[Paper Review] Portfolio Optimization with Digitized-Counterdiabatic Quantum Algorithms

N. N. Hegade, P. Chandarana+4 more|arXiv (Cornell University)|Dec 15, 2021

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes digitized-counterdiabatic quantum algorithms (DCQC and DC-QAOA) to enhance portfolio optimization in the NISQ era, using approximate counterdiabatic terms to suppress non-adiabatic transitions. It demonstrates a significant improvement in ground state success probability—up to 0.60—compared to standard QAOA, especially with optimized parameter initialization, showing promise for quantum advantage in finance applications.

ABSTRACT

We consider digitized-counterdiabatic quantum computing as an advanced paradigm to approach quantum advantage for industrial applications in the NISQ era. We apply this concept to investigate a discrete mean-variance portfolio optimization problem, showing its usefulness in a key finance application. Our analysis shows a drastic improvement in the success probabilities of the resulting digital quantum algorithm when approximate counterdiabatic techniques are introduced. Along these lines, we discuss the enhanced performance of our methods over variational quantum algorithms like QAOA and DC-QAOA.

Motivation & Objective

• Address the challenge of low success probabilities in NISQ-era quantum optimization for industrial applications.
• Overcome limitations of standard QAOA and adiabatic quantum computing, such as barren plateaus and slow evolution times.
• Apply digitized-counterdiabatic (DC) techniques to improve convergence and fidelity in quantum algorithms for financial optimization.
• Demonstrate the feasibility and performance gains of DC-QAOA over QAOA in solving the Markowitz mean-variance portfolio optimization problem.
• Explore the scalability and robustness of DC-QAOA under random parameter initialization and increasing circuit depth.

Proposed method

• Formulate the Markowitz portfolio optimization problem as a quantum spin Hamiltonian with qubit encoding.
• Implement digitized-adiabatic quantum computing (DAdQC) with discrete time evolution using Trotterization.
• Introduce approximate counterdiabatic (CD) terms as local Hamiltonians proportional to $\sigma_i^y$, derived from an operator pool to suppress non-adiabatic transitions.
• Construct a DC-QAOA ansatz by adding CD terms to the standard QAOA circuit: $U_D(\alpha) = e^{-i\alpha \sum_i h_i \sigma_i^y}$, alongside mixer and problem Hamiltonians.
• Optimize parameters $(\gamma, \beta, \alpha)$ using a classical stochastic gradient descent optimizer (Adagrad) with a fixed step size of 0.1.
• Evaluate performance using success probability $P_s$ as a metric, averaged over 20 random initializations with top 10 results reported.

Experimental results

Research questions

• RQ1 Can digitized-counterdiabatic quantum algorithms significantly improve the success probability of ground state preparation in portfolio optimization compared to standard QAOA?
• RQ2 How does the inclusion of approximate CD terms affect the convergence and robustness of hybrid quantum-classical algorithms like DC-QAOA in the presence of noisy intermediate-scale quantum devices?
• RQ3 What is the impact of parameter initialization on the performance of DC-QAOA, and how does it relate to the complexity of the energy landscape in larger systems?
• RQ4 To what extent do CD terms mitigate the challenges of barren plateaus and local minima in the optimization of financial portfolio problems?
• RQ5 Can DC-QAOA achieve higher success probabilities than QAOA across multiple random data instances of the portfolio optimization problem?

Key findings

• DC-QAOA achieves a maximum success probability of $P_s = 0.60$ for one instance with $N=6$, $n=3$, $g=2$, and $p=5$ layers, significantly outperforming standard QAOA.
• For all tested instances, DC-QAOA consistently shows higher average success probabilities than QAOA across $p=1, 3, 5$ layers, indicating improved algorithmic expressibility.
• The standard deviation of success probabilities increases with circuit depth in DC-QAOA, reflecting the sensitivity to random parameter initialization due to the complex, non-convex cost landscape.
• The success probability improvement is attributed to enhanced expressibility and suppression of non-adiabatic transitions via approximate CD terms.
• The results demonstrate that CD-assisted algorithms can drastically enhance finite-time adiabatic evolution and hybrid quantum-classical optimization for hard combinatorial problems.
• The study confirms that CD terms provide a practical path toward quantum advantage in industrial applications like finance, even with current NISQ hardware constraints.