Quantum SDP-Solvers: Better upper and lower bounds (1705.01843v4)

Abstract: Brand~ao and Svore very recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension $n$ of the problem and the number $m$ of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure. We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with $mn$ when $m\approx n$, which is the same as classical.

Quantum SDP-Solvers: Enhanced Upper and Lower Bounds

Semidefinite programs (SDPs) have become a critical tool in optimization and approximation algorithms due to their generality over linear programs (LPs) while maintaining efficient solubility. The paper presents advancements in quantum algorithms for SDPs, improving upon earlier algorithms by Brandão and Svore, specifically targeting speed enhancements through better dependence on various parameters.

Improved Quantum Algorithms

The authors propose enhanced techniques for quantum SDP-solvers. They introduce more efficient methods for estimating matrix expectations in the quantum domain, specifically focusing on Gibbs sampling, which can prepare quantum states reflecting the matrix exponential efficiently. Utilizing purified Gibbs state preparations alongside amplitude estimation allows for a refined quantization approach that significantly reduces temporal complexity, improving the polynomial dependence on key parameters like sparsity $s$, trace bound $R$, dual sum bound $r$, and error $\epsilon$.

The introduced techniques also reduce the runtime, thanks to novel methods:

• Efficient application of smooth functions over sparse Hamiltonians.
• Generalized minimum-finding procedures modifying Grover's search algorithm for use with transformed data represented in quantum states.

These adjustments lead to a quantum algorithm with complexity bounded by $$\widetilde{\mathcal O}(\sqrt{mn}s^2(Rr/\epsilon)^8 )$$. For LPs, the complexity improves further to $\widetilde{\mathcal O}(\sqrt{mn}(Rr/\epsilon)^5 )$. Such algorithms are particularly tailored for cases where $R, r,$ and $\epsilon$ are small relative to $mn$.

Theoretical Insights and Implications

This paper revises the quantum landscape for SDPs and provides valuable theoretical insights that question the lower bounds of classical approaches and highlight potential applications. A notable theoretical result is the lower bound established on the quantum query complexity of LPs for certain configurations, scaling as $\Omega\left(\sqrt{\max\{n,m\} \left( \min\{n,m\} \right)^{3/2} \right)$, indicating that polynomial improvements in parameter dependency are necessary for achieving better speeds in practical applications.

The researchers identify areas where existing SDP formulations may inherently require dense solutions, emphasizing that while current frameworks offer general solutions, specific SDP structures necessitate tailored approaches. The importance of symmetry in constraints suggests that conventional SDP-solvers may face intrinsic inefficiencies when tackling such problems, outlined by combinatorial modularity within symmetry-laden matrices.

Future prospects

Further exploration in custom quantum oracles optimized for particular SDP configurations opens new avenues for achieving significant quantum speedups in practical domains. Additionally, while theoretical query bounds suggest potential constraints, ongoing research could refine these bounds or develop more sophisticated quantum strategies, leading to practical implementations across varying scales of computational problems.

In summary, while the paper significantly advances quantum algorithm design for SDP solving, both upper and lower bounds illustrate the intricate balance between computational expenditure and problem complexity. This work provides a vital foundation upon which future quantum algorithms, optimized for specific structural SDP challenges, may be built.