Energy-Consumption Advantage of Quantum Computation

Energy consumption in solving computational problems has been gaining growing attention as a part of the performance measures of computers. Quantum computation is known to offer advantages over classical computation in terms of various computational resources; however, its advantage in energy consumption has been challenging to analyze due to the lack of a theoretical foundation to relate the physical notion of energy and the computer-scientific notion of complexity for quantum computation with finite computational resources. To bridge this gap, we introduce a general framework for studying energy consumption of quantum and classical computation based on a computational model with a black-box oracle, as conventionally used for studying query complexity in computational complexity theory. With this framework, we derive an upper bound of energy consumption of quantum computation with covering all costs, including those of initialization, control, and quantum error correction; in particular, our analysis shows an energy-consumption bound for a finite-step Landauer-erasure protocol, progressing beyond the existing asymptotic bound. We also develop techniques for proving a lower bound of energy consumption of classical computation based on the energy-conservation law and the Landauer-erasure bound; significantly, our lower bound can be gapped away from zero no matter how energy-efficiently we implement the computation and is free from the computational hardness assumptions. Based on these general bounds, we rigorously prove that quantum computation achieves an exponential energy-consumption advantage over classical computation for Simon's problem. These results provide a fundamental framework and techniques to explore the physical meaning of quantum advantage in the query-complexity setting based on energy consumption, opening an alternative way to study the advantages of quantum computation.Comment: 36 pages, 3 figure

On the Probabilistic Query Complexity of Transitively Symmetric Problems

We obtain optimal lower bounds on the nonadaptive probabilistic query complexity of a class of problems defined by a rather weak symmetry condition. In fact, for each problem in this class, given a number T of queries we compute exactly the performance (i.e., the probability of success on the worst instance) of the best nonadaptive probabilistic algorithm that makes T queries. We show that this optimal performance is given by a minimax formula involving certain probability distributions. Moreover, we identify two classes of problems for which adaptivity does not help. We illustrate these results on a few natural examples, including unordered search, Simon's problem, distinguishing one-to-one functions from two-to-one functions, and hidden translation. For these last three examples, which are of particular interest in quantum computing, the recent theorems of Aaronson, of Laplante and Magniez, and of Bar-Yossef, Kumar and Sivakumar on the probabilistic complexity of black-box problems do not yield any nonconstant lower bound

Quantum Algorithms for Attacking Hardness Assumptions in Classical and Post‐Quantum Cryptography

In this survey, the authors review the main quantum algorithms for solving the computational problems that serve as hardness assumptions for cryptosystem. To this end, the authors consider both the currently most widely used classically secure cryptosystems, and the most promising candidates for post-quantum secure cryptosystems. The authors provide details on the cost of the quantum algorithms presented in this survey. The authors furthermore discuss ongoing research directions that can impact quantum cryptanalysis in the future