Economic Decoupling Probability: A Quantum Analogy of Characterizing Bell State Errors and Noise on Real IBM Quantum Hardware

1. Introduction

This study was motivated by the inherent challenge of modeling uncertainty within economic systems, particularly concerning scenarios where established correlations between indicators or markets unexpectedly broke down, potentially impacting forecasting and risk management. Recognizing that recent explorations had considered quantum frameworks for modeling financial and economic systems due to their probabilistic nature and handling of correlations (Orrell, 2020; Herman et al., 2023; Orús et al., 2019), we employed a fundamental quantum procedure: the generation of the maximally entangled |Φ⁺⟩ Bell state using a standard circuit involving a Hadamard gate followed by a Controlled-NOT (CNOT) gate (Nielsen & Chuang, 2010). This state, defined as:

In an ideal, noiseless scenario, measurements of this state should be perfectly correlated, meaning the outcome is always |00⟩ (both qubits measured as |0⟩) or |11⟩ (both qubits measured as |1⟩) (Bell, 1964; Nielsen & Chuang, 2010). Within this quantum foundation, we proposed a conceptual mapping for analogy: the correlated outcomes |00⟩ and |11⟩ represented expected, synchronized economic scenarios ("Scenario Alpha" and "Scenario Beta," respectively), while the anti-correlated outcomes |01⟩ and |10⟩—ideally absent—were mapped to represent "Unexpected Decoupling" events, the focus of economic uncertainty. It was emphasized that this constituted a conceptual analogy (Mäki, 2009) designed to explore parallels between quantum measurement statistics and economic unpredictability. The experiment was situated within the context of Noisy Intermediate-Scale Quantum (NISQ) computing (Preskill, 2018), where processors inherently suffered from noise sources like gate errors, readout errors, and decoherence that limited accuracy. Consequently, real-world devices deviated from ideal behavior, yielding non-zero probabilities for the anti-correlated |01⟩ and |10⟩ outcomes when attempting to prepare and measure a Bell state (see e.g., Kandala et al., 2019; Temme et al., 2017).

Benchmarking such fundamental operations was crucial for characterizing these limitations (Eisert et al., 2020; Cross et al., 2019). This experiment utilized the Qiskit framework (Abraham et al., 2019) on the IBM Quantum device ibm_kyiv to investigate these deviations, addressing two primary research questions: (A1) How accurately did the device prepare and measure the correlated |Φ⁺⟩ Bell state, reflecting the analogous probability of expected economic scenarios? and (A2) How significant were the deviations from perfect correlation, specifically, what was the probability of observing the 'unexpected decoupling' outcomes (|01⟩ or |10⟩), providing an analogue for the frequency of unexpected economic events driven by noise/uncertainty factors? The paper proceeded by detailing the methods used for Bell state preparation and measurement, presenting the obtained results, and offering a discussion interpreting these findings within the economic analogy framework, followed by concluding remarks.

2. Literature Review

Quantum computing represents a paradigm shift from classical computation, leveraging quantum mechanical principles like superposition and entanglement to tackle problems previously considered intractable (Nielsen & Chuang, 2010). Entanglement, a key resource, describes correlations between quantum systems that are stronger than any classical counterpart, a concept famously debated by Einstein, Podolsky, and Rosen (1935) and later formalized by Bell (1964; Vedral, 2008). The current state of quantum hardware development is often described as the Noisy Intermediate-Scale Quantum (NISQ) era (Preskill, 2018). NISQ devices possess a limited number of qubits and are highly susceptible to noise, making the accurate characterization and benchmarking of their performance critically important (Cross et al., 2019; McKay et al., 2019).

A fundamental benchmark for quantum processors involves the generation and measurement of entangled states, particularly the Bell states. The ability to reliably create these states is a prerequisite for many quantum algorithms and protocols. However, achieving high fidelity is challenging due to various noise sources inherent in NISQ hardware. These include environmental interactions leading to decoherence (related principles discussed in Slichter, 1990), as well as imperfections in quantum gate operations and measurement readout processes. These errors cause the experimentally realized quantum state to deviate from the intended ideal state, limiting the computational power of near-term devices.

To address the impact of noise, various quantum error mitigation techniques have been developed. Strategies such as Zero Noise Extrapolation (ZNE) attempt to estimate the ideal noiseless outcome by systematically varying noise levels and extrapolating the results back to the zero-noise limit (Temme et al., 2017; Kandala et al., 2019; Qiskit Development Team, n.d.). Other methods specifically target errors occurring during the final measurement step. Among these, Matrix-Free Measurement Mitigation (M3), implemented in the mthree package, offers a scalable approach. Unlike methods requiring the construction and inversion of the full assignment matrix (which becomes computationally prohibitive for many qubits), M3 operates within a reduced subspace defined by the observed noisy measurement outcomes. This matrix-free technique significantly reduces memory requirements and computational cost, making readout mitigation feasible for larger systems compared to traditional matrix inversion methods (Nation et al., 2021). While Qiskit Runtime's EstimatorV2 primitive incorporates built-in mitigation options, SamplerV2 (used for obtaining distributions, as in this work) does not, necessitating the use of external libraries like mthree for post-processing correction. These mitigation techniques aim to improve the accuracy of results obtained from NISQ computers without the significant overhead required for full fault-tolerant quantum error correction.Beyond fundamental benchmarking, quantum computing holds potential for applications in various fields, including finance and economics. Researchers are exploring quantum algorithms for tasks like portfolio optimization (Rebentrost et al., 2014) and are investigating broader applications within quantitative finance (Orús et al., 2019; Herman et al., 2023). This burgeoning interest motivates exploring connections, even analogical ones, between quantum phenomena and complex economic systems.

The implementation and analysis of quantum experiments heavily rely on specialized software frameworks and libraries. Qiskit provides a comprehensive ecosystem for quantum computing research, development, and execution on various hardware backends, including those from IBM Quantum (Abraham et al., 2019; Qiskit Development Team, n.d.). Data processing and visualization are further supported by established scientific Python libraries such as NumPy for numerical operations (Harris et al., 2020), Matplotlib for plotting (Hunter, 2007), and Pandas for data manipulation (The pandas development team, 2024). These tools are essential for designing circuits, executing jobs, analyzing results, and communicating findings in quantum computation research.

3. Methodology

The experiment focused on generating the maximally entangled Bell state |Φ⁺⟩ = (|00⟩ + |11⟩) / √2 (Nielsen & Chuang, 2010), for which the ideal, noise-free measurement yielded perfectly correlated outcomes exclusively in the |00⟩ or |11⟩ state, corresponding to a theoretical probability of correlated outcomes P(Corr) = P(00) + P(11) = 1.0 (Bell, 1964; Nielsen & Chuang, 2010). The quantum circuit implemented to prepare this state was the standard two-qubit sequence—a Hadamard gate on the first qubit followed by a CNOT gate controlled by the first qubit targeting the second, immediately followed by measurement (Nielsen & Chuang, 2010)—defined and managed using the Qiskit framework (Abraham et al., 2019), with logical and transpiled diagrams presented in Figure 3. All simulations, hardware job submissions via the cloud, and subsequent data analysis were conducted using the Python programming language within a Miniconda environment running on a local machine equipped with an Intel Core i7 processor. Code development was performed using Visual Studio Code. Core quantum tasks relied on the Qiskit framework (Abraham et al., 2019), including the qiskit-ibm-runtime package for executing jobs on the target IBM Quantum backend (ibm_kyiv) and built-in simulators. Numerical computations utilized NumPy (Harris et al., 2020), visualization was performed with Matplotlib (Hunter, 2007), and readout error mitigation was applied using the mthree library.

Figure 2. Quantum circuit for generating the Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2 using Hadamard (H) and CNOT (CX) gates.

Figure 2. Quantum circuit for generating the Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2 using Hadamard (H) and CNOT (CX) gates.

Figure 4 visualized the measured probability distributions for the four possible computational basis outcomes ('00', '01', '10', '11') after preparing the |Φ⁺⟩ Bell state under different conditions. Ideally, the distribution showed only the correlated outcomes '00' and '11', each with a probability of 0.5. In contrast, the raw results obtained from hardware execution on both layouts ([2, 3] and [7, 8]) exhibited significant deviations, with non-zero probabilities appearing for the anti-correlated outcomes '01' and '10', and noticeable differences between the two layouts. The figure further illustrated the effect of readout error mitigation, which, when applied, reduced the erroneous anti-correlated probabilities ('01', '10') and increased the probabilities of the correct correlated outcomes ('00', '11'), bringing the mitigated distribution closer to the ideal case.

To prepare the target entangled state in this experiment, the standard logical quantum circuit for generating the Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2 using Hadamard (H) and CNOT (CX) gates, as depicted in Figure 2, was first defined. However, this abstract representation could not be directly executed on physical hardware. Therefore, before running the experiment on the ibm_kyiv processor, the logical circuit underwent a crucial adaptation process called transpilation, resulting in the transpiled quantum circuit diagram for Bell state (|Φ⁺⟩) preparation mapped to physical qubits on ibm_kyiv, shown in Figure 3. This transpilation step mapped the circuit's logical qubits (0 and 1) to specific physical qubits on the device (such as layout [2, 3] or [7, 8]) and decomposed the standard H and CX gates into the sequence of native hardware gates that ibm_kyiv could actually perform, yielding the circuit that was physically executed.

The experimental workflow began with initialization, where the Bell state circuit and key configuration parameters, including the target backend (ibm_kyiv), specific qubit layouts ([2, 3] and [7, 8]), number of runs (N=5), and shots (4096), were defined. An ideal simulation was run using AerSimulator to establish a baseline result with zero expected anti-correlated outcomes (P(Anti)=0). Connection to the IBM Quantum service was established via QiskitRuntimeService. The main execution proceeded by looping through each target qubit layout. For each layout, relevant device calibration data was retrieved and logged, the logical Bell circuit was transpiled for the specific layout and backend, and the mthree readout mitigation was calibrated. Within this loop, a second loop iterated through five independent runs.

In each run, the transpiled circuit was submitted as a job to the ibm_kyiv hardware using the SamplerV2 primitive, and the raw measurement counts were retrieved. Raw performance metrics, specifically P(Anti), were calculated from these counts. Subsequently, mthree readout mitigation was applied to the raw counts, and mitigated P(Anti) metrics were calculated. The results for each individual run, including raw and mitigated data, were stored. After complete runs for all layouts were completed, the collected data was aggregated to calculate mean and standard deviation values for P(Anti). Correlation analysis was performed to compare the aggregated raw P(Anti) against the retrieved calibration data (like readout error). Finally, visualizations comparing the results were generated, and the detailed run data and calibration logs were saved to files.

4. Discussion

4.1. Interpretation: Quantifying Variability and the Uncertainty Analogue

The experiment successfully benchmarked the preparation and measurement of the |Φ⁺⟩ Bell state across multiple runs (5 runs per layout, as shown in the results table) on two distinct qubit pairs of the ibm_kyiv device. The results revealed significant variability in raw fidelity, with layout [2, 3] achieving a mean P(Corr) of 98.45% ± 0.33%, while layout [7, 8] achieved only 90.79% ± 0.79%. Correspondingly, the raw probability of anti-correlated outcomes P(Anti), which serves as the primary output for our economic analogy, ranged from a mean of 1.55% ± 0.33% for the higher-fidelity pair [2, 3] to 9.21% ± 0.79% for the lower-fidelity pair [7, 8]. Readout error mitigation using mthree proved highly effective numerically for both pairs across the 5 runs, consistently reducing the mean P(Anti) to near-zero levels (0.09% ± 0.20% for [2, 3] and 0.00% ± 0.00% for [7, 8]), yielding mitigated P(Corr) values of approximately 99.91% and 100.00% respectively.

Viewed through the lens of the economic analogy, where P(Anti) represents the likelihood of 'unexpected decoupling', this experiment quantified a range of possibilities (~1.6% to ~9.2%) depending on the specific quantum subsystem (qubit pair) used for the simulation. This suggests that the inherent noise level, analogous to underlying factors driving economic uncertainty, can vary substantially even within the same device. The near-perfect mitigated results indicate that a large portion of the raw P(Anti) in this experiment was attributable to measurement errors (analogous perhaps to data misinterpretation or reporting errors), and correcting for these allows for probing a baseline closer to ideal correlations. This interpretation draws from ideas applying quantum concepts to economic uncertainty (Orrell, 2020) within a conceptual analogy framework (Mäki, 2009).

4.2. Noise Source Analysis and Calibration Correlation

The observed variability in raw P(Anti) between layouts [2, 3] and [7, 8] aligns strongly with the retrieved calibration data, providing insight into the contributing noise sources inherent in the NISQ hardware (Preskill, 2018). The correlation analysis revealed that higher mean raw P(Anti) was strongly correlated (Pearson r=1.000, N=2) with both higher average readout error rates and higher Hadamard gate error rates for the first qubit of the pair. Layout [7, 8], exhibiting the much higher raw P(Anti) (~9.2%), also possessed significantly worse average readout error (~8.98%) compared to layout [2, 3] (~0.76%), largely driven by a very high reported error for qubit 8 (~16.6% - detail from source text). Similarly, the Hadamard error for qubit 7 (~0.073%) was higher than for qubit 2 (~0.015%). This suggests readout errors and single-qubit gate errors were dominant contributors to the observed raw P(Anti) differences. Coherence times (T1, T2) showed smaller differences between the pairs (T1/T2: [2,3] - 322.4/144.6 µs; [7,8] - 337.0/322.1 µs) and the expected negative correlation with P(Anti) (r=-1.000, N=2), indicating that decoherence (Martinis et al., 2009) likely played a role, but perhaps less distinguishing than readout/gate errors for this short circuit. Unfortunately, CNOT error data was unavailable in the calibration report, preventing assessment of its contribution, although two-qubit gate errors are typically a primary source of entanglement fidelity loss (Chow et al., 2012). Drawing conceptual parallels (Mäki, 2009), the dominant readout errors could be seen as analogous to significant data integrity issues overshadowing other systemic factors in the economic model.

While this study proposes P(Anti) derived from quantum system noise as a quantitative analogue for unexpected correlation breakdowns, it is important to contrast this with established methods in economics and finance. Classical approaches often rely on analyzing historical time-series data using techniques such as rolling window correlations using standard coefficients (like Pearson's or Spearman's rank correlation, see e.g., Field, 2018) or other sophisticated multivariate time-series models designed to capture time-varying dependencies. These classical metrics provide valuable insights based on past observed data. The P(Anti) measure presented here differs fundamentally as it is not derived from economic time series, but rather emerges directly from the physical noise processes (readout error, gate error, decoherence) inherent in the quantum hardware simulating the correlated state, as supported by our calibration correlation analysis. Key conceptual differences include its basis in physical device noise rather than statistical properties of historical data, and its representation of an instantaneous error probability for a specific quantum operation rather than an evolved correlation coefficient derived from market prices. Therefore, while P(Anti) cannot replace classical metrics for economic forecasting, it may offer a complementary perspective – a physically grounded proxy quantifying a system's inherent susceptibility to specific types of error or 'decoupling' based on its underlying noise characteristics.

The conceptual mapping explored here also connects to a growing body of work investigating the application of quantum formalism and concepts to economics, finance, and social sciences (e.g., Orrell, 2020; Busemeyer & Bruza, 2012). Fields such as quantum cognition utilize quantum probability (including interference effects) to model decision-making paradoxes and cognitive biases (Busemeyer & Bruza, 2012), often differing from our approach which sources its uncertainty analogue (P(Anti)) directly from hardware noise rather than purely mathematical constructs. Similarly, quantum economics, as described by proponents like Orrell, often employs quantum concepts like wave functions and operators metaphorically for economic value, transactions, or price distributions (Orrell, 2020). Our work complements these theoretical approaches by providing a perspective grounded in the physical characteristics and measured error rates of contemporary quantum hardware. We utilize the actual noise measured during a fundamental quantum operation on a NISQ device as the source for our quantitative analogy, directly linking this 'uncertainty' analogue to measurable physical error rates (like readout error). This hardware-grounded analogical approach, quantifying noise via P(Anti), appears distinct from purely theoretical models using quantum mathematics or broader philosophical applications of quantum concepts to social systems and offers a novel way to leverage quantum device characterization for exploring concepts relevant to other complex domains.

4.3. Limitations

While expanding beyond a single run, this study's conclusions are still subject to limitations. The results represent performance on only two specific layouts of one device (ibm_kyiv) during a limited timeframe (April 14, 2025 - based on current date context). Quantum hardware performance fluctuates, and comprehensive benchmarking requires testing across more devices, layouts, and times (Eisert et al., 2020; Erhard et al., 2019). Although calibration data was logged and showed strong correlations, the lack of CNOT error data limits a complete error budget analysis. Furthermore, the correlation analysis is based on only two layouts, meaning the high correlation coefficients (r=±1.0) are indicative but lack statistical power. Standard statistical uncertainties due to the finite number of shots (N=4096 per run, inferred from standard practice, not explicitly in tables) still apply, captured partly by the reported standard deviations across runs (Cumming, 2014). Finally, the economic mapping remains a conceptual analogy (Mäki, 2009), useful for illustration and quantifying a quantum system's deviation, but not validated as a predictive economic model.

4.4. Implications and Future Work

This work provides updated multi-run benchmarks for Bell state fidelity on ibm_kyiv, revealing significant performance heterogeneity across qubit pairs (raw P(Corr) ~91-98%) and demonstrating highly effective readout error mitigation (mitigated P(Corr) ~99.9-100%). The range of observed raw error rates (~1.6-9.2% P(Anti)) highlights the substantial impact that device noise can have even on fundamental operations, underscoring challenges for complex algorithms requiring high fidelity (Preskill, 2018). The successful mitigation suggests readout noise is a key target for improvement. Future work should address the limitations by: extending the benchmarking to more qubit pairs and devices; performing runs over time to assess stability; ensuring CNOT and other relevant calibration data are captured and correlated (Eisert et al., 2020); and increasing the number of runs for tighter statistical bounds. Implementing and comparing other error mitigation techniques like Zero-Noise Extrapolation (Temme et al., 2017; Kandala et al., 2019) could provide further insights. The economic analogy could be explored further via controlled noise simulations representing different economic volatility levels or by investigating theoretical parallels between quantum error mitigation and economic stabilization strategies

5. Conclusion

This study successfully benchmarked the preparation and measurement fidelity of the |Φ⁺⟩ Bell state on the ibm_kyiv quantum processor, assessing performance across multiple runs (N=5) for two distinct qubit pairs ([2, 3] and [7, 8]) using the qiskit-ibm-runtime SamplerV2 primitive. The results revealed significant variability in raw device performance, with mean anti-correlated outcome probabilities (P(Anti)) ranging from ~1.6% for layout [2, 3] to ~9.2% for layout [7, 8]. Analysis demonstrated a strong correlation between this raw P(Anti) and reported device calibration metrics, particularly average readout error rates and single-qubit Hadamard gate errors, highlighting their contribution to performance differences. Furthermore, the application of mthree-based readout error mitigation proved highly effective, reducing P(Anti) to near-zero levels (≤0.1%) and achieving mean correlated state fidelities (P(Corr)) of ~99.9-100.0% for both tested layouts.

Beyond providing multi-run fidelity benchmarks and demonstrating successful error mitigation, this work utilized the experimentally measured P(Anti) as a quantitative output for a conceptual analogy mapping quantum noise to uncertainty in correlated economic systems. The observed range of raw P(Anti) illustrated how different inherent noise levels in the quantum simulation could represent varying likelihoods (~2-9%) of 'unexpected decoupling' events in the analogy, while the near-ideal mitigated results represented a baseline achievable after correcting for measurement-type errors.

While acknowledging limitations related to the specific device, timeframe, number of layouts tested, and the conceptual nature of the analogy, this research provides concrete evidence of qubit performance heterogeneity and the efficacy of readout error mitigation on ibm_kyiv. Future directions include broader benchmarking across more layouts and devices, incorporating more complete calibration data (including CNOT errors), exploring alternative error mitigation techniques, and further developing the quantum-economic analogy, potentially through controlled noise simulations. In summary, this study offers valuable benchmarks for a fundamental quantum operation and presents a quantified, noise-based quantum analogy applicable to exploring correlation breakdowns in complex systems.