We show that one can quantum simulate the dynamics of a Sachdev-Ye-Kitaev model with
N Majorana modes for time
t to precision $\ensuremath{\epsilon}$ with gate complexity $O({N}^{7/2}t+{N}^{5/2}t\phantom{\rule{0.16em}{0ex}}\mathrm{polylog}(N/\ensuremath{\epsilon}))$. In addition to scaling sublinearly in the number of Hamiltonian terms, this gate complexity represents an exponential improvement in $1/\ensuremath{\epsilon}$ and large polynomial improvement in
N and
t over prior state-of-the-art algorithms which scale as $O({N}^{10}{t}^{2}/\ensuremath{\epsilon})$. Our approach involves a variant of the qubitization technique in which we encode the Hamiltonian
H as an asymmetric projection of a signal oracle
U onto two different signal states prepared by state oracles, $A|0\ensuremath{\rangle}\ensuremath{\mapsto}|A\ensuremath{\rangle}$ and $B|0\ensuremath{\rangle}\ensuremath{\mapsto}|B\ensuremath{\rangle}$, such that $H=\ensuremath{\langle}B|U|A\ensuremath{\rangle}$. Our strategy for applying this method to the Sachdev-Ye-Kitaev model involves realizing
B using only Hadamard gates and realizing
A as a random quantum circuit.
The Sachdev-Ye-Kitaev (SYK) model is zero-dimensional model simulating quantum chaos using interacting Majorana fermions.Previously proposals have been made to realize the SYK model in fermionic systems that can support majorana zero modes. In this work, we simulate the SYK model in a Kitaev spin chain, where zero modes also exist in the ground state from the fermionic representation of spins. By adding other interactions as perturbations, we lift the degenerate majorana zero modes and obatin an effective SYK interactions. Although the majorana fermions are nonlocal in forms of spin operators, we show that the out-of-time-ordered correlator (OTOC) of the majorana fermions can be approximately simulated in forms of spin operators. Our proposal sheds light in simulating the fermonic SKY model and the related physics in purely bosonic systems.