Geometric Speed Limit of Accessible Many-Body State Preparation

We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians. We formulate a conjecture that the time integral of energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor. The conjecture implies a geometric lower bound for the quantum speed limit (QSL). We prove the conjecture for arbitrary, sufficiently slow protocols using adiabatic perturbation theory and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory. Our conjecture implies that any optimal unit-fidelity protocol, even those that drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution. When the control space includes all possible couplings, spanning the full Hilbert space, we recover the well-known Mandelstam-Tamm bound. However, using only accessible local controls to anneal in complex models such as glasses or to target individual excited states in quantum chaotic systems, the geometric bound for the quantum speed limit can be exponentially large in the system size due to a diverging geodesic length. We validate our conjecture both analytically by constructing counter-diabatic and fast-forward protocols for a three-level system, and numerically in nonintegrable spin chains and a nonlocal SYK model.

Popular Summary

Advances in quantum computing depend on the ability to reliably manipulate quantum states. One way to characterize the complexity of this ability is the “quantum speed limit,” the minimal time required to generate a quantum state from some initial condition. This time is limited by the energy fluctuations created in the system: The larger the energy fluctuations, the shorter the quantum speed limit, and the faster a state can be prepared. Since energy variance increases as more particles are added to the system, this deceptively suggests that the quantum speed limit goes down as the system size increases. Here, we resolve this conundrum by considering additional constraints that are present in experimentally realizable systems.

We present numerical and analytical evidence of a new lower bound on the energy fluctuations—and hence the quantum speed limit—that is controlled by the geodesic distance, a parameter that describes how “far apart” two quantum states are in the control space. We find that the geodesic distance between ground states typically scales as a square root of the system size, in the same way as the energy variance. In such cases, our conjecture implies that the quantum speed limit is independent of the system size. In more complex situations, the geodesic distance can increase exponentially with the system size, which means perfect fidelity is impossible in practice.

Our results imply that, beyond Heisenberg’s uncertainty, there is a fundamental constraint on the quantum speed limit set by constraints on the control.

Figure 1

Numerical justification of the conjecture (2), at the quantum speed limit of preparing the interacting ground state of $H_{3\mathrm{LS}}$ with the noninteracting fast-forward Hamiltonian $H_{2\mathrm{LS}}(t)$. The parameters are ${\lambda }_i/g=-2=-{\lambda }_{*}/g$.

Figure 2

Numerical justification of the conjecture [cf. Eq. (2)] across the quantum speed limit of preparing the interacting ground state of the Hamiltonian $H_{3\mathrm{LS}}$ following evolution generated by $H_{3\mathrm{LS}}(t)$. The inset shows numerical evidence that the conjecture is saturated in the adiabatic limit. The parameters are ${\lambda }_i/g=-2=-{\lambda }_{*}/g$. The optimal control algorithm used is GRAPE [45, 53].

Figure 3

Numerical justification of the conjecture [cf. Eq. (2)] in the high-fidelity region of the quantum state preparation problem in the many-body Hamiltonian (32) as a function of the protocol duration $T$. The parameters are ${\lambda }_i/g=-2=-{\lambda }_{*}/g$, $J/g=1$, and $L=10$. We use SD to find (nearly) optimal bang-bang protocols $\lambda (t)\in \{\pm 4\}$ of time step $\delta t=0.005J$.

Figure 4

Excited states of the Hamiltonian (33) along the adiabatic trajectory $\lambda (t)=2{\lambda }_{*}\cos \mathbf{(}\pi t/(2T)\mathbf{)}-{\lambda }_i$ in the vicinity of the adiabatically connected state (magenta) for $L=6$ (up) and $L=8$ (down).

Figure 5

Numerical verification of the geometric bound conjecture (2) for the excited states of the Hamiltonian (33).

Figure 6

Low-energy spectrum of a 14-spin disorder-free $p$-spin model for $p=51$ described by Hamiltonian (34). The model has a first-order transition with an exponentially small gap separating the two phases. In the thermodynamic limit, the geodesic length ${\ell }_{\lambda }$ jumps by $\pi /2$ at the critical point.

Figure 7

Numerical verification of the geometric bound conjecture (2) for the ground state of a disorder-free $p$-spin model described by Hamiltonian (34).

Figure 8

Probability to find the ground state of Hamiltonian (35) at a particular value of $\lambda$, in the free fermion state $\lambda =\infty$ (blue line) and the SYK ground state $\lambda =0$ (green line). The data show a single typical realization of the disorder for half-filling at $L=8$.

Figure 9

Numerical verification of the geometric bound conjecture (2) for the ground state of a fermionic SYK model described by Hamiltonian (35). The data show a single disorder realization since every realization comes with its own optimal protocols. For the presented realization, the quantum speed limit is estimated to be around $T=150$. The inset shows numerical evidence that the conjecture is saturated in the adiabatic limit. The data show a single typical realization of the disorder for half filling at $L=8$.