Reverse quantum annealing approach to portfolio optimization problems

Abstract

We investigate a hybrid quantum-classical solution method to the mean-variance portfolio optimization problems. Starting from real financial data statistics and following the principles of the Modern Portfolio Theory, we generate parametrized samples of portfolio optimization problems that can be related to quadratic binary optimization forms programmable in the analog D-Wave Quantum Annealer 2000Q^TM. The instances are also solvable by an industry-established genetic algorithm approach, which we use as a classical benchmark. We investigate several options to run the quantum computation optimally, ultimately discovering that the best results in terms of expected time-to-solution as a function of number of variables for the hardest instances set are obtained by seeding the quantum annealer with a solution candidate found by a greedy local search and then performing a reverse annealing protocol. The optimized reverse annealing protocol is found to be more than 100 times faster than the corresponding forward quantum annealing on average.

Appendices

Appendix 1. Geometric Brownian motion

A geometric Brownian motion (GBM) is a stochastic process S(t) that satisfies the following stochastic differential equation (SDE):

$$ dS(t) = \mu S(t) dt + \sigma S(t) dB(t) , $$

(10)

where t is continuous time and B(t) is a Brownian motion. GBM is widely used to model asset prices. If a unit of time is 1 year, then σ is interpreted as an annualized volatility (standard deviation) of asset’s log-returns, which are assumed to be normally distributed. The drift coefficient μ controls deterministic component of the asset price process.

Integrating the process, we obtain:

$$ S(t) = S(0) \exp \left( \left( \mu - \frac{1}{2}\sigma^2 \right)t + \sigma B(t)\right) . $$

(11)

Although GBM SDE can be used directly to simulate an asset process, it is better to use its solution to ensure that simulated asset prices do not turn negative—this may be the case for large enough time step. In our portfolio optimization example Δt = 1 month and we use the following discretization scheme for a single asset price process:

$$ S(t_{n}) = S(t_{n-1}) \exp \left( \left( \mu - \frac{1}{2}\sigma^2 \right){\Delta} t + \sigma z_{n} \sqrt{{\Delta} t} \right) , $$

(12)

where t_n = t_n− 1 + Δt and z_n is a standard normal random variable. Asset prices from the N-asset portfolio are jointly simulated using the same scheme but correlated standard normal random variables (z⁽¹⁾,…,z^(N)) are constructed via Cholesky decomposition of the correlation matrix ρ.

Appendix 2. Chimera graph of DW2000Q and embedding

In Fig. 4, we show the layout of the chip used for the experiments, belonging to the machine D-Wave 2000Q hosted at NASA Ames Research Center.

Fig. 4

Chimera Chip of DW2000Q. Each gray dot represents an active qubit (missing dots are broken qubits), the black shaded square is representative of one unit cell. The embedding for an instance (N = 42) is highlighted: blue bonds are ferromagnetic couplings set to J_F, while red and pink bonds represent logical couplings (J_ij in Eq. (6))

Appendix 3. More details on parameter setting for reverse annealing

Figure 5 displays median TTS results obtained for the mapping schemes provided by Table 1 and annealing times 1 µs and 10 µs, obtained for the first 10 instances of the benchmark ensamble on an independent set of runs with respect to the results presented in Fig. 3.¹⁰ It is clear that the choice of τ = 1 µs is the most advantageous.

Fig. 5

Time-to-solution (99% confidence level) for different t_run (median over 10 instances)

In Fig. 6, we show on an example how the optimal parameter setting is performed to generate data in Figs. 2 and 3. Scans are performed for different J_F and s_p and the best TTS is selected, instance by instance.

Fig. 6

Time-to-solution (99% confidence level) as a function of annealing parameter s_p. Results for a single instance, N = 42, τ = 1 µs. Circles point out to the best found (J_F, s_p) for these three illustrative cases