Toward Prediction of Financial Crashes with a D-Wave Quantum Annealer
, Javier Gonzalez-Conde 2,4,5,*, Lucas Lamata 6,7, José D. Martín-Guerrero 8,9, Enrique Lizaso 10, Samuel Mugel 10, Xi Chen 2,5, Román Orús 10,11,12, Enrique Solano 1,12,13 and Mikel Sanz 2,4,12,14Round 1
Reviewer 1 Report
This paper is interesting and well-written. The problems that the authors attempt to tackle using quantum annealers is very ambitious given the current abilities of annealers. As a result the authors use problem instances that are small to show that it is possible to use a combination of decomposition and annealers to solve this problem, however inefficient it may be.
My suggestion to the authors is that they discuss sizes of problem instances that would represent realistic modeling of financial crises and discuss what factor speedups would be necessary in order for customized financial annealers to handle these sizes. This would give a clearer indication of how far away such macroeconomic problems are from being effectively solve by annealers.
Author Response
We thank the Referee for the careful reading of our manuscript. We also thank him/her for raising interesting suggestions.
In order to model a realistic problem, the estimation of problem size still follows the same analysis on the example in our manuscript. It depends on the binary encoding, approximation on the nonlinearity, and hardware architecture. Even if we have a customized annealer for these problems, one can still not theoretically guarantee exponential speedup since quantum annealing is also a heuristic approach. However, there should exist speedup in experiments, at least claimed by D-Wave based on their hardwares.
Reviewer 2 Report
The manuscript presents a study of a network model which can be thought of as an assets price model. The main purpose of the proposed algorithm is to find whether the non-linear term in the model provides a higher or lower asset price. This boils down the problem as a minimization problem of continuous variables. The continuous variables are then mapped approximately by the discrete function which can be rewritten as an Ising model with only two spin interactions by introducing ancilla spins. The quantum annealer, D-wave, is then used to study the model. The manuscript is an extension of their project from the previous publications from the same group.
The paper is well written and it demonstrates the method they proposed earlier in the quantum annealer, D-wave. We recommend its publication. We hope the authors can address the following in the final version of the manuscript.
1. Is the minimization of the eq. 3 a NP-hard problem?
2. How do the number of ancilla spins and couplings scale with the q?
3. Do the testing cases reflect any 'realistic' assets?
4. Can other nonlinear terms be used in this approach? How about an exponential function?
Author Response
We would like to express our gratitude to the referee for the careful reading of our manuscript and raising interesting questions. We are pleased that the referee has recognized the significance of our work. Now, let us reply the minor concerns
- Is the minimization of the eq. 3 a NP-hard problem?
Yes, unconstrained quadratic optimization problems are generally NP-hard, meaning that finding the optimal solution is computationally infeasible for large-scale problems [Numerical Optimization (Jorge Nocedal and Stephen J. Wright), Springer, 2006.]
2. How do the number of ancilla spins and couplings scale with the q?
To our understanding, the referees refer to the number of classical bits to encode the prices in binary. Therefore, every classical bit has a one-to-one correspondence with logical qubits that encode the same problem. Additionally, we present explicit expressions in Sec. 4 for how the total number of ancillary qubits scales. Note that we require more ancillary quits and couplers if we consider the minor embedding in real devices according to their architectures.
3. Do the testing cases reflect any 'realistic' assets?
They are all synthetic but following all constraint conditions proposed in the theoretical model. We believe there is no loss of generality.
4. Can other nonlinear terms be used in this approach? How about an exponential function?
Yes, any non linear term could be approximated by the Legendre expansion, which is a universal technique to approximate a function with polynomials due to its completeness and orthogonality. The exponential function could be approximated as well.
Reviewer 3 Report
An interesting approach
Author Response
We thank the Referee for the careful reading of our manuscript.
