Realization of a Decoherence-Free Subspace Using Multiple Quantum Coherences

This Letter presents a two-dimensional nuclear magnetic resonance (NMR) approach for constructing a two-logical-qubit decoherence-free subspace (DFS) by using four multiple-quantum coherences of a ${\mathrm{CH}}_3$ spin system as logical qubits. The three protons in this spin system are magnetically equivalent and can only be used as a single qubit in one-dimensional NMR. We have experimentally demonstrated that our DFS can protect against more types of decoherences than those of the one composed of four noisy physical qubits all with different chemical shifts. This idea may provide new insights into extending qubit systems in the sense that it effectively utilizes the magnetically equivalent nuclei.

Figure 1

Pulse sequence for MQ-JRES NMR spectroscopy showing ${\rho }_1$ can protect against the error $X^1{X}^2{X}^3{Y}^4$. $(a)$ and $(b)$ are encoding and decoding processes, respectively. $(c)$ is the position to perform errors. Narrow and wide rectangles represent $\pi /2$ and $\pi$ pulses, respectively.

Figure 2

MQ NMR spectra of ${\rho }_1$ before (a) and after (b) performing the error $X^1{X}^2{X}^3{Y}^4$. The projection into $F_2$ dimension represents the ${}^1\mathrm{H}$ NMR spectrum of the ${{}^{13}\mathrm{C}\mathrm{H}}_3$ group protons with the signal from ${{}^{12}\mathrm{C}\mathrm{H}}_3$ protons being suppressed. The projection into the $F_1$ dimension shows the MQ spectrum, which is given at the right side of the 2D plot.

Figure 3

The projections in the $F_1$ dimension for MQ NMR spectra of ${\rho }_1^L\mathrm{\text{−}}{\rho }_4^L$. Rows (a)–(d) corresponding to the $F_1$ projections for the four logical states in Eq. (6), respectively. Columns (1), (2), (3) presenting the states ${\rho }_1^L-{\rho }_4^L$ without, with cnot gates, and cnot operators plus an error operator $X^1{X}^2{X}^3{X}^4$, respectively.