A New Quantum Linear System Algorithm Beyond the Condition Number and Its Application to Solving Multivariate Polynomial Systems

TL;DR

The paper tackles the bottleneck of quantum linear system (QLS) algorithms by introducing an instance-aware approach that exploits the right-hand side structure via an augmented matrix $H=[A,-\vec{b}]$ and a new instance-dependent parameter $ET=\sum p_i^2 d_i$. By constructing a graph-based walk unitary $U_{\mathcal{AB}}$ and using a right multiplication rescaling $AD$ to form a weighted Boolean Macaulay system, the authors develop a QLS algorithm whose runtime depends on $ET$ and the solution norm $\|\vec{y}\|$ rather than the traditional condition number $\kappa(A)$, enabling efficient solutions in structured regimes. They demonstrate the framework with a welded tree graph example showing exponential speedups relative to standard QLS methods and apply the approach to multivariate polynomial systems, including planted MIS, where under certain input conditions the problem solvable in polynomial time. The combination of the instance-aware QLS, the rescaling technique, and the weighted Macaulay formulation opens new avenues for quantum speedups in structured nonlinear problems and combinatorial optimization, potentially impacting GI, nonlinear differential equations, and ground-state preparation for structured Hamiltonians where input structure is exploitable.

Abstract

Given a matrix $A$ of dimension $M \times N$ and a vector $\vec{b}$, the quantum linear system (QLS) problem asks for the preparation of a quantum state $|\vec{y}\rangle$ proportional to the solution of $A\vec{y} = \vec{b}$. Existing QLS algorithms have runtimes that scale linearly with the condition number $κ(A)$, the sparsity of $A$, and logarithmically with inverse precision, but often overlook structural properties of $\vec{b}$, whose alignment with $A$'s eigenspaces can greatly affect performance. In this work, we present a new QLS algorithm that explicitly leverages the structure of the right-hand side vector $\vec{b}$. The runtime of our algorithm depends polynomially on the sparsity of the augmented matrix $H = [A, -\vec{b}]$, the inverse precision, the $\ell_2$ norm of the solution $\vec{y} = A^+ \vec{b}$, and a new instance-dependent parameter \[ ET= \sum_{i=1}^M p_i^2 \cdot d_i, \] where $\vec{p} = (AA^{\top})^+ \vec{b}$, and $d_i$ denotes the squared $\ell_2$ norm of the $i$-th row of $H$. We also introduce a structure-aware rescaling technique tailored to the solution $\vec{y} = A^+ \vec{b}$. Unlike left preconditioning methods, which transform the linear system to $DA\vec{y} = D\vec{b}$, our approach applies a right rescaling matrix, reformulating the linear system as $AD\vec{z} = \vec{b}$. As an application of our instance-aware QLS algorithm and new rescaling scheme, we develop a quantum algorithm for solving multivariate polynomial systems in regimes where prior QLS-based methods fail. This yields an end-to-end framework applicable to a broad class of problems. In particular, we apply it to the maximum independent set (MIS) problem, formulated as a special case of a polynomial system, and show through detailed analysis that, under certain conditions, our quantum algorithm for MIS runs in polynomial time.

Figures (2)

• Figure 1: Illustration of the graph $G$ constructed from $H$. Blue nodes correspond to rows ($u_i$), green nodes to columns ($v_j$), the red node represents $b$, and $w_{1,1}=1, w_{1,2}=4,w_{2,2}=4,w_{3,3}=1, w_{1,b}=4,w_{3,b}=4$.
• Figure 2: An example welded tree graph with $n=4$. The electrical flow is the solution to the incidence linear system $B\vec{y}= \vec{b}_{u,v}$, where each entry of $\vec{y}$ represents the flow along an edge. The magnitude of the flow corresponds to the amount of electrical current, and the sign is determined by the orientation specified in the vertex-edge incidence matrix $B$. In particular, these four red edges forms a cut for welded tree graph of this type.

Theorems & Definitions (51)

• Definition 2.3: Row space of $H$
• Definition 2.4: Null space of $H$
• Lemma 2.5: Fundamental Theorem of Linear Algebra Part 1 and 2 strang2022introduction
• Lemma 2.6
• proof
• Theorem 2.7: Vector Decomposition Theorem
• proof
• Remark 2.8
• Example 3.1
• Definition 3.2: Row star states