Conventional computing has many sources of heat dissipation, but one of these--the Landauer limit--poses a fundamental lower bound of 1 bit of entropy per bit erased. 'Reversible Computing' avoids this source of dissipation, but is dissipationless computation possible? In this paper, a general proof is given for open quantum systems showing that a computer thermally coupled to its environment will necessarily dissipate entropy (and hence heat). Specifically, a lower bound is obtained that corresponds to the adiabatic regime, in which the amount of entropy dissipated per computational operation is proportional to the rate of computation.
Conventional computing has many sources of heat dissipation, but one of these--the Landauer limit--poses a fundamental lower bound of 1 bit of entropy per bit erased. 'Reversible Computing' avoids this source of dissipation, but is dissipationless computation possible? In this paper, a general proof is given for open quantum systems showing that a computer thermally coupled to its environment will necessarily dissipate entropy (and hence heat). Specifically, a lower bound is obtained that corresponds to the adiabatic regime, in which the amount of entropy dissipated per computational operation is proportional to the rate of computation.
The amount of heat generated by computers is rapidly becoming one of the main problems for developing new generations of information technology. The thermodynamics of computation sets the ultimate physical bounds on heat generation. A lower bound is set by the Landauer limit, at which computation becomes thermodynamically reversible. For classical computation there is no physical principle which prevents this limit being reached, and approaches to it are already being experimentally tested. In this paper we show that for quantum computation with a set of signal states satisfying given conditions, there is an unavoidable excess heat generation that renders it inherently thermodynamically irreversible. The Landauer limit cannot, in general, be reached by quantum computers. We show the existence of a lower bound to the heat generated by quantum computing that exceeds that given by the Landauer limit, give the special conditions where this excess cost may be avoided, and provide a protocol for achieving the limiting heat cost when these conditions are met. We also show how classical computing falls within the special conditions.
Abstract Computation requires energy. Landauer’s principle establishes that erasing one bit of information irreversibly dissipates at least kBT ln 2 of thermal energy. Modern computers operate 10^6 to 10^9 times above this fundamental bound. In 1973, Bennett demonstrated that any computation can be made thermodynamically reversible through procedures that preserve information, thereby approaching the Landauer limit. However, a critical question remained unanswered for fifty years: how quickly does reversible compilation converge to the theoretical minimum, and what ensemble size achieves practical energy savings? We resolve this problem through the Fluctuation-Dissipation Compilation Theorem, which provides the first explicit convergence rate for thermodynamically optimal computing. Reversible compilation of any algorithm dissipates average energy equal to the thermal energy scale multiplied by the logical entropy change, plus corrections that decrease as the inverse square root of the ensemble size. This N^(−1/2) convergence rate, derived by combining fluctuation theorems from statistical mechanics with the central limit theorem, is provably optimal—no compilation strategy can achieve faster convergence. The proof unifies three historically independent disciplines: statistical mechanics through the Jarzynski equality and Crooks fluctuation theorem, information theory through Shannon entropy bounds, and compilation theory through Bennett’s reversible computing framework. By formulating thermodynamic compilation as a rigorous resource theory, we establish that reversible transformations exactly conserve entropy while irreversible erasure operations incur fundamental energy costs bounded by information-theoretic limits. Critically, we provide finite-ensemble performance guarantees essential for practical implementations, characterizing not just average energy dissipation but complete statistical distributions including fluctuations and tail probabilities. Numerical experiments on Grover search and quantum Fourier transforms validate our predictions. Simulations confirm the predicted N^(−1/2) convergence rate within 3% error and demonstrate energy reductions exceeding 99.9% compared to irreversible implementations. These results establish that thermodynamically optimal compilation is achievable with realistic ensemble sizes for near-term quantum hardware operating under thermal constraints. This framework bridges fundamental physics and practical computing. Applications span quantum processors cooled to millikelvin temperatures where every picojoule of dissipation limits performance, battery-powered Internet-of-Things sensors requiring decade-long autonomous operation, data centers consuming one percent of global electricity where incremental efficiency improvements yield billion-dollar savings, and deep-space missions operating on hundred-watt radioisotope generators where every watt conserved enables additional science. By establishing thermodynamic compilation as a rigorous engineering discipline with explicit finite-size guarantees, this work transforms Bennett’s asymptotic theoretical insight into a practical methodology for approaching fundamental physical limits in real computing systems
Quantum state engineering and quantum computation rely on information erasure procedures that, up to some fidelity, prepare a quantum object in a pure state. Such processes occur within Landauer's framework if they rely on an interaction between the object and a thermal reservoir. Landauer's principle dictates that this must dissipate a minimum quantity of heat, proportional to the entropy reduction that is incurred by the object, to the thermal reservoir. However, this lower bound is only reachable for some specific physical situations, and it is not necessarily achievable for any given reservoir. The main task of our work can be stated as the minimisation of heat dissipation given probabilistic information erasure, i.e., minimising the amount of energy transferred to the thermal reservoir as heat if we require that the probability of preparing the object in a specific pure state | φ 1 ⟩ be no smaller than p φ 1 max − δ . Here p φ 1 max is the maximum probability of information erasure that is permissible by the physical context, and δ ⩾ 0 the error. To determine the achievable minimal heat dissipation of quantum information erasure within a given physical context, we explicitly optimise over all possible unitary operators that act on the composite system of object and reservoir. Specifically, we characterise the equivalence class of such optimal unitary operators, using tools from majorisation theory, when we are restricted to finite-dimensional Hilbert spaces. Furthermore, we discuss how pure state preparation processes could be achieved with a smaller heat cost than Landauer's limit, by operating outside of Landauer's framework.
Irreversibility has a fundamental thermodynamic cost, erasing information inevitably generates heat. This connection is quantified by the Landauer bound, which gives the minimum dissipation needed to erase a single bit of information. While this bound applies in both classical and quantum settings, it is saturated only in idealised limits of infinite resources. Here, we provide a unified first principles description of finite-resource erasure in both classical and quantum systems. We begin by proving the communal folklore that in the idealised regime the erasure cost of a bit encoded in a quantum or classical system is the same. Despite this, we show that their practical implementation differs substantially: achieving comparable erasure quality in quantum systems requires more control, larger accessible energy gaps and longer operation times. Classical protocols can achieve the erasure of a comparable quantum protocol under far weaker constraints which we expose in trade-off relations. Our results explain why practical erasure schemes fall short of Landauer's bound and show that classical systems enjoy several fundamental thermodynamic advantages.
Landauer's principle provides a perspective on the physical meaning of information as well as on the minimum working cost of information processing. Whereas most studies have related the decrease in entropy during a computationally irreversible process to a lower bound of dissipated heat, recent efforts have also provided another lower bound associated with the thermodynamic fluctuation of heat. The coexistence of the two conceptually independent bounds has stimulated comparative studies of their close relationship or tightness; however, these studies were concerned with finite quantum systems that allowed the revival of erased information because of a finite recurrence time. We broaden these comparative studies further to open quantum systems with infinite recurrence times. By examining their dependence on the initial state, we find the independence of the thermodynamic bound from the initial coherence, whereas the entropic bound depends on both the initial coherence and the population. A crucial role is indicated by the purity of the initial state: The entropic bound is tighter when the initial condition is sufficiently mixed, whereas the thermodynamic bound is tighter when the initial state is close to a pure state. These trends are consistent with previous results obtained for finite systems.
A known aspect of the Clausius inequality is that an equilibrium system subjected to a squeezing dS < 0 of its entropy must release at least an amount the absolute value of dQ = T times the absolute value of dS of heat. This serves as a basis for the Landauer principle, which puts a lower bound T ln 2 for the heat generated by erasure of one bit of information. Here we show that in the world of quantum entanglement this law is broken. A quantum Brownian particle interacting with its thermal bath can either generate less heat or even absorb heat during an analogous squeezing process, due to entanglement with the bath. The effect exists even for weak but fixed coupling with the bath, provided that temperature is low enough. This invalidates the Landauer bound in the quantum regime, and suggests that quantum carriers of information can be more efficient than assumed so far.