Discrete frequency-modulated gates for conditional logic in trapped-ion quantum computers

Abstract

The system includes a controller configured to reconstruct the continuous waveform into a discrete analog version. The system includes a numerical optimizer configured to determine the frequency of the pulse sequence. The numerical optimizer uses the radial motion mode frequency and the desired gate time. The numerical optimizer generates the pulse sequence by closing phase space trajectories, disentangling spin and motion, and constraining Rabi frequencies for motional sideband transitions. The system also includes a display configured to depict the discrete frequency-modulated pulse sequence based on the determined frequency.

Description

[Technical field]

[0001] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT This disclosure was made with Government support under Federal Grant No. Phy-181891 awarded by the National Science Foundation. The Federal Government has certain rights in this disclosure.

[0002] CROSS-REFERENCE TO RELATED APPLICATIONS This application claims the benefit of U.S. Provisional Application No. 62 / 994,998, filed March 26, 2020, entitled "Discrete Frequency Modulated Gates For Conditional Logic In Trapped Ion Quantum Computers" (Attorney Docket No. DU7032PROV), which is incorporated herein by reference. If there is a conflict or inconsistency in language between this application and one or more of the cases incorporated by reference that may affect the interpretation of the claims of this case, the claims of this case shall be construed consistent with the language of the case.

[0003] This disclosure relates to discrete frequency modulated gates in quantum computers, and more particularly to generating discrete frequency modulated pulse sequences given measured radial motion mode frequencies and desired gate times. [Background technology]

[0004] Due to their long coherence times and high-fidelity initialization, detection, and qubit gate operation, trapped atomic ions are the leading qubit platform for the realization of quantum computers. The Molmer-Sorensen (MS) gate is a two-qubit gate that has been demonstrated with greater than 99.9% fidelity in a two-ion system using the axial mode.

[0005] In two-ion or four-ion chains, individual optical addressing of ions and modulated pulse techniques are important for MS gates. Multichannel acousto-optic modulation has been used for individual addressing of atomic qubits. In addition, acousto-electro-optic modulation and beam steering using microelectromechanical systems (MEMS) tilting mirrors have also been used to individually address atomic qubits.

[0006] It has been demonstrated that crosstalk is negligible for MEMS to individually address the qubit system. In addition, a gating scheme has been developed that is insensitive to optical phase drift between the addressing beams. Moreover, phase drift can occur that causes spin-motion entanglement of the motional modes. Phase drift can also lead to deviations in the rotation angle of the gate.

[0007] Other methods that have been used include amplitude modulation (AM) gating, frequency modulation (FM) gating, and multitone MS gating. In addition, fidelity has been demonstrated for such AM, FM, and MS gating. When using ion chains of 5 or more ions, the fidelity has been demonstrated to be typically around 97% to 98.5%. Therefore, attempts have been made to use these modulated pulses to make the gating more robust against frequency drift.

[0008] For any type of high fidelity quantum logic gate operation in larger arrays of qubits, low optical coherence must be considered. Low optical coherence between individual addressing beams can have a positive impact on the fidelity of the gate. Another factor that can affect the fidelity of the gate is crosstalk from the addressing beams to neighboring qubits. Since both factors can have a positive impact on the fidelity of the gate, the methods described above had to address low phase coherence, and crosstalk between the addressing beams and neighboring qubits.

[0009] Regarding crosstalk, several systems have demonstrated negligible crosstalk between addressing beams and neighboring qubits. For example, MEMS addressing systems and gating schemes have been developed that are insensitive to optical phase drift between the addressing beams. Various modulation pulse techniques can disentangle any internal qubit states from all collective motional modes. Disentangling the internal qubit states from all collective motional modes can increase robustness in the face of frequency drifts, which can occur at the motional mode frequencies.

[0010] Thus, methods have been developed to address individual optical addressing of ions for MS gates in chains of two or more ions. Such methods include MEMS addressing systems. Additionally, methods using PM, AM, and FM gates have been used to attempt to disentangle the internal qubit states from all collective modes of motion, where frequency drift can cause unwanted entanglement. Such methods have been used for ion chains of five or more ions.

[0011] Still, high-fidelity two-qubit gates in two- and four-ion chains need to be demonstrated, as should a system that effectively addresses drift in the motional mode frequencies that lead to entanglement and amplitude errors. Summary of the Invention [Problem to be solved by the invention]

[0012] In large-scale trapped-atom quantum computers, high-fidelity two-qubit gates need to be extended to all qubits with individual controllability. High-fidelity two-qubit gates in two-ion and four-ion chain systems using radial modes need to be characterized and realized. The fidelity of gates in two-ion and four-ion chains needs to be inferred. Proper sequences of qubit gates need to be measured. Methods need to be developed to further improve the fidelity of gates to values ​​compatible with fault-tolerant quantum computing.

[0013] Therefore, it is necessary to generate pulse sequences that are free of unwanted spin motion and entanglement caused by drifts in the motion mode frequencies. Furthermore, amplitude errors caused by phase shifts or amplitude errors need to be corrected more accurately. Therefore, a method must be in place to generate pulse sequences while taking into account drifts in the motion mode frequencies, which lead to unwanted entanglement of spin and motion of radial motion modes, and deviations in the rotation angle of the gates, which lead to amplitude errors. [Means for solving the problem]

[0014] An embodiment of the present disclosure utilizes a process for generating a symmetrically segmented discrete frequency modulated pulse sequence given a measured radial mode frequency and a desired gate time.

[0015] For high-fidelity quantum logic gate operation, loss of optical phase coherence between individual addressing beams and crosstalk from addressing beams to neighboring qubits can affect the fidelity of the gate. To overcome variations in the optical beam paths between different beams, gating schemes have been developed that are insensitive to optical phase drift between addressing beams.

[0016] To disentangle the internal qubit gates from the collective motion modes and also to increase the robustness against frequency drift, a modulated pulse technique is used. Discrete frequency modulated pulse sequences can be generated by closing the phase space trajectories of all radial motion modes, disentangling the spin and motion of the radial motion modes. Amplitude errors due to misalignment of the gate rotation angle can be corrected by intentional offset detuning instead of adjusting the laser intensity.

[0017] An illustrative embodiment of the present disclosure is a system comprising a controller configured to reconstruct a discrete analogue version from a continuous waveform. The system also includes a numerical optimizer configured to determine a frequency of a pulse sequence for the discrete analogue version, the numerical optimizer using the radial motion mode frequency and a desired gate time. The numerical optimizer generates the pulse sequence by closing a phase space trajectory for the radial motion mode, disentangling spin and motion, and constraining the Rabi frequency for the motional sideband transition. The system also includes a display configured to depict the discrete frequency modulated pulse sequence based on the determined frequency. The discrete frequency modulated pulse sequence is designed to be a sequence of equal time segments, each of the equal time segments having a constant frequency.

[0018] In some embodiments, the desired gate time is 200 microseconds or less.

[0019] In some embodiments, the numerical optimizer is configured to correct for drifts in radial motion modes that lead to spin-motion entanglement and deviations from geometric phase.

[0020] In some embodiments, the numerical optimizer is configured to correct any phase shifts that result in amplitude errors.

[0021] In some embodiments, the field programmable gate array is configured to cause frequency updates to a direct digital synthesizer (DDS) channel in real time.

[0022] An embodiment of the present disclosure is a system in which the controller is configured to obtain the predetermined gate time and the measured mode frequency. The system also includes a numerical optimizer configured to receive the predetermined gate time and the measured mode frequency to determine a frequency of the pulse sequence. The numerical optimizer removes phase space trajectories including drifts in the motional mode frequency that cause spin-motion entanglement and geometric phase shifts, and the numerical optimizer constrains the Rabi frequency of the motional sideband transition. The system also includes a display device configured to plot the pulse sequence based on the predetermined gate time, the measured mode frequency, and the frequency determined by removing the spin-motion entanglement and the shift from the geometric phase.

[0023] In some embodiments, the numerical optimizer calculates the frequency of the pulse sequence to enable the pulse sequence to have equal time segments.

[0024] In some embodiments, the numerical optimizer closes the phase space trajectories of all radial motion modes.

[0025] In some embodiments, the numerical optimizer disentangles the spin and motion of at least four motion modes.

[0026] In some embodiments, the numerical optimizer removes all amplitude errors to allow the pulse sequence to be generated with equal time segments.

[0027] Another embodiment of the present disclosure is a method comprising reconstructing a continuous waveform into a discrete analog version by a controller. The method also includes determining a frequency of a pulse sequence for the discrete analog version by a numerical optimizer, the numerical optimizer using the radial motion mode frequency and the desired gate time. The numerical optimizer generates the pulse sequence by closing a phase space trajectory for the radial motion mode, the numerical optimizer disentangling spin and motion and constraining the Rabi frequency for the motion sideband transition. The method also includes depicting a discrete frequency modulated pulse sequence by a display based on the determined frequency, the discrete frequency modulated pulse sequence designed to be a sequence of equal time segments, each of the equal time segments having a constant frequency.

[0028] In some embodiments, the overall gate error of multiple concatenated Molmer-Sorensen gates is estimated.

[0029] In some embodiments, the fidelity of the final state of successive gates is estimated, and this estimation takes into account the remaining spin and motion entanglements.

[0030] In some embodiments, the final state fidelity of successive gates with Rabi frequency shifts due to motional sideband transitions is estimated.

[0031] In some embodiments, one or more sets of microelectromechanical system (MEMS) mirrors are configured to deliver a beam that is modified by a direct digital synthesizer (DDS). [Brief description of the drawings]

[0032] [Figure 1] FIG. 1 is a block diagram of a direct digital synthesizer according to an illustrative embodiment of the present disclosure. [Diagram 2] FIG. 2 is a diagram of a display of a discrete frequency modulated pulse sequence according to an illustrative embodiment of the present disclosure. [Figure 3A] FIG. 13 illustrates phase space trajectories of motion modes according to an illustrative embodiment of the present disclosure. [Figure 3B] FIG. 13 illustrates another phase space trajectory of a motion mode according to an illustrative embodiment of the present disclosure. [Figure 3C] FIG. 13 illustrates phase space trajectories of motion modes according to an illustrative embodiment of the present disclosure. [Figure 3D] FIG. 13 illustrates another phase space trajectory of a motion mode according to an illustrative embodiment of the present disclosure. [Figure 4] 13 is a graph showing estimated total gate error for 1, 5, 13, and 21 concatenated gates under different detuning offsets in an illustrative embodiment of the present disclosure. [Diagram 5] 13 is a graph showing final state errors of 21 consecutive gates in an illustrative embodiment of the present disclosure. [Figure 6A] FIG. 13 illustrates gate fidelity for two ion chains in an illustrative embodiment of the present disclosure. [Figure 6B] FIG. 13 illustrates gate fidelity for a four-ion chain in accordance with an embodiment of the present disclosure. [Figure 7A] FIG. 1 illustrates a schematic representation of a Raman beam optics arrangement in accordance with an embodiment of the present disclosure. [Figure 7B] FIG. 2 illustrates another schematic representation of a Raman beam optics arrangement in accordance with an embodiment of the present disclosure. [Figure 7C] FIG. 2 is a schematic diagram of energy levels of the 171Yb+ ion in accordance with an embodiment of the present disclosure. [Figure 8] 1 is a flow chart illustrating a process according to an embodiment of the present disclosure. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0033] The following disclosure may be implemented for an arbitrary waveform generator (AWG) or for a direct digital synthesizer (DDS). FIG. 1 described below is not intended to limit the disclosure to a particular device such as an arbitrary waveform generator or a DDS.

[0034] FIG. 1 shows a block diagram of the DDS 100. The DDS 100 includes a system clock 110, a tuning word 120, and a phase accumulator 130. In addition, the DDS 100 includes a phase register 140. The DDS 100 also includes a phase-amplitude converter 150 and a digital-to-analog D / A converter 160. In addition, the DDS 100 includes a controller 170 and a numerical optimizer 180. The controller 170 can reconstruct a continuous waveform into a discrete analog version compatible with the DDS 100. In other embodiments, other components of the DDS 100 can also reconstruct a continuous waveform into a discrete analog version compatible with the DDS 100. In addition, as described below in later figures such as FIG. 2, the numerical optimizer 180 performs a series of steps to enable the frequency pulse modulation sequence to be displayed. The numerical optimizer 180 can close the phase space trajectories of all radial motion modes involved. The numerical optimizer 180 can also disentangle unwanted spin and motion entanglement of radial motion modes. Frequency drift of radial motion modes can lead to unwanted spin and motion of radial motion modes. Detuning errors resulting from drift of motion mode frequencies lead to dephasing of rotation angles of frequency modulation (FM) gates. Dephasing of FM gates can be considered as amplitude errors. The numerical optimizer can more accurately compensate for amplitude errors by using intentional detuning offsets instead of adjusting laser intensity.

[0035] Referring to FIG. 2, a frequency pulse modulation sequence (pulse sequence) 200 is shown in which a continuous waveform is reconstructed into a discrete analog version. Moreover, the pulse sequence 200 is designed to be a sequence of equal time segments. Each of the time segments may have a constant frequency. The numerical optimizer 180 can enable the pulse sequence 200 to be generated as the output of the DDS 100 by performing a series of steps to remove the amplitude error in the FM gate using an intentional detuning offset.

[0036] In FIG. 2, the numerical optimizer 180 can determine the frequency of the pulse sequence 200. The numerical optimizer utilizes the measured radial motion mode frequency and the desired gate time. In this disclosure, the desired gate time is shown as being 200 microseconds. In other embodiments, the desired gate time may be different. Moreover, the gate time refers to a preset period that the numerical optimizer 180 can preset to determine the frequency of the pulse sequence 200. The desired gate time may be any time less than or equal to 200 microseconds. The numerical optimizer 180 can close the phase space trajectory of all radial motion modes. In addition, the numerical optimizer 180 can disentangle unwanted spin and motion entanglement of the radial motion modes.

[0037] With reference to FIG. 2, the detuning error arises from the drift of the motional mode frequency. The drift of the motional mode frequency also leads to unwanted spin-motion entanglement. The detuning error arising from the drift of the motional mode frequency can also lead to a geometric phase shift for the Molmer-Sorensen (MS) expansion. As mentioned above, the accumulated phase shift can refer to a rotation angle shift of the FM gate. Therefore, the rotation angle shift of the FM gate can be considered to be an amplitude error. One way to correct the amplitude error is by adjusting the laser intensity. To obtain the pulse sequence 200, the numerical optimizer 180 must remove the amplitude error. In other embodiments, the amplitude error is corrected by adjusting the laser intensity. However, in those embodiments, if by adjusting the laser intensity, the detuning error changes on a timescale faster than the time between the calibration and the experimental circuit, the intensity calibration is no longer accurate and the adjustment of the laser intensity is not sufficient.

[0038] In FIG. 2, in contrast to adjusting the laser intensity, an intentional detuning offset can be applied to more accurately compensate for small amplitude errors. The remaining spin-motion entanglement and the rotation angle of the FM gate can be used with the intentional detuning offset. The intentional detuning offset can take advantage of the fact that the remaining spin-motion entanglement is negligible for any detuning error in the FM gate. The intentional detuning offset is more accurate than adjusting the laser intensity. In an embodiment, the calibration may be performed by checking the detuning offset with 21 concatenated gates, where concatenated gates refer to gates that are closely connected in a chain, in series, or in series. The calibration may be applied to state |00>. Thus, a detuning such that states |00> and |11> have equal probability indicates a perfect rotation angle in the MS gate. In an embodiment, as an example, the intentional detuning offset is applied to up to 21 consecutive MS gates to correct the amplitude error. In other embodiments, the intentional detuning offset may be applied to more or less than 21 consecutive MS gates. Instead of adjusting the laser intensity, an intentional detuning offset can eliminate the amplitude error caused by the misalignment of the gate rotation angle.

[0039] 2, a numerical optimizer 180 can correct detuning errors resulting from drift in motional mode frequencies, which leads to unwanted spin-motion entanglement and geometric dephasing of radial motional modes. Instead of adjusting the laser intensity, an intentional detuning offset can be used, which can more accurately compensate for small amplitude errors resulting from drift in motional mode frequencies.

[0040] In summary, FIG. 2 can generate a pulse sequence 200 based on the determined gate time and the measured radial motion mode frequency. The numerical optimizer can close the phase space trajectory of the radial motion mode and disentangle the spin and motion of the radial motion mode caused by the drift of the motion mode frequency. The numerical optimizer 180 can also remove the geometric dephasing of the MS evolution. The dephasing can be considered as the rotation angle deviation of the FM gate, or the amplitude error. To compensate the amplitude error more precisely, an intentional detuning offset can be used instead of adjusting the laser intensity.

[0041] 3A-3D show the phase space trajectories of the four radial motion modes 310, 320, 330, and 340. As explained in FIG. 2, the spin and motion of the radial motion modes need to be entangled to generate the pulse sequence 200 described above in FIG. 2. The numerical optimizer 180 generates the pulse sequence 200 that closes the phase space trajectories of the radial motion modes 310, 320, 330, and 340. In addition, the numerical optimizer 180 disentangles the spin and motion of the radial motion modes 310, 320, 330, and 340. The amplitude error is compensated for by the intentional detuning offset. The amplitude error is compensated for by the intentional detuning offset. The numerical optimizer 180 uses the intentional detuning offset to accurately compensate for the phase drift, or drift of the motion mode frequency that causes the unwanted spin and motion, and the geometric phase shift. The intentional detuning offset compensates for the phase drift of the motional mode frequencies. As a result, the numerical optimizer 180 can close the phase space trajectories of the radial motional modes 310, 320, 330, and 340 and disentangle the unwanted spin and motional entanglements of the radial motional modes 310, 320, 330, and 340. Thus, the pulse sequence 200 can be generated as the output of the DDS 100.

[0042] In Figures 3A-3D, the drift of the motional mode frequency leads to unwanted spin and motional entanglement for the radial motional modes 310, 320, 330, and 340. The drift of the motional mode frequency for the radial motional modes 310, 320, 330, and 340 also leads to geometric phase shift for the MS evolution. Moreover, the error due to the remaining entanglement of the radial motional modes 310, 320, 330, and 340 can be made negligible compared to the ±1 kHz detuning error in the FM gate. The phase shift or geometric phase shift is represented by the rotation angle shift of the FM gate. The rotation angle shift of the FM gate can be determined to be the amplitude error. The amplitude error of the FM gate can be corrected by using an intentional detuning offset to accurately compensate for the amplitude error. With the adjustment of the laser intensity, the intensity calibration is no longer accurate if the detuning error changes on a timescale faster than the time between the calibration and the experimental results. Therefore, rather than adjusting the laser intensity, an intentional detuning offset is preferred.

[0043] In Fig. 3A-3D, the intentional detuning offset can correct the amplitude error of the FM gate. The intentional detuning offset can be applied to consecutive gates. Moreover, the detuning offset can be probed with up to 21 concatenated gates. As a result, the corresponding calibration can greatly reduce and / or eliminate the amplitude error caused by the misalignment of the rotation angle due to the drift of the motion mode frequency. In addition, the unwanted spin and motion entanglement of the radial motion modes 310, 320, 330, and 340 can also be corrected. As a result, a pulse sequence 200 with equal time segments at a constant frequency can be generated.

[0044] FIG. 4 shows a system plot 400 in which an intentional detuning offset is applied to one MS gate, five MS gates, thirteen MS gates, and twenty-one MS gates. In other embodiments, an intentional detuning offset may be applied to more or less than twenty-one MS gates. Additionally, the system plot 400 shows an example in which an intentional detuning offset is applied to the DDS 100 of FIG. 1 to generate the pulse sequence 200 shown in FIG. 2. Estimated total gate errors of one, five, thirteen, and twenty-one concatenated gates with different detuning offsets are illustrated. Estimation of the final state fidelity after one, five, thirteen, and twenty-one consecutive MS gates is applied as a function of the detuning offset. Estimation of the final state fidelity is applied to both the remaining spin-motion entanglement of the radial motion mode and the rotation angle misalignment of the FM gate. An intentional detuning offset may be introduced to take advantage of the negligible spin-motion entanglement compared to the detuning error in the FM gate. A detuning offset of ±100 Hz can compensate for roughly ±0.8% deviation in the Rabi frequency due to the motional sideband transition. The intentional detuning offset can be used instead of the process of adjusting the laser intensity, and is a more accurate way to obtain the fidelity of the final state and to correct any amplitude errors of the FM gate.

[0045] Referring to FIG. 5, a system plot 500 of estimated final state errors of 21 consecutive MS gates with a deviation of ±0.8% in the Rabi frequency for motional sideband transitions is shown. In other embodiments, the final state errors of more or less than 21 consecutive MS gates can be estimated. The system plot 500 shows the final state errors of 21 MS gates in the DDS 100 shown in FIG. 1 to enable the pulse sequence 200 of FIG. 2 to be generated. The amplitude error caused by imperfect laser intensity can be compensated for by intentional detuning offset instead of adjusting the laser intensity. A detuning offset of ±100 Hz can compensate for the approximate deviation of ±0.8% in the Rabi frequency for motional sideband transitions.

[0046] In Fig. 5, a calibration is performed to compensate for small drifts in mode frequency and laser intensity. Furthermore, small drifts in laser intensity can be more accurately compensated for using a deliberate detuning offset as opposed to adjusting the laser intensity. The calibration can be performed by probing the detuning offset with 21 consecutive / concatenated gates applied to |00>. Moreover, a detuning such that |00> and |11> have equal probability indicates a perfect rotation angle for the MS gate. This calibration technique thus improves the fidelity of the gate as opposed to a crude calibration or adjusting the laser intensity.

[0047] Two-qubit MS gates can be demonstrated in both two-ion and four-ion chains. As shown below, the population and parity contrast can be measured to extract the fidelity of the states. Furthermore, infidelity due to population leakage and reduced parity contrast is illustrated.

[0048] FIG. 6A shows a two-qubit MS gate in a two-ion chain. A plot 600 of infidelity due to population leakage and loss of parity contrast is shown. In FIG. 6A, diamonds indicate population leakage into |01> and |10> spaces. Squares indicate loss of parity contrast. Additionally, circles indicate infidelity of the final state. First, the target qubit is initialized to the |00> state. Sequences of 1, 5, 13, and 21 MS gates result in a maximally entangled state.

number

[0049] Referring to FIG. 6B, a plot 610 of infidelity due to population leakage and reduced parity contrast in a four-ion chain is shown. As in FIG. 6A, diamonds in the figure indicate population leakage into |01> and |10> spaces. In addition, squares indicate loss of parity contrast while circles indicate infidelity of the final state. As in the two-ion chain, in the four-ion chain the target qubit is initialized to the state |00> state. 1, 5, 13, and 21 MS gate sequences are shown to result in the maximally entangled state.

number

[0050] 6A-B, the simulated error values ​​for the four-ion chain and the two-ion chain for laser dephasing, motional dephasing, Raman beam intensity fluctuations, and off-resonant coupling are the same. In addition, the simulated error values ​​for motional heating, spontaneous emission, and imperfections in the FM solution are also the same.

[0051] Using an optimized auto-calibration pipeline for ion trap type systems, a high fidelity two-qubit gate in a microelectromechanical system (MEMS) can be demonstrated. As shown in FIG. 7C, the qubits are gated at |0>≡|F=0; m with a qubit frequency division of 12.642821 GHz. F =0>and|1>≡|F=1; m F =0>, 171 Yb + In ions 2 S 1 / 2It may be encoded at the hyperfine level of the manifold. In other embodiments, the approach may be used with other atomic species and isotopes as well.

[0052] In FIG. 7A, a MEMB-based qubit addressing system 700 is illustrated. A first parallel beam 705 is incident on a focusing lens 710. The first parallel beam 705 is a pair of tightly focused individual addressing beams that can be independently steered across a quantum chain using a MEMS device. The beam 705 passes through the focusing lens 710 and is incident on a first lens 715. The beam 705 then passes through the first lens 715 and is incident on a first MEMS mirror 720 in a MEMS device. The first MEMS mirror 720 is part of a MEMS device that also includes a second MEMS mirror 730. After the beam 705 is reflected from the first MEMS mirror 720, the beam 705 is incident on a concave mirror 725. The beam 705 reflects off the concave mirror 725 and travels to a second MEMS mirror 730 in the MEMS device. From the second MEMS mirror 730, the beam 705 then travels to a second lens 735. From the second lens 735, the beam 705 reflects off a Fourier lens 740. The beam 705 travels through a Fourier plane to projection and beam combining optics 745. The beam 705 may then be incident on a dichroic mirror 750.

[0053] 7A, fiber array 760 can provide another beam 775 to another lens or detection optics 755. Beam 775 can also be incident on dichroic mirror 750 as beam 705. From dichroic mirror 750, beams 705, 775 are incident on high NA imaging lens 765. Beams 705, 775 pass through NA imaging lens 765 to qubit 770. Thus, the combination of fiber array 760 and MEMS mirrors 720, 730 leads to a clean Gaussian beam and low crosstalk to neighboring qubits 770.

[0054] 7B also shows a schematic representation of the Raman beam optics. The trapping plane 780 is shown. The trapping axis is rotated 45 degrees with respect to the tilting axes of both MEMB mirrors 720, 730 to utilize orthogonal tilting mirrors to maximize addressable qubits. The projection and beam combining optics 745 may be represented by a black box.

[0055] FIG. 7C shows 171 Yb + A schematic diagram of the energy levels of an ion 785 is shown. The red and blue lines indicate the two-photon Raman transitions for qubit operation.

[0056] Overall, the optical device may produce a clean Gaussian beam pair and low intensity crosstalk to neighboring qubits 770. Intensity crosstalk may result in gate crosstalk or a certain ratio of Rabi frequencies between the target qubit and the neighboring qubits.

[0057] FIG. 8 shows a process 800 in which a pulse modulation sequence can be generated. The numerical optimizer 180 can enable the pulse sequence 200 using the measured radial mode frequency and a desired gate time of up to 200 microseconds. The desired gate time can be 200 microseconds or less. The numerical optimizer 180 can resolve the unwanted spin and motion entanglement of the radial motion modes shown in FIG. 3A to FIG. 3D caused by the drift of the motion mode frequency. In addition, the numerical optimizer 180 also closes the phase space trajectory of the radial motion modes caused by the drift of the motion mode frequency. The accumulated phase shift or gate rotation angle shift can be considered as an amplitude error. Nevertheless, an intentional detuning offset can compensate for the amplitude error.

[0058] 8, in step 810, controller 180 within DDS 100 may reconstruct the continuous waveform into its discrete analog version. In other embodiments, other components of DDS 100 may reconstruct the continuous waveform into its discrete analog version.

[0059] 8, in step 820, numerical optimizer 180 is configured to use the radial motion mode frequency and the desired gate time to determine a frequency for pulse sequence 200. The desired gate time may be 200 microseconds or less.

[0060] In FIG. 8, in step 830, the numerical optimizer 180 generates the pulse sequence 200 by closing the phase space trajectory and disentangling the unwanted spin and motion of the radial motion modes. Drift of the motion mode frequency leads to unwanted spin and motion entanglement and phase shift of the motion modes. The numerical optimizer 180 can disentangle the unwanted spin and motion of the motion modes. In addition, the accumulated phase shift is represented by the rotation angle shift of the FM gate. The rotation angle shift of the FM gate can be determined to be an amplitude error. An intentional detuning offset can be used to precisely compensate for the amplitude error or the rotation angle shift of the FM gate.

[0061] 8, in step 840, the numerical optimizer generates the pulse sequence 200 by constraining the Rabi frequency for the motional sideband transitions. The numerical optimizer 180 constrains the Rabi frequency of the motional sideband transitions to less than 7 kHz. Moreover, in an embodiment, the required sideband Rabi frequency is 5.55 kHz for FM gates with 2 ion chains and 5.47 kHz for FM gates with 4 ion chains, respectively.

[0062] Referring to FIG. 8, in step 850, a discrete frequency modulated pulse sequence / pulse sequence 200 is displayed. The pulse sequence 200 is designed to be a sequence of equal time segments, each of which has a constant frequency. Unwanted spin entanglements and phase shifts are removed. The pulse sequence has a total gate time of 200 microseconds or less and may consist of 20 symmetrical segments.

[0063] In summary, the numerical optimizer 180 can enable the pulse sequence 200 to be generated by disentangling the unwanted spin and motional entanglement of the radial motional modes caused by the drift of the motional mode frequency. In addition, the accumulated phase fluctuation of the rotation angle of the FM gate caused by the drift of the motional mode frequency can be accurately compensated by using an intentional detuning offset. The intentional detuning offset can accurately compensate for small amplitude errors due to the misalignment of the rotation angle of the FM gate. In one embodiment, a detuning offset of ±100 Hz can compensate for approximately ±0.8% misalignment of the Rabi frequency for the motional sideband transition.

[0064] It should be understood that this disclosure teaches only some examples of embodiments according to the disclosure, that many variations of the disclosure can be readily devised by those skilled in the art after reading this disclosure, and that the scope of the disclosure should be determined by the claims which follow. [Explanation of symbols]

[0065] 110 System Clock 120 Tuning Word M 130 Phase Accumulator 140 Phase Register 150 Phase Amplitude Converter 160 D / A Converter 170 Controller 180 processors / numerical optimizers 705 First parallel beam 710 Focusing Lens 715 First Lens 720 First MEMS Mirror 725 concave mirror 730 Second MEMS Mirror 735 Second Lens 740 Fourier Lens 745 Projection and Beam Combining Optics 750 Dichroic Mirror 755 Detection optics 760 Fiber Array 765 NA imaging lens 770 qubits 775 Beam 780 Trap Surface

Claims

1. A method and apparatus for reconstructing a continuous waveform into a discrete analog version by dividing the continuous waveform into equal time intervals and setting a frequency of the continuous waveform constant over each of the equal time intervals; and a controller configured to reconstruct the continuous waveform into a discrete analog version by dividing the continuous waveform into equal time intervals and setting a frequency of the continuous waveform constant over each of the equal time intervals. a numerical optimizer configured to determine a frequency of a pulse sequence for the discrete analog version, the numerical optimizer using a radial motion mode frequency and a desired gate time, generating the pulse sequence by closing a phase space trajectory for a radial motion mode, eliminating spin-motion entanglement, and constraining Rabi frequencies for motional sideband transitions, the numerical optimizer further configured to minimize unwanted spin-motion entanglement and further eliminate amplitude errors; A system comprising: The system is configured to output a discrete frequency modulated pulse sequence generated based on the determined frequency, the discrete frequency modulated pulse sequence being designed to be a sequence of equal time segments, each of the equal time segments having a constant frequency.

2. The system of claim 1 , wherein the desired gate time is less than or equal to 200 microseconds.

3. The system of claim 1 , wherein a numerical optimizer constrains the Rabi frequency to less than 7 kHz.

4. The system of claim 1 , wherein the numerical optimizer is configured to correct drifts in the radial motion modes that lead to spin-motion entanglement and deviations from geometric phase.

5. The system of claim 1 , wherein the numerical optimizer is configured to correct any phase shifts that result in amplitude errors.

6. 10. The system of claim 1, wherein the field programmable gate array is configured to cause frequency updates for the direct digital synthesizer (DDS) channels in real time.

7. a controller configured to obtain a predetermined gate time and a measured modal frequency; a numerical optimizer configured to receive the predetermined gate times and the measured mode frequencies to determine frequencies for a discrete frequency modulated pulse sequence, the numerical optimizer configured to eliminate phase space trajectories including drifts in motional mode frequencies that cause spin-motion entanglement and geometric dephasing, and to constrain Rabi frequencies of motional sideband transitions, the numerical optimizer further configured to minimize unwanted spin-motion entanglement and further to eliminate amplitude errors; A system comprising: The system is configured to output the discrete frequency modulated pulse sequence generated based on the predetermined gate time, a measured mode frequency, and the frequency determined by removing the spin-motion entanglement and deviation from the geometric phase.

8. 8. The system of claim 7, wherein the shift in the geometric phase is for a Molmer-Sorenson (MS) two-qubit gate.

9. 8. The system of claim 7, wherein the numerical optimizer calculates the frequency for the pulse sequence to enable the pulse sequence to have equal time segments.

10. The system of claim 7 , wherein the numerical optimizer closes the phase space trajectories of all radial motion modes.

11. 8. The system of claim 7, wherein the numerical optimizer removes all amplitude errors to enable the pulse sequence to be generated with equal time segments.

12. The system of claim 7 , wherein the numerical optimizer disentangles the spin and kinematics of at least four motion modes.

13. 10. The system of claim 8, wherein the numerical optimizer determines a rotation angle deviation for a two-qubit gate.

14. reconstructing, by a controller, the continuous waveform into a discrete analog version by dividing the continuous waveform into equal time intervals and setting a frequency of the continuous waveform constant over each of the equal time intervals; determining a frequency of a pulse sequence for the discrete analog version by a numerical optimizer, the numerical optimizer using a radial motion mode frequency and a desired gate time, the numerical optimizer generating the pulse sequence by closing a phase space trajectory for a radial motion mode, the numerical optimizer disentangling spin and motion and constraining Rabi frequencies for motional sideband transitions, the numerical optimizer further configured to minimize unwanted spin and motion entanglement and further eliminate amplitude errors; and outputting a discrete frequency modulated pulse sequence generated based on the determined frequency, the discrete frequency modulated pulse sequence being designed to be a sequence of equal time segments, each of the equal time segments having a constant frequency.

15. The method of claim 14 , further comprising estimating an overall gate error of a plurality of concatenated Molmer-Sorensen gates.

16. 15. The method of claim 14, wherein the discrete frequency modulated pulse sequence is generated by closing a phase space trajectory of one or more radial motion modes.

17. 15. The method of claim 14, further comprising estimating the fidelity of the final state of successive gates, the estimation taking into account remaining spin and motion entanglement.

18. 15. The method of claim 14, further comprising estimating final state fidelity of successive gates using the Rabi frequency shift for the motional sideband transitions.

19. 15. The method of claim 14, further comprising configuring one or more sets of microelectromechanical system (MEMS) mirrors to deliver a beam that is modified by a direct digital synthesizer (DDS).

20. The method of claim 14, further comprising applying an intentional detuning offset to compensate for amplitude errors due to laser intensity.

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