Cooling a Qubit using n Others

TL;DR

The paper analyzes how to optimally cool a single qubit S by unitarily coupling it to a finite machine M composed of n thermal qubits with potentially non-identical energy gaps. It derives a core inequality that determines which machine-energy levels must swap to cool S, along with a Carnot-like bound on the allowed energy changes, and introduces reducibility criteria to quantify how many machine qubits actually participate. By representing cooling unitaries as combinations of two-level permutations and mapping the problem to minimum weight perfect matching on a bipartite graph, it provides a constructive, cost-aware method to design cooling protocols and evaluate gate complexity. The work connects to algorithmic cooling, quantum error correction, and autonomous thermal machines, and offers a versatile framework for engineering finite quantum refrigerators with explicit performance guarantees and practical circuit implementations.

Abstract

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. Firstly, we show that a set of inequalities related to the energetic structure of the $n$ qubit machine determines the optimal cooling protocol, which parts of the machine contribute to this protocol and gives rise to a Carnot-like bound. Secondly, 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.

Figures (15)

• Figure 1: Summary Illustration -- We examine unitary cooling scenarios where a qubit at temperature $\beta_S$ with gap $\omega$ is coupled to an $n$ qubit machine at temperature $\beta_M$ with energy gap structure $\Gamma = (\gamma_1 ,\gamma_2, \dots, \gamma_n)$. Since the temperature of a two-level system is determined by the ratio of its ground and excited state populations, to cool the qubit system an agent must exchange disordered populations on the global population vector from the ground state subspace $\ket{0_S i_{M}}$ to the excited state subspace $\ket{1_S j_{M}}$. We find that such exchanges only cool if the energy $E(i_M)$ is larger than $1/2(T_M\omega/T_S + E_\text{Max})$ exposing a set of inequalities which determine various properties of the cooling scenario. By finding a representation for cooling unitaries as perfect matchings on bipartite graphs we are able to optimise cooling protocols relative to a cost function by using minimum weight perfect matching techniques.
• Figure 2: In a) we have three qubits at temperature $\beta = 1$ and with gaps $\omega = \log(2), \gamma_1 = \log(3)$ and $\gamma_2 = \log(8)$ with the populations factored by $\frac{4}{9}$ for ease of readability. In orange we see a plane denoting the axis of reflection corresponding to a $\mathtt{SWAP}$ between the system qubit and the machine qubit 2 (bipartite) which results in the optimal distribution. In b) we have three qubits at temperature $\beta = 1$ but with gaps $\omega = \log(2)$ and $\gamma_1 = \log(5)$ and $\gamma_2 = \log(8)$ giving one less disordered pair, with populations factored by $\frac{81}{40}$. Here the edges in orange correspond to a series of two-level swaps or $\mathtt{Toffolis}$ in the order $\ket{1_S00} \leftrightarrow \ket{1_{S}10}$ then $\ket{0_{S}10} \leftrightarrow \ket{0_{S}11}$ and $\ket{0_{S}10} \leftrightarrow \ket{ 1_{S}10}$ (tripartite) that give the optimal distribution.
• Figure 3: In this figure we compare the change in ground-state population of $S$, $\Delta p_0$, induced by the cooling protocol for different machine energy gap structures as the number of qubits in the machine $n$ increases. We also compare the lower bounds derived to approximate the change in population, in Green we see the performance of the bound given in Eq. \ref{['eq:lower_bound_pop_change']} and in Blue we see the bound calculated algorithmically using the partial order presented in Appendix \ref{['app:swappable_set']}. We see that whilst the fixed bound Eq. \ref{['eq:lower_bound_pop_change']} performs well for machines with energetic structures that grow quickly it performs poorly for machines with fixed gaps. This is because Eq. \ref{['eq:lower_bound_pop_change']} captures a fixed set of strings to be exchanged which in the linear and exponential cases seem to capture most exchanges even as $n$ grows but in the degenerate case we see that this fixed set of strings contributes less and less to the growing number of strings which need to be exchanged as $n$ grows. In contrast, the Blue bound is adaptive constructed via the partial order and so performs better in all scenarios.
• Figure 4: Diagram of reducibility inequalities for a 3-qubit machine showing the partial order in the inequalities, where $\ket{0_S i_M} \implies \ket{0_S j_M}$ means that if $\ket{0_S i_M} \in \mathbb{S}$, then so is $\ket{0_S j_M} \in \mathbb{S}$. Although, the reverse implication does not necessarily hold.
• Figure 5: An example of a complete bipartite graph $G(K,K\oplus 1, E)$ for a cooling scenario involving a 3-qubit machine as detailed in Appendix \ref{['app:example_2']} (case 4). Each perfect matching in this bipartite graph corresponds to a choice of $U_\text{cool}$ that will lead to the same final ground state population from the system. The minimum weight perfect matching on this graph (orange) corresponds to the unitary $U^*_\text{cool}$ that minimises the Hamming weight across pairings as seen using the cost matrix Eq. \ref{['eq:cost_mat']}.

Theorems & Definitions (14)

• Lemma A.1
• proof
• Definition D.1
• Lemma H.1
• proof
• Proposition H.1
• proof
• Proposition H.2
• proof
• Definition I.1