Dynamic generation of topologically protected self-correcting quantum memory

We propose a scheme to dynamically realize a quantum memory based on the toric code. The code is generated from qubit systems with typical two-body interactions (Ising, $XY$, Heisenberg) using periodic, NMR-like, pulse sequences. It allows one to encode the logical qubits without measurements and to protect them dynamically against the time evolution of the physical qubits. A weakly coupled cavity mode mediates a long-range attractive interaction between the stabilizer operators of the toric code, thereby suppressing the creation of thermal anyons. This significantly increases the lifetime of the memory compared to the code with noninteracting stabilizers. We investigate how the fidelity, with which the toric code is realized, depends on the period length $T$ of the pulse sequence and the magnitude of possible pulse errors. We derive an optimal period $T_{\text{opt}}$ that maximizes the fidelity.

Figure 1

Schematic of the planar code Hamiltonian equation (2) on the square lattice (dotted lines). Plaquettes (P) are associated with the edges of a unit cell and stars (S) with the edges of the cells around a vertex. A circle indicates a Pauli operator acting on a qubit and the wiggly lines connect operators that have to be multiplied to obtain the corresponding stabilizer operator.

Figure 2

(a) Generic pulse (operation) sequence. The unitary operations $R_i$ (gray columns) with ${\prod }_{i=1}^n{R}_i=\mathbb{1}$ interrupt periods of (free) propagation with $H_0$. (b) Structure of a sequence of duration $T$ to generate a plaquette or star operator. A sequence of the same structure is employed to generate (each) one of the quarters of the planar code.

Figure 3

First six of seven elementary steps to generate fourth-order terms in an average Hamiltonian starting from a single-particle Hamiltonian for qubits on a quadratic lattice.

Figure 4

Numerical gate fidelity versus time of a dynamically generated planar code of $N=13$ qubits (inset) coupled to a cavity mode with three dimensional Fock space. For $\Delta ={\omega }_0$, $\delta /\Delta =0.1$, and $T={\Delta }^{-1}/8$, already the lowest-order average Hamiltonian yields $F>99.9\%$ for times $t\le 100{\Delta }^{-1}$ (solid line). Adding the second-order contribution decreases the deviation from perfect fidelity by another two orders of magnitude (dashed and dotted line, plot in inset). Based on the two leading orders of the average Hamiltonian, the exact fidelity can be approximated reasonably well, both numerically (dashed line) and analytically (dotted line).

Figure 5

Average of numerical gate fidelity $\langle F\rangle$ with imperfect pulses and $T={\Delta }^{-1}/8$. (a) shows $\langle F\rangle -1$ versus the width ${\sigma }_{\theta }$ of the Gaussian distributed pulse errors for $t={\Delta }^{-1}$, while (b) shows the fidelity versus time for ${\sigma }_{\theta }=5\times {10}^{-4}$. The solid lines are evaluated using Eq. (13) and agree well with the numerical data (crosses).

Figure 6

(a) and (b) Two steps for preparing $|\overline{0}\rangle$ from initial state $|0,...,0\rangle$ for the $L=3$ code. The modified star operators are obtained by rotating one ${\sigma }_x$ to $-{\sigma }_y$. In each of the steps, the indicated modified stabilizers are generated for a (effective) time $t=\pi /\left(4\Delta \right)$ and the ${\sigma }_y$ are chosen such that none of the stabilizers from step one acted on $y$ qubits from step two. (c) Average codeword fidelity ${\langle F\rangle }_C$ as a function of pulse errors. Numerics (crosses) agree well with the analytical result (solid line).

Figure 7

(a)–(h) Schematic illustration of the first eight of nine steps of $R_1$ to generate a quarter of the planar code Hamiltonian for a lattice of 25 qubits. When restricted to an interior unit cell (one is marked by a dashed rectangle) these steps reduce to those shown in Fig. 3. (i)–(k) Decomposition of $R_2$ into a sequence of linewise rotations. Single and double arrows represent rotations by $\pi /2$ and $\pi$, respectively.

Figure 8

Symmetric pulse sequence for the generation of the planar code. The order in which the quarters $Q_i$ are generated is reversed after every period $T$. With respect to $T$, the 24 toggling frame Hamiltonians $H_i={\mathcal{R}}_i^{†}{H}_0{\mathcal{R}}_i$ are symmetric, while the 32 operations $R_j$ are antisymmetric.