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arXiv:2203.09545 · Physical Review Letters

Third Law of Thermodynamics and the Scaling of Quantum Computers

Lorenzo Buffoni, Stefano Gherardini, Emmanuel Zambrini Cruzeiro, Yasser Omar

Published Oct 3, 2022• 25 citations
arXivHTMLPDF

Abstract

The third law of thermodynamics, also known as the Nernst unattainability principle, puts a fundamental bound on how close a system, whether classical or quantum, can be cooled to a temperature near to absolute zero. On the other hand, a fundamental assumption of quantum computing is to start each computation from a register of qubits initialized in a pure state, i.e., at zero temperature. These conflicting aspects, at the interface between quantum computing and thermodynamics, are often overlooked or, at best, addressed only at a single-qubit level. In this Letter, we argue how the existence of a small but finite effective temperature, which makes the initial state a mixed state, poses a real challenge to the fidelity constraints required for the scaling of quantum computers. Our theoretical results, carried out for a generic quantum circuit with N-qubit input states, are validated by test runs performed on a real quantum processor.

DOI
10.1103/physrevlett.129.150602

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The third law of thermodynamics, also known as the Nernst unattainability principle, puts a fundamental bound on how close a system, whether classical or quantum, can be cooled to a temperature near to absolute zero. On the other hand, a fundamental assumption of quantum computing is to start each computation from a register of qubits initialized in a pure state, i.e., at zero temperature. These conflicting aspects, at the interface between quantum computing and thermodynamics, are often overlooked or, at best, addressed only at a single-qubit level. In this Letter, we argue how the existence of a small but finite effective temperature, which makes the initial state a mixed state, poses a real challenge to the fidelity constraints required for the scaling of quantum computers. Our theoretical results, carried out for a generic quantum circuit with N-qubit input states, are validated by test runs performed on a real quantum processor.
Thermodynamics connects our knowledge of the world to our capability to manipulate and thus to control it. This crucial role of control is exemplified by the third law of thermodynamics, Nernst's unattainability principle, which states that infinite resources are required to cool a system to absolute zero temperature. But what are these resources and how should they be utilized? And how does this relate to Landauer's principle that famously connects information and thermodynamics? We answer these questions by providing a framework for identifying the resources that enable the creation of pure quantum states. We show that perfect cooling is possible with Landauer energy cost given infinite time or control complexity. However, such optimal protocols require complex unitaries generated by an external work source. Restricting to unitaries that can be run solely via a heat engine, we derive a novel Carnot-Landauer limit, along with protocols for its saturation. This generalizes Landauer's principle to a fully thermodynamic setting, leading to a unification with the third law and emphasizes the importance of control in quantum thermodynamics.
Abstract The third law of thermodynamics has two classical formulations: Nernst’s heat theorem (entropy change ΔS→0 as T→0) and the unattainability principle (absolute zero cannot be reached in a finite number of steps). Quantum statistical mechanics challenges Nernst’s theorem through residual entropy in systems such as quantum spin liquids and spin ice, which retain nonzero entropy even as T→0. Landauer’s principle connects information erasure with heat dissipation, implying that reaching absolute zero is equivalent to deleting all information from a system. This paper proposes that physical reality comprises two information-carrying sectors—the material phase (Ξ) and the information phase (Ω)—governed by strict conservation of total information. Within this framework, absolute zero is reinterpreted as the global information ground state of two-phase information settlement: it is both the absolute zero point of material-phase information capacity and the material embodiment of the minimal information potential of the information phase. Residual entropy is not an exception to the third law, but an inevitable manifestation of information conservation at the absolute zero boundary. Starting from the principle of least action incorporating two-phase coupling, we derive a quantitative condition for the unattainability of absolute zero and demonstrate that the decoherence rate of a quantum system near absolute zero is systematically modulated by the information-phase coupling parameter. The framework yields three testable predictions: a scaling relation between decoherence rate and information-phase coupling strength at ultra-low temperatures, entropy accumulation near quantum critical points corresponding to enhanced information-phase potential density, and a temperature scaling law for the coherence time of time crystals. This work unifies the third law of thermodynamics with quantum measurement, black hole information processing, dark energy cosmology, quantum entanglement, and the Higgs mechanism under a single cross-scale information settlement paradigm. Keywords: absolute zero, third law of thermodynamics, residual entropy, information conservation, Landauer’s principle, decoherence, quantum spin liquid, time crystal
We study the asymptotic dynamics of arbitrary linear quantum open systems that are periodically driven while coupled with generic bosonic reservoirs. We obtain exact results for the heat flowing from each reservoir, and these results are valid beyond the weak-coupling or Markovian approximations. We prove the validity of the dynamical third law of thermodynamics (Nernst unattainability principle), showing that the ultimate limit for cooling is imposed by a fundamental heating mechanism that dominates at low temperatures, namely the nonresonant creation of excitation pairs in the reservoirs induced by the driving field. This quantum effect, which is missed in the weak-coupling approximation, restores the unattainability principle, the validity of which was recently challenged.
One of the formulations of the third laws of thermodynamics is that a processes become more isentropic as one approaches the absolute zero temperatures. We examine this prediction by studying an operating model of a quantum refrigerator pumping heat from a cold to a hot reservoir. The working medium consists of a gas of noninteracting harmonic oscillators. The model can be solved in closed form in the quasi-static limit or numerically for general conditions. It is found that the isentropic limit for Tc → 0 is approached only on the expansion segment of the refrigeration cycle. The scaling of the cooling rate with temperature is shown to be consistent with the second law of thermodynamics. This scaling is also consistent with the unattainability principle which is an alternative formulation of the third law of thermodynamics.
We study the problem of dynamic cooling whereby a target qubit is cooled at the expense of heating up N − 1 further identical qubits by means of a global unitary operation. A standard back-of-the-envelope high-temperature estimate establishes that the target qubit temperature can be dynamically cooled by at most a factor of 1 / N . Here we provide the exact expression for the minimum temperature to which the target qubit can be cooled and reveal that there is a crossover from the high initial temperature regime, where the scaling is 1 / N , to a low initial temperature regime, where a much faster scaling of 1 / N occurs. This slow, 1 / N scaling, which was relevant for early high-temperature NMR quantum computers, is the reason dynamic cooling was dismissed as ineffectual around 20 years ago; the fact that current low-temperature quantum computers fall in the fast, 1 / N scaling regime, reinstates the appeal of dynamic cooling today. We further show that the associated work cost of cooling is exponentially more advantageous in the low-temperature regime. We discuss the implementation of dynamic cooling in terms of quantum circuits and examine the effects of hardware noise. We successfully demonstrate dynamic cooling in a three-qubit system on a real quantum processor. Since the circuit size grows quickly with N , scaling dynamic cooling to larger systems on noisy devices poses a challenge. We therefore propose a suboptimal cooling algorithm, whereby relinquishing a small amount of cooling capability results in a drastically reduced circuit complexity, greatly facilitating the implementation of dynamic cooling on near-future quantum computers. Published by the American Physical Society 2024
Landauer's limit on heat dissipation during information erasure is critical as devices shrink, requiring optimal pure-state preparation to minimize errors. However, Nernst's third law states this demands infinite resources in energy, time, or control complexity. We address the challenge of cooling quantum systems with finite resources. Using Markovian collision models, we explore resource trade-offs and present efficient cooling protocols (that are optimal for qubits) for coherent and incoherent control. Leveraging thermodynamic length, we derive bounds on heat dissipation for swap-based strategies and discuss the limitations of preparing pure states efficiently.