In this work, we introduce a new workflow to solve portfolio optimization problems on annealing platforms. We combine a classical preprocessing step with a modified unconstrained binary optimization (QUBO) model and evaluate it using simulated annealing (classical computer), digital annealing (Fujitsu’s Digital Annealing Unit), and quantum annealing (D-Wave Advantage). Starting from Markowitz’s theory on portfolio optimization, our classical preprocessing step finds the most promising assets within a set of possible assets to choose from. We then modify existing QUBO models for portfolio optimization, such that there are no limitations on the number of assets that can be invested in. Furthermore, our QUBO model enables an investor to also place an arbitrary amount of money into each asset. We apply this modified QUBO to the set of promising asset candidates we generated previously via classical preprocessing. A solution to our QUBO model contains information about what percentage of the whole available capital should be invested into which asset. For the evaluation, we have used publicly available real-world data sets of stocks of the New York Stock Exchange as well as common ETFs. Finally, we have compared the respective annealing results with randomly generated portfolios by using the return, variance, and diversification of the created portfolios as measures. The results show that our QUBO formulation is capable of creating well-diversified portfolios that respect certain criteria given by an investor, such as maximizing return, minimizing risk, or sticking to a certain budget.
Many industrially relevant problems can be deterministically solved by computers in principle, but are intractable in practice, as the seminal P/NP dichotomy of complexity theory and Cobham's thesis testify. For the many NP-complete problems, industry needs to resort to using heuristics or approximation algorithms. For approximation algorithms, there is a more refi ned classifi cation in complexity classes that goes beyond the simple P/NP dichotomy. As it is well known, approximation classes form a hierarchy, that is, FPTAS PTAS APX NPO. This classifi cation gives a more realistic notion of complexity but-unless unexpected breakthroughs happen for fundamental problems like P = NP or related questions-there is no known effi cient algorithm that can solve such problems exactly on a realistic computer. Therefore, new ways of computations are sought. Recently, considerable hope was placed on the possible computational powers of quantum computers and quantum annealing (QA) in particular. However, the precise benefi ts of such a drastic shift in hardware are still unchartered territory to a good extent. Firstly, the exact relations between classical and quantum complexity classes pose many open questions, and secondly, technical details of formulating and implementing quantum algorithms play a crucial role in real-world applications. Guided by the hierarchy of classical optimisation complexity classes, we discuss how to map problems of each class to a quantum annealer. Those problems are the Minimum Multiprocessor Scheduling (MMS) problem, the Minimum Vertex Cover (MVC) problem and the Maximum Independent Set (MIS) problem. We experimentally investigate if and how the degree of approximability infl uences implementation and run-time performance. Our experiments indicate a discrepancy between classical approximation complexity and QA behaviour: Problems
Somatic variation contributes to biological heterogeneity by modulating cellular proclivity to differentiate, expand, adapt, or die. While large-scale sequencing efforts have revealed the foundational role of somatic variants to drive human tumor evolution, our understanding of the contribution of mutations to modulate cellular fitness in non-malignant contexts remains understudied. Here, we identify a mosaic synonymous variant (m.7076A>G) in the mitochondrial DNA (mtDNA) encoded cytochrome c-oxidase subunit 1 gene (MT-CO1, p.Gly391=), which was present at homoplasmy in 47% of immune cells from a healthy donor. Using single-cell multi-omics, we discover highly specific selection against the m.7076G mutant allele in the CD8+ effector memory T cell compartment in vivo, reminiscent of selection observed for pathogenic mtDNA alleles and indicative of lineage-specific metabolic requirements. While the wildtype m.7076A allele is translated via Watson-Crick-Franklin base-pairing, the anticodon diversity of the mitochondrial transfer RNA pool is limited, requiring wobble-dependent translation of the m.7076G mutant allele. Notably, mitochondrial ribosome profiling revealed altered codon-anticodon affinity at the wobble position as evidenced by stalled translation of the synonymous m.7076G mutant allele encoding for glycine. Generalizing this observation, we provide a new ontogeny of the 8,482 synonymous variants in the human mitochondrial genome that enables interpretation of functional mtDNA variation. Specifically, via inter- and intra-species evolutionary analyses, population-level complex trait associations, and the occurrence of germline and somatic mtDNA mutations from large-scale sequencing studies, we demonstrate that synonymous variation impacting codon:anticodon affinity is actively evolving across the entire mitochondrial genome and has broad functional and phenotypic effects. In summary, our results introduce a new ontogeny for mitochondrial genetic variation and support a model where organismal principles can be discerned from somatic evolution via single-cell genomics.
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