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In the span of four decades, quantum computation has evolved from an intellectual curiosity to a potentially realizable technology. Today, small-scale demonstrations have become possible for quantum algorithmic primitives on hundreds of physical qubits and proof-of-principle error-correction on a single logical qubit. Nevertheless, despite significant progress and excitement, the path toward a full-stack scalable technology is largely unknown. There are significant outstanding quantum hardware, fabrication, software architecture, and algorithmic challenges that are either unresolved or overlooked. These issues could seriously undermine the arrival of utility-scale quantum computers for the foreseeable future. Here, we provide a comprehensive review of these scaling challenges. We show how the road to scaling could be paved by adopting existing semiconductor technology to build much higher-quality qubits, employing system engineering approaches, and performing distributed quantum computation within heterogeneous high-performance computing infrastructures. These opportunities for research and development could unlock certain promising applications, in particular, efficient quantum simulation/learning of quantum data generated by natural or engineered quantum systems. To estimate the true cost of such promises, we provide a detailed resource and sensitivity analysis for classically hard quantum chemistry calculations on surface-code error-corrected quantum computers given current, target, and desired hardware specifications based on superconducting qubits, accounting for a realistic distribution of errors. Furthermore, we argue that, to tackle industry-scale classical optimization and machine learning problems in a cost-effective manner, distributed quantum-assisted probabilistic computing with custom-designed accelerators should be considered as a complementary path toward scalability., Comment: 50 pages, 42 figures

We propose a novel technique for optimizing a modular fault-tolerant quantum computing architecture, taking into account any desired space-time trade--offs between the number of physical qubits and the fault-tolerant execution time of a quantum algorithm. We consider a concept architecture comprising a dedicated zone as a multi-level magic state factory and a core processor for efficient logical operations, forming a supply chain network for production and consumption of magic states. Using a heuristic algorithm, we solve the multi-objective optimization problem of minimizing space and time subject to a user-defined error budget for the success of the computation, taking the performance of various fault-tolerant protocols such as quantum memory, state preparation, magic state distillation, code growth, and logical operations into account. As an application, we show that physical quantum resource estimation reduces to a simple model involving a small number of key parameters, namely, the circuit volume, the error prefactors ($\mu$) and error suppression rates ($\Lambda$) of the fault-tolerant protocols, and an allowed slowdown factor ($\beta$). We show that, in the proposed architecture, $10^5$--$10^8$ physical qubits are required for quantum algorithms with $T$-counts in the range $10^6$--$10^{15}$ and logical qubit counts in the range $10^2$--$10^4$, when run on quantum computers with quantum memory $\Lambda$ in the range 3--10, for all slowdown factors $\beta \geq 0.2$., Comment: 21 pages, 6 figures