Improving Variational Quantum Optimization using CVaR (1907.04769v3)

Abstract: Hybrid quantum/classical variational algorithms can be implemented on noisy intermediate-scale quantum computers and can be used to find solutions for combinatorial optimization problems. Approaches discussed in the literature minimize the expectation of the problem Hamiltonian for a parameterized trial quantum state. The expectation is estimated as the sample mean of a set of measurement outcomes, while the parameters of the trial state are optimized classically. This procedure is fully justified for quantum mechanical observables such as molecular energies. In the case of classical optimization problems, which yield diagonal Hamiltonians, we argue that aggregating the samples in a different way than the expected value is more natural. In this paper we propose the Conditional Value-at-Risk as an aggregation function. We empirically show -- using classical simulation as well as quantum hardware -- that this leads to faster convergence to better solutions for all combinatorial optimization problems tested in our study. We also provide analytical results to explain the observed difference in performance between different variational algorithms.

• The paper introduces CVaR as an alternative to expectation minimization, accelerating convergence in variational quantum optimization.
• Experimental and theoretical analyses demonstrate that CVaR enhances solution quality for combinatorial optimization challenges.
• CVaR-modified algorithms achieve robust performance improvements on both classical simulations and noisy quantum hardware.

Improving Variational Quantum Optimization using CVaR

The paper "Improving Variational Quantum Optimization using CVaR" examines advancements in hybrid quantum/classical variational algorithms, specifically for addressing combinatorial optimization (CO) challenges. As quantum computing enters the noisy intermediate-scale quantum (NISQ) era, algorithms such as the Variational Quantum Eigensolver (VQE) and the Quantum Approximate Optimization Algorithm (QAOA) have been recognized for their potential to deliver approximate solutions to classically intractable CO problems.

Core Concept

The authors introduce the Conditional Value-at-Risk (CVaR) as an alternative to the typical expectation minimization in quantum variational methods. Traditionally, the algorithms minimize the expectation value of a problem Hamiltonian, estimated via the sample mean of numerous measurement outcomes. By switching to CVaR, which considers the mean of the worst-case outcomes, the authors propose that this alternative aggregation strategy leads to faster convergence towards superior solutions across a variety of tested CO problems.

Numerical and Theoretical Insights

The paper provides both empirical and analytical insights to bolster the use of CVaR in variational quantum algorithms. Below are some key highlights from the analysis:

1. Empirical Results: The integration of CVaR significantly enhances the performance of VQE and QAOA when evaluated using classical simulations and on actual quantum hardware. This is evidenced by a better probability distribution that results in quicker convergence to optimal solutions.
2. Algorithm Modifications: By applying CVaR, a different optimization landscape is created compared to using only the expectation value. The paper notes that difficulties can arise when using expectation value aggregation due to flat probability distributions, an issue mitigated by employing CVaR.
3. Simulation and Quantum Hardware Experiments: Across diverse trials, implementing CVaR reveals performance improvements that scale with problem sizes. Even when faced with quantum hardware's intrinsic noise, the proposed strategy demonstrated marked and robust optimization gains.
4. Analyzing Variational Form Depth: Importantly, the paper distinguishes between the depth requirements of VQE and QAOA circuits, noting that QAOA often requires deeper circuits to achieve competitive results due to its limited variational parameters at shallow circuit depths.

Implications and Future Directions

The proposition of using CVaR in variational quantum algorithms not only extends the toolkit available for quantum optimization but also provides a practical alternative that aligns more closely with the heuristic approaches often used in solving CO problems. This naturally leads to a research avenue evaluating how CVaR modifications can further leverage existing quantum algorithms and extends their applicability in solving real-world problems on noisy quantum devices.

Potential future work includes refining CVaR tuning parameters for different problem classes, integrating adaptive strategies within CVaR computation, and expanding CVaR's application spectrum. Furthermore, with ongoing advancements in quantum hardware alignment, there is an anticipation that incorporating CVaR in algorithmic strategies could significantly shorten the gap between classical and quantum optimization capacities.

Overall, the framework outlined in this paper sets the stage for more tailored quantum computing solutions in optimization, offering tangible pathways for improvement that are theoretically sound and experimentally validated.