The remainder of the paper proceeds as follows. We first present the empirical results in the “Results” section, followed by a Discussion of implications. Finally, we describe the QSW formulation and experimental design in the “Methods” section.
This section presents a comprehensive empirical validation of the QSW framework. The experimental design has two phases: (1) a recent-period parameter exploration (2018-2024) to understand and tune the model, and (2) a long-horizon, multi-universe robustness study (1990-2024) to test generalizability across regimes and universes. The first phase uses a fixed “top-100-by-market-cap” S&P 500 universe and includes both a preset-based analysis and a systematic grid search. The second phase runs 30 independent trials on dynamically maintained, point-in-time S&P 500 universes to guard against selection and survivorship bias. Across all experiments, QSW-based portfolios are compared against three benchmarks: the maximum-Sharpe MPT portfolio, the naive 1/N portfolio, and the S&P 500 index.
Our experimental framework is designed to evaluate the QSW methodology under both controlled and realistic conditions.
In Phase 1, parameter exploration (2018-2024), we use daily data from 2018 to 2024 on the top 100 S&P 500 companies by market capitalization as a fixed universe. This phase consists of two experiments. Experiment 1 (preset analysis) defines six QSW strategy presets that represent different investment philosophies, from ultra-diversified to high-activity trading styles. Each preset is evaluated across five quantum-mixing values ω ∈{0.2, 0.4, 0.6, 0.8, 1.0} to study the impact of the quantum-classical spectrum. Experiment 2 (comprehensive grid search) systematically explores a 625-point grid over the four QSW parameters (α, β, λ, ω) on the same top-100 universe. This grid search identifies robust high-performing regions of the parameter space and clarifies how the four control knobs interact.
In Phase 2, robustness validation (1990-2024), we test the generalizability of the approach over a 34-year backtest with rolling 2-year training windows. Experiment 3 (multi-universe testing) runs 30 independent trials on distinct 100-stock universes. As described in the Methods section (“Data and experimental setup”), each universe is constructed point-in-time by sampling from contemporaneous S&P 500 constituents and is dynamically maintained to avoid survivorship bias. This design validates that the QSW results are not dependent on a single, favorable asset selection.
In all experiments, QSW-based portfolios are evaluated against three benchmarks: the maximum-Sharpe MPT portfolio (classical mean-variance optimizer), the naive 1/N portfolio on the same universe, and the S&P 500 index as a market-cap-weighted passive benchmark.
This experiment evaluates 30 distinct configurations, which are derived by crossing the six parameter presets in Table 1 with five quantum-classical mixes (ω ∈{0.2, 0.4, 0.6, 0.8, 1.0}). To test the model’s sensitivity to the length of historical data, we evaluate every configuration twice: (i) using a 1-year (“short-memory”) training window, and (ii) using a 2-year (“long-memory”) training window. Both sets of models are then out-of-sample back-tested from 2018-01-02 to 2024-12-31 with quarterly rebalancing.
A combined analysis of the cumulative wealth trajectories (Fig. 1) and the detailed performance metrics (Table 2) reveals several critical insights into the QSW model’s behavior relative to the benchmarks.
First, regarding model stability, the “two starkly different regimes” visible in Fig. 1 are driven almost entirely by the instability of the MPT benchmark, not the QSW model. This highlights MPT’s extreme sensitivity to estimation error, a key limitation we identified in the Introduction. With 1-Year Training, MPT produces a poor portfolio (Sharpe 0.86) with extremely high volatility (31.61%). The short, noisy training window likely results in a highly unstable covariance matrix, leading to a flawed optimization. In contrast, all QSW strategy presets (Sharpe ≈ 0.96-0.98) are stable and easily outperform MPT. With 2-Year Training, with a more stable window, MPT’s volatility drops to 20.94% and its Sharpe ratio jumps by 58% to 1.36. The QSW and 1/N models, however, are almost completely unaffected by the change in window length, with their Sharpe (0.98) and Volatility (≈17.2%) remaining rock-solid. This demonstrates the robustness of the QSW method; its performance is not dependent on a “lucky” or perfectly stable estimation window.
Second, the QSW model converges to a “Smart 1/N”. The most striking finding is that the QSW model, across all QSW strategy presets, produces a risk and diversification profile that is nearly identical to the naive 1/N benchmark. As seen in Table 2, the QSW’s Sharpe (0.96-0.98), Volatility (≈17%), Max Drawdown (≈-19%), and HHI (≈0.01) are all in lockstep with the 1/N portfolio. This suggests the QSW’s quantum-graph dynamics naturally find the highly diversified, stable1/N state. The key difference is that QSW acts as a “smart 1/N”, initiating a very low (2-35%) but non-zero turnover, in contrast to MPT’s hyper-active 320-480% turnover.
Third, while MPT achieves the highest “paper” Sharpe ratio (1.36) in the 2-year case, Table 2 shows this is operationally unfeasible. This performance is achieved by making an extreme, concentrated bet (HHI ≥ 0.25, or ≈4-5 effective stocks) and by turning over the entire portfolio 3-5 times per year (320-480% turnover). The transaction costs from this churn would completely erase its “paper” alpha. QSW, by contrast, generates its alpha with minimal turnover and maximum diversification (HHI≈0.01, ≈90-100 effective stocks), making it a practical and cost-effective strategy.
Finally, Fig. 1 shows that in both 1-year and 2-year setups, all QSW configurations (final wealth ≈ 3.0×) and the 1/N benchmark (≈3.0×) consistently and significantly outperform the passive S&P 500 index (final wealth ≈ 2.2×).
Comparing the 1-year (Table 2a) and 2-year (Table 2b) results reveals the core weakness of MPT and the core strength of QSW. The MPT benchmark is fundamentally unstable. When the training window changes from 1-year to 2-years, MPT’s Sharpe ratio leaps 58% (from 0.86 to 1.36) and its volatility drops from 31.6% to 20.9%. This demonstrates that MPT is extremely sensitive to the estimation error in its inputs, a key classical limitation. In stark contrast, the QSW and 1/N models are rock-solid. Their key metrics (Sharpe ≈ 0.98, Vol ≈ 17.2%) are almost completely unaffected by the change in training data. This provides strong empirical evidence that our QSW framework, like the 1/N benchmark, is structurally robust and successfully insulated from the parameter instability that plagues classical MPT.
To understand the behavior within each preset, we analyze the effect of the quantum-classical mixing parameter, ω, and the portfolio’s concentration over the 2018-2024 backtest period. We present the results for the 1-year training window here; the corresponding analysis for the 2-year window, which confirms the same structural conclusions, is provided in the Supplementary Information. Figure 2 shows the performance metrics for all six QSW strategy presets as a function of the quantum-classical balance, ω. The results reveal a clear and powerful trend: the ω parameter acts primarily as a “turnover dial.” In all six QSW strategy presets, increasing ω (making the model more “classical”) leads to a near-linear increase in portfolio turnover. For example, in the “Ultra-Diversified” preset, turnover rises from 4 to 30% as ω goes from 0.2 to 1.0. Crucially, this increase in turnover has almost no impact on the Sharpe ratio or volatility. These metrics remain remarkably flat across the entire ω spectrum, and always superior to MPT’s 1-year performance. Most importantly, ω does not break the model’s diversification. The HHI remains at its 1/N floor (≈0.01) in all but the most active presets, and even then, it stays an order of magnitude more diversified than MPT. This demonstrates that the QSW model’s core benefits (high Sharpe, low volatility, and extreme diversification) are structurally inherent and robust, while the ω parameter provides a simple, interpretable knob to control the “cost” (i.e., turnover) of the strategy.
A core structural difference between QSW and MPT is revealed by analyzing their portfolio concentration. We compare the models over the 2018-2024 backtest using four standard metrics, as shown in Fig. 3. The QSW model, across all presets, behaves as a “smart 1/N”. Its concentration is tunable and directly linked to the quantum-classical mix, ω. As seen in Fig. 3, when ω is low (more quantum), the model’s HHI is locked at around the theoretical floor of 0.01. As ω increases, the model is permitted to become slightly more concentrated, and the HHI, effective number of stocks, max single stock weight, and top 5 holdings all show a small, controlled deviation from the 1/N state. However, even at its most “classical” (ω = 1.0), the QSW model remains exceptionally well-diversified. The MPT benchmark (red line) is defined by its extreme and erratic concentration. Its HHI is dangerously unstable. While its average HHI is ≈0.28, the Top 5 Holdings chart shows that MPT’s typical state is to hold ≈87-89% of the entire portfolio in just five names. This confirms MPT is a structurally unstable and highly concentrated strategy, while QSW is robustly diversified by design.
Portfolio annual turnover directly translates into implementation costs that can significantly erode theoretical performance gains. These costs compound with trading frequency, making turnover control a critical factor in real-world portfolio management. Institutional studies such as place the all-in round-trip cost of trading cash equities at: large-cap stocks (5-15 bp), mid-/small-cap (15-30 bp), and market impact (a further 10-50 bp). In this work, we apply a conservative 20 bp all-in round-trip cost for large-cap equities (10 bp commission & spread + 10 bp market impact). Table 3 converts realized turnover into ex-ante implementation drag using this rate. Even under the more forgiving 2-year training calibration, MPT forfeits ~64 bp of alpha to dealing costs versus ~7.4 bp for the QSW preset mean.
Is a 64 bp vs. 7 bp gap economically meaningful? Yes. Compare each strategy’s “paper” alpha (CAGR in excess of the S&P 500) with, and without, implementation drag (Table 4).
Even after deducting realistic dealing costs, QSW preserves virtually all of its back-tested edge, whereas MPT forfeits about 5 % of its “paper” alpha. Because the classical maximum-Sharpe solution trades eight to thirteen times more than QSW, its cost drag-while “only” 64 bp-is still an order of magnitude larger in proportional terms. QSW’s subdued trading footprint additionally mitigates price pressure, timing risk, and information leakage, making its low-turnover profile a structural implementation advantage. In Experiment 3, we show that, even when the QSW parameters are re-optimized every quarter over 1990-2024, the dynamic QSW optimizer still trades substantially less than the long-memory MPT benchmark while preserving its risk-adjusted edge.
In the previous analysis, we demonstrated that MPT is structurally unstable at the macro level: changing the training window from 1 to 2 years led to dramatic changes in performance and risk profile (a 58% jump in Sharpe). This section’s Monte Carlo test confirms this fragility on a micro level. A fundamental flaw of classical MPT is its extreme sensitivity to input parameters. A small, statistically insignificant change in the expected return vector μ can cause the optimizer to produce a radically different portfolio, leading to massive, unforced turnover. To test the robustness of our QSW model to this, we conduct a Monte Carlo sensitivity analysis. We take a single period’s data and apply a 20% random “shock” to the returns of a single, randomly chosen asset (i.e., r × random(0.8, 1.2)). We then re-run the optimization for both QSW and MPT and measure the total absolute change in portfolio weights (i.e., ∑∣w – w∣). This process is repeated 1000 times for each of the six QSW strategy presets. Figure 4 plots the 1000-run distribution of this weight change, while Table 5 quantifies the results.
The results are clear and highly suggestive. The MPT benchmark (red distributions in Fig. 4) exhibits extreme fragility. As quantified in Table 5, while MPT’s mean weight change (1.72%) is moderate, its standard deviation (7.82%) is massive-over 2× to 100× larger than any QSW preset. This highlights how MPT amplifies minor input errors into significant portfolio changes. The QSW model, by contrast, is both structurally robust and tunable. Its stability is directly governed by its parameters, which function as an accelerator and a brake: presets with high α (“Sharpe-Maximizer”, “High-Activity”) show the highest mean weight change (3.61%), as they are designed to chase returns, while presets with a high diversification penalty (β) or holding coefficient (λ) (such as “Ultra-Diversified”) show exceptionally low mean weight changes (0.07-1.90%). This test provides clear empirical evidence that the QSW framework not only overcomes the fragility of MPT but also provides an interpretable, tunable mechanism to control the model’s sensitivity to input noise. The next experiment broadens the evidence: a full 625-point grid search pinpoints globally robust parameter regions, and a 30-universe robustness study confirms that these findings generalize beyond this fixed top-100 universe.
Having established in Experiment 1 that all six QSW strategy presets behave robustly and outperform MPT’s 1-year model, we now search the full four-dimensional parameter space θ = (α, β, λ, ω) to answer three key questions: (i) Does an even better configuration exist than the presets? (ii) How sensitive is performance to each hyper-parameter? (iii) Can we extract simple design rules for practitioners?
This experiment performs a systematic grid search over the 625 parameter combinations defined by: α, β, λ ∈{0.1, 5, 50, 100, 500} and ω ∈{0.2, 0.4, 0.6, 0.8, 1.0}, yielding 625 unique combinations. All 625 combinations are evaluated using the Parameter Exploration Setup described in the “Methods” (2018-2024 backtest, top-100 universe, and quarterly rebalancing). The entire 625-point grid search is performed twice: once using the 1-year (“short-memory”) rolling training window, and a second time using the 2-year (“long-memory”) window. This results in a total of 625 × 2 = 1250 full, independent back-tests. For each of the 1250 back-tests, we record the full set of performance metrics (Sharpe ratio, volatility, MDD, turnover, efficiency , and HHI).
The grid search results provide a powerful and conclusive answer to our three research questions by revealing the QSW’s “smart” hybrid nature. The model is not a sequential “two-step” process, but a simultaneous dual-channel optimizer that intelligently balances its classical and quantum components to achieve a robust, efficient portfolio. As noted, Tables 6 and 7 provide a strong and conclusive answer to our first research question. First, QSW is structurally robust. The descriptive statistics in Table 6 show that the QSW framework is inherently stable. In the 1-year window, the worst-performing QSW configuration (Sharpe 0.85) is still comparable to the MPT benchmark (0.86). Furthermore, the median QSW Sharpe (≈0.96) is vastly superior, proving that the model delivers strong performance across the majority of its parameter space, in stark contrast to the brittle MPT solution. Second, QSW converges to a smart 1/N-like portfolio. The most critical finding comes from Table 7. In both the 1-year and 2-year windows, the Top-10 best-performing QSW configurations are effectively identical: they all have minimal return-chasing (α = 0.1) and maximal diversification penalty (β = 500). The QSW model, when allowed to search the entire parameter space, consistently identifies an almost infinitesimally traded, 1/N-like portfolio as the most robust solution in this universe. Its performance (Sharpe 0.989, HHI 0.010, Turnover < 3%) is a near-perfect match for the 1/N benchmark. Third, QSW correctly identifies MPT's "paper" victory. The difference between the 1-year and 2-year results perfectly frames the QSW's value. In the 1-Year case, MPT (0.86) fails due to estimation error. QSW (0.989) discovers that 1/N (0.980) is the superior strategy and outperforms both benchmarks. In the 2-Year case, MPT (1.36) stabilizes and finds a high-turnover, high-concentration "paper" victory. QSW, faced with the same data, still identifies the 1/N-like state as its optimal solution. It correctly avoids MPT's operationally unfeasible (320% turnover) solution and selects the most practical and efficient strategy.
Our analysis of the tables has answered our first question: the optimal strategy is the "smart 1/N" state. To visualize the global impact of each hyper-parameter, we plot the full 625-run parameter space. The 2D heatmaps (Fig. 5) and the full correlation matrix (Fig. 6) answer our second and third questions, showing how the model finds this state by using its dual-channel design. The 2-year data, which are nearly identical, are provided in the Supplementary Information and confirm that these conclusions are robust to the training window choice.
Figure 5 makes the QSW's structural behavior immediately visible. Most strikingly, the concentration panel in Fig. 5b shows that, across the entire 625-run grid, the HHI remains confined to an extremely narrow band: from a floor of 0.010 (dark blue) to a maximum of 0.014 (bright green). In other words, even the "worst-case" combination of (α, β, λ, ω) never produces anything close to a concentrated portfolio. By comparison, the MPT benchmark sits at HHI = 0.268, more than 20 × higher. This gap is not a tuning artifact but a structural property: the QSW framework is incapable of generating the kind of brittle, single-bet allocations that MPT routinely produces. The Sharpe and Efficiency panels (Fig. 5a, c) complete this picture by revealing how the hyper-parameters act as behavioral knobs rather than sources of instability. Sharpe ratios remain tightly clustered in the 0.94-0.99 range over most of the grid, indicating that performance is remarkably insensitive to moderate perturbations of (α, β, λ, ω). Efficiency, by contrast, varies much more strongly along the λ and ω axes: higher values of λ (the "holding" parameter) and ω systematically reduce by lowering Sharpe and/or increasing turnover, while leaving the underlying diversification almost untouched.
These visual impressions are quantified by the full correlation matrix in Fig. 6. Within our tested range ω ∈[0.2, 1.0], ω exhibits a strong negative correlation with Sharpe and a positive correlation with HHI, confirming that it primarily acts as a noise parameter: smaller ω values lead, on average, to more diversified, higher-quality portfolios, whereas larger ω values inject additional stochasticity that slightly raises concentration and erodes risk-adjusted returns. In contrast, λ shows a pronounced negative correlation with Efficiency while having only a mild effect on Sharpe, confirming its role as a trade-off knob between turnover and implementability rather than a driver of raw performance. Together, the heatmaps and the correlation matrix show that QSW's smart 1/N solution is not a single fine-tuned point, but a broad, structurally robust region of the parameter space.
To visually confirm these findings and answer our second research question, we analyze the global impact of each hyper-parameter. Figure 7 plots the Pearson correlations between the four control knobs (α, β, λ, ω) and our key metrics. We present the plots for both the 1-year "noisy" regime and the 2-year "stable" regime, as their comparison provides insight into the QSW model's adaptive, structural behavior.
The correlation plots provide four practical design rules for practitioners. First, ω is the main "noise" dial, with a mild impact on concentration. Among the four knobs, ω shows the strongest and most stable pattern across both horizons. Within our tested range ω ∈[0.2, 1.0], it is strongly negatively correlated with Sharpe (1Y: -0.495, 2Y: -0.556), and positively correlated with HHI (1Y: 0.225, 2Y: 0.393). In other words, higher ω injects additional stochastic noise into the walk, which slightly increases concentration and systematically erodes risk-adjusted returns. Conversely, lower ω values tend, on average, to produce more diversified, higher-quality portfolios, although the magnitude of the HHI effect is small relative to the structural diversification enforced by the QSW dynamics. Second, λ is the "turnover" knob, trading off implementability against efficiency. As expected from its construction, λ behaves as a holding/inertia parameter. Across both windows, it exhibits a moderate positive correlation with HHI (1Y: 0.225, 2Y: 0.242) and a moderate negative correlation with Efficiency (1Y: -0.199, 2Y: -0.219), while its correlation with Sharpe is weak. Forcing the model to "stay where it is" (large λ) suppresses trading and reduces turnover, but at the cost of higher concentration and lower efficiency. In other words, λ does not change the core performance level so much as it controls the trade-off between practical implementability and how aggressively the strategy rebalances back toward its preferred low-HHI state. Third, α is a regime-dependent return bias, not a standalone source of alpha. The return-preference parameter α shows only weak and inconsistent correlations with Sharpe (positive in the 1-year window, slightly negative in the 2-year window), while maintaining a tendency to reduce HHI and Efficiency when increased. This indicates that, over the 2018-2024 sample, classical return signals behave more like a noisy, regime-dependent bias than a robust source of outperformance: raising α tilts the walk toward past winners and can slightly improve Sharpe in the short-memory regime, but it does not produce a reliable, horizon-independent Sharpe uplift. Fourth, β looks neutral globally, but is decisive in the winning region. At first glance, the global bar charts in Fig. 7 suggest that the diversification penalty β has almost no correlation with HHI or Sharpe (correlations near zero). Taken in isolation, this would seem to imply that β does not matter. However, the Top-10 list in Table 7 tells a very different story: every optimal configuration sets β = 500, its maximum value. This is not a contradiction but a selection effect: once α and ω are in a favorable regime (low return-chasing, low noise), the QSW is already close to fully diversified (HHI ≈ 0.01), so further changes in β have little marginal impact and its global correlation is washed out. Conditioning on the high-Sharpe region, however, reveals the true design rule: strong diversification pressure (largeβ) is a prerequisite for ending up in the structurally robust, "smart 1/N" regime.
The grid search results provide empirical support for the dual-channel framework introduced earlier. Taken together, the Top-10 list, the full-grid statistics, and the visual diagnostics reveal a genuinely hybrid design: a classical channel that pins the portfolio to a structurally robust baseline, and a quantum channel that fine-tunes how this baseline is implemented in practice. The Top-10 list in Table 7 is the most direct piece of evidence for the classical channel acting as a stabilizer. Across both the 1-year and 2-year windows, every optimal configuration chooses the same classical settings: a minimal return-chasing coefficient (α = 0.1) and the strongest possible diversification penalty (β = 500). In other words, the model "learns" that the 2018-2024 market is effectively noisy from a classical perspective and responds by turning the accelerator α almost off, while slamming the diversification brake β to its maximum. The 2D heatmaps in Fig. 5 and the correlation plots in Fig. 7 confirm this picture: once α is kept small and β is large, the portfolio is structurally forced into an almost perfectly diversified "smart 1/N" state (HHI ≈ 0.010) across a broad region of the grid. The classical channel, therefore, acts as a stabilizer that locks in a robust risk-return profile by overpowering unstable return signals. Given this stabilized baseline, the quantum channel, controlled by ω, is then used to fine-tune how the same robust allocation is implemented. This can be seen most clearly in Table 7(a) when we fix (α, β, λ) = (0.1, 500, 0.1) and only vary ω: as ω decreases from 1.0 to 0.2, the Sharpe ratio remains essentially unchanged (0.989 → 0.988); turnover decreases (from 2.9% to 1.5%); efficiency increases (from 25.6 to 39.6); while HHI stays fixed at 0.010 in all cases. In other words, once the classical channel has pinned the portfolio to a highly diversified, smart 1/N-like state, the quantum channel does not search for a different target allocation; instead, it searches for cheaper ways to reach and maintain that allocation. Consistent with the global correlations, lower ω reduces noisy fluctuations and turnover, improving cost-efficiency without sacrificing Sharpe or diversification. This is the practical "quantum exploration" advantage: the QSW simultaneously explores multiple paths to the same robust state and selects the one with the most attractive implementation profile.
Experiment 2 shows that, for a fixed period (2018-2024), the QSW can reliably discover a structurally robust, smart 1/N-like regime and fine-tune its implementation cost through its hybrid quantum-classical channels. The natural next question is whether this behavior is truly adaptive: can the same mechanism re-learn an appropriate balance between the classical channel (α, β, λ) and the quantum channel (ω) as market conditions change? Experiment 3 addresses this question by moving from a static to a fully dynamic setting. Instead of identifying a single "best" parameter set over 2018-2024, we run the entire 625-point grid search at every quarterly rebalancing date from 1990 to 2024. At each quarter, the QSW optimizer re-estimates its optimal mix of classical and quantum parameters for that specific market regime. This 34-year, rolling experiment serves as the ultimate test of the QSW's robustness and "smartness" over multiple cycles, crises, and structural shifts.
This Phase 2 experiment stress-tests the QSW as a dynamic hybrid optimizer over a 34-year period (1990-2024) that spans multiple major crises, including the Dot-com bubble, the 2008 global financial crisis, and the 2020 COVID crash. Building directly on the insights from Phase 1, we now ask a harder question: can the same dual-channel mechanism continuously re-learn an appropriate balance between its classical channel (α, β, λ) and quantum channel (ω) as market conditions change, and can it do so robustly across different equity universes? To answer this, Experiment 3 combines a dynamic dual-channel strategy with multi-universe testing.
The experimental setup involves a dynamic adaptive model that re-optimizes channels every quarter. At each quarterly rebalancing date from 1990 to 2024, the model runs a full 625-point grid search on the most recent 2-year training window. It then selects the parameter set that best balances the classical and quantum channels (α, β, λ, ω) under that specific market regime. In contrast to Experiment 2, which identified a single "best-fit" configuration for 2018-2024, this experiment allows the QSW optimizer to re-learn its settings every quarter over 34 years. We perform multi-universe testing using 30 independent trials. To ensure that the results are not an artifact of a single, "lucky" 100-stock universe, we repeat the entire 34-year dynamic backtest 30 times. As described in the Methods ("Data and experimental setup"), each trial starts from a different, randomly sampled 100-stock subset of the point-in-time S&P 500 constituents in 1990Q1 and is dynamically maintained: at each quarterly date, delisted stocks are removed and immediately replaced by new names drawn from the contemporaneous S&P 500 membership. This multi-universe design ensures that our robustness conclusions do not rely on a particular asset selection or survivorship bias.
Table 8 reports the cross-universe means and standard deviations. Three patterns stand out. First, QSW delivers the highest average CAGR and Sharpe, with markedly lower dispersion than MPT. Second, this is not achieved by taking more risk: QSW's volatility lies between 1/N and MPT, its average maximum drawdown is smaller than both MPT and the S&P 500, and its Calmar ratio is materially higher than all benchmarks. Third, QSW attains these outcomes with less turnover than MPT, indicating that the QSW optimizer is not simply "over-trading" its way to higher returns but improves the overall risk-return-cost profile.
Figure 8 complements Table 8 by showing the full cross-universe distributions. The Sharpe violins in Fig. 8(a) show that QSW dominates not only on average but in most individual universes: the bulk of its mass lies above both 1/N and MPT, with little overlap. Figure 8(b) shows the same ordering for Calmar ratios, confirming that the advantage persists after normalizing by maximum drawdown. Finally, the final-value distribution in Fig. 8(c) illustrates how these differences compound over 34 years: starting from $1, QSW typically reaches several times the terminal wealth of 1/N and an order of magnitude more than MPT or the index. The medians and interquartile ranges closely track the means, indicating that the outperformance is systematic rather than driven by a few outlier universes.
Figure 9 examines how QSW earns its outperformance. On volatility, QSW sits between 1/N and MPT, indicating that its higher Sharpe is driven by better risk pricing rather than simply taking more overall risk. On drawdowns, QSW again occupies the favorable middle ground: its maximum drawdowns are consistently shallower than those of MPT and the index, and are either slightly better or comparable to 1/N. Finally, QSW strikes a pragmatic balance on trading activity: it trades substantially less than MPT while, of course, trading more than the nearly buy-and-hold 1/N portfolio. Taken together, these distributions show that the QSW optimizer improves the overall risk-return-cost trade-off rather than relying on extreme leverage, concentration, or turnover.
It is worth noting that the average QSW turnover in Experiment 3 (around 200% per year) is higher than in the static Phase 1 experiments (typically 20-80% for the best presets in Experiment 1 and a preset mean of about 40% in Experiment 2). This is an expected consequence of moving from a fixed parameter configuration to a fully dynamic hybrid optimizer. At each quarter, the model not only updates the portfolio weights in response to new returns, but may also switch to a different point in the (α, β, λ, ω) grid. Even if the selected configurations are individually low- or moderate-turnover, the act of re-optimizing and jumping between configurations introduces an additional layer of trading. Crucially, however, QSW still trades substantially less than the MPT benchmark (about 200% vs. 330% annual turnover on average) while delivering markedly higher Sharpe and Calmar ratios. Hence, the higher turnover in Experiment 3 reflects the cost of adaptivity rather than a loss of control.
The win-rate bars in Fig. 10a summarize the robustness of these results. QSW outperforms MPT, 1/N, and the S&P 500 in the vast majority of universes across all key metrics (Sharpe, Calmar, CAGR, maximum drawdown, and final value), with win rates close to 100% in most comparisons. Figure 10b shows the same message in risk-return space: QSW outcomes form a tight cluster along a high-Sharpe ridge, while 1/N sits on a lower ridge and MPT outcomes are both more dispersed and systematically below the QSW cloud. The index lies at the bottom-left corner, with the lowest CAGR and Sharpe. This confirms that QSW's advantage is not the result of a few extreme outliers, but a stable shift of the entire risk-return distribution.
Figure 11 shows that the shape of the QSW curves closely tracks that of the 1/N benchmark across all 30 universes. Whenever the equal-weight portfolio delivers a higher final wealth, Sharpe, or Calmar ratio in a given universe, the QSW strategy tends to sit slightly above it, with a very similar risk and drawdown profile and a moderate increase in turnover. In contrast, the MPT series is both noisier and structurally different, with large swings in volatility, drawdown, and trading activity. This behavior is exactly what one would expect from the "smart 1/N" structure identified in Experiment 2: the dynamic QSW optimizer repeatedly re-discovers an almost equal-weight allocation — driven by small α, large β, and low-to-moderate ω — and then adds a thin, data-driven tilt on top of that baseline. Across universes, QSW therefore behaves not as an unstable maximizer, but as a robust, smart version of 1/N: it inherits the diversification and resilience of the naive equal-weight portfolio, while systematically lifting its risk-adjusted performance.
While the cross-universe results establish statistical robustness, it is also instructive to inspect a single, representative universe in detail. Figure 12 illustrates the time evolution of the dynamic QSW strategy on a randomly sampled 100-stock universe, together with the corresponding path of the optimal hyper-parameters.
Several features are worth noting. First, the dynamic QSW strategy consistently stays ahead of both MPT and 1/N, with the performance gaps widening after each major crisis episode (Dot-com, 2008, and COVID-19). This confirms that the gains observed in the cross-universe statistics are not confined to a particular bull market regime, but persist across markedly different environments. Second, the bottom panel shows that the optimizer does not lock into a single parameter setting. Instead, it repeatedly re-learns a familiar pattern: α is kept low most of the time, β is frequently driven to its upper bound, and ω toggles between more classical and more quantum regimes depending on market conditions. During volatile or stressed periods, the model tends to reinforce the diversification channel (large β, low-to-moderate ω), while in calmer regimes it allows slightly more classical exposure. This behavior is fully consistent with the design rules inferred from the 2018-2024 grid search in Phase 1.
Figure 13 further shows that QSW experiences shallower and faster-recovering drawdowns than MPT and the S&P 500, while maintaining volatility in a similar range to the 1/N benchmark. The QSW optimizer, therefore, does not achieve its higher CAGR by leaning into extreme tail risk; instead, it improves the balance between return and drawdown over the full 34-year horizon. For brevity, we only report a representative subset of the 30-universe results and one detailed case study in this section. Additional diagnostic figures-including per-experiment performance panels for all 30 universes (final portfolio values, Sharpe ratios, CAGRs, Calmar ratios, maximum drawdowns, and annual turnover), as well as extended single-universe comparisons-are provided in an online supplementary image archive at https://github.com/aceest/quantum-stochastic-walk-for-Portfolio-Optimization-Theory-and-Implementation-on-Financial-Networks, in the 30_experiments subdirectory. These supplementary figures are fully consistent with the results reported here and allow interested readers to visually inspect the behavior of the QSW strategy across all individual universes and metrics.
