Energy Limits in Computation

The Landauer Principle connects the information theoretic notion of entropy to the physics of statistical mechanics. When a physical system performs a logical operation that erases or loses information, without a copy being preserved, it must transfer a minimum amount of heat, \(k_B T \log (2)\), to the environment. How can there be such a connection between the abstract idea of information and the concrete physical reality of heat? To address this question, we adopt the Jaynes approach of grounding statistical mechanics in the Shannon notion of entropy. Probability is a quantification of incomplete information. Entropy should not be conceived in terms of disorder, but rather as a measure on a probability distribution that characterizes the amount of missing information the distribution represents. The thermodynamic entropy is a special case of the Shannon entropy applied to a physical system in equilibrium with a heat bath so that its average energy is fixed. The thermal probability distribution is obtained by maximizing the Shannon entropy, subject to the physical constraints of the problem. It is then possible to naturally extend this description to include a physical memory device, which must be in a nonequilibrium long-lived metastable state. We can then explicitly demonstrate how the requirement for a fundamental minimum energy dissipation is tied to erasure of an unknown bit. Both classical and quantum cases are considered. We show that the classical thermodynamic entropy is in some situations best matched in quantum mechanics, not by the von Neumann entropy, but by a perhaps less familiar quantity—the quantum entropy of outcomes. The case of free expansion of an ideal quantum gas is examined in this context.

Quantum-dynamical proofs of dissipation bounds for Landauer erasure are presented, with emphasis on the crucial connection between conditioning in erasure protocols and fundamental limits on erasure costs. Bounds on erasure costs for conditional and unconditional erasure protocols are shown to follow from a very general and ecumenical physical description of Landauer erasure, a straightforward accounting of its energetic cost, refined definitions of what it means for physical system states to bear known and unknown data, and a transparent application of quantum dynamics and entropic inequalities. These results generalize and support the results of Landauer and Bennett for unconditional and conditional erasure, respectively, and do so using a theoretical methodology that sidesteps or otherwise withstands methodological objections that have been leveled against thermodynamic proofs and other theoretical arguments in the literature. The dissipation bounds obtained here coincide with those obtained elsewhere from an even more general approach that is based on a thoroughly physical conception of information and that clearly distinguishes information from entropy. This connection may help to clarify central issues in the debate over Landauer’s Principle, since the more general approach bounds dissipative costs of irreversible information loss in a range of scenarios that are both broader and less idealized than those typically considered in explorations of the Landauer limit.

We present a pedagogical review of the fundamental concepts in thermodynamics of information, by focusing on the second law of thermodynamics and the entropy production. Especially, we discuss the relationship among thermodynamic reversibility, logical reversibility, and heat emission in the context of the Landauer principle and clarify that these three concepts are fundamentally distinct to each other. We also discuss thermodynamics of measurement and feedback control by Maxwell’s demon. We clarify that the demon and the second law are indeed consistent in the measurement and the feedback processes individually, by including the mutual information to the entropy production.

In this chapter, we examine in detail Landauer’s influential argument that erasure of information is a fundamental source of energy dissipation. We will carefully define what we mean by general terms, such as “computation,” “reversibility,” and “entropy.” The literature contains quite a bit of confusion due to a lack of clear definitions of these terms. In particular, we will go to great length to distinguish between physical (thermodynamic) entropy and information entropy. We will be lead to conclude that Landauer’s influential argument contains a fundamental flaw in failing to distinguish between these two forms of entropy, and in using information entropy as if it were physical entropy. Erasure of information is not a fundamental source of energy dissipation in computation. Beyond Landauer’s argument, we will conclude that thermodynamics does not apply to the information-bearing degrees of freedom in a computer. The more a system can be described by thermodynamics, the less it can be used for computation. The main conclusion of this chapter is that the thermodynamics of computation is a contradiction in terms.

We summarize recent experimental and theoretical progress achieved in the physics of information. We highlight the intimate connection existing between information and energy from Maxwell’s demon and Szilard’s engine to Landauer’s erasure principle. We focus both on classical and quantum systems and conclude by discussing applications in engineering and biology.

Power dissipation is one of the most important factors limiting the development of integrated circuits. In this chapter, we will explore the limits of energy dissipation in computation with experiments and circuit designs. Our experiments show that there is no fundamental limit on energy that must be dissipated to perform computation as long as information is preserved, in agreement with the Landauer principle. The erasure of information leads to a loss of bit energy with an ultimate lower limit of k_B T ln2, sometimes incorrectly referred to as the “Landauer limit for energy dissipation in computation.” We present an experiment where a dissipation of 0.005 k_B T which is far below the limit of k_B T ln2 occurs in reversible adiabatic bit manipulation, and experimentally demonstrate that dissipation of the full bit energy occurs if information is erased. To exploit the advantages of quasi-adiabatic reversible computation, we discuss adiabatic logic systems, and present the design of a microprocessor based upon adiabatic logic. Due to their inherent leakage current, field-effect transistors have limitations in adiabatic implementations. We discuss possible devices that have a better match to adiabatic systems. Finally, we present experiments making direct measurement of the heat generated in logical operations.