Quantum Optimization of the Color-of-Money Problem on IBM Quantum Hardware: A Proof-of-Concept Study

AI summary

Overview

Research area: Quantum optimization for finance, specifically quantum approaches to treasury liquidity and invoice-funding problems (the "Color-of-Money" problem), tested on real IBM Quantum hardware.

Technical level: Intermediate. The paper assumes some familiarity with QUBO formulations, variational quantum algorithms, and penalty-based constraint encoding, but the financial problem itself is explained clearly enough for newcomers.

Scope (one sentence): A proof-of-concept study that converts a constrained treasury optimization model into a QUBO, solves it with a Pauli Correlation Encoding variational algorithm plus error mitigation, and checks whether IBM hardware reproduces the exact classical optimum on a 3-asset, 3-obligation, 2-period instance.

What This Paper Is About

Enterprises must decide which accounts or funding nodes pay which invoices, in which time period, and how to move money between nodes so that balances, eligibility rules, and payment deadlines are all respected. This "Color of Money" problem is a constrained combinatorial optimization problem that becomes harder as assets, obligations, and planning periods grow. The paper asks a narrow, practical question: can a current noisy quantum processor, using a qubit-efficient encoding, return the same optimal funding plan that an exact classical solver finds?

Key Contributions

1. A quantum-ready formulation of the CoM problem. The authors express the Invoice Funding and Liquidity Transfer Problem as a constrained Mixed-Integer Linear Programming (MILP) model covering flow balance, invoice completion, linking, eligibility, and assignment-before-due-date constraints, then transform it into a QUBO with four penalty terms (for the constraints in equations 2, 3, 4, and 6), with eligibility (equation 5) enforced directly in the variable definitions.

2. A qubit-efficient solution pipeline using Pauli Correlation Encoding (PCE). Instead of one qubit per binary variable as in standard QAOA-style formulations, PCE encodes variables in two-qubit Pauli correlation observables, reducing the qubit requirement from O(η) to O(√η) for two-body encodings. Same-axis correlations only are used, so all encoded variables can be measured with a constant number of measurement bases.

3. A hybrid workflow that separates training from hardware execution. Continuous variables (payments, transfers, balances, and a slack term) are binary-discretized via powers of two, penalty weights are chosen with the Verma and Lewis Method (VLM), variational parameters are trained on a noise-free simulator, and only the final circuit evaluation is run on hardware with Zero-Noise Extrapolation (ZNE) via circuit folding.

4. Hardware validation against an exact classical benchmark. A prototype with 3 assets, 3 obligations, and 2 time periods was executed on the ibm-aachen processor and compared with the SCIP solver, reproducing the same optimal objective value.

Main Findings

• Identical optimal objective: SCIP and the VQA-PCE implementation both returned an objective value of 151.00, and both recovered the optimal solution (reported as "Yes" in Table I).
• No runtime advantage: Reported execution time was 0.02 seconds for SCIP versus 31.00 seconds for VQA-PCE.
• Noise did not prevent convergence to the optimum: The quantum run reached the global optimum despite hardware noise and NISQ-era limitations, with ZNE applied to improve expectation-value reliability.
• Deliberately small instance: The dataset was kept to three assets, three invoices, and two time periods specifically to fit within current hardware constraints while preserving the structure of the funding and transfer problem.
• Correctness, not advantage: The authors state explicitly that the goal is to establish correctness and feasibility of the workflow, not to demonstrate quantum advantage.
• Benchmark platform details: Classical runs used SCIP on an AMD Ryzen 5 7535U processor with 12 logical cores and 16 GB of RAM; quantum runs used ibm-aachen. The number of qubits and circuit depths used are not reported.

Methodology in Plain English

The team started with a realistic treasury model: money sits in funding nodes, invoices have amounts and due dates, transfers between nodes have costs and delays, and only certain nodes are allowed to pay certain invoices. They wrote this as a standard MILP with an objective that minimizes transfer and payment costs.

To make it quantum-compatible, they converted the model into a QUBO. Money amounts are not naturally binary, so each continuous quantity was rewritten as a sum of powers of two times binary variables. Rules that cannot be broken — balance conservation, paying each invoice in full, linking payments to assignment decisions, and assigning each invoice exactly once before its due date — were turned into squared penalty terms added to the objective, with penalty weights selected using the Verma and Lewis Method.

The binary variables were then packed into pairs of qubits using Pauli Correlation Encoding: each variable corresponds to the measured correlation between two qubits, and a tanh function converts that correlation into a smooth value near 0 or 1. This lets many more variables fit on a small register than a one-qubit-per-variable approach would allow.

Training happened on a noise-free simulator, where a classical optimizer adjusted the circuit parameters to minimize the penalty-augmented objective. The trained parameters were then loaded onto the real IBM device, the circuit was run at several noise levels created by circuit folding, and the results were extrapolated back to the zero-noise limit. Finally, the measured correlations were thresholded into binary decisions to produce the funding plan, which was compared with the SCIP solution.

Why This Matters

Impact on research: Most quantum-finance work has concentrated on portfolio optimization, risk management, and collateral allocation, while treasury-oriented CoM problems have received limited attention. This paper supplies a concrete, reproducible template — MILP to QUBO, PCE encoding, simulator training, hardware execution with ZNE, and comparison against an exact solver — for a class of problems that matters to banks and corporate treasuries. It also adds to the small set of studies that actually run on hardware and check results against exact classical benchmarks rather than only simulating.

Real-world applications:

• Corporate treasury invoice funding and payment routing across multiple accounts and deadlines.
• Bank liquidity and collateral management, where restricted funds must be allocated under regulatory and operational rules.
• Multi-currency cash pooling and inter-account transfer scheduling with time-dependent settlement delays.
• Regulatory-compliant allocation of restricted or earmarked funds, which is the literal "color of money" setting.

Industry relevance: The work comes from TCS Research and targets a genuine operational pain point in financial services, where regulatory and operational complexity keeps increasing. Its honest framing — same answer as classical, far slower at this scale — sets realistic expectations for near-term adoption while identifying the encoding and error-mitigation choices that make hardware execution viable at all.

Future Directions

• Extend the framework to larger and more realistic CoM scenarios with additional assets, obligations, currencies, and planning periods, where the classical search space grows rapidly.
• Investigate more advanced optimization strategies and error-mitigation techniques to improve solution quality and execution efficiency.
• Test whether the qubit savings from PCE translate into solving instances that exact MILP solvers handle only with difficulty, since large-scale financial optimization is described as becoming computationally intensive.
• Continue benchmarking against exact classical solvers as hardware improves, to track when, if ever, quantum execution becomes competitive rather than merely correct.

Target Audience

Quantum computing researchers working on quantum finance and combinatorial optimization; treasury, liquidity, and collateral specialists curious about quantum formulations of their problems; graduate students looking for an end-to-end example of taking a constrained MILP to real quantum hardware; and industry R&D teams assessing the near-term practicality of NISQ optimization for financial workloads.

Authors’ abstract

The Color-of-Money (CoM) problem is a financial optimization challenge in which enterprises must determine how to allocate and route funds across multiple accounts, currencies, and time periods while satisfying regulatory, operational, and liquidity constraints. As the number of assets, obligations, and planning periods increases, the decision space grows rapidly, making large-scale optimization increasingly challenging for classical approaches. In this work, we present a proof-of-concept quantum implementation of the CoM problem using IBM Quantum hardware. The original optimization problem is formulated as a constrained Mixed-Integer Linear Programming (MILP) problem and transformed into a Quadratic Unconstrained Binary Optimization (QUBO) representation suitable for quantum execution. We employ a Variational Quantum Algorithm (VQA) with Pauli Correlation Encoding (PCE), a qubit efficient encoding technique and apply error-mitigation technique to improve solution fidelity on noisy quantum hardware. A prototype instance consisting of 3 assets, 3 obligations, and 2 time periods was executed on real IBM quantum hardware and benchmarked against classical exact solver SCIP. The quantum approach successfully reproduced the same optimal solution obtained classically, validating the correctness and feasibility of the proposed methodology. While current noisy quantum hardware does not outperform classical optimization on the evaluated small instance, the results demonstrate the feasibility of executing treasury optimization workloads on current quantum hardware and provide a foundation for future investigations.