Iterative Partition Search Algorithm (IPSA)

• IPSA is a variational quantum algorithm for the shortest vector problem (SVP) that leverages 1-tailed search spaces to guarantee lattice-preserving basis updates.
• It refines earlier methods by replacing broad partitioning and fixed iterations with a dynamic, stack-managed process and a hardware-efficient ansatz.
• Numerical simulations on 4- to 6-dimensional instances show at least a 73% improvement in success rate and over 35% better average solution quality compared to PSA and IQOAP.

Searching arXiv for the IPSA paper and closely related partition-search formulations. arXiv search: "Iterative Partition Search Algorithm SVP" The Iterative Partition Search Algorithm (IPSA) is a variational quantum algorithm for the Shortest Vector Problem (SVP) that was proposed as a targeted synthesis and refinement of the Partition Search Algorithm (PSA) and the Iterative Quantum Optimization with an Adaptive Problem (IQOAP) framework. It inherits PSA’s idea of “partitioning to circumvent the zero vector” and IQOAP’s “iterative lattice basis reduction” framework, but replaces broad partitioning by “1-tailed search spaces”, replaces IQOAP’s fixed iteration count by a dynamic, stack-managed process, and replaces IQOAP’s original ansatz by a hardware-efficient ansatz (HEA). The central claim is that the 1-tailed construction fundamentally ensures that every successful VQA execution yields an effective lattice basis update, thereby eliminating ineffective iterations; numerical simulations on 4- to 6-dimensional SVP instances report at least a 73\% improvement in success rate for finding optimal solutions and over a 35\% improvement in average solution quality relative to PSA and IQOAP, while maintaining comparable total circuit depth (Huang et al., 26 Aug 2025).

1. Problem class and algorithmic provenance

IPSA is formulated for lattices in $\mathbb{R}^n$. For a basis $B=[\mathbf b_1,\dots,\mathbf b_n]$, the lattice is

$$\mathcal L(B)=\left\{\sum_{i=1}^n x_i \mathbf b_i : x_i\in \mathbb Z\right\},$$

and SVP asks for the shortest nonzero vector: $$\lambda_1(\mathcal L)=\min_{\mathbf v\in \mathcal L\setminus\{0\}} \|\mathbf v\|.$$ Equivalently, one minimizes

$\left\|\sum_{i=1}^n x_i \mathbf b_i\right\|$

over $x\in\mathbb Z^n\setminus\{0\}$ (Huang et al., 26 Aug 2025).

The algorithm is explicitly motivated by limitations of two earlier VQA-based SVP methods. PSA avoids the trivial zero vector by partitioning the coefficient space, but is described as suffering from a complex optimization landscape, large qubit requirements, and the absence of iterative basis reduction. IQOAP reduces qubit count through repeated search in a small coefficient range and iterative basis updates, but is described as having no built-in zero-vector avoidance, a weak ansatz based on single-layer QAOA with constrained parameters, a fixed iteration count, and ineffective iterations in which the VQA output cannot validly replace a basis vector without changing the lattice (Huang et al., 26 Aug 2025).

A compact comparison of the three methods, using the distinctions stated in the paper, is as follows.

Method	Core idea	Stated limitation or design consequence
PSA	Partitioning to circumvent the zero vector	Complex optimization landscape; large qubit requirements
IQOAP	Iterative lattice basis reduction with small coefficient range	Zero vector remains possible; fixed iteration count; ineffective iterations
IPSA	1-tailed search spaces inside iterative basis reduction	Every successful VQA execution yields an effective basis update

Within the paper’s formulation, VQAs are used because current devices are NISQ machines that are noisy, shallow, and qubit-limited, whereas known quantum speedups for lattice problems typically assume fault-tolerant quantum computers. IPSA is therefore presented as a NISQ-oriented algorithmic design rather than as a fault-tolerant quantum cryptanalytic procedure (Huang et al., 26 Aug 2025).

2. 1-tailed search spaces and the lattice-preserving update theorem

The defining innovation of IPSA is the family of 1-tailed search spaces

$$Y_i=\{(y_1,\dots,y_{i-1},1,0,\dots,0)^T\in \mathbb Z^n\}.$$

For $Y_i$, the candidate lattice vector has the form

$$\mathbf v=\sum_{j=1}^{i-1} y_j \mathbf b_j+\mathbf b_i.$$

Thus coefficients after position $i$ are fixed to $B=[\mathbf b_1,\dots,\mathbf b_n]$0, the $B=[\mathbf b_1,\dots,\mathbf b_n]$1-th coefficient is fixed to $B=[\mathbf b_1,\dots,\mathbf b_n]$2, and only the first $B=[\mathbf b_1,\dots,\mathbf b_n]$3 coefficients vary (Huang et al., 26 Aug 2025).

This search-space design plays two roles. First, it removes the zero vector without penalties or auxiliary constraints: because the coefficient of $B=[\mathbf b_1,\dots,\mathbf b_n]$4 is fixed to $B=[\mathbf b_1,\dots,\mathbf b_n]$5, every vector in $B=[\mathbf b_1,\dots,\mathbf b_n]$6 is automatically nonzero. Second, it enforces a replaceability condition for iterative basis reduction. The paper states the following theorem: if

$B=[\mathbf b_1,\dots,\mathbf b_n]$7

and for some $B=[\mathbf b_1,\dots,\mathbf b_n]$8, $B=[\mathbf b_1,\dots,\mathbf b_n]$9, then replacing $$\mathcal L(B)=\left\{\sum_{i=1}^n x_i \mathbf b_i : x_i\in \mathbb Z\right\},$$0 by $$\mathcal L(B)=\left\{\sum_{i=1}^n x_i \mathbf b_i : x_i\in \mathbb Z\right\},$$1 yields another basis of the same lattice,

$$\mathcal L(B)=\left\{\sum_{i=1}^n x_i \mathbf b_i : x_i\in \mathbb Z\right\},$$2

Because vectors produced in $$\mathcal L(B)=\left\{\sum_{i=1}^n x_i \mathbf b_i : x_i\in \mathbb Z\right\},$$3 always have coefficient $$\mathcal L(B)=\left\{\sum_{i=1}^n x_i \mathbf b_i : x_i\in \mathbb Z\right\},$$4 on $$\mathcal L(B)=\left\{\sum_{i=1}^n x_i \mathbf b_i : x_i\in \mathbb Z\right\},$$5, every shortening update in IPSA is lattice-preserving by construction (Huang et al., 26 Aug 2025).

The paper contrasts $$\mathcal L(B)=\left\{\sum_{i=1}^n x_i \mathbf b_i : x_i\in \mathbb Z\right\},$$6 with PSA’s broader partitions

$$\mathcal L(B)=\left\{\sum_{i=1}^n x_i \mathbf b_i : x_i\in \mathbb Z\right\},$$7

in which the $$\mathcal L(B)=\left\{\sum_{i=1}^n x_i \mathbf b_i : x_i\in \mathbb Z\right\},$$8-th coefficient still varies. IPSA’s formulation is therefore described as a “highly constrained variant” of PSA partitioning, specifically tailored to IQOAP-style iteration (Huang et al., 26 Aug 2025).

This construction is the basis of the paper’s strongest algorithmic claim: if a VQA run in partition $$\mathcal L(B)=\left\{\sum_{i=1}^n x_i \mathbf b_i : x_i\in \mathbb Z\right\},$$9 returns $$\lambda_1(\mathcal L)=\min_{\mathbf v\in \mathcal L\setminus\{0\}} \|\mathbf v\|.$$0 with

$$\lambda_1(\mathcal L)=\min_{\mathbf v\in \mathcal L\setminus\{0\}} \|\mathbf v\|.$$1

then $$\lambda_1(\mathcal L)=\min_{\mathbf v\in \mathcal L\setminus\{0\}} \|\mathbf v\|.$$2 can always be replaced by $$\lambda_1(\mathcal L)=\min_{\mathbf v\in \mathcal L\setminus\{0\}} \|\mathbf v\|.$$3 without changing the lattice. In the paper’s terminology, this fundamentally ensures that every successful VQA execution yields an effective basis update (Huang et al., 26 Aug 2025).

3. Hamiltonian formulation, encoding, and iterative control flow

For partition $$\lambda_1(\mathcal L)=\min_{\mathbf v\in \mathcal L\setminus\{0\}} \|\mathbf v\|.$$4, the coefficient of $$\lambda_1(\mathcal L)=\min_{\mathbf v\in \mathcal L\setminus\{0\}} \|\mathbf v\|.$$5 is fixed and requires no qubits. Each variable coefficient $$\lambda_1(\mathcal L)=\min_{\mathbf v\in \mathcal L\setminus\{0\}} \|\mathbf v\|.$$6 for $$\lambda_1(\mathcal L)=\min_{\mathbf v\in \mathcal L\setminus\{0\}} \|\mathbf v\|.$$7 is encoded using $$\lambda_1(\mathcal L)=\min_{\mathbf v\in \mathcal L\setminus\{0\}} \|\mathbf v\|.$$8 qubits as

$$\lambda_1(\mathcal L)=\min_{\mathbf v\in \mathcal L\setminus\{0\}} \|\mathbf v\|.$$9

with encoded integer range

$\left\|\sum_{i=1}^n x_i \mathbf b_i\right\|$0

The largest partition $\left\|\sum_{i=1}^n x_i \mathbf b_i\right\|$1 therefore requires

$\left\|\sum_{i=1}^n x_i \mathbf b_i\right\|$2

qubits (Huang et al., 26 Aug 2025).

The partition-specific VQA Hamiltonian is

$\left\|\sum_{i=1}^n x_i \mathbf b_i\right\|$3

This is the squared Euclidean norm of the candidate vector restricted to $\left\|\sum_{i=1}^n x_i \mathbf b_i\right\|$4; minimizing the expectation of $\left\|\sum_{i=1}^n x_i \mathbf b_i\right\|$5 seeks the shortest vector in that partition (Huang et al., 26 Aug 2025).

The outer algorithm is iterative and stack-managed. The input basis

$\left\|\sum_{i=1}^n x_i \mathbf b_i\right\|$6

is first sorted by increasing norm,

$\left\|\sum_{i=1}^n x_i \mathbf b_i\right\|$7

The partitions $\left\|\sum_{i=1}^n x_i \mathbf b_i\right\|$8 are pushed onto a stack, with $\left\|\sum_{i=1}^n x_i \mathbf b_i\right\|$9 on top. The algorithm then repeatedly pops a partition $x\in\mathbb Z^n\setminus\{0\}$0, solves the restricted SVP subproblem with the VQA, obtains a candidate vector $x\in\mathbb Z^n\setminus\{0\}$1, and checks whether

$x\in\mathbb Z^n\setminus\{0\}$2

If so, $x\in\mathbb Z^n\setminus\{0\}$3 is replaced by $x\in\mathbb Z^n\setminus\{0\}$4, the basis is re-sorted, and if $x\in\mathbb Z^n\setminus\{0\}$5 is the new position of $x\in\mathbb Z^n\setminus\{0\}$6, then partitions $x\in\mathbb Z^n\setminus\{0\}$7 are pushed back onto the stack. When the stack becomes empty, the algorithm returns $x\in\mathbb Z^n\setminus\{0\}$8 (Huang et al., 26 Aug 2025).

Operationally, this adaptive stack replaces IQOAP’s fixed 50-iteration loop. The paper states that this prioritizes re-solving smaller partitions after updates so that the basis becomes “well-reduced” before moving on to larger partitions. A plausible implication is that IPSA treats basis reduction as a dependency-sensitive search process rather than as a fixed-budget repetition schedule (Huang et al., 26 Aug 2025).

4. Variational circuit choice and reported computational profile

The default parameterized quantum circuit in IPSA is a Hardware-Efficient Ansatz (HEA) rather than QAOA. The paper’s rationale is structural: the SVP Hamiltonian contains all-to-all interactions among encoded coefficients, so a QAOA implementation requires deep objective layers with many ZZ interactions. By contrast, the HEA used in the study is described as both more expressive and shallower for these SVP instances (Huang et al., 26 Aug 2025).

The standard IPSA implementation uses $x\in\mathbb Z^n\setminus\{0\}$9 HEA layers. For comparison, the study also implemented IPSA-QAOA with $$Y_i=\{(y_1,\dots,y_{i-1},1,0,\dots,0)^T\in \mathbb Z^n\}.$$0 QAOA layers, while IQOAP retained its original single-layer QAOA with the parameter restriction $$Y_i=\{(y_1,\dots,y_{i-1},1,0,\dots,0)^T\in \mathbb Z^n\}.$$1. All algorithms were simulated in Qiskit using StatevectorEstimator or StatevectorSampler, and the classical optimizer was SciPy’s minimize with Powell’s conjugate direction method. Initial rotation parameters were sampled uniformly from $$Y_i=\{(y_1,\dots,y_{i-1},1,0,\dots,0)^T\in \mathbb Z^n\}.$$2, with default SciPy tolerances xtol = 10^{-4} and ftol = 10^{-4} (Huang et al., 26 Aug 2025).

For IPSA and IQOAP in dimensions $$Y_i=\{(y_1,\dots,y_{i-1},1,0,\dots,0)^T\in \mathbb Z^n\}.$$3, the study used two qubits per variable coefficient, matching

$$Y_i=\{(y_1,\dots,y_{i-1},1,0,\dots,0)^T\in \mathbb Z^n\}.$$4

for this dimensional range. IPSA’s maximum qubit counts were reported as 6, 8, and 10 qubits for $$Y_i=\{(y_1,\dots,y_{i-1},1,0,\dots,0)^T\in \mathbb Z^n\}.$$5, whereas IQOAP used 8, 10, and 12 qubits. This difference follows directly from fixing one coefficient to $$Y_i=\{(y_1,\dots,y_{i-1},1,0,\dots,0)^T\in \mathbb Z^n\}.$$6 in each IPSA partition (Huang et al., 26 Aug 2025).

The internal comparison between HEA and QAOA is one of the paper’s clearest empirical findings. At $$Y_i=\{(y_1,\dots,y_{i-1},1,0,\dots,0)^T\in \mathbb Z^n\}.$$7, the reported values were:

Variant	Success rate	AAR	Total depth
IPSA-HEA	0.95	0.9835	$$Y_i=\{(y_1,\dots,y_{i-1},1,0,\dots,0)^T\in \mathbb Z^n\}.$$8
IPSA-QAOA	0.025	0.4179	$$Y_i=\{(y_1,\dots,y_{i-1},1,0,\dots,0)^T\in \mathbb Z^n\}.$$9

The corresponding total CNOT counts at $Y_i$0 were $Y_i$1 for IPSA-HEA and $Y_i$2 for IPSA-QAOA. Within this study, HEA was therefore superior in success rate, approximation quality, depth, and entangling-gate count (Huang et al., 26 Aug 2025).

5. Experimental results on benchmark and LLL-challenging instances

The evaluation used two datasets totaling 800 instances. The Benchmark Set contained 600 instances, with 200 instances each for $Y_i$3, generated by applying random unimodular transformations to reduced lattice bases. The LLL-Challenging Set contained 200 instances at $Y_i$4, selected so that classical LLL fails to find the true shortest vector and instead returns a vector only about $Y_i$5 longer on average (Huang et al., 26 Aug 2025).

The paper used four metrics. Success Rate was defined as

$Y_i$6

and Approximation Ratio for an instance as

$Y_i$7

with average over instances denoted AAR. Resource use was summarized as

$Y_i$8

Here $Y_i$9 and $$\mathbf v=\sum_{j=1}^{i-1} y_j \mathbf b_j+\mathbf b_i.$$0 are the circuit depth and CNOT count in iteration $$\mathbf v=\sum_{j=1}^{i-1} y_j \mathbf b_j+\mathbf b_i.$$1 (Huang et al., 26 Aug 2025).

On the Benchmark Set, IPSA’s reported success rates were 0.995 at $$\mathbf v=\sum_{j=1}^{i-1} y_j \mathbf b_j+\mathbf b_i.$$2, 0.985 at $$\mathbf v=\sum_{j=1}^{i-1} y_j \mathbf b_j+\mathbf b_i.$$3, and 0.95 at $$\mathbf v=\sum_{j=1}^{i-1} y_j \mathbf b_j+\mathbf b_i.$$4. IQOAP’s reported success rate was 0.26 at $$\mathbf v=\sum_{j=1}^{i-1} y_j \mathbf b_j+\mathbf b_i.$$5 and 0.01 at $$\mathbf v=\sum_{j=1}^{i-1} y_j \mathbf b_j+\mathbf b_i.$$6, while PSA variants were generally below 0.1. IPSA’s AAR remained near 1, with 0.9835 at $$\mathbf v=\sum_{j=1}^{i-1} y_j \mathbf b_j+\mathbf b_i.$$7, whereas IQOAP’s AAR declined from 0.641 at $$\mathbf v=\sum_{j=1}^{i-1} y_j \mathbf b_j+\mathbf b_i.$$8 to 0.294 at $$\mathbf v=\sum_{j=1}^{i-1} y_j \mathbf b_j+\mathbf b_i.$$9, and PSA variants were mostly in the 0.3–0.5 range (Huang et al., 26 Aug 2025).

On the LLL-Challenging Set at $i$0, the reported success rates were 0.86 for IPSA, 0.165 for 3-PSA, and 0.075 for IQOAP. The reported AAR values were 0.996 for IPSA, 0.869 for 3-PSA, 0.816 for IQOAP, and 0.978 for classical LLL. Resource use on this set was also reported: IPSA had

$i$1

while 3-PSA had

$i$2

The paper interprets this as evidence that 1-tailed partitioning is particularly effective when very similar short vectors create near-optimal distractors (Huang et al., 26 Aug 2025).

These reported results support the abstract’s headline statement that IPSA achieves at least a 73\% improvement in success rate for optimal solutions and over a 35\% improvement in average solution quality compared with PSA and IQOAP, while maintaining comparable total circuit depth (Huang et al., 26 Aug 2025).

6. Relation to the broader partition-search literature

Although the explicit acronym IPSA is used in the SVP setting above, the arXiv literature also contains several partition-search methods that are only loosely related to IPSA in nomenclature but closely related in search architecture.

An important exact partition-search result is the Minimum Information Partition method of “Fast and exact search for the partition with minimal information loss” (Hidaka et al., 2017). There the objective

$i$3

is shown to be a symmetric submodular function, so bipartition search can be solved exactly in polynomial time with Queyranne’s algorithm. This is not an iterative improvement heuristic; it is exact submodular minimization. It is therefore relevant to “partition search” in the generic sense, but not to IPSA’s VQA-based iterative lattice reduction (Hidaka et al., 2017).

“ILP-based Local Search for Graph Partitioning” presents an ILP-based large-neighborhood local search around a current graph partition, using contraction outside a kept set $i$4, symmetry breaking, and repeated improvement of an incumbent partition (Henzinger et al., 2018). The paper explicitly states that it does not use the term IPSA, but that its method is naturally interpretable as an IPSA-like iterative partition improvement method. This suggests a broader family resemblance between IPSA and partition-centered neighborhood search, even though the optimization model and domain are entirely different (Henzinger et al., 2018).

Other works extend the motif further. “Streaming Algorithms for Partitioning Integer Sequences” treats contiguous partitioning through streaming parametric search over bottleneck values $i$5, with exact feasibility testing by Probe and adaptive threshold escalation by ProbeExt (Konrad et al., 2014). “Global Search Based on Efficient Diagonal Partitions and a set of Lipschitz Constants” studies deterministic global optimization by iterative hyperinterval partitioning, lower bounds, non-dominated region selection, and a local/global search policy (Sergeyev et al., 2011). “The Kernighan-Lin Search Algorithm” extracts variable depth search from KL/LK into a general search mechanism based on temporary moves, gain sequences $i$6, and best-prefix acceptance (Dasdan, 1 Feb 2025). In a different direction, “Partition of graphs and quantum walk based search algorithms” uses equitable partitions to compress DTQW and CTQW search into invariant effective subspaces, but it is a one-shot reduction method rather than an iterative partition-refinement procedure (Ide, 2018).

A common misconception is therefore to treat IPSA as a generic label for any partition-based search method. The literature considered here does not support that usage. A more precise statement is that IPSA is the name of a specific VQA for SVP, while other partition-search methods share only selected architectural motifs such as restricted search spaces, recursive decomposition, large-neighborhood optimization, or variable-depth refinement (Huang et al., 26 Aug 2025).

7. Limitations, scope conditions, and prospective extensions

The IPSA study is explicit about several limits. The experiments are restricted to small dimensions, specifically $i$7. All results are obtained by statevector simulation in Qiskit rather than on physical quantum hardware, so noisy-hardware behavior is not tested. The paper does not present a full asymptotic or convergence proof establishing global SVP recovery in all cases. In addition, some PSA baselines could not be run in higher dimensions because of classical simulation cost, so cross-method comparison is partly constrained by simulation tractability (Huang et al., 26 Aug 2025).

These limits matter for interpretation. IPSA is best understood as a NISQ-algorithm design study showing that the combination of 1-tailed partitioning, guaranteed lattice-preserving updates, an adaptive stack schedule, and HEA-based variational optimization can improve simulated performance on small SVP instances. It is not presented as a proof of near-term practical cryptanalytic capability against lattice-based post-quantum cryptography (Huang et al., 26 Aug 2025).

The paper nevertheless identifies several concrete future directions. It suggests that IPSA could be used as a subroutine in block Korkin–Zolotarev type algorithms, and for related problems such as learning with errors. It also notes that investigating other QAOA variants is beyond scope and remains future work. This suggests that the most stable contribution of IPSA may be methodological: it shows how partitioning can be designed so that every successful restricted solve is actionable inside an iterative basis-reduction loop (Huang et al., 26 Aug 2025).