A theoretical model of a quantum device which can factorize any number N in two steps i.e. by preparing an input state and performing a measurement is discussed. The analysis reveals that the duration of state preparation and measurement is proportional to N while the energy consumption grows like log N. These results suggest the existence of Heisenberg-type relation putting limits on the efficiency of a quantum computer in terms of a total computation time, a total energy consumption and a classical complexity of the problem.
A theoretical model of a quantum device which can factorize any number N in two steps i.e. by preparing an input state and performing a measurement is discussed. The analysis reveals that the duration of state preparation and measurement is proportional to N while the energy consumption grows like log N. These results suggest the existence of Heisenberg-type relation putting limits on the efficiency of a quantum computer in terms of a total computation time, a total energy consumption and a classical complexity of the problem.
Quantum computing is considered a revolutionary technology due to its ability to solve computational problems that are beyond the capabilities of classical computers. However, quantum computing requires great amounts of energy to run. Therefore, a factor in deciding whether to use quantum computing should be not only the complexity of the problem to be solved, but also the energy required to solve it. This paper presents an empirical study developed with the aim of comparing classical and quantum computing in terms of energy efficiency to determine whether the increased power of quantum computers is offset by their higher energy consumption. To achieve this, a variety of problems with different levels of complexity were tested on both types of computers. Specifically, we used the IBM Quantum computers with a maximum of 5 qubits and an Intel i7, as a classical computer. In addition to this we have also analysed the evolution of the quantum computers, performing measurements on three time periods. Our empirical study showed that there is a variability of results obtained in the three time periods and that quantum computing is not recommended for low-complexity problems, given its high energy consumption, particularly when compared to traditional computing.
Energy consumption in solving computational problems has been gaining growing attention as one of the key performance measures for computers. Quantum computation offers advantages over classical computation in terms of various computational resources; however, proving its energy-consumption advantage has been challenging due to the lack of a theoretical foundation linking the physical concept of energy with the computer-scientific notion of complexity for quantum computation. To bridge this gap, we introduce a general framework for studying the energy consumption of quantum and classical computation, based on a computational model conventionally used for studying query complexity in computational complexity theory. Within this framework, we derive an upper bound for the achievable energy consumption of quantum computation, accounting for imperfections in implementation appearing in practice. As part of this analysis, we construct a protocol for Landauer erasure with finite precision in a finite number of steps, which constitutes a contribution of independent interest. Additionally, we develop techniques for proving a nonzero lower bound of energy consumption of classical computation, based on the energy-conservation law and Landauer’s principle. Using these general bounds, we rigorously prove that quantum computation achieves an exponential energy-consumption advantage over classical computation for solving a paradigmatic computational problem—Simon’s problem. Furthermore, we propose explicit criteria for experimentally demonstrating this energy-consumption advantage of quantum computation, analogous to the experimental demonstrations of quantum computational supremacy. These results establish a foundational framework and techniques to explore the energy consumption of computation, opening an alternative way to study the advantages of quantum computation.
How much energy does a quantum computer consume? Are they more efficient than their classical counterparts? In this work, we make a step towards answering these questions. We define the energy efficiency of a quantum computer as the ratio of the number of algorithms it can perform during a given time over the energy consumed by the hardware during this time. We analyze the most representative physical platforms currently envisioned to be used as building blocks of quantum computers: superconducting qubits, silicon spin qubits, trapped ions, neutral atoms and photonic qubits. Including insights from experts in all these technologies and taking into account algorithm compilation constraints, we discuss the advantages and inconveniences of each platform from an energy standpoint. Beyond providing concrete values of the energy consumption of current quantum computers, we lay the foundation of a framework to benchmark the energy efficiency of any future quantum computing architecture.
We have seen in the previous chapters that quantum computers seem to be more powerful than classical computers for certain problems. There are limits on the power of quantum computers, however. Since a classical computer can simulate a quantum computer, a quantum computer can only compute the same set of functions that a classical computer can. The advantage of using a quantum computer is that the amount of resources needed by a quantum algorithm might be much less than what is needed by the best classical algorithm. In Section 9.1 we briefly define some classical and quantum complexity classes and give some relationships between them. Most of the interesting questions relating classical and quantum complexity classes remain open. For example, we do not yet know if a quantum computer is capable of efficiently solving an NP-complete problem (defined later). One can prove upper bounds on the difficulty of a problem by providing an algorithm that solves that problem, and proving that it will work within in a given running time. But how does one prove a lower bound on the computational complexity of a problem? For example, if we wish to find the product of two n-bit numbers, computing the answer requires outputting roughly 2n bits and that requires Ω(n) steps (in any computing model with finite-sized gates). The best-known upper bound for integer multiplication is O(n log n log log n) steps. It has proved extremely difficult to derive non-trivial lower bounds on the computational complexity of a problem. Most of the known non-trivial lower bounds are in the ‘black-box’ model (for both classical and quantum computing), where we only query the input via a ‘black-box’ of a specific form. We discuss the black-box model in more detail in Section 9.2. We then sketch several approaches for proving black-box lower bounds. The first technique has been called the ‘hybrid method’ and was used to prove that quantum searching requires Ω(√n) queries to succeed with constant probability. The second technique is called the ‘polynomial method’. We then describe a technique based on ‘block sensitivity’, and conclude with a technique known as the ‘adversary method’.
Classical computing is approaching its physical limits, so quantum computing as an alternative model requires careful evaluation. This study compares various aspects of two systems by looking at formal algorithmic complexity, simulation-based comparisons with realistic noise models, quantitative meta-analysis, figuring out the difference between physical and logical error rates, and clearly defining contributions. A thorough study examined 24,100 papers published between 2015 and 2024 from leading scientific databases. A study of algorithmic complexity using Big-O notation confirmed that Shor's algorithm is faster than standard O(exp((64/9)²/³ (log N)²/³ (log log N)²/³)) by a super polynomial factor, Grover's algorithm is faster by a quadratic factor (O(√N) versus O(N)), and quantum simulation is faster by an exponential factor. There were Five quantum machines exhibited physical error rates ranging from 10⁻² to 10⁻⁵. Simulation-based testing revealed that the quantum advantage emerges above certain complexity thresholds: at extremely high complexity, quantum processing reached 10.3 μs versus 40.2 μs for classical computing, with better error resistance (2.3% for quantum vs. 5.9% for classical). Quantum computing has real benefits in terms of speed, mistake tolerance, and energy economy for problems that are too complicated to solve with traditional methods. However, significant challenges remain in scalability and workforce development. Hybrid quantum-classical models offer the most promising path to near-term practical benefits. On the other hand, the shift to postquantum security needs instant attention, regardless of when quantum hardware will be available.