---
title: Iterative Partition Search Algorithm (IPSA)
url: https://www.emergentmind.com/topics/iterative-partition-search-algorithm-ipsa
type: topic
---

# Iterative Partition Search Algorithm (IPSA)

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 [2508.18996].

## 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 \(x\in\mathbb Z^n\setminus\{0\}\) [2508.18996].

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 [2508.18996].

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 [2508.18996].

## 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 \(0\), the \(i\)-th coefficient is fixed to \(1\), and only the first \(i-1\) coefficients vary [2508.18996].

This search-space design plays two roles. First, it removes the zero vector without penalties or auxiliary constraints: because the coefficient of \(\mathbf b_i\) is fixed to \(1\), every vector in \(Y_i\) is automatically nonzero. Second, it enforces a replaceability condition for iterative basis reduction. The paper states the following theorem: if
\[
\mathbf v=\sum_{j=1}^n c_j \mathbf b_j,\qquad c_j\in\mathbb Z,
\]
and for some \(k\), \(|c_k|=1\), then replacing \(\mathbf b_k\) by \(\mathbf v\) yields another basis of the same lattice,
\[
B'=[\mathbf b_1,\dots,\mathbf b_{k-1},\mathbf v,\mathbf b_{k+1},\dots,\mathbf b_n].
\]
Because vectors produced in \(Y_i\) always have coefficient \(1\) on \(\mathbf b_i\), every shortening update in IPSA is lattice-preserving by construction [2508.18996].

The paper contrasts \(Y_i\) with PSA’s broader partitions
\[
X_i=\{(x_1,\dots,x_i,0,\dots,0)^T\in\mathbb Z^n:x_i\ge 1\},
\]
in which the \(i\)-th coefficient still varies. IPSA’s formulation is therefore described as a **“highly constrained variant”** of PSA partitioning, specifically tailored to IQOAP-style iteration [2508.18996].

This construction is the basis of the paper’s strongest algorithmic claim: if a VQA run in partition \(Y_i\) returns \(\mathbf v\) with
\[
\|\mathbf v\|<\|\mathbf b_i\|,
\]
then \(\mathbf b_i\) can always be replaced by \(\mathbf v\) without changing the lattice. In the paper’s terminology, this fundamentally ensures that every successful VQA execution yields an effective basis update [2508.18996].

## 3. Hamiltonian formulation, encoding, and iterative control flow

For partition \(Y_i\), the coefficient of \(\mathbf b_i\) is fixed and requires no qubits. Each variable coefficient \(y_r\) for \(r<i\) is encoded using \(\lfloor \log n\rfloor\) qubits as
\[
\hat{y_r}=\frac{1}{2}-\sum_{j=0}^{\lfloor\log n\rfloor-1}2^{j-1}\sigma^z_{r,j},
\]
with encoded integer range
\[
(-2^{\lfloor\log n\rfloor-1},\,2^{\lfloor\log n\rfloor-1}].
\]
The largest partition \(Y_n\) therefore requires
\[
(n-1)\lfloor\log n\rfloor
\]
qubits [2508.18996].

The partition-specific VQA Hamiltonian is
\[
H_i^\prime=\sum_{k=1}^{n}\left(\sum_{r=1}^{i-1}\hat{y_r}b_{r,k}+b_{i,k}\right)^2.
\]
This is the squared Euclidean norm of the candidate vector restricted to \(Y_i\); minimizing the expectation of \(H_i^\prime\) seeks the shortest vector in that partition [2508.18996].

The outer algorithm is iterative and stack-managed. The input basis
\[
B=[\mathbf b_1,\dots,\mathbf b_n]
\]
is first sorted by increasing norm,
\[
\|\mathbf b_1\|\le \|\mathbf b_2\|\le \cdots \le \|\mathbf b_n\|.
\]
The partitions \(Y_n,\dots,Y_1\) are pushed onto a stack, with \(Y_1\) on top. The algorithm then repeatedly pops a partition \(Y_i\), solves the restricted SVP subproblem with the VQA, obtains a candidate vector \(\mathbf v\), and checks whether
\[
\|\mathbf v\|<\|\mathbf b_i\|.
\]
If so, \(\mathbf b_i\) is replaced by \(\mathbf v\), the basis is re-sorted, and if \(r\) is the new position of \(\mathbf v\), then partitions \(Y_i,\dots,Y_r\) are pushed back onto the stack. When the stack becomes empty, the algorithm returns \(\mathbf b_1\) [2508.18996].

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 [2508.18996].

## 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 [2508.18996].

The standard IPSA implementation uses **\(p=2\)** HEA layers. For comparison, the study also implemented **IPSA-QAOA** with **\(p=4\)** QAOA layers, while IQOAP retained its original **single-layer QAOA** with the parameter restriction \(\beta=\gamma\). 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 \([0,\pi]\), with default SciPy tolerances `xtol = 10^{-4}` and `ftol = 10^{-4}` [2508.18996].

For IPSA and IQOAP in dimensions \(n=4,5,6\), the study used **two qubits per variable coefficient**, matching
\[
\lfloor\log n\rfloor
\]
for this dimensional range. IPSA’s maximum qubit counts were reported as 6, 8, and 10 qubits for \(n=4,5,6\), whereas IQOAP used 8, 10, and 12 qubits. This difference follows directly from fixing one coefficient to \(1\) in each IPSA partition [2508.18996].

The internal comparison between HEA and QAOA is one of the paper’s clearest empirical findings. At \(n=6\), the reported values were:

| Variant | Success rate | AAR | Total depth |
|---|---:|---:|---:|
| IPSA-HEA | 0.95 | 0.9835 | \(0.321\times 10^6\) |
| IPSA-QAOA | 0.025 | 0.4179 | \(1.638\times 10^6\) |

The corresponding total CNOT counts at \(n=6\) were \(0.242\times 10^6\) for IPSA-HEA and \(2.344\times 10^6\) for IPSA-QAOA. Within this study, HEA was therefore superior in success rate, approximation quality, depth, and entangling-gate count [2508.18996].

## 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 \(n\in\{4,5,6\}\), generated by applying random unimodular transformations to reduced lattice bases. The **LLL-Challenging Set** contained **200 instances** at \(n=6\), selected so that classical LLL fails to find the true shortest vector and instead returns a vector only about **\(2\%\)** longer on average [2508.18996].

The paper used four metrics. **Success Rate** was defined as
\[
SR = \frac{N_{\text{succ}}}{N_{\text{total}}},
\]
and **Approximation Ratio** for an instance as
\[
AR = \frac{\lambda_1(\mathcal L)}{\|v_{\text{alg}}\|},
\]
with average over instances denoted **AAR**. Resource use was summarized as
\[
D_{\text{total}}=\sum_{i=1}^{I} d_i,\qquad
C_{\text{total}}=\sum_{i=1}^{I} c_i.
\]
Here \(d_i\) and \(c_i\) are the circuit depth and CNOT count in iteration \(i\) [2508.18996].

On the Benchmark Set, IPSA’s reported success rates were **0.995** at \(n=4\), **0.985** at \(n=5\), and **0.95** at \(n=6\). IQOAP’s reported success rate was **0.26** at \(n=4\) and **0.01** at \(n=6\), while PSA variants were generally **below 0.1**. IPSA’s AAR remained near 1, with **0.9835** at \(n=6\), whereas IQOAP’s AAR declined from **0.641** at \(n=4\) to **0.294** at \(n=6\), and PSA variants were mostly in the **0.3–0.5** range [2508.18996].

On the LLL-Challenging Set at \(n=6\), 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
\[
D_{\text{total}} = 1.556\times10^5,\qquad
C_{\text{total}} = 1.177\times10^5,
\]
while 3-PSA had
\[
D_{\text{total}} = 1.476\times10^5,\qquad
C_{\text{total}} = 1.269\times10^5.
\]
The paper interprets this as evidence that 1-tailed partitioning is particularly effective when very similar short vectors create near-optimal distractors [2508.18996].

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 [2508.18996].

## 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” [1708.01444]. There the objective
\[
f(M)=I(M;V\setminus M)
\]
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 [1708.01444].

“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 \(\mathcal K\), symmetry breaking, and repeated improvement of an incumbent partition [1802.07144]. 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 [1802.07144].

Other works extend the motif further. “Streaming Algorithms for Partitioning Integer Sequences” treats contiguous partitioning through streaming **parametric search** over bottleneck values \(B\), with exact feasibility testing by **Probe** and adaptive threshold escalation by **ProbeExt** [1404.1732]. “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 [1103.2056]. “The Kernighan-Lin Search Algorithm” extracts **variable depth search** from KL/LK into a general search mechanism based on temporary moves, gain sequences \(g_i\), and best-prefix acceptance [2502.00316]. 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 [1812.06376].

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 [2508.18996].

## 7. Limitations, scope conditions, and prospective extensions

The IPSA study is explicit about several limits. The experiments are restricted to **small dimensions**, specifically \(n=4,5,6\). 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 [2508.18996].

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 [2508.18996].

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 [2508.18996].

Source: https://www.emergentmind.com/topics/iterative-partition-search-algorithm-ipsa