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Register a new userThe effect of quantization on the FCIQMC sign problem
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The sign problem in Full Configuration Interaction Quantum Monte Carlo (FCIQMC) without annihilation can be understood as an instability of the psi-particle population to the ground state of the matrix obtained by making all off-diagonal elements of the Hamiltonian negative. Such a matrix, and hence the sign problem, is basis dependent. In this paper we discuss the properties of a physically important basis choice: first versus second quantization. For a given choice of single-particle orbitals, we identify the conditions under which the fermion sign problem in the second quantized basis of antisymmetric Slater determinants is identical to the sign problem in the first quantized basis of unsymmetrized Hartree products. We also show that, when the two differ, the fermion sign problem is always less severe in the second quantized basis. This supports the idea that FCIQMC, even in the absence of annihilation, improves the sign problem relative to first quantized methods. Finally, we point out some theoretically interesting classes of Hamiltonians where first and second quantized sign problems differ, and others where they do not.
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The infamous numerical sign problem poses a fundamental obstacle to long-time stochastic Wigner simulations in high dimensional phase space. Although the existing particle annihilation via uniform mesh (PAUM) significantly alleviates the sign problem when dimensionality D $le$ 4, the setting of regular grids gives rise to another challenge in data storage when D $ge$ 6 due to the curse of dimensionality. In this paper, we propose to use a recently developed adaptive particle annihilation, termed sequential-clustering particle annihilation via discrepancy estimation (SPADE), to overcome the numerical sign problem. SPADE consists of adaptive clustering of particles via controlling their number-theoretic discrepancies and independent random matching in each group, and may learn the minimal amount of particles that can accurately capture the oscillating nature of the Wigner function. Combining SPADE with a recently proposed variance reduction technique via the stationary phase approximation, we make the first attempt to simulate the transitions of hydrogen energy levels in 6-D phase space, where the feasibility of PAUM with sample sizes about $10^9$-$10^{10}$ has also been explored as a comparison.
Micromagnetic simulations have been performed to investigate the suppression of the skyrmion Hall effect in nanotracks with their magnetic properties strategically modified. In particular, we study two categories of magnetically modified nanotracks. One of them, repulsive edges have been inserted in the nanotrack and, in the other, an attractive strip has been placed exactly on the longest axis of the nanotrack. Attractive and repulsive interactions can be generated from the engineering of magnetic properties. For instance, it is known that the skyrmion can be attracted to a region where the exchange stiffness constant is decreased. On the other hand, the skyrmion can be repelled from a region characterized by a local increase in the exchange stiffness constant. In order to provide a background for experimental studies, we vary not only the magnetic material parameters (exchange stiffness, perpendicular magnetocrystalline anisotropy and the Dzyaloshinskii-Moriya constant) but also the width of the region magnetically modified, containing either a local reduction or a local increase for each one of these magnetic properties. In the numerical simulations, the skyrmion motion was induced by a spin-polarized current and the found results indicate that it is possible to transport skyrmions around the longest axis of the nanotrack. In practice, the skyrmion Hall effect can be completely suppressed in magnetic nanotracks with strategically modified magnetic properties. Furthermore, we discuss in detail 6 ways to suppress the skyrmion Hall effect by the usage of nanotracks with repulsive edges and nanotracks with an attractive strip.
The recently proposed full configuration interaction quantum Monte Carlo method allows access to essentially exact ground-state energies of systems of interacting fermions substantially larger than previously tractable without knowledge of the nodal structure of the ground-state wave function. We investigate the nature of the sign problem in this method and how its severity depends on the system studied. We explain how cancelation of the positive and negative particles sampling the wave function ensures convergence to a stochastic representation of the many-fermion ground state and accounts for the characteristic population dynamics observed in simulations.
We present a practical analysis of the fermion sign problem in fermionic path integral Monte Carlo (PIMC) simulations in the grand-canonical ensemble (GCE). As a representative model system, we consider electrons in a $2D$ harmonic trap. We find that the sign problem in the GCE is even more severe than in the canonical ensemble at the same conditions, which, in general, makes the latter the preferred option. Despite these difficulties, we show that fermionic PIMC simulations in the GCE are still feasible in many cases, which potentially gives access to important quantities like the compressiblity or the Matsubara Greens function. This has important implications for contemporary fields of research such as warm dense matter, ultracold atoms, and electrons in quantum dots.
We demonstrate that the ability to estimate the relative sign of an arbitrary $n$-qubit quantum state (with real amplitudes), given only $k$ copies of that state, would yield a $kn$-query algorithm for unstructured search. Thus the quantum sample complexity of sign estimation must be exponential: $Omega(2^{n/2}/n)$. In particular, we show that an efficient procedure for solving the sign estimation problem would allow for a polynomial time solution to the NP-complete problem 3-SAT.
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