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We introduce double-bracket flow equations and discuss how they can be approximately implemented as quantum circuits. A direct implementation naturally takes the form of a recursion, leading to prohibitive circuit-depth requirements. We show how this obstacle can be overcome using techniques analogous to dynamic programming, suitably generalized to the quantum setting. We then demonstrate that the double-bracket flow admits a geometric interpretation as a Riemannian gradient descent that monotonically minimizes the system energy. 

As a central application, we consider quantum imaginary-time evolution (QITE), a thermodynamically motivated approach to ground-state preparation, and show that QITE arises as a solution of the double-bracket flow. This perspective enables the coherent synthesis of QITE circuits using double-bracket techniques. Finally, we briefly outline how the double-bracket framework extends beyond QITE, pointing to broader applications in quantum signal processing.