@article {2309, title = {Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning}, journal = {To appear at the 46th International Colloquium on Automata, Languages and Programming (ICALP 2019)}, year = {2018}, abstract = {

We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank r, and sparsity s. The first algorithm assumes an input model where one is given access to entries of the matrices at unit cost. We show that it has run time O~(s2(m\−\−\√ε\−10+n\−\−\√ε\−12)), where ε is the error. This gives an optimal dependence in terms of m,n and quadratic improvement over previous quantum algorithms when m\≈n. The second algorithm assumes a fully quantum input model in which the matrices are given as quantum states. We show that its run time is O~(m\−\−\√+poly(r))\⋅poly(logm,logn,B,ε\−1), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only poly-logarithmically in n and polynomially in r. We apply the second SDP solver to the problem of learning a good description of a quantum state with respect to a set of measurements: Given m measurements and copies of an unknown state ρ, we show we can find in time m\−\−\√\⋅poly(logm,logn,r,ε\−1) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as ρ on the m measurements, up to error ε. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes\&$\#$39; principle from statistical mechanics. As in previous work, we obtain our algorithm by \"quantizing\" classical SDP solvers based on the matrix multiplicative weight method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension, which could be of independent interest.

}, url = {https://arxiv.org/abs/1710.02581}, author = {Fernando G. S. L. Brand{\~a}o and Amir Kalev and Tongyang Li and Cedric Yen-Yu Lin and Krysta M. Svore and Xiaodi Wu} } @article {2001, title = {{E}ntanglement area laws for long-range interacting systems}, journal = {Physical Review Letters}, volume = {119}, year = {2017}, month = {2017/07/31}, pages = {050501}, abstract = {

We prove that the entanglement entropy of any state evolved under an arbitrary 1/rα long-range-interacting D-dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area for any α \> D + 1. We also prove that for any α \> 2D + 2, the ground state of such a Hamiltonian satisfies the entanglement area law if it can be transformed along a gapped adiabatic path into a ground state known to satisfy the area law. These results significantly generalize their existing counterparts for short-range interacting systems, and are useful for identifying dynamical phase transitions and quantum phase transitions in the presence of long-range interactions.

}, doi = {10.1103/PhysRevLett.119.050501}, url = {https://arxiv.org/abs/1702.05368}, author = {Zhe-Xuan Gong and Michael Foss-Feig and Fernando G. S. L. Brand{\~a}o and Alexey V. Gorshkov} } @article {2106, title = {Exponential Quantum Speed-ups for Semidefinite Programming with Applications to Quantum Learning}, year = {2017}, month = {2017/10/06}, abstract = {

We give semidefinite program (SDP) quantum solvers with an exponential speed-up over classical ones. Specifically, we consider SDP instances with m constraint matrices of dimension n, each of rank at most r, and assume that the input matrices of the SDP are given as quantum states (after a suitable normalization). Then we show there is a quantum algorithm that solves the SDP feasibility problem with accuracy ǫ by using \√ m log m \· poly(log n,r, ǫ \−1 ) quantum gates. The dependence on n provides an exponential improvement over the work of Brand \˜ao and Svore [6] and the work of van Apeldoorn et al. [23], and demonstrates an exponential quantum speed-up when m and r are small. We apply the SDP solver to the problem of learning a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state ρ, we show we can find in time \√ m log m \· poly(log n,r, ǫ \−1 ) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as ρ on the m measurements up to error ǫ. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes\’ principle. As in previous work, we obtain our algorithm by \“quantizing\” classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension based on the techniques developed in quantum principal component analysis, which could be of independent interest. Our quantum SDP solver is different from previous ones in the following two aspects: (1) it follows from a zero-sum game approach of Hazan [11] of solving SDPs rather than the primal-dual approach by Arora and Kale [5]; and (2) it does not rely on any sparsity assumption of the input matrices.

}, url = {https://arxiv.org/abs/1710.02581}, author = {Fernando G. S. L. Brand{\~a}o and Amir Kalev and Tongyang Li and Cedric Yen-Yu Lin and Krysta M. Svore and Xiaodi Wu} }