Double-Bracket Algorithmic Cooling

TL;DR

This work introduces Double-Bracket Algorithmic Cooling (DBAC), a dynamic quantum algorithm that coherently suppresses quantum coherence in pure states by combining density-matrix exponentiation with quantum imaginary-time evolution. Implemented on a superconducting qubit lattice, DBAC uses instruction copies of the input state to program a state-agnostic $U_{\rm DME}$ operation, enabling measurement-free cooling toward the ground state $\ket{0}$; perfect cooling requires infinitely many instruction copies, echoing the Nernst unattainability principle. The experimental results validate the DME implementation with high process fidelities across multiple circuit configurations and demonstrate progressive cooling as the number of instruction copies increases, though decoherence imposes practical limits at larger depths. Overall, DBAC showcases dynamic quantum programming as a viable route for foundational tasks in quantum thermodynamics and points to scalable extensions for exploring coherence cooling in larger quantum systems.

Abstract

Algorithmic cooling shows that it is possible to locally reduce the entropy of a qubit belonging to an isolated ensemble such as nuclear spins in molecules or nitrogen-vacancy centers in diamonds. In the same physical setting, we introduce double-bracket algorithmic cooling (DBAC), a protocol that systematically suppresses quantum coherence of pure states. DBAC achieves this by simulating quantum imaginary-time evolution through recursive unitary synthesis of Riemannian steepest-descent flows and it utilizes density-matrix exponentiation as a subroutine. This subroutine makes DBAC a concrete instance of a dynamic quantum algorithm that operates using quantum information stored in copies of the input states. Thus, the circuits of DBAC are independent of the input state, enabling the extension of algorithmic cooling from targeting entropy to quantum coherence without resorting to measurements. Akin to Nernst principle, DBAC increases the cooling performance when including more input qubits which serve as quantum instructions. Our work demonstrates that dynamic quantum algorithms are a promising route toward new protocols for foundational tasks in quantum thermodynamics.

Figures (26)

• Figure 1: Schematic of double-bracket algorithmic cooling (DBAC). The protocol applies the density-matrix exponentiation unitary $U_{\rm DME}$ between a target data qubit and input instruction copies of $|\psi\rangle$. Each application is bracketed by $e^{\pm it\hat{H}}$ echoes, producing a double-bracket step that reduces the energy of the target qubit. Because the applied unitary operations are state-agnostic, the cooling dynamics are programmed “on the fly” by the instruction copies themselves, without requiring mid-circuit measurements. DBAC thus generalizes algorithmic cooling from entropy reduction to coherent state manipulation, and -consistent with the Nernst unattainability principle- perfect cooling would require infinitely many instruction qubits. The flow is as follows: input copies of $|\psi\rangle$ are promoted to instruction copies, which together with the target qubit undergo the $U_{\rm DME}$ operation bracketed by echoes; the updated target qubit is thereby cooled, and recursion repeats with further instruction copies ($M_1,M_2$,…) as resources scale.
• Figure 2: a) Pulse sequence used to calibrate two-qubit $ZZ$ interaction via an echoed Stark-drive protocol. An off-resonant microwave drive (AC Stark tone) is applied to one qubit during free-evolution intervals of total duration $\tau$ (split into $\tau/2$ segments), while interleaved $\pi$ echo pulses on both qubits are used to cancel isolated $IZ$ and $ZI$ rates. This sequence implements the effective unitary $R_{ZZ}(\phi)=e^{-i\phi Z\otimes Z/2}$, where $\phi$ depends on both drives amplitude, frequency, and phase difference. (b) Two-dimensional sweep of Stark-drive frequency and amplitude, showing the response of the target qubit, which is used to tune up the effective $ZZ$ coupling by mapping native drive parameters to the accumulated phase $\phi$. (c) Corresponding measurement on the control qubits, capturing the differential phase accumulation due to the $ZZ$ coupling. These scans are used to select the operating points (Stark-drive drives frequency and amplitude) that implement the effective $R_{ZZ}(\phi)$ operation in the subsequent DBAC experiments.
• Figure 3: Pauli transfer matrix (PTM) characterization of DME compilation obtained via standard quantum process tomography nielsen00Emerson2007. Panels show experimental PTMs (left) alongside analytic PTMs (right) for different evolution angles: a)$\phi=0$, where the $ZZ$ interaction is not involved and deviations from theory arise from readout errors, b)$\phi=\pi/8$, c)$\phi=\pi/4$, and d)$\phi=\pi/2$. The corresponding average process fidelities are a)$92.27\%$, b)$91.78\%$, c)$91.33\%$, and d)$90.20\%$. For $\phi=\pi/2$, the DME realizes a SWAP operation (see supplementary materials for more details).
• Figure 4: DBAC as a function of initialization angle $\theta$ (i.e., apply DBAC to $\ket{\psi}=R_X(\theta)\ket 0$) for a) the minimal case with $n=2$ qubits, b) with $n=3$ qubits, and c) with $n=4$ qubits. Experimental data show that distributing DME can improve cooling, with a flattening of the energy curve. Analytic simulations predict further cooling with more DBAC steps, but decoherence places a break-even limit on our device.
• Figure 5: Circuit performing $M$ iterations of density matrix exponentiation (DME). The data qubit is denoted $q_0 (= \sigma)$, while $q_1, \dots, q_M$ represent $M$ copies of the instruction qubit $\rho$. Each iteration of DME, illustrated within the dashed box, consists of a $\delta\text{-swap}$ operation followed by a trace-out procedure. In this diagram, the trace-out is represented by a mid-circuit measurement, though the qubit can be disregarded. The circuit approximates the unitary operation $e^{-it\rho}$ acting on the data qubit $q_0$.

Theorems & Definitions (2)

• Lemma 1: Energy Change under One Step of DBAC for single-qubit
• proof