Quantum algorithms: A survey of applications and end-to-end complexities (2310.03011v1)

Abstract: The anticipated applications of quantum computers span across science and industry, ranging from quantum chemistry and many-body physics to optimization, finance, and machine learning. Proposed quantum solutions in these areas typically combine multiple quantum algorithmic primitives into an overall quantum algorithm, which must then incorporate the methods of quantum error correction and fault tolerance to be implemented correctly on quantum hardware. As such, it can be difficult to assess how much a particular application benefits from quantum computing, as the various approaches are often sensitive to intricate technical details about the underlying primitives and their complexities. Here we present a survey of several potential application areas of quantum algorithms and their underlying algorithmic primitives, carefully considering technical caveats and subtleties. We outline the challenges and opportunities in each area in an "end-to-end" fashion by clearly defining the problem being solved alongside the input-output model, instantiating all "oracles," and spelling out all hidden costs. We also compare quantum solutions against state-of-the-art classical methods and complexity-theoretic limitations to evaluate possible quantum speedups. The survey is written in a modular, wiki-like fashion to facilitate navigation of the content. Each primitive and application area is discussed in a standalone section, with its own bibliography of references and embedded hyperlinks that direct to other relevant sections. This structure mirrors that of complex quantum algorithms that involve several layers of abstraction, and it enables rapid evaluation of how end-to-end complexities are impacted when subroutines are altered.

• The paper presents a comprehensive survey of quantum algorithms by mapping applications in fields like quantum chemistry, physics, and optimization to their end-to-end complexities.
• It employs a modular methodology that dissects algorithmic primitives and highlights resource costs in qubit mapping and state preparation for practical quantum simulations.
• The paper identifies key challenges such as efficient state preparation, robust quantum error correction, and realistic assessment of quantum speedup over classical methods.

Quantum Algorithms: A Survey of Applications and End-to-End Complexities

The paper "Quantum algorithms: A survey of applications and end-to-end complexities" provides a comprehensive overview of current research in quantum algorithms, specifically focusing on diverse applications and their complexities across both practical and theoretical domains. The paper emphasizes a modular structure, thereby allowing researchers to navigate various sections independently and efficiently assess algorithms pertinent to their specific interests.

Overview of Quantum Algorithms

The anticipated applications of quantum computing are vast, spanning scientific fields such as quantum chemistry, many-body physics, optimization, finance, and machine learning. To develop quantum solutions in these areas, researchers often combine multiple quantum algorithmic primitives into a comprehensive quantum algorithm. This paper investigates the intricacies involved in understanding how specific applications might benefit from quantum computing. Quantum error correction and fault tolerance play critical roles in accurately implementing these algorithms on quantum hardware.

Survey Findings

1. Applications of Quantum Algorithms:
   • Quantum Chemistry and Materials: The challenges primarily lie in the difficulty of preparing the ground state of the Fermi-Hubbard model and extrapolating measured finite properties to the thermodynamic limit. Experimental studies demonstrate increased interest in understanding non-equilibrium properties and the potential to simulate these on NISQ devices.
   • Computational Models: Simulating models like the SYK model promises insight into quantum chaos and strongly coupled systems. The difficulty of determining properties analytically makes these models suitable candidates for quantum simulation.
   • Spin Models as Quantum Challenges: Spin systems offer significant exploration potential for quantum simulators and a proving ground for quantum computing's advantage over classical algorithms. Notably, simulating spin models could deliver beyond-classical computational tasks on quantum computers.
2. Dominant Resource Costs:
   • Mapping to Qubits: Efficiently mapping problem domains to qubits remains a foundation for quantum computation. Simulating condensed matter systems, for example, requires access techniques such as block-encoding or Hamiltonian simulation.
   • State Preparation and Measurement: Preparing desired states, whether ground or thermal, involves considerable computational resources. These are crucial steps for leveraging quantum simulations' full potential, and accurate preparation limits the overall complexity.
3. Challenges and Evolution:
   • Technical Caveats: Preparing states efficiently, understanding overlaps between initial and target states, and the extrapolation of finite-size calculations present ongoing challenges within quantum algorithms.
   • Classical vs. Quantum Complexity: Classical methods for computing specific model properties, while efficient at scale, face significant challenges from quantum algorithms rooted in quantum supremacy.
   • Speedup Assessment: The assessment of quantum speedups, often necessitates considering experimental data and realistically competing technologies such as analog quantum simulation.

Implications and Future Directions

The paper highlights various ways quantum algorithms might surpass classical approaches, especially relevant when the domain's inherent complexity gains classical methods' limits. Yet, realizing practical quantum advantages will require overcoming substantial technical and theoretical challenges alone and integrating quantum technologies within existing computing infrastructures.

Looking forward, advancements in quantum error correction, continued refinement of quantum hardware, and inter-disciplinary collaboration will play vital roles in shaping quantum computing's trajectory. As quantum technologies mature, their applications will expand, increasing academic and commercial interests.

In summary, "Quantum algorithms: A survey of applications and end-to-end complexities" illuminates the nuanced landscape of quantum computing, intricately detailing the complexities, frameworks, and advancements required to harness quantum computing's potential across various fields. The paper proves to be an invaluable guide for navigating the intricate field of quantum algorithms, providing researchers with essential insights crucial for further exploration and development within this dynamic area.