Titans-QFWP: A Regime-Aware Hybrid Quantum Fast Weight Programmer for Portfolio Optimization

# A Regime-Aware Hybrid Quantum Fast Weight Programmer for Portfolio Optimization
Source: [https://arxiv.org/html/2608.29093](https://arxiv.org/html/2608.29093)
## Titans\-QFWP: A Regime\-Aware Hybrid Quantum Fast Weight Programmer for Portfolio OptimizationThanks:The views expressed in this article are those of the authors and do not represent the views of Wells Fargo\. This article is for informational purposes only\. Nothing contained in this article should be construed as investment advice\. Wells Fargo makes no express or implied warranties and expressly disclaims all legal, tax, and accounting implications related to this article\.Thanks:†These authors contributed equally to this work\.Thanks:\*Corresponding author\.

[https://orcid.org/0009-0002-5141-7909](https://orcid.org/0009-0002-5141-7909)Ming\-Kai Hung†Affiliation:Dept\. of Technology Application and Human Resource Development National Taiwan Normal University Taipei, Taiwan[https://orcid.org/0009-0000-3296-1824](https://orcid.org/0009-0000-3296-1824)Jun\-Hao Chen†Affiliation:Dept\. of Bioenvironmental Systems Engineering National Taiwan University Taipei, Taiwan[https://orcid.org/0000-0003-0114-4826](https://orcid.org/0000-0003-0114-4826)Samuel Yen\-Chi Chen\*Affiliation:Wells Fargo USAAffiliation:

###### Abstract

We propose Titans\-QFWP, a hybrid reinforcement learning architecture integrating a Quantum Fast Weight Programmer with Titans\-style memory \(Persistence, Surprise, and Forgetting\) for adaptive portfolio optimization\. To address high\-dimensional market features, we introduce an enhanced A3C2framework with Hungarian\-alignedKK\-means clustering and scaled log\-return rewards\. Evaluated on 468 S&P 500 stocks under an Equal\-Parameter\-Count \(EPC\) benchmark with approximately 3,000 trainable parameters, Titans\-QFWP achieves strong performance \(median ARR0\.42600\.4260, Calmar8\.55048\.5504, IR0\.84270\.8427\)\. Ablation results reveal that quantum gating fundamentally reshapes memory component roles, with Persistence supporting drawdown control, Surprise contributing to return generation, and Forgetting providing additional stabilization\. By stabilizing these quantum representations, the model enables defensive allocation during market drawdowns while preserving upside potential\.

###### Index Terms:

Deep Reinforcement Learning, Fast Weight Programmer, Portfolio Optimization, Quantum Machine Learning, Regime\-Aware Architecture

## IIntroduction

Financial markets are inherently time\-varying\[[1](https://arxiv.org/html/2608.29093#bib.bib1)\]and non\-stationary\[[2](https://arxiv.org/html/2608.29093#bib.bib2)\], driven by regime shifts and secular changes\[[3](https://arxiv.org/html/2608.29093#bib.bib3)\]that undermine traditional portfolio optimization\[[4](https://arxiv.org/html/2608.29093#bib.bib5),[5](https://arxiv.org/html/2608.29093#bib.bib4)\]\. Deep Reinforcement Learning \(DRL\) addresses these challenges through sequential policy optimization\[[6](https://arxiv.org/html/2608.29093#bib.bib6),[7](https://arxiv.org/html/2608.29093#bib.bib7),[8](https://arxiv.org/html/2608.29093#bib.bib8)\], but it often struggles with high computational costs\[[9](https://arxiv.org/html/2608.29093#bib.bib9)\]and high\-dimensional state spaces\[[10](https://arxiv.org/html/2608.29093#bib.bib10)\]\. To manage such expansive spaces, static clustering is typically employed; however, it remains fundamentally incapable of capturing abrupt market transitions\[[11](https://arxiv.org/html/2608.29093#bib.bib11)\]\. In the quantum domain, Variational Quantum Circuits \(VQC\) can exploit high\-dimensional Hilbert spaces\[[12](https://arxiv.org/html/2608.29093#bib.bib12)\], yet they lack adaptability due to fixed parameterization\[[13](https://arxiv.org/html/2608.29093#bib.bib13),[14](https://arxiv.org/html/2608.29093#bib.bib14),[15](https://arxiv.org/html/2608.29093#bib.bib15)\]\. Although the Quantum Fast Weight Programmer \(QFWP\) introduces dynamic weights\[[16](https://arxiv.org/html/2608.29093#bib.bib16),[17](https://arxiv.org/html/2608.29093#bib.bib17)\]to overcome this, its update mechanisms often degrade memory stability\. To address these challenges, we propose a dynamically adaptive hybrid architecture with two key contributions:

1. 1\.Titans\-QFWP Architecture\.Upgrading the original Q\-A3C2\[[18](https://arxiv.org/html/2608.29093#bib.bib20)\], we replace the VQC with Titans\-QFWP\. This architecture integrates QFWP with Titans\-style memory \(Persistence, Surprise, Forgetting\)\[[19](https://arxiv.org/html/2608.29093#bib.bib18)\]\.
2. 2\.Enhanced A3C2Framework\.We redesign the A3C2reinforcement learning environment\[[18](https://arxiv.org/html/2608.29093#bib.bib20)\], incorporating Hungarian\-alignedKK\-means clustering, a defensive cash action, and a scaled log\-return reward\[[20](https://arxiv.org/html/2608.29093#bib.bib19)\]\.

## IIMethodology

Fig\. 1:System architecture of the hybrid Titans\-QFWP agent integrated with the Enhanced A3C2continuous\-reward portfolio simulation environment\.### II\-AState Representation and Temporal Alignment

Using a 60\-day lookback, we rebalance everyD=20D=20market days\. For each assetiiat timett, the 5\- and 20\-day moving averages and 20\-day volatility of log\-returns \(Ri,t\(5\),Ri,t\(20\),Vi,t\(20\)R\_\{i,t\}^\{\(5\)\},R\_\{i,t\}^\{\(20\)\},V\_\{i,t\}^\{\(20\)\}\) are cross\-sectionallyzz\-scored to form𝐅t∈ℝN×3\\mathbf\{F\}\_\{t\}\\in\\mathbb\{R\}^\{N\\times 3\}\(N=468N=468\), and then partitioned intoK=10K=10clusters viaKK\-means\. To maintain temporal consistency, the Hungarian algorithm aligns centroids by minimizingL2L\_\{2\}drift:

πt∗=arg⁡min⁡∑k=0K−1π⁡‖𝐜k,t−1−𝐜~π⁡\(k\),t‖2\.\\pi\_\{t\}^\{\*\}=\\arg\\min\_\{\\pi\}\\sum\_\{k=0\}^\{K\-1\}\\\|\\mathbf\{c\}\_\{k,t\-1\}\-\\tilde\{\\mathbf\{c\}\}\_\{\\pi\(k\),t\}\\\|\_\{2\}\.\(1\)To filter short\-term market noise for the next time step, the matched centroids are smoothed with EMA \(λ=0\.8\\lambda=0\.8\) via𝐜k,t=λ​𝐜k,t−1\+\(1−λ\)​𝐜~πt∗​\(k\),t\\mathbf\{c\}\_\{k,t\}=\\lambda\\mathbf\{c\}\_\{k,t\-1\}\+\(1\-\\lambda\)\\tilde\{\\mathbf\{c\}\}\_\{\\pi\_\{t\}^\{\*\}\(k\),t\}, initialized based on the 20\-day log\-return componentck,0\(R\(20\)\)c\_\{k,0\}^\{\(R^\{\(20\)\}\)\}\. Each cluster𝒞k\\mathcal\{C\}\_\{k\}yields five features: meansck,t\(X\)c\_\{k,t\}^\{\(X\)\}forX∈\{R\(5\),R\(20\),V\(20\)\}X\\in\\\{R^\{\(5\)\},R^\{\(20\)\},V^\{\(20\)\}\\\}, relative size\|𝒞k\|/N\|\\mathcal\{C\}\_\{k\}\|/N, and momentum dispersion\(\|𝒞k\|−1​∑i∈𝒞k\(Ri,t\(20\)−ck,t\(R\(20\)\)\)2\)1/2\(\|\\mathcal\{C\}\_\{k\}\|^\{\-1\}\\sum\_\{i\\in\\mathcal\{C\}\_\{k\}\}\(R\_\{i,t\}^\{\(20\)\}\-c\_\{k,t\}^\{\(R^\{\(20\)\}\)\}\)^\{2\}\)^\{1/2\}\. Concatenating these5​K5Kfeatures with three S&P 500 index features \(Rs​p​x,t\(5\),Rs​p​x,t\(20\),Vs​p​x,t\(20\)R\_\{spx,t\}^\{\(5\)\},R\_\{spx,t\}^\{\(20\)\},V\_\{spx,t\}^\{\(20\)\}\) forms the 53\-dimensional state𝐬t\\mathbf\{s\}\_\{t\}\(empty clusters zero\-padded\)\.

### II\-BPortfolio Construction and Reward Function

At each 20\-day steptt, let𝐰target,t−1\\mathbf\{w\}\_\{\\mathrm\{target\},t\-1\}be the prior target weight vector and𝐆t=exp⁡\(∑τ=120𝐫τ\)\\mathbf\{G\}\_\{t\}=\\exp\(\\sum\_\{\\tau=1\}^\{20\}\\mathbf\{r\}\_\{\\tau\}\)the compound gross return vector derived from daily log\-returns𝐫τ\\mathbf\{r\}\_\{\\tau\}\. The drifted weights are calculated as𝐰drift,t=\(𝐰target,t−1⊙𝐆t\)/\(𝐰target,t−1⊤​𝐆t\)\\mathbf\{w\}\_\{\\mathrm\{drift\},t\}=\(\\mathbf\{w\}\_\{\\mathrm\{target\},t\-1\}\\odot\\mathbf\{G\}\_\{t\}\)/\(\\mathbf\{w\}\_\{\\mathrm\{target\},t\-1\}^\{\\top\}\\mathbf\{G\}\_\{t\}\), where⊙\\odotdenotes the element\-wise multiplication\. The new weights are then assigned following an inverse volatility strategy,𝐰target,t=𝝈t−1/‖𝝈t−1‖1\\mathbf\{w\}\_\{\\mathrm\{target\},t\}=\\bm\{\\sigma\}\_\{t\}^\{\-1\}/\\\|\\bm\{\\sigma\}\_\{t\}^\{\-1\}\\\|\_\{1\}, using the 60\-day historical volatility vector𝝈t\\bm\{\\sigma\}\_\{t\}\. If the cash actionat=Ka\_\{t\}=Kis chosen,𝐰target,t=𝟎\\mathbf\{w\}\_\{\\mathrm\{target\},t\}=\\mathbf\{0\}\. Rebalancing to these target weights at a cost ratec=0\.0015c=0\.0015yields a net survival fractionμt∈\(0,1\]\\mu\_\{t\}\\in\(0,1\], which is solved by successive substitution:μt=1−c​‖𝐰target,t−μt​𝐰drift,t‖1\\mu\_\{t\}=1\-c\\\|\\mathbf\{w\}\_\{\\mathrm\{target\},t\}\-\\mu\_\{t\}\\mathbf\{w\}\_\{\\mathrm\{drift\},t\}\\\|\_\{1\}\. With the gross portfolio returnRportfolio,t=𝐰target,t−1⊤​\(𝐆t−𝟏\)R\_\{\\mathrm\{portfolio\},t\}=\\mathbf\{w\}\_\{\\mathrm\{target\},t\-1\}^\{\\top\}\(\\mathbf\{G\}\_\{t\}\-\\mathbf\{1\}\), the net realized return is given byRrealized,t=μt​\(1\+Rportfolio,t\)−1R\_\{\\mathrm\{realized\},t\}=\\mu\_\{t\}\(1\+R\_\{\\mathrm\{portfolio\},t\}\)\-1\. To ensure stability, the reward floors this return at10−610^\{\-6\}and scales it by 100, treated as a hyperparameter:

rt=100⋅ln⁡\(max⁡\(1\+Rrealized,t,10−6\)\)\.r\_\{t\}=100\\cdot\\ln\\Bigl\(\\max\\bigl\(1\+R\_\{\\mathrm\{realized\},t\},\\,10^\{\-6\}\\bigr\)\\Bigr\)\.\(2\)

### II\-CTitans\-QFWP Architecture and Optimization

As illustrated in Fig\.[1](https://arxiv.org/html/2608.29093#S2.F1), the policy fuses Titans\-style memory with the Quantum Fast\-Weight Programmer \(QFWP\)\. First, we apply twotanh\\tanhactivations to compress the state𝐬t∈ℝ53\\mathbf\{s\}\_\{t\}\\in\\mathbb\{R\}^\{53\}into an encoded vector𝐱enc,t∈ℝnq\\mathbf\{x\}\_\{\\mathrm\{enc\},t\}\\in\\mathbb\{R\}^\{n\_\{q\}\}\(nq=8n\_\{q\}=8\)\. This is augmented by a trainable matrix𝐏∈ℝnq×np\\mathbf\{P\}\\in\\mathbb\{R\}^\{n\_\{q\}\\times n\_\{p\}\}\(np=4n\_\{p\}=4\) through an attention mechanism\. The attention weights are computed as𝐚t=softmax⁡\(𝐖attn​𝐱enc,t\)∈ℝnp\\mathbf\{a\}\_\{t\}=\\operatorname\{softmax\}\(\\mathbf\{W\}\_\{\\mathrm\{attn\}\}\\mathbf\{x\}\_\{\\mathrm\{enc\},t\}\)\\in\\mathbb\{R\}^\{n\_\{p\}\}with𝐖attn∈ℝnp×nq\\mathbf\{W\}\_\{\\mathrm\{attn\}\}\\in\\mathbb\{R\}^\{n\_\{p\}\\times n\_\{q\}\}, yielding the augmented representation:

𝐱in,t=𝐱enc,t\+𝐏𝐚t\.\\mathbf\{x\}\_\{\\mathrm\{in\},t\}=\\mathbf\{x\}\_\{\\mathrm\{enc\},t\}\+\\mathbf\{P\}\\mathbf\{a\}\_\{t\}\.\(3\)A slow\-program layer projects𝐱in,t\\mathbf\{x\}\_\{\\mathrm\{in\},t\}to𝐮t∈ℝhslow\\mathbf\{u\}\_\{t\}\\in\\mathbb\{R\}^\{h\_\{\\mathrm\{slow\}\}\}\(hslow=23h\_\{\\mathrm\{slow\}\}=23\)\. From𝐮t\\mathbf\{u\}\_\{t\}, three parallel softmax heads generate layer \(𝐩t\(l\)∈ℝL\\mathbf\{p\}\_\{t\}^\{\(l\)\}\\in\\mathbb\{R\}^\{L\},L=2L=2\), qubit \(𝐩t\(q\)∈ℝnq\\mathbf\{p\}\_\{t\}^\{\(q\)\}\\in\\mathbb\{R\}^\{n\_\{q\}\}\), and axis \(𝐩t\(a\)∈ℝ3\\mathbf\{p\}\_\{t\}^\{\(a\)\}\\in\\mathbb\{R\}^\{3\}for PauliX,Y,ZX,Y,Z\) distributions, whose normalized outer product forms the Surprise Tensor𝒮t=𝚵t/\(‖𝚵t‖F\+ε\)\\mathcal\{S\}\_\{t\}=\\bm\{\\Xi\}\_\{t\}/\(\\\|\\bm\{\\Xi\}\_\{t\}\\\|\_\{F\}\+\\varepsilon\), where𝚵t=𝐩t\(l\)⊗𝐩t\(q\)⊗𝐩t\(a\)\\bm\{\\Xi\}\_\{t\}=\\mathbf\{p\}\_\{t\}^\{\(l\)\}\\otimes\\mathbf\{p\}\_\{t\}^\{\(q\)\}\\otimes\\mathbf\{p\}\_\{t\}^\{\(a\)\}andε=10−8\\varepsilon=10^\{\-8\}\. Conditioned on𝐱in,t\\mathbf\{x\}\_\{\\mathrm\{in\},t\}, learned sigmoid gates for memory retention \(1−αt1\-\\alpha\_\{t\}\), momentum \(ηt\\eta\_\{t\}\), and surprise theta \(θt\\theta\_\{t\}\) dictate the evolution of fast\-weight memory from𝐌0=𝚯0=𝟎\\mathbf\{M\}\_\{0\}=\\bm\{\\Theta\}\_\{0\}=\\mathbf\{0\}:

𝐌t=ηt​𝐌t−1−θt​𝒮t,𝚯t=\(1−αt\)​𝚯t−1\+𝐌t\.\\mathbf\{M\}\_\{t\}=\\eta\_\{t\}\\mathbf\{M\}\_\{t\-1\}\-\\theta\_\{t\}\\mathcal\{S\}\_\{t\},\\quad\\bm\{\\Theta\}\_\{t\}=\(1\-\\alpha\_\{t\}\)\\bm\{\\Theta\}\_\{t\-1\}\+\\mathbf\{M\}\_\{t\}\.\(4\)To serve as valid rotation angles, the raw memory is bounded via𝚯^t=clamp⁡\(𝚯t,−π,π\)∈\[−π,π\]L×nq×3\\hat\{\\bm\{\\Theta\}\}\_\{t\}=\\operatorname\{clamp\}\(\\bm\{\\Theta\}\_\{t\},\-\\pi,\\pi\)\\in\[\-\\pi,\\pi\]^\{L\\times n\_\{q\}\\times 3\}to parameterize the variational quantum circuit\. Starting from uniform superposition\|ψ0⟩=H⊗nq​\|0⟩⊗nq\|\\psi\_\{0\}\\rangle=H^\{\\otimes n\_\{q\}\}\|0\\rangle^\{\\otimes n\_\{q\}\}, a feature encoding layer\|ψenc⟩=⨂q=1nqRY​\(π​xin,t,q\)​\|ψ0⟩\|\\psi\_\{\\mathrm\{enc\}\}\\rangle=\\bigotimes\_\{q=1\}^\{n\_\{q\}\}R\_\{Y\}\\bigl\(\\pi\\,x\_\{\\mathrm\{in\},t,q\}\\bigr\)\|\\psi\_\{0\}\\rangleprecedesLLlayers of CNOT entanglersUentU\_\{\\mathrm\{ent\}\}and parameterized rotationsUrot\(l\)​\(𝚯^t\)=⨂q=1nq\[RZ​\(Θ^t,l,q,2\)​RY​\(Θ^t,l,q,1\)​RX​\(Θ^t,l,q,0\)\]U\_\{\\mathrm\{rot\}\}^\{\(l\)\}\\bigl\(\\hat\{\\bm\{\\Theta\}\}\_\{t\}\\bigr\)=\\bigotimes\_\{q=1\}^\{n\_\{q\}\}\\\!\\Bigl\[R\_\{Z\}\\bigl\(\\hat\{\\Theta\}\_\{t,l,q,2\}\\bigr\)R\_\{Y\}\\bigl\(\\hat\{\\Theta\}\_\{t,l,q,1\}\\bigr\)R\_\{X\}\\bigl\(\\hat\{\\Theta\}\_\{t,l,q,0\}\\bigr\)\\Bigr\]\. Finally, the output state\|ψout⟩=∏l=1L\(Urot\(l\)​Uent\)​\|ψenc⟩\|\\psi\_\{\\mathrm\{out\}\}\\rangle=\\prod\_\{l=1\}^\{L\}\(U\_\{\\mathrm\{rot\}\}^\{\(l\)\}U\_\{\\mathrm\{ent\}\}\)\|\\psi\_\{\\mathrm\{enc\}\}\\rangleyields the expectation vector𝐳t∈\[−1,1\]nq\\mathbf\{z\}\_\{t\}\\in\[\-1,1\]^\{n\_\{q\}\}, composed of the Pauli\-Z expectation values⟨ψout\|Zq\|ψout⟩\\langle\\psi\_\{\\text\{out\}\}\|Z\_\{q\}\|\\psi\_\{\\text\{out\}\}\\ranglefor each qubit\. This vector is subsequently processed through a ReLU activation function to generate actor and critic heads for cluster selection, with the constituent stocks weighted by inverse volatility\.

## IIIExperimental Setup

Dataset\.Daily log\-returns of S&P 500 stocks \(Jan\. 2015–Apr\. 2026\) are used\. Assets with more than 10% missing observations are excluded, yielding 468 stocks\. All features are constructed using only the information available at timett\. The data are divided into 2,262 training days \(2015\-01\-06 to 2023\-12\-29\), 252 validation days \(2024\-01\-02 to 2024\-12\-31\), and 332 test days \(2025\-01\-02 to 2026\-04\-30\)\.

Equal\-Parameter\-Count \(EPC\) Benchmark\.For fair comparison, all 11 architectures are limited to approximately 3,000 trainable parameters\. Inspired by Titans \(Memory\-as\-Context\)\[[19](https://arxiv.org/html/2608.29093#bib.bib18)\], our proposed Titans\-QFWP \(Full\) is evaluated alongside its variants \(allocations detailed in Table[I](https://arxiv.org/html/2608.29093#S3.T1)\)\.

TABLE I:Parameter configuration for the EPC benchmark\.ArchitectureTotalQuantumClassicalClassical ArchitecturesTitans \(Memory\-as\-Context\)2,99702,997FWP3,00003,000Titans\-FWP \(Full\)2,99902,999Titans\-FWP \(−\-Forgetting\)2,99002,990Titans\-FWP \(−\-Persistence\)2,93102,931Titans\-FWP \(−\-Surprise\)2,98102,981Quantum ArchitecturesQFWP3,0012612,740Titans\-QFWP \(Full\)3,0001,2531,747Titans\-QFWP \(−\-Forgetting\)2,9911,2531,738Titans\-QFWP \(−\-Persistence\)2,9321,2531,679Titans\-QFWP \(−\-Surprise\)2,9821,2531,729
## IVExperimental Results and Discussion

### IV\-AOverall Analysis

Results over 10 random seeds \(1–10\), reported as median \[IQR\] in Table[II](https://arxiv.org/html/2608.29093#S4.T2), are compared with the market benchmark \(S&P 500 Buy & Hold\)\. Titans\-QFWP \(Full\) achieves higher Annualized Rate of Return \(ARR;0\.42600\.4260vs\.0\.25280\.2528\), Calmar Ratio \(Calmar;8\.55048\.5504vs\.7\.16387\.1638\), and the highest Information Ratio \(IR;0\.84270\.8427\)\. In contrast, the market benchmark exhibits lower Maximum Drawdown \(MDD;−0\.0353\-0\.0353\) and higher Sortino Ratio \(Sortino;8\.54058\.5405\), partly due to the exclusion of transaction costs\. These results demonstrate that Titans\-QFWP effectively captures market trends under the strong bull\-market regime\.

TABLE II:S&P 500 out\-of\-sample performance across asset allocation baselines \(median \[IQR\], seeds 1–10\)\.ArchitectureARRMDDSortinoCalmarIRMarket BenchmarkS&P 500 \(Buy & Hold\)0\.25280\.2528−0\.0353\-0\.03538\.54058\.54057\.16387\.1638N/ABaseline & Classical ArchitecturesTitans \(Memory\-as\-Context\)0\.2557​\[0\.2649\]0\.2557\\;\\;\[0\.2649\]−0\.0512​\[0\.0355\]\\mathbf\{\-0\.0512\}\\;\\;\[\\mathbf\{0\.0355\}\]3\.3645​\[4\.4568\]3\.3645\\;\\;\[4\.4568\]3\.9077​\[7\.2651\]3\.9077\\;\\;\[7\.2651\]0\.1374​\[1\.8125\]0\.1374\\;\\;\[1\.8125\]FWP0\.3423​\[0\.1904\]\\mathbf\{0\.3423\}\\;\\;\[\\mathbf\{0\.1904\}\]−0\.0918​\[0\.1046\]\-0\.0918\\;\\;\[0\.1046\]3\.3498​\[6\.1209\]3\.3498\\;\\;\[6\.1209\]3\.9429​\[7\.7964\]3\.9429\\;\\;\[7\.7964\]0\.4447​\[1\.3117\]\\mathbf\{0\.4447\}\\;\\;\[\\mathbf\{1\.3117\}\]Titans\-FWP \(Full\)0\.3175​\[0\.2013\]0\.3175\\;\\;\[0\.2013\]−0\.0606​\[0\.0648\]\-0\.0606\\;\\;\[0\.0648\]4\.9382​\[3\.6665\]\\mathbf\{4\.9382\}\\;\\;\[\\mathbf\{3\.6665\}\]6\.2776​\[5\.6673\]\\mathbf\{6\.2776\}\\;\\;\[\\mathbf\{5\.6673\}\]0\.3891​\[0\.9531\]0\.3891\\;\\;\[0\.9531\]Titans\-FWP \(−\-Forgetting\)0\.2752​\[0\.1713\]0\.2752\\;\\;\[0\.1713\]−0\.0790​\[0\.0453\]\-0\.0790\\;\\;\[0\.0453\]3\.5281​\[3\.4122\]3\.5281\\;\\;\[3\.4122\]3\.4519​\[4\.0856\]3\.4519\\;\\;\[4\.0856\]0\.1993​\[1\.0045\]0\.1993\\;\\;\[1\.0045\]Titans\-FWP \(−\-Persistence\)0\.2841​\[0\.2500\]0\.2841\\;\\;\[0\.2500\]−0\.0702​\[0\.0540\]\-0\.0702\\;\\;\[0\.0540\]4\.1309​\[2\.8475\]4\.1309\\;\\;\[2\.8475\]4\.5385​\[3\.4245\]4\.5385\\;\\;\[3\.4245\]0\.1927​\[1\.2757\]0\.1927\\;\\;\[1\.2757\]Titans\-FWP \(−\-Surprise\)0\.1907​\[0\.1834\]0\.1907\\;\\;\[0\.1834\]−0\.0774​\[0\.0731\]\-0\.0774\\;\\;\[0\.0731\]2\.8587​\[3\.8253\]2\.8587\\;\\;\[3\.8253\]2\.5238​\[6\.5771\]2\.5238\\;\\;\[6\.5771\]−0\.4547​\[1\.1375\]\-0\.4547\\;\\;\[1\.1375\]Quantum ArchitecturesQFWP0\.1806​\[0\.1437\]0\.1806\\;\\;\[0\.1437\]−0\.1017​\[0\.0887\]\-0\.1017\\;\\;\[0\.0887\]1\.8849​\[1\.1011\]1\.8849\\;\\;\[1\.1011\]1\.5299​\[2\.2969\]1\.5299\\;\\;\[2\.2969\]−0\.4002​\[0\.8815\]\-0\.4002\\;\\;\[0\.8815\]Titans\-QFWP \(Full\)0\.4260​\[0\.4197\]\\mathbf\{0\.4260\}\\;\\;\[\\mathbf\{0\.4197\}\]−0\.0500​\[0\.0212\]\\mathbf\{\-0\.0500\}\\;\\;\[\\mathbf\{0\.0212\}\]6\.9018​\[6\.5930\]\\mathbf\{6\.9018\}\\;\\;\[\\mathbf\{6\.5930\}\]8\.5504​\[6\.0697\]\\mathbf\{8\.5504\}\\;\\;\[\\mathbf\{6\.0697\}\]0\.8427​\[1\.3992\]\\mathbf\{0\.8427\}\\;\\;\[\\mathbf\{1\.3992\}\]Titans\-QFWP \(−\-Forgetting\)0\.1965​\[0\.1276\]0\.1965\\;\\;\[0\.1276\]−0\.0520​\[0\.0415\]\-0\.0520\\;\\;\[0\.0415\]3\.0186​\[1\.3646\]3\.0186\\;\\;\[1\.3646\]3\.3316​\[2\.0293\]3\.3316\\;\\;\[2\.0293\]−0\.3887​\[0\.9337\]\-0\.3887\\;\\;\[0\.9337\]Titans\-QFWP \(−\-Persistence\)0\.1929​\[0\.1297\]0\.1929\\;\\;\[0\.1297\]−0\.1010​\[0\.0400\]\-0\.1010\\;\\;\[0\.0400\]2\.0905​\[1\.1939\]2\.0905\\;\\;\[1\.1939\]2\.1663​\[1\.7055\]2\.1663\\;\\;\[1\.7055\]−0\.3136​\[0\.6757\]\-0\.3136\\;\\;\[0\.6757\]Titans\-QFWP \(−\-Surprise\)0\.1586​\[0\.1749\]0\.1586\\;\\;\[0\.1749\]−0\.0782​\[0\.0318\]\-0\.0782\\;\\;\[0\.0318\]2\.6949​\[1\.5092\]2\.6949\\;\\;\[1\.5092\]2\.3728​\[1\.3064\]2\.3728\\;\\;\[1\.3064\]−0\.5138​\[1\.1810\]\-0\.5138\\;\\;\[1\.1810\]

### IV\-BAblation Diagnostics

#### IV\-B1Classical Ablation \(Titans\-FWP\)

In classical variants \(Table[III](https://arxiv.org/html/2608.29093#S4.T3)\), removing Surprise produces the largest degradation in ARR, Sortino, Calmar, and IR\. Removing Forgetting results in the largest deterioration in MDD\. Removing Persistence leads to smaller performance changes across most metrics\.

TABLE III:Performance metrics for Titans\-FWP ablation variants \(median \[IQR\], seeds 1–10\)\.ArchitectureARRMDDSortinoCalmarIRClassical Ablation Study: Titans\-FWPFWP \(Baseline\)0\.3423​\[0\.1904\]\\mathbf\{0\.3423\}\\;\\;\[\\mathbf\{0\.1904\}\]−0\.0918​\[0\.1046\]\-0\.0918\\;\\;\[0\.1046\]3\.3498​\[6\.1209\]3\.3498\\;\\;\[6\.1209\]3\.9429​\[7\.7964\]3\.9429\\;\\;\[7\.7964\]0\.4447​\[1\.3117\]\\mathbf\{0\.4447\}\\;\\;\[\\mathbf\{1\.3117\}\]Titans\-FWP \(Full\)0\.3175​\[0\.2013\]0\.3175\\;\\;\[0\.2013\]−0\.0606​\[0\.0648\]\\mathbf\{\-0\.0606\}\\;\\;\[\\mathbf\{0\.0648\}\]4\.9382​\[3\.6665\]\\mathbf\{4\.9382\}\\;\\;\[\\mathbf\{3\.6665\}\]6\.2776​\[5\.6673\]\\mathbf\{6\.2776\}\\;\\;\[\\mathbf\{5\.6673\}\]0\.3891​\[0\.9531\]0\.3891\\;\\;\[0\.9531\]Titans\-FWP \(−\-Forgetting\)0\.2752​\[0\.1713\]0\.2752\\;\\;\[0\.1713\]−0\.0790​\[0\.0453\]\-0\.0790\\;\\;\[0\.0453\]3\.5281​\[3\.4122\]3\.5281\\;\\;\[3\.4122\]3\.4519​\[4\.0856\]3\.4519\\;\\;\[4\.0856\]0\.1993​\[1\.0045\]0\.1993\\;\\;\[1\.0045\]Titans\-FWP \(−\-Persistence\)0\.2841​\[0\.2500\]0\.2841\\;\\;\[0\.2500\]−0\.0702​\[0\.0540\]\-0\.0702\\;\\;\[0\.0540\]4\.1309​\[2\.8475\]4\.1309\\;\\;\[2\.8475\]4\.5385​\[3\.4245\]4\.5385\\;\\;\[3\.4245\]0\.1927​\[1\.2757\]0\.1927\\;\\;\[1\.2757\]Titans\-FWP \(−\-Surprise\)0\.1907​\[0\.1834\]0\.1907\\;\\;\[0\.1834\]−0\.0774​\[0\.0731\]\-0\.0774\\;\\;\[0\.0731\]2\.8587​\[3\.8253\]2\.8587\\;\\;\[3\.8253\]2\.5238​\[6\.5771\]2\.5238\\;\\;\[6\.5771\]−0\.4547​\[1\.1375\]\-0\.4547\\;\\;\[1\.1375\]

#### IV\-B2Quantum Ablation \(Titans\-QFWP\)

In quantum variants \(Table[IV](https://arxiv.org/html/2608.29093#S4.T4)\), removing Persistence causes the largest degradation in the median MDD, Sortino, and Calmar, while removing Surprise leads to the lowest median ARR and IR\. These patterns suggest that quantum gating alters memory roles, with greater expressive capacity favoring Persistence over Forgetting for representation preservation\.

TABLE IV:Performance metrics for Titans\-QFWP ablation variants \(median \[IQR\], seeds 1–10\)\.ArchitectureARRMDDSortinoCalmarIRQuantum Ablation Study: Titans\-QFWPQFWP \(Baseline\)0\.1806​\[0\.1437\]0\.1806\\;\\;\[0\.1437\]−0\.1017​\[0\.0887\]\-0\.1017\\;\\;\[0\.0887\]1\.8849​\[1\.1011\]1\.8849\\;\\;\[1\.1011\]1\.5299​\[2\.2969\]1\.5299\\;\\;\[2\.2969\]−0\.4002​\[0\.8815\]\-0\.4002\\;\\;\[0\.8815\]Titans\-QFWP \(Full\)0\.4260​\[0\.4197\]\\mathbf\{0\.4260\}\\;\\;\[\\mathbf\{0\.4197\}\]−0\.0500​\[0\.0212\]\\mathbf\{\-0\.0500\}\\;\\;\[\\mathbf\{0\.0212\}\]6\.9018​\[6\.5930\]\\mathbf\{6\.9018\}\\;\\;\[\\mathbf\{6\.5930\}\]8\.5504​\[6\.0697\]\\mathbf\{8\.5504\}\\;\\;\[\\mathbf\{6\.0697\}\]0\.8427​\[1\.3992\]\\mathbf\{0\.8427\}\\;\\;\[\\mathbf\{1\.3992\}\]Titans\-QFWP \(−\-Forgetting\)0\.1965​\[0\.1276\]0\.1965\\;\\;\[0\.1276\]−0\.0520​\[0\.0415\]\-0\.0520\\;\\;\[0\.0415\]3\.0186​\[1\.3646\]3\.0186\\;\\;\[1\.3646\]3\.3316​\[2\.0293\]3\.3316\\;\\;\[2\.0293\]−0\.3887​\[0\.9337\]\-0\.3887\\;\\;\[0\.9337\]Titans\-QFWP \(−\-Persistence\)0\.1929​\[0\.1297\]0\.1929\\;\\;\[0\.1297\]−0\.1010​\[0\.0400\]\-0\.1010\\;\\;\[0\.0400\]2\.0905​\[1\.1939\]2\.0905\\;\\;\[1\.1939\]2\.1663​\[1\.7055\]2\.1663\\;\\;\[1\.7055\]−0\.3136​\[0\.6757\]\-0\.3136\\;\\;\[0\.6757\]Titans\-QFWP \(−\-Surprise\)0\.1586​\[0\.1749\]0\.1586\\;\\;\[0\.1749\]−0\.0782​\[0\.0318\]\-0\.0782\\;\\;\[0\.0318\]2\.6949​\[1\.5092\]2\.6949\\;\\;\[1\.5092\]2\.3728​\[1\.3064\]2\.3728\\;\\;\[1\.3064\]−0\.5138​\[1\.1810\]\-0\.5138\\;\\;\[1\.1810\]

### IV\-CPortfolio Dynamics and Asset Selection

As shown in Fig\.[2](https://arxiv.org/html/2608.29093#S4.F2)\(Seed 1\), Titans\-QFWP achieves a cumulative return of \+30\.33%, slightly outperforming the S&P 500 \(\+28\.46%\), driven by allocations to higher\-return clusters \(e\.g\., Clusters 7 and 9\)\. The model adaptively reallocates to cash during market downturns\.

Averaged across 10 seeds \(1–10\), the top\-20 selected stocks exhibit concentrated outperformance in technology sub\-sectors, including data storage, optical networking, photonics, and memory semiconductors \(Fig\.[3](https://arxiv.org/html/2608.29093#S4.F3)\)\. Measured by active return \(excess simple return over the S&P 500\), WDC and CIEN outperform the benchmark at every time step, with peak active returns of \+42\.7% and \+51\.6%\. In addition, COHR, LITE, and MU also contribute substantial excess returns\.

Fig\. 2:Dynamic cluster selection of Titans\-QFWP vs\. the S&P 500 index\.![Refer to caption](https://arxiv.org/html/2608.29093v1/titans_qfwp_heatmap_active_return_vs_sp500.png)Fig\. 3:Top\-20 selected stocks of Titans\-QFWP vs\. the S&P 500 index\.

## VLimitations

Limitations include survivorship bias, a short out\-of\-sample period, seed variability, simplified transaction costs, and evaluation of quantum components usinglightning\.qubitquantum simulation\. Risk\-adjusted metrics are reported descriptively rather than as statistically conclusive evidence\.

## VIConclusion

Integrating QFWP with Titans\-style memory provides a robust framework for addressing market non\-stationarity\. Crucially, the expressive capacity of quantum gating fundamentally alters learning dynamics: it shifts the mechanism of representation preservation, relying more on Persistence rather than Forgetting for drawdown control\. Although frictionless passive benchmarks exhibit a marginal downside advantage during bull markets, Titans\-QFWP mitigates this through regime\-aware allocation, dynamically transitioning between defensive and high\-performing asset clusters\. These findings illustrate how quantum\-classical memory architectures reorganize representation learning, advancing adaptive financial reinforcement learning\.

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