Inexact Quantum Linear Programming Solver

• The paper introduces a hybrid inexact quantum approach that uses QLSA to compute approximate Newton directions in linear programming.
• It demonstrates quadratic convergence with an iteration bound of O(√n log(nμ₀/ζ)) by controlling approximation errors.
• The framework integrates iterative refinement and explicit error management to preserve feasibility and bolster computational efficiency.

An inexact quantum linear programming solver is a hybrid algorithmic framework for linear optimization in which the Newton systems arising from a barrier or interior-point method are solved by a Quantum Linear System Algorithm (QLSA), while the resulting search direction is treated as approximate because quantum state preparation, finite-precision inversion, and tomography introduce error. In "A quantum dual logarithmic barrier method for linear optimization," this framework is instantiated as an inexact-feasible quantum variant of the dual logarithmic barrier method (DLBM): a QLSA is used at each barrier iteration, the algorithm converges quadratically toward the central path with inexact directions, and it retains the best-known $\mathcal{O}(\sqrt{n}\log(n\mu_0/\zeta))$ iteration complexity for the stated setting (Wu et al., 2024).

1. Standard linear program and dual barrier geometry

The underlying optimization problem is the standard primal-dual pair

$$(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,$$

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

For a barrier parameter $\mu>0$, the dual logarithmic barrier is

$$\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).$$

Writing $S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$, its gradient and Hessian are

$$\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1}, \qquad \nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.$$

This formulation places the linear-algebraic core of the method in the symmetric positive-definite matrix $A S^{-2} A^T$. In the DLBM, the iterates are organized around the dual central path through the barrier parameter $\mu$ and the slack matrix $S$; consequently, the quality of each Newton solve directly determines whether the method preserves positivity and retains fast local convergence (Wu et al., 2024).

2. Exact dual logarithmic barrier method

At a dual iterate $$(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,$$0, the residual is

$$(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,$$1

The exact Newton system for $$(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,$$2 is the normal-equation form

$$(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,$$3

Once $$(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,$$4 is obtained, the slack correction is

$$(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,$$5

With the full Newton update

$$(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,$$6

the local progress measure

$$(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,$$7

satisfies

$$(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,$$8

provided that $$(\mathrm{P})\quad \min_x c^T x \;\;\text{s.t.}\;\; Ax=b,\;x\ge 0,$$9. The exact DLBM therefore exhibits quadratic contraction of the proximity measure. One shows that $$(\mathrm{D})\quad \max_{y,s} b^T y \;\;\text{s.t.}\;\; A^T y+s=c,\;s\ge 0.$$0 iterations suffice for gap at most $$(\mathrm{D})\quad \max_{y,s} b^T y \;\;\text{s.t.}\;\; A^T y+s=c,\;s\ge 0.$$1 (Wu et al., 2024).

This exact scheme is the benchmark against which the inexact quantum version is measured. The central question is whether the quadratic local behavior and short-step iteration bound survive once $$(\mathrm{D})\quad \max_{y,s} b^T y \;\;\text{s.t.}\;\; A^T y+s=c,\;s\ge 0.$$2 is no longer computed exactly.

3. Inexact-feasible variant and preservation of positivity

In the quantum setting, the Newton direction is only available approximately. The DLBM is therefore reformulated as an inexact-feasible method by writing

$$(\mathrm{D})\quad \max_{y,s} b^T y \;\;\text{s.t.}\;\; A^T y+s=c,\;s\ge 0.$$3

The key feasibility result is that positivity and quadratic local convergence persist under explicit inexactness bounds. In particular, Theorem 3.1 states that if

$$(\mathrm{D})\quad \max_{y,s} b^T y \;\;\text{s.t.}\;\; A^T y+s=c,\;s\ge 0.$$4

then

$$(\mathrm{D})\quad \max_{y,s} b^T y \;\;\text{s.t.}\;\; A^T y+s=c,\;s\ge 0.$$5

and

$$(\mathrm{D})\quad \max_{y,s} b^T y \;\;\text{s.t.}\;\; A^T y+s=c,\;s\ge 0.$$6

For global complexity, the algorithm adopts

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

Then Theorem 3.3 shows that the inexact-feasible DLBM still terminates in

$$(\mathrm{D})\quad \max_{y,s} b^T y \;\;\text{s.t.}\;\; A^T y+s=c,\;s\ge 0.$$8

full Newton-step iterations, achieving a duality gap

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

(Wu et al., 2024).

This establishes the defining feature of an inexact quantum LP solver in the DLBM setting: the approximation error is not treated as an implementation nuisance but as a first-class analytical object. A plausible implication is that the viability of the quantum acceleration hinges less on abstract QLSA asymptotics alone than on the compatibility between approximation error, feasibility, and barrier-path contraction.

4. Quantum linear-system subroutine and tomography-induced error

Each barrier iteration requires solving

$\mu>0$0

The quantum subroutine proceeds by block-encoding $\mu>0$1 in QRAM as a unitary $\mu>0$2 and applying QSVT, in the Childs-Kothari-Somma style, to solve the Hermitian system

$\mu>0$3

The quantum solver outputs a normalized state $\mu>0$4 with error $\mu>0$5 in $\mu>0$6-norm. This state is then rescaled through

$\mu>0$7

to form the approximate classical direction

$\mu>0$8

A final tomography step converts $\mu>0$9 into a classical vector within error $$\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).$$0.

Tomography is analytically nontrivial because it feeds directly into the slack error. To maintain feasibility, the paper imposes

$$\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).$$1

which guarantees

$$\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).$$2

Theorem 4.6 gives the QRAM-query complexity of each QLSA-plus-tomography call as

$$\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).$$3

with classical overhead $$\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).$$4 per iteration (Wu et al., 2024).

The dependence on tomography error is central. This suggests that the solver is intrinsically hybrid: the quantum component prepares an approximate direction efficiently in amplitude form, but the classical optimization loop ultimately depends on extracting that direction with enough fidelity to satisfy a barrier-method neighborhood condition.

5. Iterative refinement and end-to-end complexity

A direct dependence on the target accuracy $$\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).$$5 can enter through the conditioning of $$\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).$$6. To avoid a linear $$\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).$$7 factor in $$\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).$$8, the algorithm embeds the low-precision quantum IPM inside an iterative-refinement outer loop.

At iterative-refinement step $$\mathcal{L}(y;\mu)=-b^T y-\mu\sum_{i=1}^n \log\!\bigl(c_i-(A^T y)_i\bigr).$$9, one solves a refining dual LP

$S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$0

with barrier parameter

$S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$1

to constant accuracy $S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$2. The resulting increment is scaled back by $S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$3 and used to update the main iterate. By Lemmas 3.9–3.11, each iterative-refinement call requires $S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$4 quantum-IPM steps, and only

$S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$5

outer iterative-refinement calls are needed to reach final tolerance $S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$6 (Wu et al., 2024).

Combining the inexact-feasible DLBM analysis with iterative refinement yields the total QRAM-query complexity

$S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$7

where

$S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$8

is the condition number of the initial Newton system. The classical arithmetic cost is

$S=\mathrm{Diag}(s)=\mathrm{Diag}(c-A^T y)$9

A particularly notable regime is specified explicitly: if the LP has $$\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1}, \qquad \nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.$$0 constraints and $$\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1}, \qquad \nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.$$1 variables with

$$\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1}, \qquad \nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.$$2

for some constant $$\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1}, \qquad \nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.$$3, then

$$\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1}, \qquad \nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.$$4

so the QRAM-query cost scales sublinearly in the dimension $$\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1}, \qquad \nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.$$5 (Wu et al., 2024).

6. Related formulations, misconceptions, and practical assessment

The broader literature on quantum LP solvers includes several closely related formulations that differ primarily in how they preserve feasibility, how they manage condition-number growth, and whether they rely on Newton linearization at every step.

Paper	Core formulation	Stated property
"An Inexact Feasible Interior Point Method for Linear Optimization with High Adaptability to Quantum Computers" (Mohammadisiahroudi et al., 2023)	Orthogonal Subspaces System (OSS)	Inexact solve preserves primal and dual feasibility; $$\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1}, \qquad \nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.$$6 iteration complexity
"An Inexact Feasible Quantum Interior Point Method for Linearly Constrained Quadratic Optimization" (Wu et al., 2023)	Feasible QIPM via orthogonal-subspace reduction	For LP specialization, $$\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1}, \qquad \nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.$$7 iterations and per-iteration cost $$\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1}, \qquad \nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.$$8
"Efficient Use of Quantum Linear System Algorithms in Interior Point Methods for Linear Optimization" (Mohammadisiahroudi et al., 2022)	Inexact infeasible QIPM	$$\nabla \mathcal{L}(y)=-b+\mu\,A\,(c-A^T y)^{-1}, \qquad \nabla^2 \mathcal{L}(y)=\mu\,A\,S^{-2}A^T.$$9 iterations; iterative refinement recovers exactness in polynomial time
"A preconditioned inexact infeasible quantum interior point method for linear optimization" (Wu et al., 2024)	Preconditioned reduced augmented system	Improves condition number from $A S^{-2} A^T$0 to $A S^{-2} A^T$1
"Improvements to Quantum Interior Point Method for Linear Optimization" (Mohammadisiahroudi et al., 2023)	Preconditioned modified normal equations	Uses preconditioning with iterative refinement and $A S^{-2} A^T$2 qubit overhead
"A quantum central path algorithm for linear optimization" (Augustino et al., 2023)	One-shot nonlinear central-path simulation	Avoids Newton iteration by simulating the full complementarity system

One common misconception is that inexactness necessarily destroys feasibility. The feasible variants show otherwise. In the OSS-based approach, one sets $A S^{-2} A^T$3 with $A S^{-2} A^T$4 spanning $A S^{-2} A^T$5 and $A S^{-2} A^T$6, so even an inexact solve preserves

$A S^{-2} A^T$7

exactly for any $A S^{-2} A^T$8 (Mohammadisiahroudi et al., 2023). The DLBM-based construction reaches the same objective from the dual side by imposing explicit bounds on $A S^{-2} A^T$9 and on the tomography error (Wu et al., 2024).

A second misconception is that asymptotic QLSA speedups are sufficient to establish a practical LP advantage. The practical lower-bound study "Practical lower bounds for hybrid quantum interior point methods in linear programming" evaluates a standard hybrid QIPM pipeline against the classical open-source solver HiGHS on a broad collection of LP instances and concludes that, across all instances and for any realistic quantum cycle duration, the quantum runtime lower bounds already exceed the classical runtimes (Binkowski, 27 Apr 2026). This does not refute the asymptotic theory; rather, it isolates tomography, state loading, and end-to-end hybrid overhead as decisive obstacles.

A related lesson appears outside LP in "Approximate Quantum Linear Solvers for Hybrid CFD: End-to-End Analysis with a Chebyshev-LCU Approach." There, convergence of the outer nonlinear workflow is preserved for a non-exact quantum solver with only a moderate overhead in iteration count, provided that the high-frequency components of the linear system are resolved with sufficient accuracy; the approximate Cheb-LCU implementation reduces the required number of single-qubit rotations by over an order of magnitude relative to a QSVT-based solver while requiring only a modest increase in iteration count (Goldfriend et al., 31 May 2026). This suggests that end-to-end analysis, rather than isolated linear-solver complexity, is likely to remain central for assessing inexact quantum LP solvers as well.

In this landscape, the inexact quantum linear programming solver is best understood as a family of hybrid barrier and interior-point methods in which feasibility, neighborhood control, and precision management are co-designed with the quantum linear-system primitive. The DLBM-based solver of Wu, Sampourmahani, Mohammadisiahroudi, and Terlaky provides a particularly sharp realization of that program by coupling inexact-feasible analysis, QLSA-plus-tomography implementation, and iterative refinement into a single convergence theory with short-step iteration complexity (Wu et al., 2024).