1$06 quantum complexity (Orsucci et al., 2021). Classical preconditioning does not automatically improve the effective condition number in block-encoding-based QLS (Pechan et al., 10 Aug 2025). HHL remains historically central, but QSVT-based functional solvers are the stronger practical candidate in current query-count studies (Lefterovici et al., 27 Mar 2025). And broad claims of “exponential quantum advantage” for QLS are usually conditional on oracle assumptions, state preparation, and information-extraction requirements, which remain the decisive practical bottlenecks in scientific applications (Pechan et al., 10 Aug 2025).