MathDB

Frobenius light-cone conjecture for the shift unitary

Let H(t)H(t) be a 2-local Hamiltonian on a ring with power-law decaying interactions of exponent α\alpha, and let UU be the unitary it generates in time TT, while UshU_{\mathrm{sh}} is the shift unitary. The currently established bound requires

T≤{C′L−C α≥4C′L(α−1)/3−C3<α<4C′L(α−2)(α−1)/(2α−3)−ϵ2+1/2<α≤3C′L1/2−ϵ−C2≤α<2+1/2C′L(α−1)/2−C1<α<2.T \le \left\lbrace \begin{array}{ll} C^\prime L-C &\ \alpha\ge 4 \\ C^\prime L^{(\alpha-1)/3} - C & 3<\alpha<4 \\ C^\prime L^{(\alpha-2)(\alpha-1)/(2\alpha-3)-\epsilon} & 2+1/\sqrt{2}<\alpha\le 3 \\ C^\prime L^{1/2-\epsilon} - C & 2\le\alpha<2+1/\sqrt{2} \\ C^\prime L^{(\alpha-1)/2} - C & 1<\alpha<2 \end{array}\right..

Frobenius light-cone conjecture. The bound above can be replaced by

T≤{C′L−Cα≥2C′Lα−1−C1<α<2.T \le \left\lbrace \begin{array}{ll} C^\prime L-C & \alpha\ge 2 \\ C^\prime L^{\alpha-1} - C & 1<\alpha<2 \end{array}\right..

for arbitrarily small ϵ>0\epsilon>0 and constants C,C′C,C^\prime independent of LL. This conjecture would give a qualitatively tighter lower bound on the time required to implement the shift unitary, and asserts that the Frobenius light cone controls the hardness of implementing the shift for all power-law exponents in the stated range.

Sources & referencesView supporting material

Primary source

Chao Yin, Andrew Lucas and David T. Stephen, “Frobenius light cone and the shift unitary”, arXiv:2402.07990 (2024).

long-range-interactionsmany-body-physicsquantum-information

Progress summary

Refreshed
Open

The conjecture remains open: a 2024 paper proves weaker bounds but gives no proof or counterexample for the proposed improvement.

The conjecture asks whether the Frobenius light cone determines the difficulty of implementing the shift unitary for every power-law exponent α>1\alpha>1. The directly relevant 2024 paper presents this as Conjecture 1.2, not as an established theorem.

Known results

  • The 2024 paper proves only the stated weaker, piecewise time bounds and shows ∥U−Ush∥F≥18\lVert U-U_{\mathrm{sh}}\rVert_{\mathrm F}\geq\frac18 under those restrictions.
  • Chen and Lucas (2021) established an optimal Frobenius light-cone scaling, up to logarithmic factors, with time scale t∼rmin⁡(α−1,1)t\sim r^{\min(\alpha-1,1)} for α>1\alpha>1.
  • Related Lieb-Robinson work compares information-propagation bounds but does not address the conjecture directly.

2024 conjecture statement

The published and arXiv versions explicitly leave Conjecture 1.2 open. The retrieved sources report no claimed proof, counterexample, verification, withdrawal, or retraction.

Current status (as of August 2026): The conjecture is open; the weaker piecewise bounds are proved, but no source reports progress establishing or refuting the proposed all-α\alpha improvement.

Sources

Solutions 0

No solutions have been posted yet.