Initializing a permutation-invariant quantum error-correction code

Recently, there has been growing interest in using quantum error correction in practical devices. A central issue in quantum error correction is the initialization of quantum data into a quantum error-correction code. Most studies have concentrated on generating quantum codes based on their encoding quantum circuits. However, this often leads to a large number of steps required in the initialization, and hence this process can be prone to errors. The purpose of this work is to demonstrate that permutation-invariant quantum error-correction codes can be created with high fidelity by exploiting their underlying symmetry. The code is initialized on multiple qubits that mutually interact or are themselves coupled to a quantum harmonic oscillator. We show that the so-called selective resonant interaction is derivable on such physical systems. By utilizing the selective resonant interaction, these highly symmetric codes may be rapidly generated with excellent fidelity. We also discuss the potential of initializing permutation-invariant quantum error-correction codes based on the state-of-art experimental techniques.

Figure 1

Population of $|9,n\rangle$ (where $n=0,\cdots ,9$) with initial states prepared in (a) $|9,0\rangle$, (b) $|9,1\rangle$, (c) $|9,2\rangle$, (d) $|9,3\rangle$, (e) $|9,4\rangle$, (f) $|9,5\rangle$, (g) $|9,6\rangle$, (h) $|9,7\rangle$, and (i) $|9,8\rangle$, respectively, by appropriately selecting parameters. Time $t\mathrm{s}$ are dependent on $g_n^{\mathrm{eff}}$ where $n=0$ for (a), $n=1$ for (b), $n=2$ for (c), $n=3$ for (d), $n=4$ for (e), $n=5$ for (f), $n=6$ for (g), $n=7$ for (h), and $n=8$ for (i).

Figure 2

Population of states $|9,n\rangle |\varnothing \rangle$ (where $n=0,\cdots ,9$) with initial states prepared in (a) $|9,0\rangle |\varnothing \rangle$, (b) $|9,1\rangle |\varnothing \rangle$, (c) $|9,2\rangle |\varnothing \rangle$, (d) $|9,3\rangle |\varnothing \rangle$, (e) $|9,4\rangle |\varnothing \rangle$, (f) $|9,5\rangle |\varnothing \rangle$, (g) $|9,6\rangle |\varnothing \rangle$, (h) $|9,7\rangle |\varnothing \rangle$, and (i) $|9,8\rangle |\varnothing \rangle$, respectively, by properly choosing parameters. Time $t\mathrm{s}$ are dependent on $g_{n,\mathrm{eff}}$ with $n=0$ for (a), $n=1$ for (b), $n=2$ for (c), $n=3$ for (d), $n=4$ for (e), $n=5$ for (f), $n=6$ for (g), $n=7$ for (h), and $n=8$ for (i).

Figure 3

Schematic of circuit QED design to generate the permutation-invariant QECC code. An array of charge qubits array are placed inside a transmission line resonator.