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It was a rainy night, bro was learning QAOA and stuck on such a problem. For example, 100 CAD is exchanged for USD, USD is exchanged for EUR, and EUR is then exchanged back to CAD. The final value in CAD is calculated as 100 × (1/1.05) × (1/1.094) × 1.18 ≈ 102.72, so you get 2 CAD more after this circle.

My thought is seting the rate R[i][j] between currency i and j,we can use bitstring as the solution of the path(like '0110' means R[0][1]R[1][0]), so we can build the objective function and calculate the expectation value now.

My question is, do we need to add constraints to the objective function to compute the expected value of only reasonable paths? Like some path, '1110', it doesn't make sense: you cannot sell your money twice, can we just remove such a path?

Furthur more, the path should be a closed circle(after all we want our money back), do we need to check all the paths and remove all the non-circle paths? That's kinda ridiculous, if we check all the paths why don't we calculate the expectation value together.

It's been a whole night it still confused me, thanks for any help.

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  • $\begingroup$ You're bigger issue might be, how to make such an objective function quadratic? $\endgroup$
    – Ethan Davies
    Commented Jun 19 at 10:09
  • $\begingroup$ True, thanks for clarify, that's the point. $\endgroup$
    – Budget
    Commented Jun 20 at 3:06

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This problem is known as the Currency Arbitrage Problem. This white paper shows how to formulate it as a QUBO.

The white paper is talking about solving the problem using a quantum annealer. Of course, you can use this QUBO with QAOA.

You may find this paper also useful. It shows how to solve this problem using Qiskit.

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  • $\begingroup$ Thanks for help! After read the papers I found that I know nothing about QUBO problem. It seems that I need to: (1) get the QUBO format (2) encoding the QUBO function into the Hamiltonian (like, in Pauli operators?) (3) build the QAOA circuit and solve the problem. $\endgroup$
    – Budget
    Commented Jun 20 at 15:13

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