Quantitative Group Testing (QGT)

• Quantitative Group Testing (QGT) is a model where each pooled test returns the precise count of defectives, enabling exact support recovery under sparse conditions.
• It employs a blend of combinatorial, probabilistic, and coding-theoretic methods including adaptive designs, sparse-graph codes, and AMP algorithms.
• Advanced techniques like LDPC constructions and reinforcement learning strategies enhance performance in both noiseless and noisy testing environments.

Quantitative Group Testing (QGT) is the group-testing model in which each pooled test returns the exact number of defectives in the tested subset. In its standard linear form, with pooling matrix $A\in\{0,1\}^{m\times n}$ and defective indicator $x\in\{0,1\}^n$, the observation vector is $y=Ax$, so each measurement is an integer count rather than a binary positive/negative label (Feige et al., 2020). The central task is to recover $x$, or equivalently its support, from $m\ll n$ pooled tests (Grampurohit et al., 8 Dec 2025). The literature spans adaptive and non-adaptive designs, combinatorial and probabilistic signal models, noiseless and noisy observation channels, and both algorithmic and information-theoretic analyses (Li et al., 16 Jan 2026).

1. Formal model and relation to neighboring testing paradigms

A common combinatorial formulation assumes exactly $k$ defectives, so $x\in\{0,1\}^n$ with $\|x\|_0=k$, and the goal is exact support recovery from pooled counts (Feige et al., 2020). Other works adopt a probabilistic sparse prior, for example $x_j\sim \mathrm{Bernoulli}(K/N)$ independently so that $\mathbb E[\|x\|_0]=K$, or a linear-prevalence regime in which the empirical defective fraction converges to a constant $x\in\{0,1\}^n$0 (Grampurohit et al., 8 Dec 2025, Tan et al., 2023). These choices change both the performance criteria and the decoder design: worst-case combinatorial guarantees, average-case high-probability recovery, and thresholded error metrics all coexist in the literature.

Although many QGT papers use binary pooling matrices, the broader adder-channel view also allows non-binary test matrices in which an entry specifies the amount or concentration of a sample placed into a test. In that formulation, QGT is the exact adder model, and semi-quantitative models arise by quantizing the adder output (Emad et al., 2012). This is useful when the physical assay is naturally quantitative but not fully precise.

The sharpest conceptual relation is with Semi-Quantitative Group Testing (SQGT). In the saturation-threshold model, if the exact QGT count in test $x\in\{0,1\}^n$1 is $x\in\{0,1\}^n$2, the SQGT output is $x\in\{0,1\}^n$3 with $x\in\{0,1\}^n$4. Thus SQGT is literally “QGT + output quantization” (Cheraghchi et al., 2021). When $x\in\{0,1\}^n$5, one recovers classical binary group testing; when $x\in\{0,1\}^n$6 is large enough to distinguish all feasible counts, SQGT essentially becomes QGT (Cheraghchi et al., 2021). A related capped-response formulation defines $x\in\{0,1\}^n$7, interpolating between binary testing at $x\in\{0,1\}^n$8 and full quantitative testing at $x\in\{0,1\}^n$9 (Kowalski et al., 2021).

2. Recovery criteria and information-theoretic scales

The most stringent target is exact recovery: given $y=Ax$0 and $y=Ax$1, output $y=Ax$2 such that $y=Ax$3 in the asymptotic regime of interest (Li et al., 16 Jan 2026). In adaptive minimax formulations, the objective is instead to minimize the worst-case number of tests over all defective sets of a given size (Wang et al., 2014). Noisy and adversarial settings often weaken the goal to bounded support error, false positive/false negative control, or exact recovery with vanishing probability of error under a random model (Tan et al., 2023, Li et al., 2021).

For noiseless non-adaptive QGT, one cited benchmark is the information-theoretic order $y=Ax$4 (Li et al., 16 Jan 2026). In constructive algorithmic regimes, many explicit schemes operate at $y=Ax$5 tests, which is near-optimal in sparse settings but still above the counting bound by a $y=Ax$6-type factor (Karimi et al., 2019, Karimi et al., 2019). Deterministic capped-quantitative constructions attain

$y=Ax$7

queries, within polylogarithmic factors of a lower bound

$y=Ax$8

and for full QGT, $y=Ax$9, this becomes $x$0 (Kowalski et al., 2021).

Noise substantially changes the scaling. In additive Gaussian QGT with

$x$1

the least-squares estimator (LSE) achieves

$x$2

and the converse matches in order (Li et al., 16 Jan 2026). In the noisy Z-channel model, the LSE achieves

$x$3

while the converse remains of order $x$4 with a $x$5-dependent constant (Li et al., 16 Jan 2026). For adversarially perturbed combinatorial QGT with bounded additive noise $x$6, $x$7, and tolerated output error $x$8, the optimal non-adaptive pooling complexity is

$x$9

up to a constant factor independent of $m\ll n$0, in the regime $m\ll n$1 (Li et al., 2021).

3. Adaptive and non-adaptive combinatorial methods

In adaptive combinatorial QGT, the classical exact result concerns nested noiseless test plans. For $m\ll n$2 items with exactly $m\ll n$3 defectives, the optimal nested minimax cost is

$m\ll n$4

where

$m\ll n$5

and the optimal first split size is

$m\ll n$6

for $m\ll n$7 (Wang et al., 2014). This yields a closed-form recursive strategy: test a subset of size $m\ll n$8, observe the exact defective count in that subset, and recurse on the two induced subproblems. If the total number of defectives is unknown, the optimal nested plan first tests the entire population, because one quantitative test immediately reveals $m\ll n$9 (Wang et al., 2014). For fixed $k$0, the nested optimum is order-optimal among all adaptive plans, with

$k$1

(Wang et al., 2014).

A different non-adaptive viewpoint is the reduction of QGT to the intermediate problem “Subset Select.” Instead of recovering $k$2 directly, one first finds a set $k$3 of size $k$4 containing $k$5. If the induced square submatrix $k$6 is full rank or nearly full rank, then exact recovery reduces to solving

$k$7

in polynomial time (Feige et al., 2020). This decomposition separates combinatorial localization from algebraic disambiguation and shows that in random Bernoulli designs, exact support identification is not always the fundamental bottleneck (Feige et al., 2020).

A recent adaptive development is learning-based. One approach recursively partitions the $k$8-dimensional QGT problem into repeated $k$9-dimensional integer-vector identification tasks and then uses offline reinforcement learning with Decision Transformers to choose adaptive subset-sum queries (Soleymani et al., 1 Sep 2025). If $x\in\{0,1\}^n$0 is the number of reduced-dimension queries used at stage $x\in\{0,1\}^n$1, the total number of QGT queries is

$x\in\{0,1\}^n$2

and if $x\in\{0,1\}^n$3, the average becomes

$x\in\{0,1\}^n$4

(Soleymani et al., 1 Sep 2025). Empirically, this method matches the adaptive lower bound for $x\in\{0,1\}^n$5 and, for $x\in\{0,1\}^n$6, reduces the average number of queries below the classical non-adaptive information-theoretic bound (Soleymani et al., 1 Sep 2025). The paper presents this as an experimental breakthrough rather than a theorem-driven one.

4. Sparse-graph, LDPC, and coding-theoretic constructions

A major line of QGT research uses sparse graph codes. In one construction, items are left nodes of a bi-regular bipartite graph, each right node corresponds to a bundle of quantitative tests, and each bundle carries a short signature matrix built from a binary $x\in\{0,1\}^n$7-error-correcting BCH code (Karimi et al., 2019). A right node is $x\in\{0,1\}^n$8-resolvable if it contains at most $x\in\{0,1\}^n$9 defectives; its first test reveals exactly how many defectives it contains, and the BCH syndrome part identifies which local positions are active. A peeling decoder then iteratively removes recovered defectives from neighboring bins. The resulting test complexity is

$\|x\|_0=k$0

with exact recovery with probability approaching one in the sub-linear regime $\|x\|_0=k$1, and for $\|x\|_0=k$2 the encoding and decoding complexities are

$\|x\|_0=k$3

respectively (Karimi et al., 2019). The numerical density-evolution analysis identifies $\|x\|_0=k$4 as the choice minimizing the number of tests (Karimi et al., 2019).

The irregular sparse-graph extension replaces the regular left degree by an optimized degree profile. With maximum left degree $\|x\|_0=k$5, average left degree $\|x\|_0=k$6, and a constant $\|x\|_0=k$7 determined by density evolution, the test count becomes

$\|x\|_0=k$8

again with exact recovery of all defectives with probability approaching one (Karimi et al., 2019). For $\|x\|_0=k$9, the paper states testing and recovery complexities

$x_j\sim \mathrm{Bernoulli}(K/N)$0

and shows that irregular designs outperform both the earlier regular sparse-graph scheme and the contemporaneous method of Gebhard et al. in test count (Karimi et al., 2019).

LDPC-based QGT reframes the pooling matrix as the parity-check matrix of a sparse Tanner graph. One variant replaces BCH component codes by simple single-parity-check constraints and then applies spatial coupling (Mashauri et al., 2024). In that model, a check resolves all attached items whenever its residual count is $x_j\sim \mathrm{Bernoulli}(K/N)$1 or equals its current degree, so the quantitative output itself supplies strong local information. The uncoupled LDPC rate is $x_j\sim \mathrm{Bernoulli}(K/N)$2, whereas the earlier GLDPC/BCH design uses

$x_j\sim \mathrm{Bernoulli}(K/N)$3

tests per item (Mashauri et al., 2024). At fixed $x_j\sim \mathrm{Bernoulli}(K/N)$4, the spatially coupled LDPC-QGT thresholds reported in Table III rise from $x_j\sim \mathrm{Bernoulli}(K/N)$5 to $x_j\sim \mathrm{Bernoulli}(K/N)$6, depending on $x_j\sim \mathrm{Bernoulli}(K/N)$7, and the LDPC-based scheme substantially outperforms the original BCH-based design (Mashauri et al., 2024).

A further extension groups items into bundles of size $x_j\sim \mathrm{Bernoulli}(K/N)$8 and introduces hidden non-binary variables

$x_j\sim \mathrm{Bernoulli}(K/N)$9

representing the number of defectives within each bundle (Mashauri et al., 2024). The physical tests remain ordinary QGT tests, but the decoder passes lower and upper bounds on these bundle counts, in a manner inspired by counter braids. At a test rate of $\mathbb E[\|x\|_0]=K$0, the asymptotic threshold improves from $\mathbb E[\|x\|_0]=K$1 for the binary baseline $\mathbb E[\|x\|_0]=K$2 to $\mathbb E[\|x\|_0]=K$3 for $\mathbb E[\|x\|_0]=K$4 and $\mathbb E[\|x\|_0]=K$5 for $\mathbb E[\|x\|_0]=K$6 in the reported table (Mashauri et al., 2024). The paper interprets this as evidence that decoder-side non-binary structure can materially strengthen sparse-graph QGT without changing the measurement process itself.

For the binary LDPC-based construction, soft-decision belief propagation also helps. Instead of peeling only on deterministic implications, a sum-product decoder treats each test as an exact cardinality factor

$\mathbb E[\|x\|_0]=K$7

and passes soft probabilities (Xhemrishi et al., 21 Jan 2025). In simulation, the misdetection rate is consistently lower than that of peeling; at target $\mathbb E[\|x\|_0]=K$8, the reported prevalence gains range from about $\mathbb E[\|x\|_0]=K$9 to about $x\in\{0,1\}^n$00, with the largest gains at larger blocklengths (Xhemrishi et al., 21 Jan 2025).

5. Statistical inference, AMP, and learned decoders

Beyond combinatorial decoding, QGT admits statistical estimators based on approximate inference. A rigorous Approximate Message Passing (AMP) theory has been developed for pooled-data problems with Bernoulli pooling matrices, with QGT as the two-category special case (Tan et al., 2023). After centering and scaling the Bernoulli design,

$x\in\{0,1\}^n$01

the QGT model becomes

$x\in\{0,1\}^n$02

and AMP state evolution applies despite the non-Gaussian design (Tan et al., 2023). In the large-system linear-prevalence regime, the effective scalar channel for each coordinate is asymptotically Gaussian, and for thresholded AMP the paper derives exact limiting formulas

$x\in\{0,1\}^n$03

(Tan et al., 2023). This supplies a precise operating-curve description rather than only a success/failure threshold.

A complementary line studies explicit estimators under several noise models. For non-adaptive QGT with Bernoulli design and exact sparsity $x\in\{0,1\}^n$04, one paper analyzes both a correlation-based linear estimator and a combinatorial LSE in the noiseless, additive Gaussian, and noisy Z-channel settings (Li et al., 16 Jan 2026). In the Gaussian case, the LSE succeeds with

$x\in\{0,1\}^n$05

and the converse matches in order; the simpler correlation decoder requires

$x\in\{0,1\}^n$06

(Li et al., 16 Jan 2026). The same paper shows that in the noiseless model the linear decoder succeeds with

$x\in\{0,1\}^n$07

tests, which is polynomial-time but not information-theoretically optimal (Li et al., 16 Jan 2026).

Learned inversion has also been explored directly. In a non-adaptive QGT model with sparse bounded integer perturbations,

$x\in\{0,1\}^n$08

a fully connected neural network can be trained to map measurements to defect indicators (Grampurohit et al., 8 Dec 2025). The most distinctive claim is structural verifiability: the Jacobian of the trained network with respect to the input can be used to recover an estimate $x\in\{0,1\}^n$09 of the hidden pooling matrix (Grampurohit et al., 8 Dec 2025). Under

$x\in\{0,1\}^n$10

moderate architectures recovered the pooling matrix with errors as low as $x\in\{0,1\}^n$11, while a Level 5 model achieved $x\in\{0,1\}^n$12 and success rate $x\in\{0,1\}^n$13 (Grampurohit et al., 8 Dec 2025). The same paper is explicit that these are empirical rather than theorem-backed guarantees.

6. Extensions, applications, and neighboring measurement models

QGT is often idealized relative to physical assays. In PCR-based pooled testing, for example, the instrument produces cycle-threshold values rather than exact counts, so semiquantitative models are often a better measurement abstraction (Cheraghchi et al., 2021). In that sense, SQGT provides a practically motivated intermediate layer: binary testing is too coarse, exact QGT may be too optimistic, and coarse quantized counts are often the natural compromise (Cheraghchi et al., 2021).

One qPCR screening study implements this viewpoint through a two-stage adaptive semiquantitative protocol that quantizes $x\in\{0,1\}^n$14 values into three bins using thresholds $x\in\{0,1\}^n$15 (Nambiar et al., 2023). In simulations of $x\in\{0,1\}^n$16 individuals at prevalence $x\in\{0,1\}^n$17, the reported optimum uses group size $x\in\{0,1\}^n$18 and requires $x\in\{0,1\}^n$19 tests, compared with Dorfman’s $x\in\{0,1\}^n$20 and $x\in\{0,1\}^n$21 tests, a reduction of about $x\in\{0,1\}^n$22; the corresponding false negative rates are about $x\in\{0,1\}^n$23 and $x\in\{0,1\}^n$24 (Nambiar et al., 2023). These are SQGT rather than QGT results, but they illustrate how count-valued pooled information is translated into deployable laboratory workflows.

QGT also appears in monitoring and networking. In adaptive combinatorial QGT, heavy-hitter detection maps naturally to testing aggregated flows, where a quantitative test returns the number of heavy hitters in a subset of flows (Wang et al., 2014). The same work reports simulation examples showing orders of magnitude improvement over two prevailing sampling-based approaches in detection accuracy and counter consumption, and explicitly mentions anomaly detection and wideband spectrum sensing in cognitive radio systems as further applications (Wang et al., 2014).

Structured priors can further change the role of quantitative information. For a single burst of positives in an ordered population, studied in a semiquantitative model, the minimum number of tests is shown to depend essentially only on the largest threshold $x\in\{0,1\}^n$25, and an order-optimal scheme recovers any burst of length $x\in\{0,1\}^n$26 using roughly

$x\in\{0,1\}^n$27

measurements (Li et al., 2023). This suggests that when support structure is strong, saturation level can matter more than fine quantization. A plausible implication is that analogous structural priors in QGT can be at least as consequential as improvements in raw count resolution, though the interval result itself is proved in SQGT rather than exact QGT (Li et al., 2023).

Across these developments, QGT occupies a central position: it is the exact-count model from which quantized, saturated, and application-specific variants are derived, and it remains a reference point for both information-theoretic benchmarks and algorithmic design.