Quantum Error Correction for Phase Errors

Currently envisaged scalable quantum computers require the use of quantum error correction to enable relatively error-free computation on a platform of physical systems that are inherently error-prone. For this reason, some of the most commonly used ``subroutines'' in quantum computers will be associated with maintaining information in encoded forms. This observation motivates experimental realizations of quantum error-correction to determine whether adequate control can be achieved in order to implement these subroutines and to see in a practical setting that error-correction has the desired effects. Experiments to date have included realizations of a version of the three-qubit repetition code [24] and of the five-qubit one-error-correcting code (the shortest possible such code) [25]. In this section, we discuss the experimental implementation of the former.

In NMR, one of the primary sources of error is phase decoherence of the nuclear spins due to both systematic and random fluctuations in the field along the $z$-axis. At the same time, using gradient pulses and diffusion, phase decoherence is readily induced artificially and in a controlled way. The three-bit quantum repetition code (see [26]) can be adapted to protect against phase errors to first order. Define $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{+}\mbox{$\rangle\hspace*{-... ...$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$})$ and $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{-}\mbox{$\rangle\hspace*{-... ...$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$})$. The code we want is defined by the logical states

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfr... ...mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}.$$ (30)

It is readily seen that the three one-qubit phase errors, ${\sigma_z}^{({\mathsf {1}})},{\sigma_z}^{({\mathsf {2}})},{\sigma_z}^{({\mathsf {3}})}$ and ``no error'' ( ${\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}$) unitarily map the code to orthogonal subspaces. It follows that this set of errors is correctable. See the introduction to quantum error-correction [26]. The simplest way to use this code is to encode one qubit's state into it, wait for some errors to happen, and then decode to an output qubit. Success is indicated by the output qubit's state being significantly closer to the input qubit's state after error correction. Without errors between encoding and decoding, the output state should be the same as the input state, provided that the encoding and decoding procedures are implemented perfectly. Therefore, in this case, the experimentally determined difference between input and output gives a measurement of how well the procedures were implemented.

To obtain the phase-correcting repetition code from the standard repetition code, Hadamard transforms or $90^\circ$ $y$-rotations are applied to each qubit. The quantum network shown in Fig. 14 was obtained in this fashion from the network given in [26].

FIG. 14: Quantum network for the three-qubit phase-error-correcting repetition code. The bottom qubit is encoded with two controlled-nots and three $y$-rotations. In the experiment, either physical or controlled noise is allowed to act. The encoded information is then decoded. For the present purposes, it is convenient to separate the decoding procedures into two steps: The first is the inverse of the encoding procedure, the second consists of a Toffoli gate that uses the error information in the syndrome qubits (the top two) to restore the encoded information. The Toffoli gate in the last step flips the output qubit conditionally on the syndrome qubits' state being $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{1}\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. This gate can be realized with NMR-pulses and delays by using more sophisticated versions of the implementation of the controlled-not. The syndrome qubits can be ``dumped'' at the end of the procedure. The behavior of the network is shown for a generic state in which the bottom qubit experiences a $\sigma_z$ error. See also [26].

To determine the behavior and the quality of the implementation for various $\sigma_z$-error models in an actual NMR realization, one can use as initial states labeled pseudopure states with deviations $\sigma_u\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}\mathf... ...}{\mathfrak{0}\mathfrak{0}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$ for $u=x,y,z$. Without error, the total output signal on spin $\mathsf {1}$ along $\sigma_u$ for each $u$ should be the same as the input signal. Some of the data reported in [24] is shown in Fig. 15.

FIG. 15: Experimentally obtained fidelities for the error-correction experiment. The inset bar graph shows fidelities for explicitly applied errors. The fidelities $f$ (technically, the ``entanglement'' fidelities) are an average of the signed ratios $f_u$ of the input to the output signals for the initial deviations $\sigma_u\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}\mathf... ...}{\mathfrak{0}\mathfrak{0}}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$ with $u=x,y,z$. Specifically, $f={1\over 4}(1+f_x+f_y+f_z)$. The reduction from $1$ of the green bars (showing fidelity for the full procedure) is due to errors in our implementation of the pulses and from relaxation processes. The red bars are the fidelity for the output before the last error-correction step, and they contain the effects of the errors. The main graph shows the fidelities for the physical relaxation process. Here, the evolution consisted of a delay varying up to $1000{\mathchoice{\mbox{ms}}{\mbox{ms}}{\mbox{\small ms}}{\mbox{\tiny ms}}}\,$. The red curve is the fidelity of the output qubit before the final Toffoli gate that corrects the errors based on the syndrome. The green curve is the fidelity of the output after the Toffoli gate. The effect of error-correction can be seen by a significant flattening of the curve because correction of first-order (that is, single) phase errors implies that residual, uncorrected (that is, double or triple) phase errors increase quadratically in time. The green curve starts lower than the red one because of additional errors incurred by the implementation of the the Toffoli gate. The dashed curves are obtained by simulation using estimated phase relaxation rates with half times of $2{\mathchoice{\mbox{s}}{\mbox{s}}{\mbox{\small s}}{\mbox{\tiny s}}}\,$ (proton), $0.76{\mathchoice{\mbox{s}}{\mbox{s}}{\mbox{\small s}}{\mbox{\tiny s}}}\,$ (first carbon) and $0.42{\mathchoice{\mbox{s}}{\mbox{s}}{\mbox{\small s}}{\mbox{\tiny s}}}\,$ (second carbon). Errors in the data points are approximately $0.05$. The molecule used was TCE. For a more thorough implementation and analysis of a three-qubit phase-error correcting code, see [27].

Work on benchmarking error-control methods using liquid-state NMR is continuing. Other experiments include the implementation of a two-qubit code with an application to phase-errors [28] and the verification of the shortest non-trivial noiseless subsystem on three qubits [29]. The latter demonstrates that for some physically realistic noise models, it is possible to store quantum information in such a way that it is completely unaffected by the noise.