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arXiv:2506.10059 · PRX Quantum

Cooling a Qubit Using n Others

Jake Xuereb, Benjamin Stratton, Alberto Rolandi, Jinming He, Marcus Huber, Pharnam Bakhshinezhad

Published Oct 29, 2025• 2 citations
arXivHTMLPDF

Abstract

In the task of unitarily cooling a quantum system with access to a larger quantum system, known as the machine or reservoir, ? 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.

DOI
10.1103/hrph-dbv7

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In the task of unitarily cooling a quantum system with access to a larger quantum system, known as the machine or reservoir, ? 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.
Cooling quantum systems is arguably one of the most important thermodynamic tasks connected to modern quantum technologies and an interesting question from a foundational perspective. It is thus of no surprise that many different theoretical cooling schemes have been proposed, differing in the assumed control paradigm and complexity, and operating either in a single cycle or in steady state limits. Working out bounds on quantum cooling has since been a highly context dependent task with multiple answers, with no general result that holds independent of assumptions. In this Letter we derive a universal bound for cooling quantum systems in the limit of infinite cycles (or steady state regimes) that is valid for any control paradigm and machine size. The bound only depends on a single parameter of the refrigerator and is theoretically attainable in all control paradigms. For qubit targets we prove that this bound is achievable in a single cycle and by autonomous machines.
The erasure of quantum information is governed by two fundamental thermodynamic bounds: Landauer’s principle, which establishes the absolute minimum energetic cost of resetting a state, and Nernst’s unattainability principle, which dictates that perfect cooling to absolute zero entropy requires infinite resources. Under finite constraints, conventional cooling protocols utilizing thermal baths composed of n non-interacting particles are bottlenecked by a linear dissipation barrier (O(1/n)). In this thesis, we establish the ultimate thermodynamic limits of finite-resource multiqubit erasure and introduce the most thermodynamically efficient cooling protocol currently known. Building upon the max-cooling framework—a global unitary operation that systematically sorts the eigenvalues of the joint system-bath state to pack the highest probability mass into the target’s ground state—we engineer a bath Hamiltonian with specific, target-dependent degeneracies for arbitrary m-qubit systems. This collective protocol achieves an optimal asymptotic entropy production scaling of Σ = O(mˆ2/nˆ2), successfully saturating the scaling of the fundamental theoretical lower bound. This degeneracy-optimized approach strictly outperforms both standard non-interacting reservoirs and the best-known protocol for the independent cooling of single qubits using a reservoir of interacting particles (O(mˆ3/nˆ2)). Furthermore, we establish that surpassing conventional dissipation barriers requires an operational threshold of just n ≥ 2mˆ2 bath particles to achieve this optimal scaling.
Practical implementations of quantum technologies require preparation of states with a high degree of purity---or, in thermodynamic terms, very low temperatures. Given finite resources, the third law of thermodynamics prohibits perfect cooling; nonetheless, attainable upper bounds for the asymptotic ground-state population of a system repeatedly interacting with quantum thermal machines have been derived. These bounds apply within a memoryless (Markovian) setting, in which each refrigeration step proceeds independently of those previous. Here, we expand this framework to study the effects of memory on quantum cooling. By introducing a memory mechanism through a generalized collision model that permits a Markovian embedding, we derive achievable bounds that provide an exponential advantage over the memoryless case. For qubits, our bound coincides with that of heat-bath algorithmic cooling, which our framework generalizes to arbitrary dimensions. We lastly describe the adaptive stepwise optimal protocol that outperforms all standard procedures.
Cooling a quantum system with NNN thermal machines costs more than Landauer's bound. For swap-based collision models the excess is set by thermodynamic length, which is optimal for qubits and was suggested by numerics to be optimal in general. We show that for qudits, permutations of energy eigenstates that use several resonant transitions at once do better for every spectrum, reducing the excess by up to 83% near the ground state. To second order, coherent collisions do no better, and at infinite temperature dMd_MdM​-level machines reduce the excess by at most a factor dM−1d_M-1dM​−1.
We introduce a class of thermal operations based on the collision model, where the system sequentially interacts with uncorrelated bath molecules via energy-preserving unitaries. To ensure finite complexity, each molecule is constrained to be no larger than the system. We identify a necessary condition for cooling below the bath temperature via a single collision: The system must initially lack a well-defined effective temperature, even a negative one. By constructing an iterative protocol, we demonstrate that sub-bath cooling is achievable without a machine under these restricted thermal operations. Moreover, introducing a qubit machine further enhances both the cooling limit and energy efficiency. These findings contribute to the broader study of cooling with finite resources.
Recent technological advances have allowed the construction of many-body quantum systems that operate below the thermodynamic limit. This has given rise to the field of quantum thermodynamics (QTD), which describes heat, work, and entropy while accounting for quantum effects. Within the framework of QTD, numerous theoretical and experimental quantum thermal machines (QTMs) have been proposed, aiming to exploit quantum effects as a resource. Hence, understanding the relationship between quantum effects and thermodynamic properties is essential for creating efficient QTMs. We explore the use of Nth-root controlled-not gates (NRCGs) in QTD, modelling our system as multiple QTMs. These gates allow for a fractional application of two-qubit operations, facilitating access to intermediary states inaccessible by full controlled-not (CNOT) gates. The system presented is a closed, out-of-equilibrium multi-qubit quantum system in which we employ NRCGs in three distinct protocols, comparing them using regime-specific performance measures. First, we consider a quantum battery, where we optimise initial parameters to maximise performance measures related to ergotropy and charging potential. Our results show that, across all protocols considered, initial quantum coherence in a single qubit improves the battery’s performance. Moreover, some NRCG protocols demonstrate better performance than CNOTs. We then characterise our system as a quantum heat engine, where initial quantum coherence increases extractable work which is achieved at high efficiencies. Furthermore, we uncover a strong linear correlation between work production and many-body correlations. Finally, we demonstrate the system can operate as different thermal machines without changes to protocol design or system size, with regime selection dictated by the initialisation parameters of a single qubit. Two uniquely interesting regimes are accessed: a work producing refrigerator and reverse flow heat engine, both of which produce work and provide cooling simultaneously. These regimes could reduce the energetic cost of cooling in quantum computer architecture.