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Spatially uniform single-qubit gate operations with near-field microwaves and composite pulse compensation

We present a microfabricated surface-electrode ion trap with a pair of integrated waveguides that generate a standing microwave field resonant with the 171Yb+ hyperfine qubit. The waveguides are engineered to position the wave antinode near the center of the trap, resulting in maximum field amplitude and uniformity along the trap axis. By calibrating the relative amplitudes and phases of the waveguide currents, we can control the polarization of the microwave field to reduce off-resonant coupling to undesired Zeeman sublevels. We demonstrate single-qubit pi-rotations as fast as 1 us with less than 6 % variation in Rabi frequency over an 800 um microwave interaction region. Fully compensating pulse sequences further improve the uniformity of X-gates across this interaction region.


Introduction
The future development of a fault-tolerant quantum processor will require the storage and manipulation of large numbers of qubits and the ability to execute quantum gates with high fidelity [1]. Some proposed architectures use trapped atomic ions to store and transport quantum information, and recent years have witnessed significant progress in the control of these systems [2][3][4][5][6]. Most trapped-ion experiments utilize laser fields to control internal and motional quantum states [7]; however, the fidelity of laser-mediated gates suffers from errors induced by unavoidable spontaneous emission [8][9][10], pointing instabilities, and variations in the laser frequency, phase, and power [11]. The systems required for laser-mediated gates are also large and complex, making them difficult to scale to the large numbers of qubits required to execute algorithms or simulations of interest. Alternatively, microwaves can be used to control the ground electronic hyperfine state of the ions, and several experiments have demonstrated high-fidelity single-qubit gates that take advantage of the long natural lifetime of hyperfine qubits and the stability of microwave sources [12][13][14][15]. The advent of microfabricated ion traps facilitates the use of near-field microwaves, since microwave structures can be integrated directly into the trap. These structures can enable single-ion addressing and multi-qubit gate operations by providing a mechanism to generate field gradients across significantly sub-wavelength inter-ion distances [13,14,[16][17][18][19][20]. In addition, ions trapped in the near-field can experience a large field amplitude resulting in fast gate operations.
Here we present a microfabricated surface-electrode ion trap with integrated microwave waveguides fabricated with standard very large-scale integration (VLSI) techniques. The waveguides are designed to place an antinode of a 12.64 GHz standing microwave field, suitable for coupling the ground state hyperfine levels of 171 Yb + , at the center of the ion trap. The microwaves are polarization tunable, so that we can optimize coupling to a desired Zeeman transition while suppressing off-resonant coupling to neighboring transitions. With the ion located 59 µm above the trap surface, a 0.037 mT field amplitude allows execution of sub-microsecond π-rotations. The intrinsic position dependence of the standing microwave field causes the Rabi frequency to vary by less than 6% over an 800 µm linear span. We correct for this residual non-uniformity by executing gates with fully compensating broadband pulse sequences.

Trapping structures
The trap conforms to a symmetric five-wire surface-electrode Paul trap geometry [21] fabricated on a 11 × 11 mm 2 silicon die (figure 1a) similar to the designs reported in [22][23][24]. Electrodes etched into three sputtered aluminum layers are separated by insulating silicon dioxide films. Radio-frequency (RF) potentials applied to two parallel electrodes provide radial ion confinement (in the x-y plane, figure 1b) 59 µm above the Metal 2 layer (figure 1c). Quasi-static potentials applied to segmented DC electrodes confine and transport ions along z. The RF electrodes are 30 µm wide along x with an inner edge-to-edge separation of 92 µm. The segmented DC electrodes are 56 µm wide along z except for six 100 µm wide electrodes bordering the loading slot. Two additional DC electrodes traversing the entire length of the trapping region are used to apply uniform x-y fields and to rotate the radial principal axes. Each electrode is separated from neighboring conductors by 4 µm gaps. Each DC electrode incorporates a 60 pF plate capacitor (1 mm 2 area) to filter unwanted RF pickup [22]. A loading slot allows a thermal beam of neutral Yb to reach the trapping volume from an oven located below the trap.

Integrated waveguides
The trap includes a pair of conductor-backed coplanar waveguides that generate local microwave magnetic fields. Each waveguide includes a 40 µm wide electrode with 4 µm gaps to neighboring conductors, and 10 µm of SiO 2 separate the coplanar layer from the ground plane below (figure 1b). The waveguides support a ω mw = 2π × 12.64 GHz quasi-TEM guided mode resonant with the hyperfine splitting between the F = 0 and F = 1 manifolds in the 2 S 1/2 ground state of 171 Yb + (see figure 3a). In the ideal case, currents in each waveguide generate a magnetic field along the trapping axis where k = 1, 2 is an index for the two waveguides, φ k is the phase of the microwave current source, I k (z) is the current in each waveguide, and β x,1 = β x,2 0.08 mT/A and β y,1 = −β y,2 0.17 mT/A are properties of the waveguide mode. β y,1 = −β y,2 due to the symmetric placement of the waveguides around the trapping axis. The waveguides terminate in an open circuit at a position that is approximately a quarter-wavelength from the trap center, which produces a standing wave field with maximum amplitude and uniformity in the gate region. A smaller traveling wave component also exists due to on-chip attenuation that generates an amplitude difference between forward and backward propagating waves.
Far from the trap center, the waveguides meander to fit a complete wavelength on the chip and then terminate on wirebond pads at the edge of the chip. Extending the waveguides to a full wavelength in this way places a current node at the wirebond pads and reduces the potential for resistive power loss in the connections. A series of 25.4 µm diameter aluminum wirebonds connect the chip waveguides to two PCB waveguides that route microwaves from the edge of the trap package (figure 2). Connections between the PCB top level ground and the on-chip Metal 1 and Metal 2 ground planes are symmetric about each microwave electrode. Quarter-wave transformers match the 50 ohm impedance of the PCB waveguide to the 27 ohm on-chip characteristic impedance. The PCB is fabricated using a 254 µm thick Rogers 4350B substrate with two 18 µm thick copper foil conductive layers and a 3-6 µm electroless nickel immersion gold (ENIG) finish. The skin depth in the PCB at ω mw is comparable to the thickness of the lossy nickel layer resulting in ≈ 3 dB of power loss between the microwave connector and wirebonds. Much of this power loss could be recovered by replacing the ENIG finish with an electroplated gold surface.

Trap performance
3.1. Microwave control of 171 Yb + hyperfine qubits Figure 3a shows the transitions in the ground state hyperfine manifold of 171 Yb + addressed by the microwave field. A static 0.74 mT field along y defines the quantization axis and lifts the degeneracy of the F = 1 triplet. We select the clock states | 2 S 1/2 , F = 1, m F = 0 ≡ |↑ and | 2 S 1/2 , F = 0, m F = 0 ≡ |↓ as the qubit states. We observe Rabi oscillations at a frequency Ω = 2π × 0.49 MHz (figure 3b) by optically pumping into |↓ , applying microwave power for a variable interval of time, and measuring the resulting population transfer into the F = 1 manifold through state-selective fluorescence of the 369.5 nm cycling transition (figure 3a) [25]. Currents in each waveguide generate an oscillating magnetic field which contains both a π−polarized component that couples to the ∆m F = 0 clock transition and also a transversepolarized component that couples to the ∆m F = ±1 transitions. Figure 3c shows the hyperfine spectrum measured with microwave power applied to a single waveguide. From the relative Rabi frequencies of the transitions, we estimate the ratio of polarization components for each waveguide as |β x,k |/|β y,k | ≈ 0.46.
The polarization of the near-field microwaves may be controlled by adjusting the relative amplitude and phase of the microwave currents in the two waveguides. In particular, the polarization may be aligned along the quantization axis, thereby maximizing the qubit transition Rabi frequency while also suppressing off-resonant ∆m F = ±1 transitions. The active and passive microwave components supplying energy to these electrodes are not perfectly power-balanced and phase-matched. We calibrate the microwave sources by first driving each waveguide independently to map the relationship between source power and Rabi frequency (figure 4a). Once the field amplitudes from the waveguides have been equalized, the relative phase between microwave currents φ r can be varied to produce an arbitrary linear polarization in the x-y plane. Figure 4b shows the resonant Rabi frequency for each of the three transitions as φ r is varied, demonstrating the desired suppression of the ∆m F = ±1 transitions at φ r = π. We suspect that the mismatch between the ∆m F = ±1 curves in figure 4b is caused by a frequency dependence of the microwave system output power.

Microwave field uniformity and compensation
The standing wave current in the waveguides produces a microwave field with non-uniform amplitude along the trap axis. Figure 5 shows the measured Rabi frequency of the qubit transition at several positions along the trap axis. We observe a maximum Rabi frequency of 2π × 0.52 MHz, corresponding to a field amplitude of 0.037 mT, located z 0 = 957 µm from the loading slot center. Finite element calculations predict an antinode location at z = 895 µm in reasonable agreement with the experiment. These models indicate that the maximum field corresponds to a local current in each electrode of I z (z 0 ) ≈ 0.1 A.
The microwave field non-uniformity appears as an effective amplitude error when implementing global single-qubit rotations on multiple ions located at different positions  in the trap. To improve single-qubit gate uniformity, we implement global gates using broadband compensating pulse sequences [26,27]. This technique replaces a simple pulse with a sequence of pulses whose phases are chosen to nearly cancel the effective microwave amplitude error. The excitation profiles of such pulse sequences enable global rotations on many qubits, although the microwave amplitude may differ significantly between distant ions.
Here we demonstrate the error-canceling properties of composite X-gates constructed from the first-order SK1 [28,29] and second-order BB1 [30] sequences. The experiments prepare the qubit in |↓ , apply a logical X-gate, and then measure the population in the F = 1 manifold. To simulate the effect of systematic over/under rotations, we uniformly scale the pulse areas of every pulse in the sequence by adjusting the pulse duration.  plots the measured excitation profiles produced by compensated gates, overlayed on the signal predicted by theory. Assuming the field non-uniformity is the sole source of error, we calculate theoretical fidelities of X-gates. For the 6% amplitude deviation observed at z = 300 µm, a simple rotation performs a global X-gate with a minimum fidelity F ≥ 0.995, whereas SK1 and BB1 perform the same gate with minimum fidelities of F ≥ 1 − 1.5 × 10 −4 and F ≥ 1 − 2.2 × 10 −7 respectively.
As a demonstration of uniform global gates, we perform an experiment where n sequential logical X-gates are applied to a qubit initialized in |↓ . We calibrate gate times so that an ion located at the microwave amplitude maximum (z 0 = 957 µm) experiences nearly perfect rotations. Qubits displaced from the field maximum rotate at lower Rabi frequencies, acquiring an under-rotation error that accumulates as n increases. We measure the F = 1 population as a function of ion axial position and number of sequential X-gates. For X-gates implemented by simple rotations we observe fringes (figure 7a) arising from the local qubit falling behind by an entire Rabi cycle relative to the maximal Rabi frequency. Instead, when implementing SK1 or BB1 pulses (figures 7b and 7c) the error accumulates so slowly that the excitation profile remains flat over the trapping region after n = 55 logical X-gates. Our ability to resolve fringe structure in these cases is currently limited by systematic statepreparation and measurement errors and by the number of simple pulse operations we can implement.
We analytically calculate the fidelity scaling of the sequential logical X-gates as a function of the microwave field strength. For simple rotations, the fidelity drops as

Conclusion
We developed a microfabricated surface-electrode ion trap with integrated microwave waveguides that performs arbitrary single-qubit gates on the 171 Yb + hyperfine qubit. The polarization of the local microwave field can be tuned to minimize off-resonant coupling to adjacent Zeeman sub-levels. This suppression of off-resonant coupling will be most useful when Rabi frequencies are large compared to the Zeeman splittings, thus causing reduced spectral resolution of neighboring transitions. We demonstrate the use of fully compensating pulse sequences to reduce single-qubit gate variation caused by a spatially non-uniform microwave amplitude. Similar passband and narrowband pulse sequences [27] could be used to exploit microwave amplitude variations to enable single-ion addressing without requiring perfect field suppression at the location of neighboring ions. Schematic of microwave delivery system connected to the outside of the UHV chamber. The two DDS sources are mixed with a stable local oscillator and provide independent control over the amplitude, phase, and frequency (ω mw = ω lo ± ω dds ) of the microwave signal applied to the trap waveguides, though during typical operation ω dds1 = ω dds2 . DDS = direct digital synthesizer, LO = local oscillator, LNA = low noise amplifier, HPA = high power amplifier, MW=microwave.
approximately 300 MHz detuned from the qubit resonant frequency near 12.64 GHz. Delivery of this signal is controlled by an American Microwave SW-218 high-speed RF switch. The signal is split to supply the local oscillator ports of two frequency mixers. Two separate direct digital synthesis (DDS) boards with synchronized clocks supply the intermediate frequency (IF) signals near 300 MHz. The DDS outputs have independently controllable amplitude, phase, and frequency and these signals are amplified prior to mixing with the LO signal. The mixers reject the carrier frequency and produce two RF sidebands at ω mw = ω lo ± ω dds where one of these sidebands is tuned to the frequency of the desired hyperfine transition, while the other is far off resonance. Both signals are amplified in separate Mini-Circuits ZVE-3W-183+ amplifiers and routed to a coaxial feedthrough port on the UHV chamber. Inside the chamber the microwaves are carried by two coaxial lines with Kapton dielectric and braided conductor, selected for UHV compatibility. The combined power loss in the feedthrough and Kapton cables is ≈ 5.3 dB.

Trap packaging
We mount chips to a gold electroplated stainless steel plate that also serves as a structurally rigid platform for mounting other components, including microwave connectors, printed circuit board (PCB), grounded screen, and a 100-pin ceramic pin-grid array (CPGA). The chip is bonded to the interface plate with electrically conductive epoxy (Epoxy Technology H21D) which accommodates the differential thermal expansion between the stainless steel mounting plate and the silicon chip during vacuum bake-out. Wire bonds at the chip perimeter establish electrical connections to the CPGA to provide both DC and RF trapping potentials, while wire bonds to the PCB supply microwave energy to the chip (figure 2b). To minimize scattered laser light, wire bonds are excluded from regions where cross-chip laser access is required.