2020-07-24

# Improvements in Quantum SDP-Solving with Applications

## Publication

### Publication

*Presented at the 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) (July 2019)*

Following the first paper on quantum algorithms for SDP-solving by Brandão and Svore [Brandão and Svore, 2017] in 2016, rapid developments have been made on quantum optimization algorithms. In this paper we improve and generalize all prior quantum algorithms for SDP-solving and give a simpler and unified framework. We take a new perspective on quantum SDP-solvers and introduce several new techniques. One of these is the quantum operator input model, which generalizes the different input models used in previous work, and essentially any other reasonable input model. This new model assumes that the input matrices are embedded in a block of a unitary operator. In this model we give a O~((sqrt{m}+sqrt{n}gamma)alpha gamma^4) algorithm, where n is the size of the matrices, m is the number of constraints, gamma is the reciprocal of the scale-invariant relative precision parameter, and alpha is a normalization factor of the input matrices. In particular for the standard sparse-matrix access, the above result gives a quantum algorithm where alpha=s. We also improve on recent results of Brandão et al. [Fernando G. S. L. Brandão et al., 2018], who consider the special case when the input matrices are proportional to mixed quantum states that one can query. For this model Brandão et al. [Fernando G. S. L. Brandão et al., 2018] showed that the dependence on n can be replaced by a polynomial dependence on both the rank and the trace of the input matrices. We remove the dependence on the rank and hence require only a dependence on the trace of the input matrices. After we obtain these results we apply them to a few different problems. The most notable of which is the problem of shadow tomography, recently introduced by Aaronson [Aaronson, 2018]. Here we simultaneously improve both the sample and computational complexity of the previous best results. Finally we prove a new Omega~(sqrt{m}alpha gamma) lower bound for solving LPs and SDPs in the quantum operator model, which also implies a lower bound for the model of Brandão et al. [Fernando G. S. L. Brandão et al., 2018].

Additional Metadata | |
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doi.org/10.4230/LIPIcs.ICALP.2019.99 | |

Quantum algorithms and applications (nwo/680.91.034) | |

46th International Colloquium on Automata, Languages, and Programming (ICALP 2019) | |

This work was funded by the The Netherlands Organisation for Scientific Research (NWO); grant id nwo/680.91.034 - Quantum algorithms and applications | |

van Apeldoorn, J.T.S, & Gilyén, A.P. (2020). Improvements in Quantum SDP-Solving with Applications. In
Proceedings of 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019. doi:10.4230/LIPIcs.ICALP.2019.99 |