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The accuracy of quantum dynamics simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. Our result extends that of [Jahnke, Lubich, BIT Numer. Math. 2000] to the time-dependent setting. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for time-dependent Hamiltonian simulations.
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We present a quantum algorithm for approximating the real time evolution $e^{-iHt}$ of an arbitrary $d$-sparse Hamiltonian to error $epsilon$, given black-box access to the positions and $b$-bit values of its non-zero matrix entries. The complexity of our algorithm is $mathcal{O}((tsqrt{d}|H|_{1 rightarrow 2})^{1+o(1)}/epsilon^{o(1)})$ queries and a factor $mathcal{O}(b)$ more gates, which is shown to be optimal up to subpolynomial factors through a matching query lower bound. This provides a polynomial speedup in sparsity for the common case where the spectral norm $|H|ge|H|_{1 rightarrow 2}$ is known, and generalizes previous approaches which achieve optimal scaling, but with respect to more restrictive parameters. By exploiting knowledge of the spectral norm, our algorithm solves the black-box unitary implementation problem -- $mathcal{O}(d^{1/2+o(1)})$ queries suffice to approximate any $d$-sparse unitary in the black-box setting, which matches the quantum search lower bound of $Omega(sqrt{d})$ queries and improves upon prior art [Berry and Childs, QIP 2010] of $tilde{mathcal{O}}(d^{2/3})$ queries. Combined with known techniques, we also solve systems of sparse linear equations with condition number $kappa$ using $mathcal{O}((kappa sqrt{d})^{1+o(1)}/epsilon^{o(1)})$ queries, which is a quadratic improvement in sparsity.
We present a quantum algorithm for the dynamical simulation of time-dependent Hamiltonians. Our method involves expanding the interaction-picture Hamiltonian as a sum of generalized permutations, which leads to an integral-free Dyson series of the time-evolution operator. Under this representation, we perform a quantum simulation for the time-evolution operator by means of the linear combination of unitaries technique. We optimize the time steps of the evolution based on the Hamiltonians dynamical characteristics, leading to a gate count that scales with an $L^1$-norm-like scaling with respect only to the norm of the interaction Hamiltonian, rather than that of the total Hamiltonian. We demonstrate that the cost of the algorithm is independent of the Hamiltonians frequencies, implying its advantage for systems with highly oscillating components, and for time-decaying systems the cost does not scale with the total evolution time asymptotically. In addition, our algorithm retains the near optimal $log(1/epsilon)/loglog(1/epsilon)$ scaling with simulation error $epsilon$.
Given the Hamiltonian, the evaluation of unitary operators has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated at each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The mean square error consists of a variance term and a bias term, arising respectively from the random sampling of the Hamiltonian terms and the operator splitting error. Leveraging on the bias/variance trade-off, the error can be minimized by balancing the two. The concentration bound provides an estimate on the number of gates. The estimates are verified by using numerical experiments on classical computers.
We present faster high-accuracy algorithms for computing $ell_p$-norm minimizing flows. On a graph with $m$ edges, our algorithm can compute a $(1+1/text{poly}(m))$-approximate unweighted $ell_p$-norm minimizing flow with $pm^{1+frac{1}{p-1}+o(1)}$ operations, for any $p ge 2,$ giving the best bound for all $pgtrsim 5.24.$ Combined with the algorithm from the work of Adil et al. (SODA 19), we can now compute such flows for any $2le ple m^{o(1)}$ in time at most $O(m^{1.24}).$ In comparison, the previous best running time was $Omega(m^{1.33})$ for large constant $p.$ For $psimdelta^{-1}log m,$ our algorithm computes a $(1+delta)$-approximate maximum flow on undirected graphs using $m^{1+o(1)}delta^{-1}$ operations, matching the current best bound, albeit only for unit-capacity graphs. We also give an algorithm for solving general $ell_{p}$-norm regression problems for large $p.$ Our algorithm makes $pm^{frac{1}{3}+o(1)}log^2(1/varepsilon)$ calls to a linear solver. This gives the first high-accuracy algorithm for computing weighted $ell_{p}$-norm minimizing flows that runs in time $o(m^{1.5})$ for some $p=m^{Omega(1)}.$ Our key technical contribution is to show that smoothed $ell_p$-norm problems introduced by Adil et al., are interreducible for different values of $p.$ No such reduction is known for standard $ell_p$-norm problems.
In this paper we propose the time & fitness-dependent Hamiltonian form of human biomechanics, in which total mechanical + biochemical energy is not conserved. Starting with the Covariant Force Law, we first develop autonomous Hamiltonian biomechanics. Then we extend it using a powerful geometrical machinery consisting of fibre bundles, jet manifolds, polysymplectic geometry and Hamiltonian connections. In this way we derive time-dependent dissipative Hamiltonian equations and the fitness evolution equation for the general time & fitness-dependent human biomechanical system. Keywords: Human biomechanics, configuration bundle, Hamiltonian connections, jet manifolds, time & fitness-dependent dynamics
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