Hamiltonian and double-bracket flow formulations of quantum measurements

TL;DR

The paper develops a unified framework that treats quantum measurement, Hamiltonian dynamics, and double-bracket gradient flows within a stochastic-Hamiltonian formalism.It derives explicit SB and DB generator processes in a Stratonovich setting, linking monitored quantum dynamics to gradient flows on the unitary orbit and clarifying how measurement back-action manifests as both drift and diffusion in a geometric picture.The authors then leverage this viewpoint to design feedback schemes that realize deterministic DB flows for ground-state preparation and propose state-agnostic feedback strategies for preparing arbitrary pure states, with stability analyses and exponential convergence in key examples.Overall, the work offers both conceptual unification and practical protocol recipes for measurement-based quantum control, potentially informing experimental implementations of gradient-flow-inspired state engineering.

Abstract

We introduce a framework that unifies quantum measurement dynamics, Hamiltonian dynamics, and double-bracket gradient flows. We do so by providing explicit expressions for stochastic Hamiltonians that produce state dynamics identical to those that happen during continuous quantum measurements. When such dynamical processes are integrated over sufficiently long time intervals, they yield the same results and statistics as during wavefunction collapse. That is, wavefunction collapse can be interpreted as coarse-grained (stochastic) Hamiltonian dynamics. Alternatively, wavefunction collapse can be interpreted as double-bracket gradient flows determined by derivatives of (stochastic) potentials defined in terms of observables with direct physical interpretations. The gradient flows minimize the variance of the monitored observable. Our derivations hold for general monitoring described by non-Hermitian jump processes. We show that such reinterpretations of measurement dynamics facilitate the design of feedback processes. In particular, we introduce feedback processes that yield deterministic double-bracket flow equations, which prepare ground states of a target Hamiltonian, and feedback processes for state preparation. We conclude by re-interpreting feedback processes as gradient flows with tilted fixed points.

Figures (6)

• Figure 1: Illustration of the Hamiltonian evolution given by \ref{['eq:sme_strato']}. The evolution takes place on the unitary orbit ${\cal O}$ of the density operator. In the tangent space ${\cal T} _\rho {\cal O}$ at point $\rho$, the increment $d\rho$ is composed by two (not necessarily orthogonal) contributions, $d\rho^\text{SB} = -i[dH_t^\text{SB}, \rho_t]$ and $d\rho^\text{DB} = [[dH_t^\text{DB}, \rho_t], \rho_t]$, corresponding to the single-bracket and double-bracket Hamiltonians, respectively.
• Figure 2: Schematic representation of the gradient flow formulation in Theorem \ref{['th:grad_flow']}. We illustrate the embedding of the state's orbit ${\cal O} _\rho$ (circumference), in the in the unitary group $U(N)$ (sphere). The tangent vectors to the orbit correspond to vectors in the Lie algebra that are orthogonal to the stabilizer algebra of $\rho$, which is generated by vectors $X$ that commute with the $\rho$. In particular, the gradient flow generates displacements $\delta \rho_t = \nabla F_t \delta t$ tangent to the orbit.
• Figure 3: Illustration of the pure-measurement case $C = A$ with $H_0 = 0$. The increment $d\rho$ is composed only of the double-bracket term $d\rho^\text{DB}$, with drift dictated by the Riemannian gradient of the the variance $V_t(\rho_t) = \text{Tr}(\Delta A_t^2 \rho_t)$. The variance vanishes when $\rho_t$ corresponds to an eigenstate $\ket{a}$ of $A$.
• Figure 4: Energy errors for the 1-D Heisenberg model. We monitor the error between the energy expectation $\text{Tr}\left( H_0 \rho_t \right)$ and the true ground state energy $E_0$ obtained from exact diagonalization. We do this for a total time $T=2$ and $300$ time steps. The damping rate is $\gamma = 4.0e-1$. To ensure a positive spectrum, we shift the system Hamiltonian by adding a constant offset of $18$ for all qubit numbers. The $y$-axis is in logarithmic scale, showcasing the exponential convergence of the DB dynamics with errors reaching machine precision.
• Figure 5: Measurement-based feedback state preparation and stabilization for the single-qubit. We display (top panel) the time evolution of the Bloch coordinates $m_i(t) = \text{Tr}\left( \sigma_i\rho_t \right)\,(i=x,\, y,\, z)$ of $50$ trajectories along their ensemble averages $\overline{m}_x,\,\overline{m}_y,\, \overline{m}_z$. The parameters are $\gamma_- = 1$ and $\gamma_+ = 4$, with final time $T=8$, for a total of $800$ time steps. All trajectories stabilize relatively fast around the target, with a final average infidelity (bottom panel) $1- \overline{\text{Tr}(Q \rho_t)} \sim 10^{-6}$, and median $\sim 10^{-10}$.

Theorems & Definitions (5)

• Theorem 1
• Proposition 1
• Theorem 2
• Theorem 3
• Corollary 1