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Physical Review A

Pareto-front shape in multiobservable quantum control

Qiuyang Sun, Re-Bing Wu, HERSCHEL A. RABITZ

Published Mar 17, 2017• 5 citations
Publisher

Abstract

Many scenarios in the sciences and engineering require simultaneous optimization of multiple objective functions, which are usually conflicting or competing. In such problems the Pareto front, where none of the individual objectives can be further improved without degrading some others, shows the tradeoff relations between the competing objectives. This paper analyzes the Pareto-front shape for the problem of quantum multiobservable control, i.e., optimizing the expectation values of multiple observables in the same quantum system. Analytic and numerical results demonstrate that with two commuting observables the Pareto front is a convex polygon consisting of flat segments only, while with noncommuting observables the Pareto front includes convexly curved segments. We also assess the capability of a weighted-sum method to continuously capture the points along the Pareto front. Illustrative examples with realistic physical conditions are presented, including NMR control experiments on a $^{1}\mathrm{H}\ensuremath{-}^{13}\mathrm{C}$ two-spin system with two commuting or noncommuting observables.

DOI
10.1103/physreva.95.032319

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Many scenarios in the sciences and engineering require simultaneous optimization of multiple objective functions, which are usually conflicting or competing. In such problems the Pareto front, where none of the individual objectives can be further improved without degrading some others, shows the tradeoff relations between the competing objectives. This paper analyzes the Pareto-front shape for the problem of quantum multiobservable control, i.e., optimizing the expectation values of multiple observables in the same quantum system. Analytic and numerical results demonstrate that with two commuting observables the Pareto front is a convex polygon consisting of flat segments only, while with noncommuting observables the Pareto front includes convexly curved segments. We also assess the capability of a weighted-sum method to continuously capture the points along the Pareto front. Illustrative examples with realistic physical conditions are presented, including NMR control experiments on a 1H\ensuremath−13C^{1}\mathrm{H}\ensuremath{-}^{13}\mathrm{C}1H\ensuremath−13C two-spin system with two commuting or noncommuting observables.
We describe algorithms and experimental strategies for the Pareto optimal control problem of simultaneously driving an arbitrary number of quantum observable expectation values to their respective extrema. Conventional quantum optimal control strategies are less effective at sampling points on the Pareto frontier of multiobservable control landscapes than they are at locating optimal solutions to single observable control problems. The present algorithms facilitate multiobservable optimization by following direct paths to the Pareto front, and are capable of continuously tracing the front once it is found to explore families of viable solutions. The numerical and experimental methodologies introduced are also applicable to other problems that require the simultaneous control of large numbers of observables, such as quantum optimal mixed state preparation.
An important task in multi-objective optimization is generating the Pareto front -- the set of all Pareto-optimal compromises among multiple objective functions applied to the same set of variables. Since this task can be computationally intensive even for small problems, it is a natural target for quantum optimization. Indeed, this problem was recently approached using the quantum approximate optimization algorithm (QAOA) on an IBM gate-model processor. Here we compare these QAOA results with quantum annealing on the same two input problems, using the same methodology. We find that quantum annealing vastly outperforms not just QAOA run on the IBM processor, but all classical and quantum methods analyzed in the previous study. On the harder problem, quantum annealing improves upon the best known Pareto front. This small study reinforces the promise of quantum annealing in multi-objective optimization.
The goal of multi-objective optimization is to understand optimal trade-offs between competing objective functions by finding the Pareto front, that is, the set of all Pareto-optimal solutions, where no objective can be improved without degrading another one. Multi-objective optimization can be challenging classically, even if the corresponding single-objective optimization problems are efficiently solvable. Thus, multi-objective optimization represents a compelling problem class to analyze with quantum computers. Here we use a low-depth quantum approximate optimization algorithm to approximate the optimal Pareto front of certain multi-objective weighted maximum-cut problems. We demonstrate its performance on an IBM Quantum computer, as well as with matrix product state numerical simulation, and show its potential to outperform classical approaches. This study explores the use of quantum computing to address multi-objective optimization challenges. By using a low-depth quantum approximate optimization algorithm to approximate the optimal Pareto front of multi-objective weighted max-cut problems, the authors demonstrate promising results—both in simulation and on IBM Quantum hardware—surpassing classical approaches.
Multi-objective combinatorial optimization requires identifying Pareto-optimal trade-off solutions among conflicting objectives, often making it more demanding than its single-objective counterpart. Although quantum multi-objective optimization methods have begun to emerge, most existing quantum optimization workflows are still built around single-objective or fixed-scalarization settings. Building on existing weighted-sum QAOA approaches to quantum multi-objective optimization, we propose QEMOO, a quantum-enhanced multi-objective optimization framework that combines Pareto-based selection and warm-started QAOA sampling in a multi-round protocol under the same total shot budget. We further introduce a PBI-inspired adaptive direction-update scheme to improve coverage in strongly conflicting benchmark regimes. Across three benchmark stages, QEMOO improves Pareto-front hypervolume over the single-pass weighted-sum QAOA baseline under matched shot budgets, suggesting a practical route toward shot-efficient quantum-assisted multi-objective optimization and its future applications.
Generating a unitary transformation in the shortest possible time is of practical importance to quantum information processing because it helps to reduce decoherence effects and improve robustness to additive control field noise. Many analytical and numerical studies have identified the minimum time necessary to implement a variety of quantum gates on coupled-spin qubit systems. This work focuses on exploring the Pareto front that quantifies the tradeoff between the competitive objectives of maximizing the gate fidelity F\mathcal{F}F and minimizing the control time TTT. In order to identify the critical time T∗{T}^{*}T∗ below which the target transformation is not reachable, as well as to determine the associated Pareto front, we introduce a numerical method of Pareto front tracking (PFT). We consider closed two- and multiqubit systems with constant interqubit coupling strengths and each individual qubit controlled by a separate time-dependent external field. Our analysis demonstrates that unit fidelity (to a desired numerical accuracy) can be achieved at any $T\ensuremath{\ge}{T}^{*}$ in most cases. However, the optimization search effort rises superexponentially as TTT decreases and approaches T∗{T}^{*}T∗. Furthermore, a small decrease in control time incurs a significant penalty in fidelity for T<T∗T<{T}^{*}T<T∗, indicating that it is generally undesirable to operate below the critical time. We investigate the dependence of the critical time T∗{T}^{*}T∗ on the coupling strength between qubits and the target gate transformation. Practical consequences of these findings for laboratory implementation of quantum gates are discussed.
We investigate obtaining intermediate quantum states during the quantum annealing process to address the limitation of the linear weighted sum method in multi-objective optimization, which inherently fails to reach non-convex regions of the Pareto front. We validate this approach through physical experiments utilizing quench-based readout and numerical simulations assuming ideal mid-anneal measurements. Both methods consistently demonstrate a clear trade-off where earlier timing enhances diversity of the solutions, whereas later timing ensures convergence to non-dominated solutions. Notably, a practical compromise timing balances both metrics. The qualitative agreement between practical quench and ideal simulation indicates the potential of accessing the intermediate states for comprehensive Pareto front exploration.