The third law of thermodynamics forbids cooling a physical system to absolute zero in a finite number of operational steps. Although this unattainability principle has been quantified for specific state-to-state transitions, a universal, dynamics-independent bound for implementing a state-agnostic reset map remains elusive. In this work, we unveil the fundamental limits of physical map implementation by deriving a trade-off relation based on geometric complexity. By analyzing continuous paths of maps on a geometric manifold, we prove that the geometric complexity of any classical stochastic map or quantum channel is bounded from below by its execution error. As a consequence, we show that achieving zero error in a state-reset operation requires a divergent geometric complexity -- a unified measure that naturally incorporates disparate physical resources, including infinite time, energetic cost, or control bandwidth. This unattainability principle holds universally across both classical and quantum regimes, establishing a strict geometric limit on the physical realization of reset operations in thermodynamic control and quantum computation.
The third law of thermodynamics forbids cooling a physical system to absolute zero in a finite number of operational steps. Although this unattainability principle has been quantified for specific state-to-state transitions, a universal, dynamics-independent bound for implementing a state-agnostic reset map remains elusive. In this work, we unveil the fundamental limits of physical map implementation by deriving a trade-off relation based on geometric complexity. By analyzing continuous paths of maps on a geometric manifold, we prove that the geometric complexity of any classical stochastic map or quantum channel is bounded from below by its execution error. As a consequence, we show that achieving zero error in a state-reset operation requires a divergent geometric complexity -- a unified measure that naturally incorporates disparate physical resources, including infinite time, energetic cost, or control bandwidth. This unattainability principle holds universally across both classical and quantum regimes, establishing a strict geometric limit on the physical realization of reset operations in thermodynamic control and quantum computation.
Elucidating fundamental limitations inherent in physical systems is a central subject in physics. For important thermodynamic operations such as information erasure, cooling, and copying, resources like time and energetic cost must be expended to achieve the desired outcome within a predetermined error margin. In the context of cooling, the unattainability principle of the third law of thermodynamics asserts that infinite “resources” are needed to reach absolute zero. However, the precise identification of relevant resources and how they jointly constrain achievable error remains unclear within the frameworks of stochastic and quantum thermodynamics. In this work, we introduce the concept of separated states, which consist of fully unoccupied and occupied states, and formulate the corresponding thermokinetic cost and error, thereby establishing a unifying framework for a broad class of thermodynamic operations. We then uncover a three-way trade-off relation between time, cost, and error for thermodynamic operations aimed at creating separated states, simply expressed as τ C ϵ τ ≥ 1 − η . This fundamental relation is applicable to diverse thermodynamic operations, including information erasure, cooling, and copying. It provides a profound quantification of the unattainability principle in the third law of thermodynamics in a general form. Building upon this relation, we explore the quantitative limitations governing cooling operations, the preparation of separated states, and a no-go theorem for exact classical copying. Furthermore, we extend these findings to the quantum regime, encompassing both Markovian and non-Markovian dynamics. Specifically, within Lindblad dynamics, we derive a similar three-way trade-off relation that quantifies the cost of achieving a pure state with a given error. The generalization to general quantum dynamics involving a system coupled to a finite bath implies that the dissipative cost becomes infinite as the quantum system is exactly cooled down to the ground state or perfectly reset to a pure state, thereby resolving an open question regarding the thermodynamic cost of information erasure.
Quantum complexity measures the difficulty of realizing a quantum process, such as preparing a state or implementing a unitary. We present an approach to quantifying the thermodynamic resources required to implement a process if the process’s complexity is restricted. We focus on the prototypical task of information erasure, or Landauer erasure, wherein an n -qubit memory is reset to the all-zero state. We show that the minimum thermodynamic work required to reset an arbitrary state in our model, via a complexity-constrained process, is quantified by the state’s . The complexity entropy therefore quantifies a trade-off between the work cost and complexity cost of resetting a state. If the qubits have a nontrivial (but product) Hamiltonian, the optimal work cost is determined by the . The complexity entropy quantifies the amount of randomness a system appears to have to a computationally limited observer. Similarly, the complexity relative entropy quantifies such an observer’s ability to distinguish two states. We prove elementary properties of the complexity (relative) entropy. In a random circuit—a simple model for quantum chaotic dynamics—the complexity entropy transitions from zero to its maximal value around the time corresponding to the observer’s computational-power limit. Also, we identify information-theoretic applications of the complexity entropy. The complexity entropy quantifies the resources required for data compression if the compression algorithm must use a restricted number of gates. We further introduce a , which arises naturally in a complexity-constrained variant of information-theoretic decoupling. Assuming that this entropy obeys a conjectured chain rule, we show that the entropy bounds the number of qubits that one can decouple from a reference system, as judged by a computationally bounded referee. Overall, our framework extends the resource-theoretic approach to thermodynamics to integrate a notion of , as quantified by .
Elucidating fundamental limitations inherent in physical systems is a central subject in physics. For important thermodynamic operations such as information erasure, cooling, and copying, resources like time and energetic cost must be expended to achieve the desired outcome within a predetermined error margin. In the context of cooling, the unattainability principle of the third law of thermodynamics asserts that infinite “resources” are needed to reach absolute zero. However, the precise identification of relevant resources and how they jointly constrain achievable error remains unclear within the frameworks of stochastic and quantum thermodynamics. In this work, we introduce the concept of separated states, which consist of fully unoccupied and occupied states, and formulate the corresponding thermokinetic cost and error, thereby establishing a unifying framework for a broad class of thermodynamic operations. We then uncover a three-way trade-off relation between time, cost, and error for thermodynamic operations aimed at creating separated states, simply expressed as 𝜏𝒞𝜖𝜏≥1−𝜂. This fundamental relation is applicable to diverse thermodynamic operations, including information erasure, cooling, and copying. It provides a profound quantification of the unattainability principle in the third law of thermodynamics in a general form. Building upon this relation, we explore the quantitative limitations governing cooling operations, the preparation of separated states, and a no-go theorem for exact classical copying. Furthermore, we extend these findings to the quantum regime, encompassing both Markovian and non-Markovian dynamics. Specifically, within Lindblad dynamics, we derive a similar three-way trade-off relation that quantifies the cost of achieving a pure state with a given error. The generalization to general quantum dynamics involving a system coupled to a finite bath implies that the dissipative cost becomes infinite as the quantum system is exactly cooled down to the ground state or perfectly reset to a pure state, thereby resolving an open question regarding the thermodynamic cost of information erasure.
Nielsen's geometric approach to quantum circuit complexity provides a Riemannian framework for quantifying the cost of implementing unitary closed-system dynamics. For open evolutions, the reduced dynamics is described by time-dependent families of quantum channels {Λt}t≥0 admitting many inequivalent Stinespring realizations, so any meaningful complexity notion must specify which microscopic resources are counted and which transformations are treated as gauge. We develop a dilation-based geometric complexity framework that distinguishes implementation-dependent complexity, attached to a specific realization, from intrinsic channel complexity obtained by minimizing over a physically admissible class of dilations. The proposed functional is subtractive: it removes purely environmental contributions from the geometric cost of the joint unitary realization. The subtraction is derived from concise structural postulates --- closed-system consistency, environment-only neutrality, and invariance under dilation gauge transformations --- and admits a natural variational and geometric interpretation. We also define noise complexity as a companion quantity measuring the geometric deviation of a noisy implementation from a prescribed ideal closed-system evolution. In the Hilbert-Schmidt setting, we establish coherence-based lower bounds for unitary and noise complexity, prove linear time scaling under time-homogeneous dilation models, and obtain dissipator-controlled bounds in the Markovian GKSL regime. We illustrate the framework on canonical single-qubit benchmark channels: dephasing, amplitude damping, and Pauli/depolarizing noise.
We establish a unified framework for understanding the fundamental physical limits of computation by developing a geometric theory of resource trade-offs. Our approach treats computation as a physical process constrained by thermodynamics, quantum mechanics, and relativity, leading to a multidimensional capacity region in resource space. We prove that any algorithm achieving error ε on problem class P must satisfy joint constraints across five dimensions: space S, time T, bandwidth H, energy E, and coherence C. The capacity region is characterized by fundamental physical constants: Landauer’s principle (kBT ln 2 per bit erasure), the Margolus-Levitin quantum speed limit (2E/(π~) operations per second), the Bekenstein bound (maximum information density), and statistical sampling limits. We provide explicit constructions showing these bounds are achievable within constant factors for canonical problems including matrix multiplication, covariance estimation, and Gaussian process inference. This framework unifies classical complexity theory with quantum information and thermodynamics, providing the first general theory of multi-resource computational limits with applications to algorithm design, hardware optimization, and fundamental physics.
Quantum reservoir computing provides a framework for processing complex temporal data, yet its fundamental computational and energetic limits remain unresolved. Here, we establish a non-equilibrium thermodynamic framework that links the macroscopic predictive performance of driven open quantum systems to their microscopic energetic costs. By mapping Holevo capacities onto the Bogoliubov-Kubo-Mori geometric manifold, we analytically prove that the computational peak within the quantum critical region originates from a spectral resonance: the closing of the intrinsic energy gap forces the reservoir's internal transition frequencies to align with the chaotic drive. To evaluate the associated thermodynamic costs, we introduce quantum informational dissipation to quantify the non-predictive historical data retained by the reservoir. This allows us to derive a generalized Landauer bound for continuous temporal processing, which reveals a fundamental thermodynamic trade-off: the critical resonance that maximizes predictive capacity simultaneously maximizes informational dissipation and the irreversible work required for environmental erasure. Furthermore, coherence decomposition demonstrates that quantum coherences amplify predictive capacity without demanding additional mechanical work. These findings establish the fundamental energetic limits of quantum learning devices, providing theoretical principles for designing energy-efficient quantum neuromorphic hardware.