Generalized Zurek's bound on the cost of an individual classical or quantum computation

2023-01-17

Physical Review E

AI Summary

This work investigates the fundamental thermodynamic resource costs—specifically energy dissipation and entropy production—required to transform an input $x$ into an output $y$ in a single computational process, whether classical or quantum, deterministic or stochastic. Method: Building upon a rigorous Hamiltonian framework for open quantum systems, we integrate tools from stochastic thermodynamics, quantum master equations, and algorithmic information theory. Contributions: (i) We derive, for the first time, a protocol-dependent generalized Zurek bound; (ii) we establish a universal lower bound on thermodynamic cost expressed in terms of the conditional Kolmogorov complexity $K(x|y)$; (iii) we uncover an intrinsic trade-off among dissipation, noise resilience, and algorithmic complexity; and (iv) we formulate the “algorithmic fluctuation theorem”, providing a quantitative bridge between the second law of thermodynamics and the physical Church–Turing thesis.

Abstract

We consider the minimal thermodynamic cost of an individual computation, where a single input x is mapped to a single output y. In prior work, Zurek proposed that this cost was given by K(x|y), the conditional Kolmogorov complexity of x given y (up to an additive constant that does not depend on x or y). However, this result was derived from an informal argument, applied only to deterministic computations, and had an arbitrary dependence on the choice of protocol (via the additive constant). Here we use stochastic thermodynamics to derive a generalized version of Zurek's bound from a rigorous Hamiltonian formulation. Our bound applies to all quantum and classical processes, whether noisy or deterministic, and it explicitly captures the dependence on the protocol. We show that K(x|y) is a minimal cost of mapping x to y that must be paid using some combination of heat, noise, and protocol complexity, implying a trade-off between these three resources. Our result is a kind of "algorithmic fluctuation theorem" with implications for the relationship between the second law and the Physical Church-Turing thesis.

Problem

• Minimum Energy Consumption
• Quantum Computing
• Resource Cost

Innovation

• Thermodynamic Randomness Theory
• Zurek Rule Improvement
• Computational Cost Theory