We solve the closed-system problem of cooling $M$ qubits, selected from $N$ identical thermal qubits, to the lowest common local temperature allowed by unitarity. The optimal protocol consists of two conceptually distinct steps. First, a passive rearrangement assigns the largest eigenvalues of the initial state to target sectors of lowest Hamming weight, thereby minimizing the total target energy. Second, a target-only complex-Hadamard transformation within each fixed-Hamming-weight subspace equalizes the one-qubit target marginals without changing any target-sector probability or the total energy. Consequently, imposing a common local temperature costs neither cooling depth nor additional work: the constrained optimum coincides with the unconstrained passive minimum for every $N>M$ and every initial temperature. The complex-Hadamard correction may nevertheless be costly at the circuit level. We therefore derive an exact arithmetic criterion for when the same optimum can be attained by a temperature-independent computational-basis permutation alone, and exhaustively classify the resulting finite-size islands of feasibility for $M+2\leq N\leq 128$. At isolated temperatures, further optimal permutations can arise through numerical cancellations between different thermal eigenvalue shells. These alternative realizations may reduce implementation complexity, but they cannot improve the cooling curve already attained by the universal protocol. We also derive the exact cooling curve, prove that at least two ancillary qubits are necessary and sufficient for nontrivial cooling, and show that joint many-target cooling can strictly outperform parallel single-target strategies.
We solve the closed-system problem of cooling M qubits, selected from N identical thermal qubits, to the lowest common local temperature allowed by unitarity. The optimal protocol consists of two conceptually distinct steps. First, a passive rearrangement assigns the largest eigenvalues of the initial state to target sectors of lowest Hamming weight, thereby minimizing the total target energy. Second, a target-only complex-Hadamard transformation within each fixed-Hamming-weight subspace equalizes the one-qubit target marginals without changing any target-sector probability or the total energy. Consequently, imposing a common local temperature costs neither cooling depth nor additional work: the constrained optimum coincides with the unconstrained passive minimum for every N>M and every initial temperature. The complex-Hadamard correction may nevertheless be costly at the circuit level. We therefore derive an exact arithmetic criterion for when the same optimum can be attained by a temperature-independent computational-basis permutation alone, and exhaustively classify the resulting finite-size islands of feasibility for M+2≤N≤128. At isolated temperatures, further optimal permutations can arise through numerical cancellations between different thermal eigenvalue shells. These alternative realizations may reduce implementation complexity, but they cannot improve the cooling curve already attained by the universal protocol. We also derive the exact cooling curve, prove that at least two ancillary qubits are necessary and sufficient for nontrivial cooling, and show that joint many-target cooling can strictly outperform parallel single-target strategies.
We study the problem of dynamic cooling whereby a target qubit is cooled at the expense of heating up N − 1 further identical qubits by means of a global unitary operation. A standard back-of-the-envelope high-temperature estimate establishes that the target qubit temperature can be dynamically cooled by at most a factor of 1 / N . Here we provide the exact expression for the minimum temperature to which the target qubit can be cooled and reveal that there is a crossover from the high initial temperature regime, where the scaling is 1 / N , to a low initial temperature regime, where a much faster scaling of 1 / N occurs. This slow, 1 / N scaling, which was relevant for early high-temperature NMR quantum computers, is the reason dynamic cooling was dismissed as ineffectual around 20 years ago; the fact that current low-temperature quantum computers fall in the fast, 1 / N scaling regime, reinstates the appeal of dynamic cooling today. We further show that the associated work cost of cooling is exponentially more advantageous in the low-temperature regime. We discuss the implementation of dynamic cooling in terms of quantum circuits and examine the effects of hardware noise. We successfully demonstrate dynamic cooling in a three-qubit system on a real quantum processor. Since the circuit size grows quickly with N , scaling dynamic cooling to larger systems on noisy devices poses a challenge. We therefore propose a suboptimal cooling algorithm, whereby relinquishing a small amount of cooling capability results in a drastically reduced circuit complexity, greatly facilitating the implementation of dynamic cooling on near-future quantum computers. Published by the American Physical Society 2024
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.
Heat-Bath Algorithmic Cooling is a set of techniques for producing highly pure quantum systems by utilizing a surrounding heat-bath and unitary interactions. These techniques originally used the thermal environment only to fully thermalize ancillas at the environment temperature. Here we extend HBAC protocols by optimizing over the thermalization strategy. We find, for any d -dimensional system in an arbitrary initial state, provably optimal cooling protocols with surprisingly simple structure and exponential convergence to the ground state. Compared to the standard ones, these schemes can use fewer or no ancillas and exploit memory effects to enhance cooling. We verify that the optimal protocols are robusts to various deviations from the ideal scenario. For a single target qubit, the optimal protocol can be well approximated with a Jaynes-Cummings interaction between the system and a single thermal bosonic mode for a wide range of environmental temperatures. This admits an experimental implementation close to the setup of a micromaser, with a performance competitive with leading proposals in the literature. The proposed protocol provides an experimental setup that illustrates how non-Markovianity can be harnessed to improve cooling. On the technical side we 1. introduce a new class of states called maximally active states and discuss their thermodynamic significance in terms of optimal unitary control, 2. introduce a new set of thermodynamic processes, called β-permutations , whose access is sufficient to simulate a generic thermalization process, 3. show how to use abstract toolbox developed within the resource theory approach to thermodynamics to perform challenging optimizations, while combining it with open quantum system dynamics tools to approximate optimal solutions within physically realistic setups.
The Unruh response thermalizes accelerated detectors, but the entanglement remaining in a register does not by itself specify what distributed measurements can recover. We study this distinction for a four-qubit cluster state whose two assisting nodes independently undergo generalized amplitude damping, while the target nodes remain isolated and the environments are inaccessible. We derive the exact joint-assisted concurrence and the exact one-way value for arbitrary local measurements. The joint optimum equals the bipartition negativity, and computational-basis measurements attain the one-way optimum. A polar-decomposition argument, together with compactness, establishes a necessary and sufficient condition for separable measurements to attain the joint optimum and proves a strict gap whenever this condition fails. For assisting detectors at equal Unruh temperature with relaxation-rate ratio r>1, joint advantage exists precisely when the thermal excitation factor satisfies q<1/r. At finite positive temperature it ends at a unique finite time, while the remaining entanglement is nonzero. For r=2, both the transition time and remaining resource have closed expressions. These results identify when joint assistance is necessary within the independent weak-coupling thermalization model.
The paper has two components. Parts I–IV develop a conditional operational theory of reservoir-assisted cooling and exact reset at the level of channels; Part V is a self-contained Hamiltonian-level case study of exact reset embeddings in a D1–D5 CFT reservoir. The two notions of “exact reset” are different and are kept distinct throughout. We separate three logically distinct questions in reservoir-assisted cooling: what the reservoir entropy implies thermodynamically, which low-temperature ancillas are accessible, and what extra dynamical resource exact ground-state preparation needs. Convex microcanonical entropy is neither sufficient nor necessary for a microscopic cooling operation; it is a thermodynamic route to an ancilla supply, under a stated local-Gibbsian hypothesis. For a resonant qubit target we prove that every energy-conserving collision with a qubit ancilla acts as p' = p + s(q − p), 0 ≤ s ≤ 1, so cooling to error ε is possible exactly when ancillas with excited population at most ε can be supplied. We then remove the two-level restriction: for an arbitrary energy-incoherent sink, within the stated collision model, the least reachable excited population is a sector-wise majorization (sorting) problem with an explicit closed-form solution. Its consequences are: (a) a cooling criterion: a sink can cool a target of population p if and only if some pair of its levels one target gap apart has population ratio below p/(1−p); (b) a Gibbs sink at any finite temperature leaves the exact floor min{p, (1−p)e^{−β E_sys}} > 0, and this floor is stable under arbitrary sequences of fresh sinks at the same temperature (the sequence extension is proved by a concavity lemma); (c) the floor is a single-temperature statement: a hot and a cold thermal sink used together cool every p ∈ (0,1), and iterating fresh pairs drives p → 0 with no work store, measurement or record, while sequential single-sink collisions already reach n̄_c / (1 + n̄_c + n̄_h); (d) sharpness of the energy support, or one classical record of entropy h_2(p), is what buys exactness in finitely many steps. Exact reset has two independent resource obstructions in the stated model. The optimal residual excitation of any energy-conserving reset is the sink weight on directions that lack spectral room (the room floor), with a criterion for measurement-free exact reset; against limited control, any circuit of m gates with rank(G − I) ≤ ρ leaves the target excited with probability at least the sink-spectrum tail beyond ρm, so, within this gate model, the number of gates needed for exactness is at least the corresponding tail rank divided by ρ, with the operational integer requirement expressed by the ceiling where a finite target error is specified. Within the shell-ancilla model, an explicit dimension requirement dim ≥ 1/ε follows, with the corresponding bound attained by the stated construction. The room floor and the rank–error bound are energy-conserving, sector-wise and gate-rank refinements of the Ticozzi–Viola purification bound, and are presented as such. We also give a conditional zero-designated-work SWAP construction with N = O(log log(1/ε)) collisions for a quadratic-entropy family, a dimension-counting reset criterion with an exact ladder-sink cost–error curve, an error budget for measurement–feedback completion, and a finite birth–death noise model with a positive spectral gap. Black holes, Na⁺_147 nanoclusters, structured-vacuum engines and Casimir experiments are treated as delimited case studies, not demonstrations. The contribution is a conditional operational characterisation with falsifiable requirements; it is not a claim that the third law has been circumvented. Finite instances of the principal formulas were checked by exact arithmetic and numerical sampling (random energy-conserving unitaries against the closed forms, and exact evaluation of tabulated values; Section 27); these checks are consistency tests, not substitutes for the proofs. Part V. The case study proves two conditional obstructions to exact spectral reset embeddings in a D1–D5 CFT reservoir. The first is a sector-specific capacity bound: under explicit component-string, single-strand, graded-orbifold and constant-character hypotheses, the maximal-left-charge Ramond-ground-state sector has dimension 4N, so a larger source cannot embed into it. The second is a finite-dimensional perturbative criterion: for an exactly degenerate source block transported by a regular family, the second-order effective Hamiltonian must act as a scalar on the transported block, equivalently the block must lie in a single second-order eigenspace, which gives an eigenspace-multiplicity bound for a prescribed target sector. At level four, under an explicit Hermiticity and perturbative-matching hypothesis, fifteen long-multiplet primaries with distinct second-order lifts give a fixed-label non-scalar target block. All obstructions are conditional and local, no universal no-go theorem for the full D1–D5 Hilbert space is claimed, and no microscopic reset dynamics are constructed.
The minimal sampling overhead of the quadratic collective phase e−iγK2 over one layer of single-qubit rotations is decided by the continued fraction of γ/π. Here K is a collective observable with spectrum {0,…,n}, such as the permutation-symmetric Hamming weight. This gate is the cardinality-penalty layer of constrained optimization, the one-axis-twisting gate of spin squeezing and the Kerr phase of a bosonic mode. Instead of compiling it to two-qubit gates, we sample it as an LCU over such layers at overhead Γ. Our results give an optimality theory for this overhead. Ancilla-free sampling reproduces the target's outcome probabilities up to the minimal Γ, and no smaller factor works for every input; for permutation-symmetric K, independent per-qubit angles or one layer of arbitrary single-qubit gates give no further reduction. Rational angles πp/q in lowest terms cost exactly q once n≥2q−2, attained uniquely by a uniform q-point sampler; the cost is Θ(n) if and only if γ/π is badly approximable, and bounded in n if and only if it is rational. A second result shows what the symmetry buys: without it, the squaring phase of Fourier arithmetic on m qubits costs at least 20.557m−O(1) over single-qubit gates, beyond the 2m/2 of every cut bound, by a new certificate-lifting argument. For constructions known only through upper bounds, it gives the optimum: the Fourier-based LCU of Carrera Vazquez, Egger and Woerner is, at fixed n, optimal up to a constant at typical angles and beaten by (n+1)/q at rational ones with an n-independent sampler; the Kerr decomposition of Upreti, Quesada and Chabaud is the unique optimum over phase shifters at rational parameters without photon-number cutoff; and it identifies the rational couplings where one-axis twisting admits optimal cat decompositions.