Complexity-constrained quantum thermodynamics (2403.04828v2)

Abstract: 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, via a complexity-constrained process, is quantified by the state's complexity entropy. 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 complexity relative entropy. 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 and determine the complexity entropy's behavior under random circuits. 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 complexity conditional entropy, 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 time, as quantified by complexity.

• The paper demonstrates how integrating computational complexity with thermodynamic principles quantifies the minimal work needed to reset n-qubit memories, generalizing Landauer erasure.
• It introduces novel measures—complexity entropy and complexity relative entropy—that leverage hypothesis-testing relative entropy and semidefinite programming to assess transformation costs.
• The framework offers insights for optimizing quantum computing under hardware limits and explores potential links to quantum gravity and the efficiency of cooling processes.

Complexity-Constrained Quantum Thermodynamics: An Overview

The paper "Complexity-constrained quantum thermodynamics" incorporates quantum complexity considerations into thermodynamic analysis, focusing on quantifying the resources necessary for state transformations when constrained by computational complexity. This intersection of quantum complexity and thermodynamics is not only intriguing but also essential in understanding processes on quantum scales subjected to real-world computational restrictions.

Core Contributions and Results

A primary subject investigated is the minimum thermodynamic work required to reset an n-qubit memory to a predetermined state, (e.g., the all-zero state |0⟩), when the resetting process is constrained by complexity. This scenario generalizes the concept of Landauer erasure to settings where computational capabilities are limited, providing new insights into thermodynamic costs linked to complexity.

Two major theoretical constructs are introduced: complexity entropy and complexity relative entropy. These measures stem from the hypothesis-testing relative entropy and incorporate limitations on an observer’s computational power, essentially quantifying a system's apparent randomness and distinguishability from a reference state under complexity constraints. For an n-qubit state, this complexity entropy provides a lower bound on the work required for erasure, revealing a direct interplay between work and complexity.

The paper generalizes these concepts to arbitrary state transformations. In scenarios where observers have limited computational power (either extremely high or low), it defines frameworks for identifying feasible transformations based on the thermodynamic work cost. For high computational power, the methodology leverages semidefinite programming (SDP) to numerically evaluate the transformations' work costs through hypothesis-testing relative entropy.

Implications and Future Directions

This framework for analyzing complexity-limited quantum thermodynamics opens pathways to explore quantum systems' operational properties both in quantum computing and condensed matter physics. In quantum computing, this approach has practical ramifications for near-term devices where complexity constraints naturally arise due to hardware limitations. It provides a theoretical foundation for optimizing computations thermodynamically under real-world constraints involving finite circuit depths and noise.

The connection drawn between complexity theory and thermodynamics also offers intriguing insights into issues such as efficiency of quantum cooling processes and the entropy associated with quantum states perceived by bounded observers. Future research might further explore regimes of quantum gravity, particularly through the lens of the AdS/CFT correspondence, potentially shedding light on the thermodynamics of black holes and the nature of quantum spacetime itself.

Speculatively, understanding complexity-constrained transformations might also illuminate relationships between classical computational tasks and their quantum counterparts, offering novel frameworks for assessing computational complexity's role in physical theories.

Conclusion

By intertwining computational complexity with foundational thermodynamic principles, this work enriches our comprehension of quantum systems' operational limits and opens new inquiries in theoretical and practical contexts. The quantified trade-offs between thermodynamic work and computational complexity are poised to advance both the theoretical underpinnings and the applied methodologies in quantum information science. With rapid advancements in quantum technologies, complexity-constrained approaches will likely play a pivotal role in refining the operational frameworks for future quantum devices.