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False traps on quantum-classical optimization landscapes

Xiaozhen Ge, Shuming Cheng, Guofeng Zhang, Re-Bing Wu

TL;DR

This work investigates optimization landscapes of quantum optimization problems, and obtains that the parameter sufficiency is not enough to ensure the absence of false traps, and presents a complete framework for analyzing critical features of optimization landscapes.

Abstract

Optimization is ubiquitous in quantum information science and technology, however, the corresponding optimization landscape can encounter false traps, i.e., local but not global optima, likely to prevent used optimizers from finding optimal solutions. Such traps are believed to arise from parameter insufficiency and are expected to disappear when tunable parameters are sufficiently abundant. In this work, we investigate optimization landscapes of quantum optimization problems, and especially obtain that the parameter sufficiency is not enough to ensure the absence of false traps. First, we present a complete framework for analyzing critical features of optimization landscapes, by deriving necessary and sufficient conditions to identify all critical points and to classify them as local maxima, minima, or saddles, under some assumptions. Then, we show that false traps can still emerge on landscapes even with sufficient parameters, implying their appearance cannot be solely attributed to parameter insufficiency. Moreover, a close connection between landscape topology and quantum distinguishability is revealed that the emergence of false traps is linked to the loss of distinguishability among states or operators in the objective function. Finally, implications of our results are noted. Our work not only provides a deeper understanding of the intrinsic complexity of quantum-classical optimization, but also provides practical guidance for solving quantum-classical optimization problems, thus significantly aiding the progress in witnessing quantum advantages of the underlying quantum information processing tasks.

False traps on quantum-classical optimization landscapes

TL;DR

This work investigates optimization landscapes of quantum optimization problems, and obtains that the parameter sufficiency is not enough to ensure the absence of false traps, and presents a complete framework for analyzing critical features of optimization landscapes.

Abstract

Optimization is ubiquitous in quantum information science and technology, however, the corresponding optimization landscape can encounter false traps, i.e., local but not global optima, likely to prevent used optimizers from finding optimal solutions. Such traps are believed to arise from parameter insufficiency and are expected to disappear when tunable parameters are sufficiently abundant. In this work, we investigate optimization landscapes of quantum optimization problems, and especially obtain that the parameter sufficiency is not enough to ensure the absence of false traps. First, we present a complete framework for analyzing critical features of optimization landscapes, by deriving necessary and sufficient conditions to identify all critical points and to classify them as local maxima, minima, or saddles, under some assumptions. Then, we show that false traps can still emerge on landscapes even with sufficient parameters, implying their appearance cannot be solely attributed to parameter insufficiency. Moreover, a close connection between landscape topology and quantum distinguishability is revealed that the emergence of false traps is linked to the loss of distinguishability among states or operators in the objective function. Finally, implications of our results are noted. Our work not only provides a deeper understanding of the intrinsic complexity of quantum-classical optimization, but also provides practical guidance for solving quantum-classical optimization problems, thus significantly aiding the progress in witnessing quantum advantages of the underlying quantum information processing tasks.
Paper Structure (25 sections, 7 theorems, 86 equations, 6 figures)

This paper contains 25 sections, 7 theorems, 86 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The optimization landscape
  4. The landscape equivalence
  5. Identification and Classification of Critical Points
  6. The framework for optimization landscapes
  7. The optimization landscape for $M=1$
  8. Discovery of false traps
  9. An Example
  10. Sufficient condition for reconcilable critical points
  11. Necessary and sufficient condition for false traps
  12. Projective measurements
  13. General measurements
  14. Distinguishability v.s. false traps
  15. Discussions
  16. ...and 10 more sections

Key Result

Theorem 1

$U$ is a critical point of the objective function $F(U)$ in (Uloss), if and only if where $[\cdot, \cdot]$ denotes the commutator. Moreover, $\boldsymbol{\theta}^\prime$ is a critical point of $F(\boldsymbol{\theta})$ in (fun1), if and only if the above critical-point condition holds with $U=U(\boldsymbol{\theta}^\prime)$, under Assumptions assm1 and assm2.

Figures (6)

  • Figure 1: Quantum optimization problems with the objective function (\ref{['fun1']}) are ubiquitous in optimal control, state and machine learning, hypothesis testing, and simulation and computation, and etc.
  • Figure 2: (a) An example of the objective function $F(\boldsymbol{\theta})$ with parameters $\boldsymbol{\theta}=(\theta_1, \theta_2)^\top$; (b) False traps emerge as local optima but not global optimal points on the optimization landscape $\mathcal{L}_{\boldsymbol{\theta}}(F)$ generated by $F(\boldsymbol{\theta})$, likely to prevent algorithmic optimizers from successfully finding global optima. We show that false traps are unavoidable to optimize the objective (\ref{['fun1']}) with $M>1$. This is surprising, as it is different from the $M=1$ case, under the same assumptions.
  • Figure 3: Numerical simulation on the example (\ref{['opt1']}). A gradient-ascend optimizer is utilized to find its maximal value, and $100$ experiments are implemented. It is found that the search for global optimum can be trapped at the critical point $U_2=|00\rangle\langle 11|+|01\rangle\langle 01|+|10\rangle\langle 00|+|11\rangle\langle 10|$, leading to $F(U_2)=0.36$ and thus forming a FT.
  • Figure 4: The element exchange among different blocks of $\hat{\sigma}_m$ in Eq. (\ref{['mainform']}) induced by reconcilable critical points. $\hat{\Lambda}_i^m\rightarrow\hat{\Lambda}_j^m$ denotes the event that at least one diagonal element of the block $\hat{\Lambda}_i^m$ is transferred to the block $\hat{\Lambda}_i^m$, and it is proven in Theorem \ref{['theo3']} that FTs emerge on the optimization landscape if and only if there exist critical points which induce directed cycles.
  • Figure 5: The numerical simulation of gradient-ascent and gradient-descent searches for the example in Appendix \ref{['appendixE']}. 100 independent simulations are performed with randomly generated initial seeds, and the results demonstrate that the maximal or minimal landscape values can be always achieved.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Corollary 1
  • Theorem 3
  • Corollary 2
  • Theorem 4
  • Conjecture 1