QIC Journalπ#call4reading #highlycitedpaper
βοΈA #quantumalgorithm for simulating non-sparse #Hamiltonians #by Chunhao Wang and Leonard Wossnig
π10.26421/QIC20.7-8-5 (#arXiv:1803.08273)
π#call4reading
βοΈ#Coherence generation with #Hamiltonians #by Manfredi Scalici Moein Naseri and Alexander Streltsov
π https://doi.org/10.26421/QIC24.7-8-2 (#arXiv:2402.17567)
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RTW#Hamiltonians are #mathematical functions or operators that describe the total energy of a system in #physics. They play a crucial role in both classical and #quantum mechanics.
The Hamiltonian formulation provides an alternative to #Newton's equations of #motion & is particularly useful in studying the dynamics of #complexsystems with constraints.
https://www.sciencedirect.com/science/article/abs/pii/S0010854514003397
KilleansRow πΊπ² πΊπ¦πSometimes the shiny brand new thing is just as shiny a year and a half later..
Via arXiv:
Learning quantum #Hamiltonians at any temperature in #polynomial time
https://arxiv.org/abs/2310.02243
Learning quantum Hamiltonians at any temperature in polynomial time
We study the problem of learning a local quantum Hamiltonian $H$ given copies of its Gibbs state $Ο= e^{-Ξ²H}/\textrm{tr}(e^{-Ξ²H})$ at a known inverse temperature $Ξ²>0$. Anshu, Arunachalam, Kuwahara, and Soleimanifar (arXiv:2004.07266) gave an algorithm to learn a Hamiltonian on $n$ qubits to precision $Ξ΅$ with only polynomially many copies of the Gibbs state, but which takes exponential time. Obtaining a computationally efficient algorithm has been a major open problem [Alhambra'22 (arXiv:2204.08349)], [Anshu, Arunachalam'22 (arXiv:2204.08349)], with prior work only resolving this in the limited cases of high temperature [Haah, Kothari, Tang'21 (arXiv:2108.04842)] or commuting terms [Anshu, Arunachalam, Kuwahara, Soleimanifar'21]. We fully resolve this problem, giving a polynomial time algorithm for learning $H$ to precision $Ξ΅$ from polynomially many copies of the Gibbs state at any constant $Ξ²> 0$. Our main technical contribution is a new flat polynomial approximation to the exponential function, and a translation between multi-variate scalar polynomials and nested commutators. This enables us to formulate Hamiltonian learning as a polynomial system. We then show that solving a low-degree sum-of-squares relaxation of this polynomial system suffices to accurately learn the Hamiltonian.
Jens EisertThis is not really a paper, but rather a note that stresses that if one implements a #variationalprinciple like in #VQA for translationally invariant #Hamiltonians, one better obtains energy densities that scale better than a small constant.
A note on lower bounds to variational problems with guarantees
Variational methods play an important role in the study of quantum many body problems, both in the flavour of classical variational principles based on tensor networks as well as of quantum variational principles in near-term quantum computing. This brief pedagogical note stresses that for translationally invariant lattice Hamiltonians, one can easily derive efficiently computable lower bounds to ground state energies that can and should be compared with variational principles providing upper bounds. As small technical results, it is shown that (i) the Anderson bound and a (ii) common hierarchy of semi-definite relaxations both provide approximations with performance guarantees that scale like a constant in the energy density for cubic lattices. (iii) Also, the Anderson bound is systematically improved as a hierarchy of semi-definite relaxations inspired by the marginal problem.