Despite the scientific and engineering challenges facing the development of quantum computers, considerable progress is being made toward applying the technology to commercial applications. In this article, we discuss the solutions that some companies are already building using quantum hardware. Framing these as examples of combinatorics problems, we illustrate their application in four industry verticals: cybersecurity, materials and pharmaceuticals, banking and finance, and advanced manufacturing. While quantum computers are not yet available at the scale needed to solve all of these combinatorics problems, we identify three types of near-term opportunities resulting from advances in quantum computing: quantum-safe encryption, material and drug discovery, and quantum-inspired algorithms.

We discuss designer Hamiltonians—lattice models tailored to be free from sign problems (“de-signed”) when simulated with quantum Monte Carlo (QMC) methods but which still host complex many-body states and quantum phase transitions of interest in condensed matter physics. We focus on quantum spin systems in which competing interactions lead to nonmagnetic ground states. These states and the associated quantum phase transitions can be studied in great detail, enabling direct access to universal properties and connections with low-energy effective quantum field theories. As specific examples, we discuss the transition from a Néel antiferromagnet to either a uniform quantum paramagnet or a spontaneously symmetry-broken valence-bond solid (VBS) in SU(2) and SU(N) invariant spin models. We also discuss anisotropic (XXZ) systems harboring topological Z₂ spin liquids and the XY* transition. We briefly review recent progress on QMC algorithms, including ground-state projection in the valence-bond basis and direct computation of the Renyi variants of the entanglement entropy.

Trapped ions have emerged as one of the highest quality platforms for the quantum simulation of interacting spin models of interest to various fields of physics. In such simulators, two effective spins can be made to interact with arbitrary strengths by coupling to the collective vibrational or phonon states of ions, controlled by precisely tuned laser beams. However, the task of determining laser control parameters required for a given spin–spin interaction graph is a type of inverse problem , which can be highly mathematically complex. In this paper, we adapt a modern machine learning technique developed for similar inverse problems to the task of finding the laser control parameters for a number of interaction graphs. We demonstrate that typical graphs, forming regular lattices of interest to physicists, can easily be produced for up to 50 ions using a single GPU workstation. The scaling of the machine learning method suggests that this can be expanded to hundreds of ions with moderate additional computational effort.

A quantum computer exhibits quantum advantage when it can perform a calculation that a classical computer is unable to complete. It follows that a company with a quantum computer would be a monopolist in the market for such a calculation if its only competitor was a company with a classical computer. Conversely, economic outcomes are unclear if quantum computers do not exhibit a quantum advantage, but classical and quantum computers have different cost structures. We model a Cournot duopoly where a quantum computing company competes against a classical computing company. The model features an asymmetric variable cost structure between the two companies and the potential for an asymmetric fixed cost structure, where each firm can invest in scaling its hardware to expand its respective market. We find that even if (1) the companies can complete identical calculations, and thus there is no quantum advantage, and (2) it is more expensive to scale the quantum computer, the quantum computing company may be more profitable and also invest more in market creation due to efficiency gains from using quantum algorithms. Finally, we provide examples of settings where the classical computer can also perform a calculation, but not in a cost-effective enough manner to be commercially viable. In such a setting, the quantum computing company becomes a monopolist despite exhibiting no quantum advantage. Taken together, quantum computers may not need to display a quantum advantage to be able to generate a quantum economic advantage for the companies that deploy them. This paper was accepted by D. J. Wu, information systems. Funding: R. G. Melko is supported by the Natural Sciences and Engineering Research Council of Canada, Canada Research Chair program, and the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. A. Goldfarb is supported by the Sloan Foundation and the Social Sciences and Humanities Council of Canada. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2022.4578 .