Practical trapped-ion protocols for universal qudit-based quantum computing

The notion of universal quantum computation can be generalized to multilevel qudits, which offer advantages in resource usage and algorithmic efficiencies. Trapped ions, which are pristine and well-controlled quantum systems, offer an ideal platform to develop qudit-based quantum information processing. Previous work has not fully explored the practicality of implementing trapped-ion qudits accounting for known experimental error sources. Here, we describe a universal set of protocols for state preparation, single-qudit gates, a generalization of the Mølmer-Sørensen gate for two-qudit gates, and a measurement scheme which utilizes shelving to a metastable state. We numerically simulate known sources of error from previous trapped-ion experiments, and show that there are no fundamental limitations to achieving fidelities above $99\%$ for three-level qudits encoded in ${}^{137}\mathrm{Ba}^{+}$ ions. Our methods are extensible to higher-dimensional qudits, and our measurement and single-qudit gate protocols can achieve $99\%$ fidelities for five-level qudits. We identify avenues to further decrease errors in future work. Our results suggest that three-level trapped-ion qudits will be a useful technology for quantum information processing.

Figure 1

Examples of qudit encodings. (a) We prefer a zigzag configuration because it simplifies laser manipulations. (b) A bunched configuration minimizes decoherence due to magnetic field fluctuations but requires more polarization control of lasers. (c) Disconnected configurations are not preferred due to the experimental complexity of transferring population among all possible states.

Figure 2

The shelving procedure for a three-level qudit measurement. (1) Map states $|1\rangle ,|2\rangle$ to the metastable state. (2) Fluoresce on the cycling transition $S_{1/2}\leftrightarrow P_{1/2}$. (3) If no fluorescence is detected, return one state from the metastable state to the ground state. (4) Measure it with fluorescence.

Figure 3

(a) The fidelity of population transfer [Eq. (12)] plotted for various applied Rabi frequencies and passage times $t=\frac{2\mathrm{\Delta }}{\alpha }$. The gray curve follows the optimal parameters. (b) The measurement time and fidelity [Eq. (13)] for different prime-dimensional qudits. We use the optimal Rabi frequency for each measurement time. Fluorescence time is included in the measurement duration, and we assume that the amount of adiabatic passages needed is $2d-3$ (the maximum amount of transfers needed for an arbitrary measurement).

Figure 4

Schematics of laser perturbations applied to ${}^{137}\mathrm{Ba}^{+}$ for (a) three-level qudit and (b) five-level qudit for qudit MS gate. ${\omega }_{R,n}$ denotes the $n\mathrm{th}$ laser frequency applied for the gate. (c) Illustration of physical implementation of laser beams and frequencies for the qudit MS gate. Cyan circles represent the ions.

Figure 5

(a) Illustration of evolution of three-level qudits in position-momentum phase space during the qudit MS gate. The displacement of all states returns to the original position at the end of the gate. The phase gained by each basis state is proportional to the area enclosed by the trajectory in phase space. (b) State probability evolution of the three-level qudit MS gate. The gate is set to complete a loop in the position-momentum phase space every $100\mu \mathrm{s}$. ${\theta }_0$ is set to ${\theta }_0=-\frac{\pi }{4}$. After one loop, at $t=\frac{2\pi }{{\omega }_C-\mu }=100\mu \mathrm{s}$ (dashed line), multilevel entanglement is achieved.

Figure 6

${}^{137}\mathrm{Ba}^{+}$ energy structure [82]. The $6S_{1/2}\leftrightarrow 6P_{1/2}$ optical transition is used for optical pumping, Doppler cooling, and fluorescence measurement. The $6S_{1/2}\leftrightarrow 5D_{5/2}$ transition is used to shelve qudit states. The $5D_{3/2}\leftrightarrow 6P_{1/2}$ transition is used to repump dark states back into the cooling/fluorescence cycle. The $5D_{5/2}\leftrightarrow 6P_{3/2}$ transition is used to empty the $5D_{5/2}$ state. Because of its nuclear spin of $\frac{3}{2}$, each level is split into hyperfine levels: the frequency differences between these levels are shown [83, 84, 85].

Figure 7

Absolute frequency spectrum for shelving five-level qudits. Shaded areas show the laser sweep with $1.3\mathrm{MHz}$ detuning. We use a $-1130$ MHz laser carrier frequency. Text on top of transitions is denoted as $(F,m_F)\leftrightarrow {(F,m_F)}^{\prime }$, where the prime denotes the shelving state.

Figure 8

Measurement sequence for a five-level qudit. Solid blue and purple straight arrows are desired transitions and dashed orange arrows are unwanted transitions. Black straight arrows are microwave or Raman transitions used to hide population. Line sizes denote the order of the transition: carrier transitions are thickest (leftmost purple arrow) and first order are thinner (orange arrow driving from $|0\rangle$). Noncarrier transitions also have white arrows. Symbols on lines denote that a transition's frequency is within an encoded transition's laser frequency sweep (with the same symbol) and it will be driven. Large black numbers denote the step number and the small arrows pointing up (down) denote shelving (deshelving) and hiding. Red vector states denote the encoded states. Wiggly teal arrows denote fluorescence measurement of an encoded state.

Figure 9

Measurement sequence for a seven-level qudit. Red vector states denote the encoded states. (a) Solid blue and purple arrows are desired transitions and dashed red and orange arrows are unwanted transitions. The line sizes denote the order of the transition: carrier transitions are thickest (leftmost purple arrow) and first order are thinner (orange arrow driving from $|0\rangle$). Noncarrier transitions also have white arrows. Symbols on lines denote that a transition's frequency is within an encoded transition's laser frequency sweep (with the same symbol) and it will be driven. (b), (c) Large black numbers denote the step number and small arrows pointing up and down denote moving population, either shelving, deshelving, or hiding. (b) Black transitions are microwave or Raman transitions used to hide population. (c) Wiggly teal arrows denote fluorescence measurement of an encoded state.

Figure 10

Simulation histograms of gate errors for all Pauli gates (${\widehat{X}}_d, {\widehat{Y}}_d, {\widehat{Z}}_d$), the generalized Hadamard gate ${\mathcal{H}}_d$, and $\pi /8$ gates (${\widehat{T}}_d$) [50] for (a) three- and (b) five-level qudits. Each gate has a sample size of 500, where the initial qudit state is a randomized superposition of the encoded states. A magnetic field offset is set at the standard deviation of $\sqrt{\langle \mathrm{\Delta }B{\left(t\right)}^2\rangle }\approx 2.7\mathrm{pT}$ and off-resonant coupling to other states is incorporated into the simulations. The Rabi frequency is set at 10 kHz for the data sets in this figure.

Figure 11

Schematic for laser frequencies applied to implement a five-level qudit entangling gate for ${}^{137}\mathrm{Ba}^{+}$. Black arrows indicate the desired frequencies to be applied to the energy levels. Red arrows are the (unwanted) off-resonant frequencies.

Figure 12

Estimations of error values from magnetic field noise for single- and two-qudit gates as a function of magnetic field standard deviation. Legend: 1QG3D(M), abbreviation for one-qudit gate three-dimensional (microwave transition); 1QG5D(R), one-qudit gate five-dimensional (Raman transition). The rest of the abbreviations of the legends can be extrapolated from the information presented. Black dashed line marks the error value of ${10}^{-4}$.