Logical Irreversibility Research Articles

Pipelined information flow in molecular mechanical circuits leads to increased error and irreversibility.

Pipelining is a design technique for logical circuits that allows for higher throughput than circuits in which multiple computations are fed through the system one after the other. It allows for much faster computation than architectures in which inputs must pass through every layer of the circuit before the next computation can begin (phased chaining). We explore the hypothesis that these advantages may be offset by a higher error rate, logical irreversibility, and greater thermodynamic costs by simulating pipelined molecular mechanical circuits using an explicit physical model. We observe the emergent logical irreversibility, and see that the simultaneous action of multiple components indeed leads to a higher error rate than in phase-chained circuits. The thermodynamic costs of operating the gates are much larger than in equivalent phase-chained circuits, and these costs do not appear to tend to zero in the limit of slowgate operation. Redesigning the gates to eliminate errors and artificially enforcing logical reversibility reduces the thermodynamic costs and recovers thermodynamically reversible behavior in the limit of slow gate operation. The breakdown of logical reversibility and accuracy are both associated with a breakdown of the digital behavior of the device, likely contributing to thermodynamic costs that are large relative to the scale of the information being processed.

Network analysis for the steady-state thermodynamic uncertainty relation.

We perform network analysis of a system described by the master equation to estimate the lower bound of the steady-state current noise, starting from the level 2.5 large deviation function and using the graph theory approach. When the transition rates are uniform, and the system is driven to a nonequilibrium steady state by unidirectional transitions, we derive a noise lower bound, which accounts for fluctuations of sojourn times at all states and is expressed using mesh currents. This bound is applied to the uncertainty in the signal-to-noise ratio of the fluctuating computation time of a schematic Brownian computation plus reset process described by a graph containing one cycle. Unlike the mixed and pseudo-entropy bounds that increase logarithmically with the length of the intended computation path, this bound depends on the number of extraneous predecessors and thus captures the logical irreversibility.

Landauer Bound in the Context of Minimal Physical Principles: Meaning, Experimental Verification, Controversies and Perspectives

The physical roots, interpretation, controversies, and precise meaning of the Landauer principle are surveyed. The Landauer principle is a physical principle defining the lower theoretical limit of energy consumption necessary for computation. It states that an irreversible change in information stored in a computer, such as merging two computational paths, dissipates a minimum amount of heat kBTln2 per a bit of information to its surroundings. The Landauer principle is discussed in the context of fundamental physical limiting principles, such as the Abbe diffraction limit, the Margolus–Levitin limit, and the Bekenstein limit. Synthesis of the Landauer bound with the Abbe, Margolus–Levitin, and Bekenstein limits yields the minimal time of computation, which scales as τmin~hkBT. Decreasing the temperature of a thermal bath will decrease the energy consumption of a single computation, but in parallel, it will slow the computation. The Landauer principle bridges John Archibald Wheeler’s “it from bit” paradigm and thermodynamics. Experimental verifications of the Landauer principle are surveyed. The interrelation between thermodynamic and logical irreversibility is addressed. Generalization of the Landauer principle to quantum and non-equilibrium systems is addressed. The Landauer principle represents the powerful heuristic principle bridging physics, information theory, and computer engineering.

On the Supposed Mass of Entropy and That of Information.

In the theory of special relativity, energy can be found in two forms: kinetic energy and rest mass. The potential energy of a body is actually stored in the form of rest mass, the interaction energy too, but temperature is not. Information acquired about a dynamical system can be potentially used to extract useful work from it. Hence, the "mass-energy-information equivalence principle" that has been recently proposed. In this paper, it is first recalled that for a thermodynamic system made of non-interacting entities at constant temperature, the internal energy is also constant. So, the energy involved in a variation in entropy (TΔS) differs from a change in the potential energy stored or released and cannot be associated to a corresponding variation in mass of the system, even if it is expressed in terms of the quantity of information. This debate gives us the opportunity to deepen the notion of entropy seen as a quantity of information, to highlight the difference between logical irreversibility (a state-dependent property) and thermodynamical irreversibility (a path-dependent property), and to return to the nature of the link between energy and information that is dynamical.

Thermodynamical versus Logical Irreversibility: A Concrete Objection to Landauer's Principle.

Landauer's principle states that the logical irreversibility of an operation, such as erasing one bit, whatever its physical implementation, necessarily implies its thermodynamical irreversibility. In this paper, a very simple counterexample of physical implementation (that uses a two-to-one relation between logic and thermodynamic states) is given that allows one bit to be erased in a thermodynamical quasistatic manner (i.e., one that may tend to be reversible if slowed down enough).

Spontaneous Fluctuation-Symmetry Breaking and the Landauer Principle

We study the problem of the energetic cost of information erasure by looking at it through the lens of the Jarzynski equality. We observe that the Landauer bound, langle W rangle ge kT ln 2, on average dissipated work langle W rangle associated to an erasure process, literally emerges from the underlying second law bound as formulated by Kelvin, langle W rangle ge 0, as consequence of a spontaneous breaking of the Crooks–Tasaki fluctuation-symmetry, that accompanies logical irreversibility. We illustrate and corroborate this insight with numerical simulations of the process of information erasure performed on a 2D Ising ferromagnet.

Dependence of integrated, instantaneous, and fluctuating entropy production on the initial state in quantum and classical processes.

We consider the additional entropy production (EP) incurred by a fixed quantum or classical process on some initial state ρ, above the minimum EP incurred by the same process on any initial state. We show that this additional EP, which we term the "mismatch cost of ρ," has a universal information-theoretic form: it is given by the contraction of the relative entropy between ρ and the least-dissipative initial state φ over time. We derive versions of this result for integrated EP incurred over the course of a process, for trajectory-level fluctuating EP, and for instantaneous EP rate. We also show that mismatch cost for fluctuating EP obeys an integral fluctuation theorem. Our results demonstrate a fundamental relationship between thermodynamic irreversibility (generation of EP) and logical irreversibility (inability to know the initial state corresponding to a given final state). We use this relationship to derive quantitative bounds on the thermodynamics of quantum error correction and to propose a thermodynamically operationalized measure of the logical irreversibility of a quantum channel. Our results hold for both finite- and infinite-dimensional systems, and generalize beyond EP to many other thermodynamic costs, including nonadiabatic EP, free-energy loss, and entropy gain.

The mass-energy-information equivalence principle

Landauer’s principle formulated in 1961 states that logical irreversibility implies physical irreversibility and demonstrated that information is physical. Here we formulate a new principle of mass-energy-information equivalence proposing that a bit of information is not just physical, as already demonstrated, but it has a finite and quantifiable mass while it stores information. In this framework, it is shown that the mass of a bit of information at room temperature (300K) is 3.19 × 10-38 Kg. To test the hypothesis we propose here an experiment, predicting that the mass of a data storage device would increase by a small amount when is full of digital information relative to its mass in erased state. For 1Tb device the estimated mass change is 2.5 × 10-25 Kg.

The physics of implementing logic: Landauer's principle and the multiple-computations theorem

This paper makes a novel linkage between the multiple-computations theorem in philosophy of mind and Landauer's principle in physics. The multiple-computations theorem implies that certain physical systems implement simultaneously more than one computation. Landauer's principle implies that the physical implementation of “logically irreversible” functions is accompanied by minimal entropy increase. We show that the multiple-computations theorem is incompatible with, or at least challenges, the universal validity of Landauer's principle. To this end we provide accounts of both ideas in terms of low-level fundamental concepts in statistical mechanics, thus providing a deeper understanding of these ideas than their standard formulations given in the high-level terms of thermodynamics and cognitive science. Since Landauer's principle is pivotal in the attempts to derive the universal validity of the second law of thermodynamics in statistical mechanics, our result entails that the multiple-computations theorem has crucial implications with respect to the second law. Finally, our analysis contributes to the understanding of notions, such as “logical irreversibility,” “entropy increase,” “implementing a computation,” in terms of fundamental physics, and to resolving open questions in the literature of both fields, such as: what could it possibly mean that a certain physical process implements a certain computation.

Micro electro-mechanical logic device at fundamental energy limit

In 1961 Landauer argued that reset a bit of information requires a minimum energy expenditure of k B T ln 2. This quantity is directly related to the entropy reduction in the system. This famous claim was then extended to all logic operations when the entropy at the inputs is larger than the entropy at the outputs, i.e. logically irreversible operations. However, in certain situations, a reduction of information in terms of input/output does not lead to a reduction of the physical entropy in the device. In this article we show the implementation of a storage unit and a logic gate with a micro-electro-mechanical device and measure the energy dissipated during their operation. We show that while an entropy reduction in the device implies a finite heat production, logic irreversibility it is not always associated with physical irreversibility.

Reversibility and energy dissipation in adiabatic superconductor logic

Reversible computing is considered to be a key technology to achieve an extremely high energy efficiency in future computers. In this study, we investigated the relationship between reversibility and energy dissipation in adiabatic superconductor logic. We analyzed the evolution of phase differences of Josephson junctions in the reversible quantum-flux-parametron (RQFP) gate and confirmed that the phase differences can change time reversibly, which indicates that the RQFP gate is physically, as well as logically, reversible. We calculated energy dissipation required for the RQFP gate to perform a logic operation and numerically demonstrated that the energy dissipation can fall below the thermal limit, or the Landauer bound, by lowering operation frequencies. We also investigated the 1-bit-erasure gate as a logically irreversible gate and the quasi-RQFP gate as a physically irreversible gate. We calculated the energy dissipation of these irreversible gates and showed that the energy dissipation of these gate is dominated by non-adiabatic state changes, which are induced by unwanted interactions between gates due to logical or physical irreversibility. Our results show that, in reversible computing using adiabatic superconductor logic, logical and physical reversibility are required to achieve energy dissipation smaller than the Landauer bound without non-adiabatic processes caused by gate interactions.

The Physics of Forgetting: Thermodynamics of Information at IBM 1959–1982

The origin and history of Landauer’s principle is traced through the development of the thermodynamics of computation at IBM from 1959 to 1982. This development was characterized by multiple conceptual shifts: memory came to be seen not as information storage, but as delayed information transmission; information itself was seen not as a disembodied logical entity, but as participating in the physical world; and logical irreversibility was connected with physical, thermodynamic, irreversibility. These conceptual shifts were characterized by an ambivalence opposing strong metaphysical claims to practical considerations. Three sorts of practical considerations are discussed. First, these conceptual shifts engaged materials central to IBM’s business practice. Second, arguments for metaphysical certainties were made with reference to the practical functioning of typical computers. Third, arguments for metaphysical certainties were made in the context of establishing the thermodynamics of information as a sub-discipline of physics.

On the Origin of Ambiguity in Efficient Communication

This article studies the emergence of ambiguity in communication through the concept of logical irreversibility and within the framework of Shannon’s information theory. This leads us to a precise and general expression of the intuition behind Zipf’s vocabulary balance in terms of a symmetry equation between the complexities of the coding and the decoding processes that imposes an unavoidable amount of logical uncertainty in natural communication. Accordingly, the emergence of irreversible computations is required if the complexities of the coding and the decoding processes are balanced in a symmetric scenario, which means that the emergence of ambiguous codes is a necessary condition for natural communication to succeed.

Heat Dissipation in Nanocomputing: Lower Bounds From Physical Information Theory

Computing circuits that irreversibly discard information unavoidably dissipate heat. Dissipative costs resulting from information loss, while insignificant in CMOS technology, may be dominant or even prohibitive in some dense, high-speed post-CMOS nanocomputing circuits that employ logically irreversible operations. In transistor-based paradigms, dissipation costs associated with logical irreversibility may be supplemented by additional unavoidable costs associated with particle supply required to maintain the computational “working substance.” These considerations motivate determination of fundamental lower bounds on the dissipative cost of computation that can be applied to concrete nanocomputing technology proposals. In this paper, we present a methodology for the determination of such bounds and illustrate its application to half-adder circuits implemented in the quantum cellular automata and nano-wire-based nano-application specific-integrated circuit paradigms. The resulting bounds reflect fundamental costs inherent in the underlying computational strategies employed by these circuits. Prospective use of this methodology as an assessment tool for post-CMOS nanocomputing technology proposals is discussed.

On the physical implementation of logical transformations: Generalized [formula omitted]-machines

Any account of computation in a physical system, whether an artificial computing device or a natural system considered from a computational point of view, invokes some notion of the relationship between the abstract-logical and concrete-physical aspects of computation. In a recent paper, James Ladyman explored this relationship using a “hybrid physical–logical entity”–the L -machine–and the general account of computation that it supports [J. Ladyman, What does it mean to say that a physical system implements a computation?, Theoretical Computer Science 410 (2009) 376–383]. The underlying L -machine of Ladyman’s analysis is, however, classical and highly idealized, and cannot capture essential aspects of computation in important classes of physical systems (e.g. emerging nanocomputing devices) where logical states do not have faithful physical representations and where noise and quantum effects prevail. In this work we generalize the L -machine to allow for generally unfaithful and noisy implementations of classical logical transformations in quantum mechanical systems. We provide a formal definition and physical-information-theoretic characterization of generalized quantum L -machines (QLMs), identify important classes of QLMs, and introduce new efficacy measures that quantify the faithfulness and fidelity with which logical transformations are implemented by these machines. Two fundamental issues emphasized by Ladyman–realism about computation and the connection between logical and physical irreversibility–are reconsidered within the more comprehensive account of computation that follows from our generalization of the L -machine.

What does it mean to say that a physical system implements a computation?

When we are concerned with the logical form of a computation and its formal properties, then it can be theoretically described in terms of mathematical and logical functions and relations between abstract entities. However, actual computation is realised by some physical process, and the latter is of course subject to physical laws and the laws of thermodynamics in particular. An issue that has been the subject of much controversy is that of whether or not there are any systematic connections between the logical properties of computations considered abstractly and the thermodynamical properties of their concrete physical realizations. Landauer [R. Landauer, Irreversibility and heat generation in the computing process, IBM Journal of Research and Development 5 (1961) 183–191. Reprinted in Leff and Rex (1990)] proposed such a general connection, known as Landauer’s Principle. To resolve this matter an analysis of the notion of the implementation of a computation by a physical system is clearly required. Another issue that calls for an analysis of implementation is that of realism about computation. The account of implementation presented here is based on the notion of an L - machine. This is a hybrid physical-logical entity that combines a physical device, a specification of which physical states of that device correspond to various logical states, and an evolution of that device which corresponds to the logical transformation L . The most general form of Landauer’s Principle can be precisely stated in terms of L -machines, namely that the logical irreversibility of L implies the thermodynamic irreversibility of every corresponding L -machine.

The connection between logical and thermodynamic irreversibility

There has recently been a good deal of controversy about Landauer's Principle, which is often stated as follows: the erasure of one bit of information in a computational device is necessarily accompanied by a generation of kT ln 2 heat. This is often generalised to the claim that any logically irreversible operation cannot be implemented in a thermodynamically reversible way. Norton [2005. Eaters of the lotus: Landauer's principle and the return of Maxwell's demon. Studies in History and Philosophy of Modern Physics, 36, 375–411] and Maroney [2005. The (absence of a) relationship between thermodynamic and logical reversibility. Studies in History and Philosophy of Modern Physics, 36, 355–374] both argue that Landauer's Principle has not been shown to hold in general, and Maroney offers a method that he claims instantiates the operation Reset in a thermodynamically reversible way. In this paper we defend the qualitative form of Landauer's Principle, and clarify its quantitative consequences (assuming the second law of thermodynamics). We analyse in detail what it means for a physical system to implement a logical transformation L, and we make this precise by defining the notion of an L-machine. Then we show that logical irreversibility of L implies thermodynamic irreversibility of every corresponding L-machine. We do this in two ways. First, by assuming the phenomenological validity of the Kelvin statement of the second law, and second, by using information-theoretic reasoning. We illustrate our results with the example of the logical transformation ‘Reset’, and thereby recover the quantitative form of Landauer's Principle.

Irreversibility and heat generation in the computing process

It is argued that computing machines inevitably involve devices which perform logical functions that do not have a single-valued inverse. This logical irreversibility is associated with physical irreversibility and requires a minimal heat generation, per machine cycle, typically of the order of kT for each irreversible function. This dissipation serves the purpose of standardizing signals and making them independent of their exact logical history. Two simple, but representative, models of bistable devices are subjected to a more detailed analysis of switching kinetics to yield the relationship between speed and energy dissipation, and to estimate the effects of errors induced by thermal fluctuations.