Effective quantum-memory Hamiltonian from local two-body interactions

In Phys. Rev. A 88, 062313 (2013) we proposed and studied a model for a self-correcting quantum memory in which the energetic cost for introducing a defect in the memory grows without bounds as a function of system size. This positive behavior is due to attractive long-range interactions mediated by a bosonic field to which the memory is coupled. The crucial ingredients for the implementation of such a memory are the physical realization of the bosonic field as well as local five-body interactions between the stabilizer operators of the memory and the bosonic field. Here, we show that both of these ingredients appear in a low-energy effective theory of a Hamiltonian that involves only two-body interactions between neighboring spins. In particular, we consider the low-energy, long-wavelength excitations of an ordered Heisenberg ferromagnet (magnons) as a realization of the bosonic field. Furthermore, we present perturbative gadgets for generating the required five-spin operators. Our Hamiltonian involving only local two-body interactions is thus expected to exhibit self-correcting properties as long as the noise affecting it is in the regime where the effective low-energy description remains valid.

Figure 1

An excerpt of a toric code. Black dots are code qubits. Stabilizer operators $W_p$ involve operators acting on the four qubits around a white or a dark plaquette. Operators acting on the four qubits around a white plaquette are of the form $W_p={\left({\sigma }^x\right)}^{\otimes 4}$, while operators acting on the four qubits around a dark plaquette are of the form $W_p={\left({\sigma }^z\right)}^{\otimes 4}$.

Figure 2

A 2D toric code [whose contours, lying in an $xy$ plane, are sketched by solid and dashed (red) lines] is embedded in a 3D Heisenberg ferromagnet (blue) which is ordered in the $z$ direction. The stabilizer operators of the toric code $W_p$ (illustrated in Fig. 1) couple to the $x$ component of an adjacent spin ${\mathbf{S}}_p$ of the ferromagnet. The linear size of the planar toric code $\left(L\right)$ is assumed to be much smaller than that of the ferromagnet $\left(\Lambda \right)$, i.e., $L\ll \Lambda$.

Figure 3

Auxiliary qubits $f$, $g$, and $u$ mediate a five-body interaction $W_p\otimes S_p^x$ between qubits $a$, $b$, $c$, and $d$, and the operator $S_p^x$ (spin of the FM). This interaction emerges from local two-body interactions only. The excited state of the auxiliary qubits is penalized by an energy $\Delta$, which is the dominant energy scale in the system. The interactions which are indicated by solid lines produce the actual five-body interaction. Interactions which are indicated by dashed lines allow one to tune the strength of two-, three-, and four-body terms without changing the strength of the five-body term. Choosing $\delta$ and $\tau$ appropriately allows one to counter undesired two- and three-body terms. Finally, the parameter $\beta$ allows one to tune the strength of the four-qubit interaction $W_p$ independently of the five-body interaction $W_p\otimes O_p$.

Figure 4

A graph of $\langle S_i^x\rangle$ (in the middle of the code) against $L$ for the classical Heisenberg FM with $J=1$. The data were obtained numerically by using the metropolis algorithm. The magnetic length is $L_h=L^2$ and the FM size is $\Lambda =2L_h$. The data show agreement with the relation $S_i^x(t\to \infty )\propto LA/J$, obtained from Eq. (22). On the same graph we plot the total $z$ magnetization $\frac{1}{N_s}{\sum }_i\langle S_i^z\rangle$ against $L$, demonstrating that the backaction is only a localized effect. The scaling chosen here is different from that in the main text. This different choice is motivated only by the difficulty of simulating the classical Heisenberg ferromagnet with a large number of spins. This does not alter the analysis since the chosen scaling satisfies the necessary requirements $E_z\gg E_x$ and $L_h\gg L$.