Quantum state tomography of large nuclear spins in a semiconductor quantum well: Optimal robustness against errors as quantified by condition numbers

We discuss methods of quantum state tomography for solid-state systems with a large nuclear spin $I=3/2$ in nanometer-scale semiconductors devices based on a quantum well. Due to quadrupolar interactions, the Zeeman levels of these nuclear-spin devices become nonequidistant, forming a controllable four-level quantum system (known as quartit or ququart). The occupation of these levels can be selectively and coherently manipulated by multiphoton transitions using the techniques of nuclear magnetic resonance (NMR) [Yusa et al., Nature (London) 434, 1001 (2005)]. These methods are based on an unconventional approach to NMR, where the longitudinal magnetization $M_z$ is directly measured. This is in contrast to the standard NMR experiments and tomographic methods, where the transverse magnetization $M_{xy}$ is detected. The robustness against errors in the measured data is analyzed by using the condition number based on the spectral norm. We propose several methods with optimized sets of rotations yielding the highest robustness against errors, as described by the condition number equal to 1, assuming an ideal experimental detection. This robustness is only slightly deteriorated, as given by the condition number equal to 1.05, for a more realistic “noisy” $M_z$ detection based on the standard cyclically ordered phase sequence (CYCLOPS) method.

Figure 1

Schematic diagram of the energy levels of a spin $I=3/2$ nucleus in an external magnetic field $B_0$ due to Zeeman interactions with additional shifts originating from the first- and second-order quadrupole interactions. The second-order shifts completely remove the degeneracy between the levels, which is desirable for QST. Unfortunately, the second-order shifts are negligible in the nanoscale device studied here and thus will be omitted hereafter.

Figure 2

A quartit (a spin-3/2 nucleus) is formally equivalent to two qubits (two spin-1/2 nuclei), as shown here via their energy levels and their corresponding frequencies. Single-photon ($\hslash {\omega }_{01},\hslash {\omega }_{12},\hslash {\omega }_{23}$), two-photon ($\hslash {\omega }_{02}/2,\hslash {\omega }_{13}/2$), and three-photon ($\hslash {\omega }_{03}/3$) transitions drive single, double, and triple quantum coherent oscillations between two levels separated by one ($\mathrm{\Delta }m=1$), two ($\mathrm{\Delta }m=2$), and three ($\mathrm{\Delta }m=3$) quanta of angular momentum, respectively.

Figure 3

Schematic diagram of the implementation of a coherently and selectively controllable solid-state system with nuclear spins-3/2 for quantum information processing and quantum tomography. This is an on-chip monolithic semiconductor device integrated with a point-contact channel, fabricated by Yusa et al. [42]. The point-contact channel (indicated with an arrow), as defined by a pair of Schottky split gates, is composed of a small ensemble of nuclear quadrupolar spins-3/2 of isotopes ${}^{69}\mathrm{Ga}, {}^{71}\mathrm{Ga}$, and ${}^{75}\mathrm{As}$. The nuclear spins in this device can exhibit extremely long decoherence times (a few milliseconds). The antenna gate, which is insulated from the Schottky gates, can locally irradiate the channel with an RF field. The ensemble of spins-3/2 in the channel can be selectively and coherently polarized and controlled by NMR techniques. The resistance of the channel changes after an RF pulse, which directly corresponds to the change of the longitudinal magnetization $M_z$ induced by the change in the population of nuclear-spin states. Thus, the discussed tomographic methods, which are based on the detection of the longitudinal magnetization of nuclear spins, can be effectively implemented in such devices. This device is based on a quantum-well GaAs structure, but other semiconductors can also be used. For example, by using the semiconductor ${\mathrm{FeSb}}_2$, instead of GaAs, the coherent control of nuclear quadrupolar spins-7/2 for the isotope ${}^{123}\mathrm{Sb}$ would enable quantum information processing with a quoctit (eight-level qudit) corresponding to three virtual qubits. This figure is based on Fig. 1 in Ref. [42].

Figure 4

Set of rotations $R_{\mathrm{diag}}^{\mathrm{opt}1}$, given in Eq. (47), which enables the optimal reconstruction of all the diagonal elements of $\rho$, by measuring only the first peak of the $M_z$ spectra, i.e., the peak corresponding to the transition $|0\rangle \leftrightarrow |1\rangle$. The operation marked by $\times -\times$ denotes the SWAP-like operation $S_{nm}\equiv {\mathcal{Y}}_{nm}\left(\pi \right)$ between the levels $|n\rangle$ and $|m\rangle$. Reading pulses (which are not shown here) are applied only after applying these SWAP-like gates. The set $R_{\mathrm{diag}}^{\mathrm{opt}1}$ corresponds to the coefficient matrix $A\equiv A_{\mathrm{diag}}^{\mathrm{opt}1}$, given in Eq. (46), yielding the smallest condition number $\kappa \left(A^{T}A\right)=1$. Note that the last row in $A$ corresponds to the normalization condition, while the first row corresponds to applying a reading pulse without the SWAP. For brevity, this trivial case corresponding to the identity operation is not presented here.

Figure 5

Set of rotations $R_{\mathrm{diag}}^{\mathrm{opt}2}$, given in Eq. (48). Same as in Fig. 4, but the coefficient matrix $A_{\mathrm{diag}}^{\mathrm{opt}2}$ corresponds to measuring only the central peak of the $M_z$ spectra, i.e., that corresponding to the transition $|1\rangle \leftrightarrow |2\rangle$.

Figure 6

Set of rotations $R_{\mathrm{offdiag}}^{\mathrm{temp}}$ given in Eq. (58), for the reconstruction of all the off-diagonal elements of $\rho$ in the 12 series of measurements by applying the $\pi /2$ pulses: $X_{nm}={\mathcal{X}}_{nm}(\pi /2)$ and $Y_{nm}={\mathcal{Y}}_{nm}(\pi /2)$, which are in both cases marked by the red double arrows. In the theoretical approach, the off-diagonal elements can directly be obtained by Eq. (61), which implies that the method is optimal with $\kappa =1$. However, this is not the optimal method assuming the experimental approaches, where only some of the off-diagonal elements can directly be measured, according Eq. (63), while other elements are calculated indirectly, which results in $\kappa >1$.

Figure 7

Set of rotations $R_{\mathrm{offdiag}}^{\mathrm{opt}0}$ given in Eq. (64) for the optimal reconstruction of all the off-diagonal elements of $\rho$. The optimal coefficient matrix $A_{\mathrm{offdiag}}^{\mathrm{opt}0}$ is obtained by measuring only a properly chosen single peak of each $M_z$ spectrum, as indicated in Eqs. (63) and (65).

Figure 8

Set of rotations $R_{\mathrm{offdiag}}^{\mathrm{opt}1}$ given in Eq. (69) for the optimal reconstruction of all the off-diagonal elements of $\rho$. Same as in Fig. 6, but the optimal matrix $A_{\mathrm{offdiag}}^{\mathrm{opt}1}$ is obtained by measuring only the first peak of the $M_z$ spectra.

Figure 9

Set of rotations $R_{\mathrm{offdiag}}^{\mathrm{opt}2}$ given in Eq. (71). Same as in Fig. 8, but the optimal matrix $A_{\mathrm{offdiag}}^{\mathrm{opt}2}$ is obtained by measuring only the central peak of the $M_z$ spectra.