High-Fidelity Preparation, Gates, Memory, and Readout of a Trapped-Ion Quantum Bit

We implement all single-qubit operations with fidelities significantly above the minimum threshold required for fault-tolerant quantum computing, using a trapped-ion qubit stored in hyperfine “atomic clock” states of ${}^{43}{\mathrm{Ca}}^{+}$. We measure a combined qubit state preparation and single-shot readout fidelity of 99.93%, a memory coherence time of $T_2^{*}=50\sec$, and an average single-qubit gate fidelity of 99.9999%. These results are achieved in a room-temperature microfabricated surface trap, without the use of magnetic field shielding or dynamic decoupling techniques to overcome technical noise.

Figure 1

The ion trap and the qubit. (a) Schematic diagram of the surface ion trap, showing (left-hand inset) central electrode layout. Microwave (m.w., 3.2 GHz) signals are combined with the trap radio frequency voltage (rf, 40 MHz) via filters, as indicated here for the lower axial electrode. Also shown are laser beam directions and polarizations with respect to the static magnetic field $B_0=146\mathrm{G}$. The violet 397 nm Doppler-cooling beam is elliptically polarized such that it contains only $\pi$ and ${\sigma }^{+}$ polarizations, and copropagates with linearly (${\sigma }^{\pm }$) polarized infrared repumping beams. Circularly ${\sigma }^{+}$ and linearly $\pi$ polarized beams are used for state preparation and readout (the $\pi$ beam is at $\sim 45°$ to the plane of the figure and reflects off the trap surface). To load the trap, neutral Ca atoms effusing from the oven are ionized by laser beams at 423 and 389 nm which copropagate with the Doppler-cooling beams. Ion fluorescence is collected by an imaging system perpendicular to the plane of the trap; the right-hand inset shows an image of a single ${}^{43}{\mathrm{Ca}}^{+}$ ion, which is trapped $75\mu \mathrm{m}$ above the electrodes. (b) ${}^{43}{\mathrm{Ca}}^{+}$ level structure, showing optical transitions used for Doppler cooling, qubit state preparation, and readout. The inset shows part of the ground level hyperfine structure, labeled by quantum numbers $F$ and $M$, with the qubit $|\downarrow ⟩$ and $|\uparrow ⟩$ states, the 3.2 GHz qubit transition (red), and the auxiliary transitions (green) used for state preparation and readout. The Zeeman splittings between adjacent $M$ states are $\approx 50\mathrm{MHz}$.

Figure 2

(a) Microwave spectroscopy of the qubit transition, varying the static magnetic field $B$ through the field-independent point $B_0=146.094\mathrm{G}$. At each field value, the qubit transition frequency $f$ was measured by Ramsey spectroscopy, to a precision $\approx 0.1\mathrm{Hz}$. The field-independent qubit transition is at $f_0=3199941077\mathrm{Hz}$ after adjusting for a $-5\mathrm{Hz}$ shift due to the trap (see text). The solid line shows the expected frequency calculated using the Breit-Rabi formula assuming the known zero-field hyperfine splitting [20] and a nuclear magnetic moment [21] of ${\mu }_I=-1.31535{\mu }_N$. (b) Qubit coherence time measurements. At each value of the Ramsey free precession time $t_R$ the phase of the second $\pi /2$ pulse was varied to produce a set of Ramsey fringes. The contrast of the fringes is fitted with an exponential decay, giving a coherence time $T_2^{*}=50(10)\sec$.

Figure 3

Qubit state preparation and measurement (SPAM) results. We prepared and measured the $|\downarrow ⟩$ (blue histogram) and $|\uparrow ⟩$ (gray histogram) qubit states 150 000 times each. For each measurement the likelihood $p_{\downarrow }$ (or $p_{\uparrow }$) that the state was $|\downarrow ⟩$ (or $|\uparrow ⟩$) was calculated from the time-resolved photon counts detected on a photomultiplier (see Ref. [8]); if $p_{\uparrow }>p_{\downarrow }$, we infer that the qubit was in the $|\uparrow ⟩$ state, and vice versa. If the state inferred disagrees with the state prepared, either a preparation or a measurement error has occurred; thus, the fraction of experiments in which an error occurred is given by the sum of the blue events above the $p_{\uparrow }=p_{\downarrow }$ threshold ($2\times 10^{-4}$) and the gray events below threshold ($5\times 10^{-4}$).

Figure 4

Randomized benchmarking of single-qubit gates. The start of an example pseudorandom sequence is shown: each computational gate is composed of three physical $\pi /2$ pulses (absent in the case of identity $I$ operations or $z$ rotations). Additional $\pi /2$ pulses at the end of the sequence rotate the qubit into the ($|\downarrow ⟩$,$|\uparrow ⟩$) basis for measurement. If the measured state disagrees with that expected, an error is recorded. For each sequence length, 32 distinct sequences are used, each one being repeated the same number of times (typically 100) to measure the total error. Results are shown in the plot, where repeated runs have been offset horizontally for clarity. The gradient of a weighted straight line fit gives an average error per gate of $1.0(3)\times 10^{-6}$, while the intercept agrees with the independently measured SPAM error of $6.8(5)\times 10^{-4}$ (dashed line). Error bars represent 68% confidence intervals, assuming binomial statistics. The fit gives a reduced ${\chi }^2\approx 0.84$.