The increasing number of genetic association studies conducted in multiple populations provides an unprecedented opportunity to study how the genetic architecture of complex phenotypes varies between populations, a problem important for both medical and population genetics. Here, we have developed a method for estimating the transethnic genetic correlation: the correlation of causal-variant effect sizes at SNPs common in populations. This methods takes advantage of the entire spectrum of SNP associations and uses only summary-level data from genome-wide association studies. This avoids the computational costs and privacy concerns associated with genotype-level information while remaining scalable to hundreds of thousands of individuals and millions of SNPs. We applied our method to data on gene expression, rheumatoid arthritis, and type 2 diabetes and overwhelmingly found that the genetic correlation was significantly less than 1. Our method is implemented in a Python package called Popcorn.

Schizophrenia is a debilitating psychiatric disorder with approximately 1% lifetime risk globally. Large-scale schizophrenia genetic studies have reported primarily on European ancestry samples, Users may view, print, copy, and download text and data-mine the content in such documents, for the purposes of academic research, subject always to the full Conditions of use:

79 80 * These authors contributed equally to the work 81 §These authors jointly supervised the work 82 †Lists of participants and their affiliations appear in the Supplementary Information 83 84 85 86Finally, polygenic risk score (PRS) prediction is emerging as a useful tool for studying 129 the effects of genetic liability, identifying more homogeneous phenotypes, and stratifying 130 patients, but the applicability of training data from EUR studies to those of non-European 131 ancestry has not been fully assessed, leaving us with an uncertainty as to the biological 132 implications and utility in non-Europeans 20 . 133 134 Schizophrenia genetic associations in the East Asian populations 135To systematically examine the genetic architecture of schizophrenia in individuals of East Asian 136 ancestry (EAS), we compiled 22,778 schizophrenia cases and 35,362 controls from 20 samples 137

Recent studies have examined the genetic correlations of single‐nucleotide polymorphism (SNP) effect sizes across pairs of populations to better understand the genetic architectures of complex traits. These studies have estimated ρ g, the cross‐population correlation of joint‐fit effect sizes at genotyped SNPs. However, the value of ρ g depends both on the cross‐population correlation of true causal effect sizes ( ρ b) and on the similarity in linkage disequilibrium (LD) patterns in the two populations, which drive tagging effects. Here, we derive the value of the ratio ρ g / ρ b as a function of LD in each population. By applying existing methods to obtain estimates of ρ g, we can use this ratio to estimate ρ b. Our estimates of ρ b were equal to 0.55 ( SE = 0.14) between Europeans and East Asians averaged across nine traits in the Genetic Epidemiology Research on Adult Health and Aging data set, 0.54 ( SE = 0.18) between Europeans and South Asians averaged across 13 traits in the UK Biobank data set, and 0.48 ( SE = 0.06) and 0.65 ( SE = 0.09) between Europeans and East Asians in summary statistic data sets for type 2 diabetes and rheumatoid arthritis, respectively. These results implicate substantially different causal genetic architectures across continental populations.

We study the computational difficulty of computing the ground state degeneracy and the density of states for local Hamiltonians. We show that the difficulty of both problems is exactly captured by a class which we call #BQP, which is the counting version of the quantum complexity class QMA. We show that #BQP is not harder than its classical counting counterpart #P, which in turn implies that computing the ground state degeneracy or the density of states for classical Hamiltonians is just as hard as it is for quantum Hamiltonians.Understanding the physical properties of correlated quantum many-body systems is a problem of central importance in condensed matter physics. The density of states, defined as the number of energy eigenstates per energy interval, plays a particularly crucial role in this endeavor. It is a key ingredient when deriving many thermodynamic properties from microscopic models, including specific heat capacity, thermal conductivity, band structure, and (near the Fermi energy) most electronic properties of metals. Computing the density of states can be a daunting task however, as it in principle involves diagonalizing a Hamiltonian acting on an exponentially large space, though other more efficient approaches which might take advantage of the structure of a given problem are not a priori ruled out.In this Letter, we precisely quantify the difficulty of computing the density of states by using the powerful tools of quantum complexity theory. Quantum complexity aims at generalizing the well-established field of classical complexity theory to assess the difficulty of tasks related to quantum mechanical problems, concerning both the classical difficulty of simulating quantum systems as well as the fundamental limits to the power of quantum computers. In particular, quantum complexity theory has managed to explain the difficulty of computing ground state properties of quantum spin systems in various settings, such as two-dimensional (2D) lattices [1] and even one-dimensional (1D) chains [2], as well as fermionic systems [3].We will determine the computational difficulty of two problems: First, computing the density of states of a local Hamiltonian, and second, counting the ground state degeneracy of a local gapped Hamiltonian; in both cases, the result holds even if the Hamiltonian is restricted to act on a 2D lattice of qubits, or on a 1D chain. To this end, we will introduce the quantum counting class #BQP (sharp BQP), which constitutes the natural counting version of the class QMA (Quantum Merlin Arthur) which itself captures the difficulty of computing the ground state energy of a local Hamiltonian [4,5]. Vaguely speaking, #BQP counts the number of possible "quantum solutions" to a quantum problem that can be verified using a quantum computer. We show that both problems, computing the density of states and counting the ground state degeneracy, are complete problems for the class #BQP, i.e., they are among the hardest problems in this class.Having quantified the difficulty of computing the density...