Quantum Interior Point Method

• Quantum interior point methods are hybrid algorithms that integrate quantum linear system solvers into classical IPM iterations for convex optimization.
• They employ techniques such as state tomography, iterative refinement, and preconditioning to ensure feasibility under inexact quantum solves.
• Recent developments extend QIPM to linear, semidefinite, and cone programming, balancing quantum speedups with challenges like conditioning and readout overhead.

Quantum interior point methods (QIPMs) are quantum or hybrid quantum-classical variants of interior point methods in which the Newton linear system, or the barrier quantities needed to define it, are accelerated by quantum subroutines. In the dominant line of work, a quantum linear system algorithm (QLSA) replaces the classical linear solver used at each iteration of a primal-dual or predictor-corrector IPM, while state tomography or related readout is used to recover a classical search direction for subsequent updates (Kerenidis et al., 2018, Casares et al., 2019). The modern literature treats QIPM not as a single algorithm but as a family of formulations for linear, semidefinite, second-order cone, and linearly constrained quadratic optimization, with the central technical issues being feasibility under inexact quantum solves, conditioning of Newton systems, tomography overhead, and the role of QRAM (Augustino et al., 2021, Mohammadisiahroudi et al., 6 Dec 2025).

1. Classical interior-point structure and quantum embedding

Interior point methods solve convex optimization problems by following a central path defined by barrier or perturbed KKT conditions. For linear programming, the canonical primal-dual pair is

$$\min_{x\in \mathbb{R}^n} \ c^T x \quad \text{s.t.}\quad Ax=b,\ x\ge 0,$$

with dual

$$\max_{(y,s)} \ b^T y \quad \text{s.t.}\quad A^T y+s=c,\ s\ge 0,$$

and each iteration requires solving a Newton system derived from the barrier equations (Mohammadisiahroudi et al., 6 Dec 2025). In classical dense regimes this linear algebra is the dominant per-iteration cost, motivating the substitution of the Newton solve by a QLSA.

Early QIPM proposals retained the classical outer IPM structure and quantized the linear algebra bottleneck. In the 2018 LP/SDP framework of Kerenidis and Prakash, the method used block-encoded linear algebra, QRAM-style data access, and tomography, with worst-case running time $$\widetilde{O}\!\left(\frac{n^{1.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$ for LPs and $$\widetilde{O}\!\left(\frac{n^{2.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$ for SDPs under the stated conditioning assumptions (Kerenidis et al., 2018). The 2019 predictor-corrector line similarly quantized the classical predictor-corrector method and emphasized that the algorithm returns a classical description of the solution vector and the value of the optimal solution, rather than only a quantum state (Casares et al., 2019).

A distinct line of research does not simply insert a QLSA into a standard Newton solve. For tall linear programs, one quantum algorithm based on the Lee-Sidford IPM approximates the Hessian and gradient directly, using spectral approximation of $A^T A$, leverage score sampling, Grover search, and multivariate mean estimation, and runs in time $$\sqrt{n}\cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$$ for $n$ inequality constraints on $d$ variables (Apers et al., 2023). This places QIPM in two partially distinct traditions: QLSA-plus-tomography hybridization of classical Newton systems, and quantum approximation of barrier primitives themselves.

2. Newton-system formulations

The choice of linear system is structurally central in QIPM because dimension, symmetry, feasibility preservation, and condition number enter directly into the quantum cost model. A common linear-programming formulation is the normal equation system (NES),

$AD^2A^T \Delta y = Ax - \beta \mu A S^{-1} e,$

with $D=S^{-1/2}X^{1/2}$, where $$\max_{(y,s)} \ b^T y \quad \text{s.t.}\quad A^T y+s=c,\ s\ge 0,$$0 and $$\max_{(y,s)} \ b^T y \quad \text{s.t.}\quad A^T y+s=c,\ s\ge 0,$$1 are diagonal matrices of the current primal and dual variables (Mohammadisiahroudi et al., 2023, Mohammadisiahroudi et al., 6 Dec 2025). NES is attractive because it yields an $$\max_{(y,s)} \ b^T y \quad \text{s.t.}\quad A^T y+s=c,\ s\ge 0,$$2 symmetric positive definite system, but its conditioning can deteriorate sharply near optimality.

To preserve feasibility under inexact quantum solves, several papers replace the conventional Newton system by formulations that restrict the search direction to feasible subspaces. In the orthogonal subspaces system (OSS), a null-space basis $$\max_{(y,s)} \ b^T y \quad \text{s.t.}\quad A^T y+s=c,\ s\ge 0,$$3 of $$\max_{(y,s)} \ b^T y \quad \text{s.t.}\quad A^T y+s=c,\ s\ge 0,$$4 is used so that $$\max_{(y,s)} \ b^T y \quad \text{s.t.}\quad A^T y+s=c,\ s\ge 0,$$5 and $$\max_{(y,s)} \ b^T y \quad \text{s.t.}\quad A^T y+s=c,\ s\ge 0,$$6, leading in linear optimization to

$$\max_{(y,s)} \ b^T y \quad \text{s.t.}\quad A^T y+s=c,\ s\ge 0,$$7

which ensures primal and dual feasibility by construction even when the linear system is solved inexactly (Mohammadisiahroudi et al., 2023). The same nullspace principle appears in semidefinite optimization, where the feasible scheme writes the search direction in nullspace and row-space coordinates so that residuals affect complementarity rather than feasibility (Augustino et al., 2021).

A later development is the modified normal equation system (MNES), which keeps the small symmetric positive definite character of NES while correcting the step so that feasibility is preserved. In this formulation the system solved at iteration $$\max_{(y,s)} \ b^T y \quad \text{s.t.}\quad A^T y+s=c,\ s\ge 0,$$8 is

$$\max_{(y,s)} \ b^T y \quad \text{s.t.}\quad A^T y+s=c,\ s\ge 0,$$9

with

$$\widetilde{O}\!\left(\frac{n^{1.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$0

and a corresponding right-hand side $$\widetilde{O}\!\left(\frac{n^{1.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$1 built from $$\widetilde{O}\!\left(\frac{n^{1.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$2, $$\widetilde{O}\!\left(\frac{n^{1.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$3, $$\widetilde{O}\!\left(\frac{n^{1.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$4, $$\widetilde{O}\!\left(\frac{n^{1.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$5, and the centering parameter $$\widetilde{O}\!\left(\frac{n^{1.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$6 (Mohammadisiahroudi et al., 2023). The paper emphasizes that this system is $$\widetilde{O}\!\left(\frac{n^{1.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$7 and symmetric positive definite, which reduces qubit and gate requirements relative to larger non-symmetric formulations.

The almost-exact framework of 2025 introduces yet another Newton system design centered on constructing and solving the dual log-barrier Newton system entirely on quantum hardware while restricting classical work to vector additions and ancillary calculations. In the review formulation this is written as

$$\widetilde{O}\!\left(\frac{n^{1.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$8

with $$\widetilde{O}\!\left(\frac{n^{1.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$9 and $$\widetilde{O}\!\left(\frac{n^{2.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$0 (Mohammadisiahroudi et al., 6 Dec 2025).

3. Inexactness, feasibility, iterative refinement, and preconditioning

The central methodological shift in QIPM occurred when the literature stopped assuming exact Newton steps. Quantum solvers are inexact, and tomography adds further error; as a result, a naive quantum replacement of a classical linear solve can violate primal or dual feasibility and invalidate the standard iteration-complexity theory. This issue is explicit in later surveys and comparative papers, which note that early quantum IPMs assumed exact Newton steps, an assumption invalid under quantum inexactness (Mohammadisiahroudi et al., 2023, Mohammadisiahroudi et al., 6 Dec 2025).

One response is the inexact infeasible QIPM (II-QIPM). In the 2022 linear optimization framework, the Newton system

$$\widetilde{O}\!\left(\frac{n^{2.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$1

is solved approximately by QLSA and quantum tomography, and the method controls residuals so that convergence of the inexact infeasible IPM is retained (Mohammadisiahroudi et al., 2022). A related semidefinite variant likewise allows infeasible iterates and mirrors classical inexact infeasible IPMs, but obtains no clear practical advantage over classical methods because quantum errors affect feasibility and complementarity simultaneously (Augustino et al., 2021).

A more influential response is the inexact feasible QIPM (IF-QIPM). In linear optimization, the OSS-based IF-IPM preserves $$\widetilde{O}\!\left(\frac{n^{2.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$2 and $$\widetilde{O}\!\left(\frac{n^{2.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$3 exactly because the search direction is parameterized inside the null space and row space of $$\widetilde{O}\!\left(\frac{n^{2.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$4, so the only error enters the complementarity condition (Mohammadisiahroudi et al., 2023). The linearly constrained quadratic optimization version makes the same point explicitly: regardless of inexactness in solving OSS, the updates maintain $$\widetilde{O}\!\left(\frac{n^{2.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$5 and dual feasibility, and the only error incurs in the complementarity condition (Wu et al., 2023). For semidefinite optimization, the feasible quantum scheme similarly uses a nullspace representation of the Newton system to guarantee feasibility even with inexact search directions (Augustino et al., 2021).

Iterative refinement (IR) is the second major stabilization device. In the linear-optimization literature, IR is used to improve the solution accuracy of an inexact quantum solve without solving a severely ill-conditioned system to extremely high precision in one shot (Mohammadisiahroudi et al., 2022, Mohammadisiahroudi et al., 2023). The 2023 MNES work states that iterative refinement can exponentially improve the dependence on precision and condition number in the overall algorithmic complexity, and its experiments report feasible precision improvements down to $$\widetilde{O}\!\left(\frac{n^{2.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$6 via nested refinement over 100 problem instances (Mohammadisiahroudi et al., 2023). The 2025 almost-exact framework pushes this further by placing iterative refinement both within IPM iterations and outside them, so that low-precision quantum solves are wrapped into an almost-exact hybrid algorithm with logarithmic precision dependence (Mohammadisiahroudi et al., 4 Dec 2025).

Preconditioning is the third major mechanism. In the preconditioned MNES framework, a basis $$\widetilde{O}\!\left(\frac{n^{2.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$7 is selected and the resulting preconditioned system can have condition number bounded by a problem-dependent constant $$\widetilde{O}\!\left(\frac{n^{2.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$8 independent of iteration count or proximity to optimality, under the stated conditions (Mohammadisiahroudi et al., 2023). In the preconditioned II-QIPM based on optimal partition estimation, the condition number of the Newton systems is improved from quadratic dependence on the reciprocal of the duality gap to linear, namely from $$\widetilde{O}\!\left(\frac{n^{2.5}}{\xi^2}\mu\kappa^3\log(1/\epsilon)\right)$$9 to $A^T A$0 (Wu et al., 2024). Taken together, these developments recast QIPM as a numerically structured algorithmic framework rather than a direct application of HHL to a Newton step.

4. Complexity landscape and asymptotic regimes

QIPM complexity results are model-dependent, and direct comparison is delicate because the literature mixes QRAM-query complexity, quantum gate complexity, tomography cost, and classical post-processing. Four recurrent regimes nevertheless appear.

Framework	Representative complexity claim	Citation
Dense LP/SDP QIPM	LP: $A^T A$1; SDP: $A^T A$2	(Kerenidis et al., 2018)
Predictor-corrector LP	$A^T A$3	(Casares et al., 2019)
Tall-LP quantum IPM	$A^T A$4	(Apers et al., 2023)
Almost-exact dense LO	$A^T A$5 QRAM queries and $A^T A$6 classical arithmetic operations	(Mohammadisiahroudi et al., 6 Dec 2025, Mohammadisiahroudi et al., 4 Dec 2025)

The earliest dense-matrix QIPMs exposed the characteristic QIPM tradeoff: improved dimension dependence paired with explicit dependence on condition number, approximation quality, and tomography precision (Kerenidis et al., 2018, Casares et al., 2019). The 2019 predictor-corrector algorithm, for example, achieves a speedup in the number of variables for dense LPs relative to classical interior-point algorithms, but its cost scales as $A^T A$7 because readout requires full classical reconstruction of the direction vector (Casares et al., 2019).

The feasible and preconditioned lines changed this balance. The OSS-based IF-QIPM for linearly constrained quadratic optimization attains the optimal feasible-IPM iteration complexity $A^T A$8, and for $A^T A$9-norm soft margin SVM the overall quantum complexity has $$\sqrt{n}\cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$$0 dependence on dimension (Wu et al., 2023). The MNES-plus-IR-plus-preconditioning framework yields quantum complexity

$$\sqrt{n}\cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$$1

with classical post-processing cost $$\sqrt{n}\cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$$2, and the paper’s comparison table reports an $$\sqrt{n}\cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$$3 system and condition-number bound $$\sqrt{n}\cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$$4 for the new method (Mohammadisiahroudi et al., 2023).

The 2025 almost-exact line shifts the comparison again by moving all matrix-vector operations to quantum hardware. In that framework the total time in QRAM queries is $$\sqrt{n}\cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$$5 and the classical time is $$\sqrt{n}\cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$$6, with the stated claim that this gives optimal worst-case scalability with respect to dimension for fully dense linear optimization (Mohammadisiahroudi et al., 6 Dec 2025). A closely related formulation reports $$\sqrt{n}\cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$$7 queries to QRAM and $$\sqrt{n}\cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$$8 classical arithmetic operations (Mohammadisiahroudi et al., 4 Dec 2025). This suggests that recent QIPM theory is increasingly concerned with eliminating classical matrix-vector bottlenecks, not only with accelerating the Newton solve itself.

5. Extensions beyond standard linear programming

Although linear programming is the canonical setting, the term QIPM also covers semidefinite, second-order cone, and linearly constrained quadratic optimization. In semidefinite optimization, two quantum IPMs were proposed in 2021: an inexact-infeasible scheme close to classical practice, and a feasible nullspace-based scheme that guarantees feasibility even with inexact search directions (Augustino et al., 2021). The same paper states that the feasible scheme obtains a speedup over classical algorithms in terms of the dimension of the problem $$\sqrt{n}\cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$$9, but has worse dependence on other numerical parameters.

A different semidefinite line uses a robust IPM framework. The 2022 robust quantum SDP algorithm quantizes a robust classical second-order method and reports high-accuracy in both optimality and feasibility, with running time depending on $n$0 on well-conditioned instances (Huang et al., 2022). Its stated significance is that prior quantum SDP solvers either had polynomial error dependence or low accuracy in feasibility, whereas the robust framework outputs a classical, feasible, high-accuracy solution (Huang et al., 2022).

For second-order cone programming, QIPM has been analyzed in an end-to-end resource model guided by portfolio optimization. That work provides a complete quantum circuit-level description of the algorithm from problem input to problem output, including logical qubits and $n$1-gate counts (Dalzell et al., 2022). It also studies feasible and infeasible QIPM variants for SOCP and gives explicit resource scaling in instance-specific parameters such as the condition number of Newton systems (Dalzell et al., 2022).

Linearly constrained quadratic optimization provides another natural extension because the KKT structure remains close to linear programming while permitting direct applications such as $n$2-norm soft margin support vector machines. The IF-QIPM for LCQO uses OSS, maps the system to a Hermitian block-encoded form, reconstructs a classical solution by quantum tomography, and claims the best known dimensional scaling for a quantum algorithm that produces a classical SVM solution (Wu et al., 2023).

6. Practical status, misconceptions, and current outlook

A persistent misconception is that a QLSA-based speedup for one linear system automatically implies a practically faster interior-point method. The literature now treats this as false for two independent reasons. First, Newton systems in IPMs become ill-conditioned near optimality unless a feasibility-preserving formulation, preconditioning, or iterative refinement is used (Mohammadisiahroudi et al., 2022, Mohammadisiahroudi et al., 2023). Second, any hybrid QIPM that requires a full classical search direction pays a tomography cost that can dominate the asymptotics (Binkowski, 27 Apr 2026).

Simulator studies do show that the newer formulations improve algorithmic behavior. The MNES paper implemented its algorithms in Python with IBM Qiskit’s HHL implementation and reports that the modified NES formulation solved an LOP with one million variables and 16 constraints on the quantum simulator; it also states that in nondegenerate problems preconditioning makes NES systems’ condition number nearly constant, while in degenerate cases iterative refinement keeps condition numbers in check by bounding the linear system size before restart (Mohammadisiahroudi et al., 2023). These results support the claim that formulation choice materially affects quantum resource usage even before hardware is considered.

The strongest practical counterpoint is the 2026 lower-bound analysis against HiGHS. Under highly benevolent assumptions—Chebyshev-based QLSA, copy-access tomography bounds, one IPM iteration, one refinement step, and a quantum cycle duration around the current speed record of $n$3—the lower bounds already exceed the actual classical runtimes across all tested LP instances and for both MNES and OSS (Binkowski, 27 Apr 2026). The paper identifies tomography as the main bottleneck and states that extracting a $n$4-dimensional, amplitude-encoded classical solution requires at least $n$5 repetitions of the QLSA (Binkowski, 27 Apr 2026).

The circuit-level SOCP resource analysis reaches a similar conclusion from a different angle. For a moderate, classically-easy portfolio optimization instance with $n$6, it reports approximately $n$7 million logical qubits, $n$8 $n$9-gates, $d$0 $d$1-depth, and $d$2 quantum circuit repetitions (Dalzell et al., 2022). This does not negate the asymptotic theory, but it sharply limits claims of near-term utility.

The current outlook is therefore bifurcated. On the theoretical side, QIPM has developed from exact-step proposals into a numerically aware family of methods with feasibility maintenance, iterative refinement, preconditioning, and almost-exact hybridization (Mohammadisiahroudi et al., 2023, Wu et al., 2024, Mohammadisiahroudi et al., 6 Dec 2025). On the practical side, resource estimates and lower bounds indicate that tomography, conditioning, QRAM assumptions, and constant factors remain decisive obstacles for stand-alone classical-output solvers (Dalzell et al., 2022, Binkowski, 27 Apr 2026). A plausible implication is that future progress will depend either on architectures that avoid repeated full classical readout, or on use cases in which only limited global properties of the quantum output are required rather than the full interior-point iterate.