Cooling a Qubit Using n Others

In the task of unitarily cooling a quantum system with access to a larger quantum system, known as the machine or reservoir, how does the structure of the machine impact an agent’s ability to cool and the complexity of their cooling protocol? Focusing on the task of cooling a single qubit given access to 𝑛 separable, thermal qubits with arbitrary energy structure, we answer these questions by giving two new perspectives on this task. First, we show that a set of inequalities related to the energetic structure of the 𝑛-qubit machine determines the protocol which cools the qubit to the coldest reachable state, which parts of the machine contribute to this protocol and give rise to a Carnot-like bound. Second, we show that cooling protocols can be represented as perfect matchings on bipartite graphs enabling the optimization of cost functions, e.g., gate complexity or dissipation. Our results generalize the algorithmic cooling problem, establish new fundamental bounds on quantum cooling and offer a framework for designing novel autonomous thermal machines and cooling algorithms.