Quantum signal processing in Hilbert space fragmented systems

Abstract

Quantum signal processing (QSP), originally developed for composite pulse sequences in nuclear magnetic resonance systems, has recently attracted attention as a unified framework for quantum algorithms. A pioneering study applied QSP to nonequilibrium control in integrable many-body systems, enabling the realization of nonequilibrium dynamics with greater flexibility than Floquet engineering. However, extending QSP to nonintegrable systems faces fundamental obstacles arising from the limited number of conserved quantities and thermalization. In this work, we propose a protocol that leverages QSP in systems exhibiting Hilbert space fragmentation (HSF). Specifically, we consider a pair-hopping model with four-fold periodic potentials that exhibits an HSF structure, thereby providing integrable and nonintegrable sectors within a single system. We analytically show that nonequilibrium dynamics can be flexibly designed through QSP engineered by these potentials in the integrable sectors. In contrast, we numerically identify signatures of thermalization in the nonintegrable sectors. Remarkably, by inserting domain walls, we achieve parallel control of multiple quantum dynamics within a single system. This approach sheds light on the control of nonequilibrium dynamics from the perspective of quantum computation by extending the scope of QSP to nonintegrable systems.

Figures (5)

• Figure 1: Schematic illustration of our proposal. Depending on the initial pseudospin configurations shown at the top, the whole system is divided into integrable sectors (white boxes with light-blue outlines), nonintegrable sectors (white boxes with red outlines), and frozen regions (purple rectangles). In this paper, we analytically show that nonequilibrium dynamics can be independently controlled through QSP in the integrable sectors. In contrast, we also demonstrate signatures of thermalization in the nonintegrable sectors.
• Figure 2: (a) Illustration of QSP. In the gate structure, the orange blocks labeled by $a$ denote the signal operators and the navy blocks labeled by $\vec{\phi}=(\phi_0,\phi_1,\dots,\phi_d)$ denote the signal-processing operators. The QSP sequence (blue curve) emulates the target unitary dynamics (red curve) for a given input state $\ket{\psi_{\rm in}}$ and output state $\ket{\psi_{\rm out}}$. (b) Schematic picture of Hilbert space fragmentation. In addition to the global symmetries of the Hamiltonian, kinetic constraints cause the Hilbert space to decompose into a direct sum of Krylov subspaces. The dimension of some fragments can grow with the increase in system size. The Hamiltonian restricted to each fragment can be integrable (blue block) or nonintegrable (red block).
• Figure 3: Transition probabilities for each BdG sector and their dependence on the phase sequences. The blue curve depicts the transition probability for the trivial sequence $h\vec{t} = (0, 0)$. The orange curve depicts the transition probability for the BB1 pulse sequence $h\vec{t} = (\pi/2, -\chi, 2\chi, 0, -2\chi, \chi)$, where $\chi = \frac{1}{2} \cos^{-1} (-1/4)$.
• Figure 4: Stroboscopic time evolution (a) for an initial state in the integrable sector $\ket{\uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow}$ , and (b) for an initial state in the nonintegrable sector $\ket{\uparrow \downarrow \uparrow \downarrow \uparrow -++-\downarrow \uparrow \downarrow\uparrow \downarrow}$. The horizontal axis denotes the site index $m$ in the pseudospin representation, and the vertical axis denotes the cycle number $l$ of the BB1 sequences. The color scale shows the expectation value of the pseudospin-$z$ operator.
• Figure 5: Comparison of the expectation values of the pseudospin operator $\sigma_{m}^{z}$ at the final time (Final), time-averaged expectation values (Time avg), and ensemble-averaged expectation values over the Krylov subspace (Ensemble avg), for initial states (a) in the integrable sector $\ket{\uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow \uparrow \downarrow}$ and (b) the nonintegrable sector $\ket{\uparrow \downarrow \uparrow \downarrow \uparrow -++-\downarrow \uparrow \downarrow\uparrow \downarrow}$ under the repitition of BB1 sequences. The horizontal axis denotes the site index $m$ in the pseudospin representation, and the vertical axis denotes the expectation value of the pseudospin-$z$ operator.