A new unbiased global optimization approach is proposed, based on quantum staging path integral Monte Carlo sampling and local minimization of individual imaginary time slices. This algorithm uses the quantum tunneling effect to speed up the crossing of energy barriers. This method differs in important ways from previous work on quantum annealing and is able to find all the global minima of Lennard-Jones clusters of size up to N=100, except for N=76, 77, and 98. The comparison between this new algorithm and several other classes of algorithms is presented.
Path integrals are a powerful method for calculating real time, finite temperature, and ground state properties of quantum systems. By exploiting some remarkable properties of the symmetric Trotter formula and the discrete Fourier transform, we arrive at a high-accuracy method for removing ``time slice'' errors in Trotter-approximated propagators. We provide an explicit demonstration of the method applied to the two-body density matrix of
4He. Our method is simultaneously fast, high precision, and computationally simple and can be applied to a wide variety of quantum propagators.
We propose an approximate classical algorithm for estimating quantum circuits' transition amplitudes,
⟨Ψf∣U^∣Ψ0⟩, based on Monte Carlo sums over bidirectionally sampled Feynman paths. The algorithm generates Feynman paths, i.e., sequences of intermediate computational basis states, by combining information from random walks initiated at both the input and output states,
Ψ0 and
Ψf, respectively. It further combines this bidirectional approach with Multiple Importance Sampling, a powerful variance reduction technique that avoids variance peaks caused by divisions by the very low sampling probabilities associated with hard to reach paths. Empirical data demonstrate the superior convergence rate exhibited by the bidirectional algorithm over a unidirectional, forward-only sampling approach. This superiority is due to the effectiveness of the proposed algorithm in finding and weighting paths with non-zero amplitude in an extremely sparse path space. Sum of paths algorithms exhibit very favourable space complexity. Monte Carlo replaces exhaustive enumeration of the exponentially large path space by stochastic sampling, the computational cost being determined by the variance of the estimator. This work demonstrates these two features by presenting results on very large quantum circuits, with up to 4096 qubits.
Randomized product formulas such as qDrift offer a resource-efficient alternative to deterministic Trotter--Suzuki decompositions for Hamiltonian simulation, removing their polynomial dependence on the number of Hamiltonian terms. qDrift, however, is intrinsically limited to first order in the evolution time, so its query complexity remains linear in the inverse of the target accuracy,
1/ε. We introduce Pathwise Random Hamiltonian Simulation (PRHS), which extends qDrift to arbitrary order by subdividing each time step into
M correlated slices, each evolving under a term sampled from a quasi-probability distribution that we construct in closed form and prove unique, with a bias decaying factorially in
M. Optimizing jointly over the number of slices
M and the number of independent blocks
N interpolates between the standard qDrift protocol at long times and a high-precision regime where the query cost grows slower than any power of
1/ε, without requiring ancillary qubits. Numerical simulations of five molecular Hamiltonians confirm this advantage, with PRHS achieving accuracies two to four orders of magnitude beyond qDrift at equal query cost.
An Ising spin model interacting with neighboring spins has been eagerly studied along with the progress in Quantum Computing (QC). Quantum Monte Carlo (QMC) is often used in simulation, but it is such a heuristic approach that it has great difficulty in that it takes a long time to solve the problem. It mainly depends on the existence of a trotter layer derived from Suzuki-Trotter decomposition. In my previous work, I proposed a new approach to compute it much faster by changing the order in which all energies over the trotter layer are summed and minimized. In the algorithm, a plural of trotter layers could be treated in parallel and Time to Solution (TTS) could be reduced as well as that of Simulated Annealing (SA), although it requires a lot of computational resources.In this paper, in order to balance them, a new approach is proposed, where the number of inner loops is not changed but the number of the trotter layers is smaller. With such a small trotter loop repeated in several times, the proposed method works as so-called partial optimization and it leads to very fast convergence. Moreover, it doesn’t affect the result whether the updating spins are the same or not during the partial optimization. This fact also indicates that the new method would open another door to parallel computation. These results are verified by solving MaxCut problem, GSET.