FQS: Disambiguation in Diverse Research Fields

• FQS is a polysemous term that represents distinct concepts, including a fidelity quantum similarity loss in hybrid quantum-classical learning, a free quaternion optimizer for quantum circuits, and a unitarity criterion in conformal field theory.
• FQS also names curated data resources, such as a fundus quality dataset for biomedical imaging and the Forgotten Quadrant Survey for mapping Galactic molecular gas.
• Its application across domains enhances model convergence in quantum algorithms, refines quality assessments in imaging, and supports rigorous classification in mathematical physics.

FQS is a polysemous research abbreviation rather than a single concept. In current arXiv literature it denotes at least five distinct objects: the "Fidelity Quantum Similarity" loss used in hybrid quantum-classical remote-sensing representation learning; "Free Quaternion Selection" for sequential optimization of parameterized quantum circuits; the Friedan–Qiu–Shenker unitarity criterion for the Neveu–Schwarz algebra; the Fundus Quality Score dataset for fundus image quality assessment; and the Forgotten Quadrant Survey of Galactic molecular gas (Hossain et al., 29 May 2026, Pankkonen et al., 26 Mar 2025, Palcoux, 2010, Gong et al., 2024, Benedettini et al., 2019). A common source of confusion is that these usages are unrelated despite sharing the same abbreviation; in nearby PDE notation, $\fqs$ instead denotes the fractional $(q,s)$-Laplacian and is not an acronym (Dhanya et al., 2024).

1. Polysemy and field-specific meaning

Within machine learning for Earth observation, FQS denotes a differentiable fidelity-based loss defined through a SWAP test and incorporated into HQ-JEPA. Within variational quantum algorithms, FQS denotes a gradient-free optimizer that updates one single-qubit gate at a time by solving a constrained quadratic problem in quaternion coordinates. In mathematical physics, the FQS criterion refers to the Friedan–Qiu–Shenker unitarity classification for highest-weight modules of the Neveu–Schwarz algebra. In biomedical imaging, FQS is the name of a fundus-image quality dataset with continuous Mean Opinion Scores and three-level labels. In radio astronomy, FQS is the Forgotten Quadrant Survey, an outer-Galaxy $^{12}\mathrm{CO}$ and $^{13}\mathrm{CO}$ mapping program (Hossain et al., 29 May 2026, Pankkonen et al., 26 Mar 2025, Palcoux, 2010, Gong et al., 2024, Benedettini et al., 2019).

This distribution of meanings suggests that “FQS” is best treated as a disambiguation term. Technical interpretation depends entirely on domain context: in one setting it is a loss, in another an optimizer, in another a unitarity criterion, and elsewhere a dataset or survey name.

2. Fidelity Quantum Similarity in HQ-JEPA

In "HQ-JEPA: Hybrid Quantum Joint-Embedding Predictive Architecture for Cross-Modal Remote Sensing Representation Learning" (Hossain et al., 29 May 2026), FQS denotes the differentiable SWAP-test-based Fidelity Quantum Similarity loss. HQ-JEPA extends JEPA-style masked latent prediction to paired Sentinel-1 and Sentinel-2 imagery by predicting masked target representations from visible context regions while aligning heterogeneous modality features in a shared embedding space. The framework combines four complementary objectives: latent token prediction, cross-modal token alignment, SIGReg-based Gaussian regularization in the fused latent space, and the FQS loss.

The starting point is quantum-state fidelity for pure states:

$$F(|\psi\rangle,|\phi\rangle)=|\langle \psi|\phi\rangle|^2.$$

HQ-JEPA encodes a student latent summary $z_s$ into a parameterized multi-qubit state $U_s(z_s)|0\ldots 0\rangle$, and a teacher latent target summary $z_t$ into $U_t(z_t)|0\ldots 0\rangle$. The SWAP-test circuit estimates fidelity through the ancilla expectation

$$\langle Z_{\mathrm{anc}}\rangle = \mathrm{Prob}(\mathrm{ancilla}=0)-\mathrm{Prob}(\mathrm{ancilla}=1),$$

with

$(q,s)$0

The FQS loss is then defined as the infidelity

$(q,s)$1

The derivation follows the standard SWAP-test construction: prepare the ancilla in $(q,s)$2; place the student state in register $(q,s)$3 and the teacher state in register $(q,s)$4; apply controlled-SWAP gates conditioned on the ancilla; apply a final Hadamard to the ancilla; and measure $(q,s)$5. One can show that

$(q,s)$6

so that $(q,s)$7.

Several mathematical properties are explicit. The range is $(q,s)$8, hence $(q,s)$9. Because $^{12}\mathrm{CO}$0 and $^{12}\mathrm{CO}$1 are parameterized by continuous rotations such as $^{12}\mathrm{CO}$2, $^{12}\mathrm{CO}$3, and $^{12}\mathrm{CO}$4, $^{12}\mathrm{CO}$5 is differentiable almost everywhere. Its gradients satisfy

$^{12}\mathrm{CO}$6

and PennyLane’s adjoint-differentiation is used for efficient gradients with respect to both encoder parameters and the latent-to-rotation mapping. The paper also emphasizes that fidelity captures full Hilbert-space overlap, not just first- or second-order statistics.

The circuit implementation uses 11 qubits total. Teacher and student registers are encoded with layers of $^{12}\mathrm{CO}$7, $^{12}\mathrm{CO}$8, and $^{12}\mathrm{CO}$9 entangling gates; the student register also receives additional predictor layers parameterized by $^{13}\mathrm{CO}$0, with $^{13}\mathrm{CO}$1, $^{13}\mathrm{CO}$2, $^{13}\mathrm{CO}$3, and $^{13}\mathrm{CO}$4. Controlled-SWAP operations are then applied between the two registers, followed by a final Hadamard and measurement of the ancilla. In HQ-JEPA, this routine is called once per masked target block or pooled summary and is back-propagated through the classical encoders and rotation mappings.

FQS is integrated into the overall self-supervised objective as

$^{13}\mathrm{CO}$5

with default weights $^{13}\mathrm{CO}$6, $^{13}\mathrm{CO}$7, $^{13}\mathrm{CO}$8, and $^{13}\mathrm{CO}$9. Here $$F(|\psi\rangle,|\phi\rangle)=|\langle \psi|\phi\rangle|^2.$$0 is SmoothL1 between student-predicted tokens and EMA teacher tokens, $$F(|\psi\rangle,|\phi\rangle)=|\langle \psi|\phi\rangle|^2.$$1 is SmoothL1 between projected S1 tokens and predicted S2 tokens, and $$F(|\psi\rangle,|\phi\rangle)=|\langle \psi|\phi\rangle|^2.$$2 is SIGReg Gaussian characteristic-function matching in the fused cross-modal latent space.

The empirical effects reported for FQS are specific. $$F(|\psi\rangle,|\phi\rangle)=|\langle \psi|\phi\rangle|^2.$$3 converges fastest, reaching near zero by epoch $$F(|\psi\rangle,|\phi\rangle)=|\langle \psi|\phi\rangle|^2.$$4. On m-BigEarthNet, the token-only variant yields $$F(|\psi\rangle,|\phi\rangle)=|\langle \psi|\phi\rangle|^2.$$5 $$F(|\psi\rangle,|\phi\rangle)=|\langle \psi|\phi\rangle|^2.$$6 under linear probing and fine-tuning, while adding $$F(|\psi\rangle,|\phi\rangle)=|\langle \psi|\phi\rangle|^2.$$7 only raises this to $$F(|\psi\rangle,|\phi\rangle)=|\langle \psi|\phi\rangle|^2.$$8; adding $$F(|\psi\rangle,|\phi\rangle)=|\langle \psi|\phi\rangle|^2.$$9 only gives $z_s$0; and the full model reaches $z_s$1, described as the best single setting on nearly every downstream task. Including $z_s$2 in any combination consistently adds $z_s$3–$z_s$4 points absolute over the corresponding no-quantum baseline. Gains are especially strong on challenging segmentation tasks and on small-data settings; for m-cashew, Jaccard improves from $z_s$5 under linear probing and from $z_s$6 under fine-tuning when FQS is added.

3. Free Quaternion Selection for parameterized quantum circuits

In variational quantum-circuit optimization, FQS denotes Free Quaternion Selection, a gradient-free sequential optimizer for single-qubit gates (Pankkonen, 26 Mar 2026, Pankkonen et al., 9 Oct 2025). The central idea is to replace restricted angle-only updates by a full $z_s$7 update represented as a unit quaternion. For an isolated gate on qubit $z_s$8,

$z_s$9

with $U_s(z_s)|0\ldots 0\rangle$0 and $U_s(z_s)|0\ldots 0\rangle$1.

After fixing all other gates, the local cost is written as

$U_s(z_s)|0\ldots 0\rangle$2

Expanding the trace yields the quadratic form

$U_s(z_s)|0\ldots 0\rangle$3

where $U_s(z_s)|0\ldots 0\rangle$4 is real symmetric. Its entries can be reconstructed from 10 circuit evaluations. Minimizing $U_s(z_s)|0\ldots 0\rangle$5 over the unit sphere is equivalent to the eigenvalue problem

$U_s(z_s)|0\ldots 0\rangle$6

so the optimal update is the normalized eigenvector with the lowest eigenvalue. This gives a closed-form greedy decrease of the local cost for each gate subproblem.

The method is “free” in the sense that it optimizes all three axis-degrees plus the rotation angle simultaneously, rather than fixing the generator. This is the key contrast with Rotosolve and Fraxis. Rotosolve adjusts only the angle around a fixed axis and requires three evaluations per gate. Fraxis adjusts one axis degree of freedom plus angle and requires six evaluations per gate. FQS requires ten circuit evaluations and one $U_s(z_s)|0\ldots 0\rangle$7 eigenproblem per gate, but it is described as strictly more expressive and as returning the global optimum on $U_s(z_s)|0\ldots 0\rangle$8 for each gate subproblem.

The same papers present FQS as particularly relevant for noisy intermediate-scale quantum settings because it is gradient-free and gate-local. A practical implication is that it can escape minima that angle-only or restricted-axis updates may not resolve, but the higher per-gate evaluation cost and greater sensitivity to measurement noise create an immediate expressivity-versus-sampling trade-off.

4. Hybrid, two-gate, and gate-freezing extensions of Free Quaternion Selection

Subsequent work develops FQS along three directions: hybridization with cheaper sequential optimizers, simultaneous optimization of two gates, and selective freezing of nearly converged gates (Pankkonen et al., 26 Mar 2025, Pankkonen, 26 Mar 2026, Pankkonen et al., 10 Jul 2025).

The hybrid line couples FQS to Rotosolve–Haar. In Rotosolve–Haar, each Pauli-$U_s(z_s)|0\ldots 0\rangle$9 generator is first conjugated by a Haar-random $z_t$0, and the angle is optimized one gate at a time by sampling the cost at three angles $z_t$1 and solving the sinusoidal minimum analytically. Two hybrid schemes are described. The cycle-specific hybrid performs a full Rotosolve–Haar pass on cycles with $z_t$2 and an FQS pass on cycles with $z_t$3. The gate-specific hybrid samples $z_t$4 for each gate and chooses either a Rotosolve–Haar update or an FQS update according to a mixing probability $z_t$5. Numerical benchmarks use energy expectation $z_t$6 and trace-distance to a random target state versus circuit evaluations. The reported trade-off is consistent: Rotosolve–Haar converges fastest in the first steps because it uses 3 evaluations per gate, but it typically plateaus at a higher energy; FQS is the most expressive single-quaternion method and reaches the best final energy, but its 10-evaluation cost slows early progress. In almost all tested Hamiltonians, the best mean performance versus circuit evaluations is achieved by a gate-specific or cycle-specific hybrid rather than by Rotosolve, Fraxis, or FQS alone.

Two-Gate FQS, abbreviated TGFQS, extends the same principle to pairs of arbitrary single-qubit gates. Instead of a quadratic local cost in one quaternion, it constructs an exact quartic local cost in two quaternions:

$z_t$7

All coefficients are reconstructed from 100 circuit evaluations per gate pair, and a constrained classical optimizer such as SLSQP minimizes the quartic form under $z_t$8. The measurement overhead is therefore much larger than for single-gate FQS: 100 runs per two-gate update, or an average of 50 per gate, versus 10 for FQS. Empirically, however, TGFQS frequently achieves lower final relative error or infidelity than FQS. On spin Hamiltonians, molecular Hamiltonians, and quantum state preparation, random and half-shifted gate-pairing strategies perform best in many tested settings. The reported summary includes improvements such as up to two orders of magnitude lower final relative energy error in Fermi–Hubbard and large gains on transverse-field Ising instances.

Gate freezing addresses a different bottleneck: many gates become nearly static after a few iterations. The freezing rule measures the change between successive quaternion updates by the quaternion-sphere distance

$z_t$9

If $U_t(z_t)|0\ldots 0\rangle$0, gate $U_t(z_t)|0\ldots 0\rangle$1 is frozen for the next $U_t(z_t)|0\ldots 0\rangle$2 iterations. Frozen gates keep their last optimized quaternion and incur only counter updates rather than fresh circuit evaluations. In the reported Fermi–Hubbard example, with $U_t(z_t)|0\ldots 0\rangle$3 and $U_t(z_t)|0\ldots 0\rangle$4, baseline FQS reaches $U_t(z_t)|0\ldots 0\rangle$5 in $U_t(z_t)|0\ldots 0\rangle$6 iterations, whereas gate freezing reaches $U_t(z_t)|0\ldots 0\rangle$7 in $U_t(z_t)|0\ldots 0\rangle$8 iterations and shrinks the interquartile range by $U_t(z_t)|0\ldots 0\rangle$9. On larger problems, the freezing extension can reduce the number of costly circuit evaluations by up to 50%.

Taken together, these extensions do not change the core FQS update rule; they change how often it is invoked, whether it is interleaved with cheaper methods, and whether the local subproblem is one-gate or two-gate. This suggests that the primary design variable is no longer only expressivity, but expressivity under a fixed measurement budget.

5. Friedan–Qiu–Shenker criterion for the Neveu–Schwarz algebra

In the operator-algebraic and conformal-field-theoretic literature, FQS denotes the Friedan–Qiu–Shenker criterion for unitarity of highest-weight modules of the Neveu–Schwarz algebra (Palcoux, 2010). The criterion classifies exactly when a highest-weight Verma module $$\langle Z_{\mathrm{anc}}\rangle = \mathrm{Prob}(\mathrm{ancilla}=0)-\mathrm{Prob}(\mathrm{ancilla}=1),$$0, with central charge $$\langle Z_{\mathrm{anc}}\rangle = \mathrm{Prob}(\mathrm{ancilla}=0)-\mathrm{Prob}(\mathrm{ancilla}=1),$$1 and lowest $$\langle Z_{\mathrm{anc}}\rangle = \mathrm{Prob}(\mathrm{ancilla}=0)-\mathrm{Prob}(\mathrm{ancilla}=1),$$2-eigenvalue $$\langle Z_{\mathrm{anc}}\rangle = \mathrm{Prob}(\mathrm{ancilla}=0)-\mathrm{Prob}(\mathrm{ancilla}=1),$$3, admits a positive-definite Hermitian form.

The criterion has two branches. The continuous series is

$$\langle Z_{\mathrm{anc}}\rangle = \mathrm{Prob}(\mathrm{ancilla}=0)-\mathrm{Prob}(\mathrm{ancilla}=1),$$4

The discrete series is indexed by integers $$\langle Z_{\mathrm{anc}}\rangle = \mathrm{Prob}(\mathrm{ancilla}=0)-\mathrm{Prob}(\mathrm{ancilla}=1),$$5 and pairs $$\langle Z_{\mathrm{anc}}\rangle = \mathrm{Prob}(\mathrm{ancilla}=0)-\mathrm{Prob}(\mathrm{ancilla}=1),$$6 satisfying

$$\langle Z_{\mathrm{anc}}\rangle = \mathrm{Prob}(\mathrm{ancilla}=0)-\mathrm{Prob}(\mathrm{ancilla}=1),$$7

with

$$\langle Z_{\mathrm{anc}}\rangle = \mathrm{Prob}(\mathrm{ancilla}=0)-\mathrm{Prob}(\mathrm{ancilla}=1),$$8

The statement is that $$\langle Z_{\mathrm{anc}}\rangle = \mathrm{Prob}(\mathrm{ancilla}=0)-\mathrm{Prob}(\mathrm{ancilla}=1),$$9 is unitary if and only if either $(q,s)$00 and $(q,s)$01, or $(q,s)$02 is one of the discrete points $(q,s)$03. There are no other unitary highest-weight modules with $(q,s)$04.

The proof structure in Palcoux’s exposition combines Kac-determinant analysis with the GKO coset construction. The determinant of the Shapovalov form at level $(q,s)$05 factorizes over vanishing curves $(q,s)$06, and these curves stratify the $(q,s)$07-plane for $(q,s)$08. On every open region bounded by consecutive curves, one finds a level with negative determinant, producing a ghost. The “first-intersection” argument then isolates the discrete minimal-model points as the only ghost-free locations on the curves. Sufficiency is proved by the GKO construction, where multiplicity spaces $(q,s)$09 inherit unitarity from an ambient tensor product of $(q,s)$10-modules and a free Majorana fermion.

A technical subtlety emphasized in the paper is the handling of first intersections, especially the distinction between $(q,s)$11 and $(q,s)$12, and the role of transversality in excluding spurious ghost-free points. At true minimal-model points, exactly two distinct singular vectors appear, reproducing the expected minimal-model character structure.

6. FQS as dataset name and astronomical survey

In medical-image quality assessment, FQS stands for Fundus Quality Score (Gong et al., 2024). The dataset contains 2,246 color fundus images of size $(q,s)$13, with 92% drawn from routine clinical examinations and 8% from adolescent myopia screening. Each image has two labels: a continuous Mean Opinion Score in $(q,s)$14 and a three-level quality label. The class counts are Good $(q,s)$15, Usable $(q,s)$16, and Reject $(q,s)$17. Annotation was performed by six ophthalmologist graders, three senior and three junior, with final MOS computed by a weighted average

$(q,s)$18

Inter-rater consistency is reported through the standard deviation distribution, with 25% of images having $(q,s)$19, 50% having $(q,s)$20, and 75% having $(q,s)$21. The accompanying FTHNet model is a Transformer-based Hypernetwork with a Transformer backbone, distortion perception network, parameter hypernetwork, and five-layer target regressor trained with Smooth-L1 loss. On the FQS dataset, average 10-fold cross-validation yields PLCC $(q,s)$22 and SRCC $(q,s)$23; the reported model variants are FTHNet-S with 5.66 M parameters and FTHNet-L with 14.88 M parameters.

In Galactic astronomy, FQS denotes the Forgotten Quadrant Survey (Benedettini et al., 2019). This ESO large project used the Arizona Radio Observatory 12 m antenna to map the Galactic Plane over $(q,s)$24 and $(q,s)$25 in both $(q,s)$26 and $(q,s)$27. The survey achieved a beam HPBW of approximately $(q,s)$28 at 115 GHz, used On-The-Fly mapping in $(q,s)$29 tiles, and delivered data cubes rebinned to $(q,s)$30 pixels and 1 km s$(q,s)$31 channels, with 0.3 km s$(q,s)$32 products also available. The first release includes a catalogue of 263 molecular clouds extracted from the $(q,s)$33 cube using SCIMES. Distances span 1–8.6 kpc and cluster into Local, Perseus, and Outer arm groupings. The survey identifies two structural classes: Class I clouds with $(q,s)$34 pc and often $(q,s)$35, and Class II sub-parsec structures with $(q,s)$36–2 pc and $(q,s)$37. The derived mean CO-to-$(q,s)$38 conversion factors are $(q,s)$39 for $(q,s)$40 and $(q,s)$41 for $(q,s)$42.

These latter uses illustrate a final distinction: in fundus imaging and Galactic surveys, FQS names curated data resources rather than losses, optimizers, or mathematical criteria. That contrast is easy to miss when the abbreviation is encountered without surrounding disciplinary context.