Qubit Regularization of Asymptotic Freedom

We provide strong evidence that the asymptotically free ($1+1$)-dimensional nonlinear $O(3)$ sigma model can be regularized using a quantum lattice Hamiltonian, referred to as the “Heisenberg comb,” that acts on a Hilbert space with only two qubits per spatial lattice site. The Heisenberg comb consists of a spin-half antiferromagnetic Heisenberg chain coupled antiferromagnetically to a second local spin-half particle at every lattice site. Using a world-line Monte Carlo method, we show that the model reproduces the universal step-scaling function of the traditional model up to correlation lengths of 200000 in lattice units and argue how the continuum limit could emerge. We provide a quantum circuit description of the time evolution of the model and argue that near-term quantum computers may suffice to demonstrate asymptotic freedom.

Figure 1

A pictorial representation of the Heisenberg comb whose Hamiltonian is given in Eq. (2). The asymptotically free QFT described by Eq. (1) is reproduced in the limit $J/J_p\to \infty$.

Figure 2

The value of $\beta$ chosen for each value of $L$ during calculations of $\xi (L)$ in the Heisenberg comb at various values of $J$. The inset shows a similar plot for the symmetric ladder (circles) and asymmetric ladder (squares). While $\beta (L)$ depends on $J$ in the Heisenberg comb, it does not depend on $J_p$ for the parameters we have explored in the symmetric and asymmetric ladders.

Figure 3

Universal step-scaling function $F(z)=\xi (2L)/\xi (L)$ as a function of $z=\xi (L)/L$. The dark line in both plots is this function reproduced from [40], where it was calculated using the traditional model defined by Eq. (3). The dashed line is the function $F(z)=2(1-0.0276/z^2-0.00258/z^4)$, computed in Ref. [40] using perturbation theory near the asymptotically free fixed point. The left plot shows results for (1) a symmetric ladder ($J_1=J_2=1$; $J_p=0.10$, 0.20) and (2) an asymmetric ladder ($J_1=1$, $J_2=0.5$; $J_p=0.05$, 0.10). The right plot shows results for the Heisenberg comb at $J_1=J=3$, 5, 10; $J_2=0$;$J_p=1$. Each data point is constructed using two calculations, one at lattice size $L$ and another at $2L$, with $12\le L\le 256$.

Figure 4

Quantum circuit implementing $e^{-iJ[H_{(x,a),(x,b)}+1/4]t}$ on two qubits $(x,a)$ and $(y,b)$ with exactly three entangling gates and one single-qubit gate. The first and second lines represent the qubits $(x,a)$ and $(y,b)$. The phase rotation $P$ and the $X$-rotation $R_X(\varphi )$ are defined in Eq. (11). The angle $\varphi =J_{p}t/2$ for terms in $H_1$ and $\varphi =Jt/2$ for terms in $H_2$ and $H_3$.