Neutral atom quantum information processor

Abstract

The invention relates to a neutral atom quantum information processor. Systems and methods involve arranging atoms in a 1D and / or 2D array; exciting the atoms into a Rydberg state and evolving the array of atoms, such as using laser manipulation techniques and high fidelity laser systems described herein; and observing the resulting final state. Further, improvements may be made, such as providing high fidelity and coherence control of the assembled atomic array. Exemplary problems may be solved using systems and methods for arrangement and control of atoms.

Description

[0001] This case is a divisional application of the patent application with application number 201880059408.2 and title “Neutral Atom Quantum Information Processor” filed by the present applicant on July 13, 2018. The entire contents of the parent case are incorporated into this divisional application by reference.

[0002] CROSS-REFERENCE TO RELATED APPLICATIONS

[0003] This application claims the benefit of priority to U.S. Provisional Application No. 62 / 531,993, filed on July 13, 2017, entitled “PROBING MANY-BODY DYNAMICS ON A PROGRAMABLE 51-ATOM QUANTUM SIMULATOR,” the disclosure of which is incorporated herein by reference in its entirety. This application also claims the benefit of priority to U.S. Provisional Application No. 62 / 589,716, filed on November 22, 2017, entitled “RYDBERG QUANTUM OPTIMIZER FOR MAXIMUM INDEPENDENT SET,” the disclosure of which is incorporated herein by reference in its entirety.

[0004] Copyright Notice

[0005] This patent disclosure may contain material that is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the U.S. Patent and Trademark Office patent file or records, but otherwise reserves any and all copyright rights. Technical Field

[0006] This patent relates to quantum computing, and more specifically, to preparing and evolving arrays of atoms. Summary of the invention

[0007] According to some embodiments, the systems and methods described herein relate to: arranging atoms into 1D and / or 2D arrays; exciting atoms into Rydberg states and evolving the atomic arrays, for example, using laser manipulation techniques and high-fidelity laser systems as described herein; and observing the resulting final state. In addition, the systems and methods described herein can be improved, such as providing high-fidelity and coherent control of the assembled atomic arrays. Exemplary problems that can be solved using the systems and methods described herein are discussed.

[0008] In one or more embodiments, a method includes: forming an array of atoms in a first array state, wherein the forming includes: exciting a crystal with a plurality of discrete adjustable acoustic tone frequencies; passing a laser through the crystal to produce a plurality of confinement regions, wherein each acoustic tone frequency corresponds to a separate confinement region for a single atom; trapping at least two atoms in at least two of the plurality of confinement regions; associating with the discrete adjustable acoustic tone frequencies to identify the confinement regions containing the trapped atoms; and adjusting the spacing between at least two of the trapped atoms by sweeping at least one associated adjustable acoustic tone frequency; evolving the plurality of atoms in the first array state into a plurality of atoms in a second array state by subjecting at least some of the trapped atoms to photon energy to convert at least some of the trapped atoms into an excited state; and observing the plurality of atoms in the second array state.

[0009] In one or more embodiments, the excited state is a Rydberg state.

[0010] In one or more embodiments, the plurality of atoms in the first array state includes 7 to 51 atoms.

[0011] In one or more embodiments, evolving the plurality of atoms includes preparing at least some of the atoms in the first array state to a Zeeman sublevel of a ground state prior to subjecting at least some of the atoms to photon energy.

[0012] In one or more embodiments, preparing the atoms in the first array state to a Zeeman sublevel of the ground state includes optical pumping in a magnetic field.

[0013] In one or more embodiments, subjecting at least some of the atoms to photon energy comprises applying light having two different wavelengths, and wherein said converting the at least some of the atoms to an excited state comprises a two-photon transition.

[0014] In one or more embodiments, the two different wavelengths are approximately 420 nm and approximately 1013 nm.

[0015] In one or more embodiments, the method further includes applying a phase gate having a third wavelength.

[0016] In one or more embodiments, the third wavelength is approximately 809 nm.

[0017] In one or more embodiments, subjecting the at least some of the atoms to photon energy comprises applying two half-π pulses.

[0018] In one or more embodiments, subjecting the at least some of the atoms to photon energy further comprises applying a π pulse between the two half π pulses.

[0019] In one or more embodiments, trapping the at least two atoms includes capturing at least two atoms from an atom cloud, and dispersing atoms from the atom cloud that are not trapped in one of the plurality of confinement regions.

[0020] In one or more embodiments, the crystal and the laser constitute a first controlled acousto-optic deflector (AOD), and wherein trapping the at least two atoms comprises trapping atoms from a holding well array having at least three wells spaced apart in two dimensions.

[0021] In one or more embodiments, the holding well array is generated by at least one of holding an AOD, a spatial light modulator (SLM), and an optical lattice.

[0022] In one or more embodiments, the method also includes a second control AOD, which is configured to be in a crossed relationship with the first control AOD, and wherein: associating with the discrete adjustable acoustic tone frequency to identify the confined area containing the trapped atoms includes associating with the discrete adjustable acoustic tone frequency of the first control AOD and the second control AOD, and adjusting the spacing between the at least two of the trapped atoms includes sweeping the adjustable acoustic tone frequency associated with at least one of the first control AOD or the second control AOD.

[0023] In one or more embodiments, adjusting the spacing between the at least two of the trapped atoms further comprises adjusting positions of a plurality of atoms in a row.

[0024] In one or more embodiments, the method further includes: forming a second atomic array adjacent to the first atomic array in a third array state, wherein the forming includes: exciting a second crystal with a plurality of second discrete adjustable acoustic tone frequencies; passing a second laser through the second crystal to form a plurality of second confinement regions, wherein each second acoustic tone frequency corresponds to a separate second confinement region for a single atom; capturing at least two second atoms in at least two of the plurality of second confinement regions; associating the second discrete adjustable acoustic tone frequencies to identify the second confinement region containing the captured atoms; and adjusting the captured first confinement region by sweeping at least one second associated adjustable acoustic tone frequency. spacing between at least two of the two atoms; wherein evolving the plurality of atoms in the first array state into a plurality of atoms in the second array state by subjecting at least some of the trapped atoms to photon energy so as to transform the at least some of the trapped atoms into the excited state also includes evolving the plurality of second atoms in the third array state into a plurality of second atoms in a fourth array state by subjecting at least some of the second trapped atoms to photon energy so as to transform at least some of the second trapped atoms into the excited state; and wherein observing the plurality of atoms in the second array state also includes observing the plurality of second atoms in the fourth array state.

[0025] In one or more embodiments, adjusting the spacing between at least two of the trapped atoms by sweeping at least one second associated adjustable acoustic tone frequency includes encoding a quantum computing problem; evolving the plurality of atoms in a first array state to the plurality of atoms in the second array state produces a solution to the quantum computing problem; and observing the plurality of atoms in the second array state includes reading out the solution to the quantum computing problem.

[0026] In one or more embodiments, the quantum computing problem includes at least one of an Ising problem and a maximum independent set (MIS) optimization problem.

[0027] In one or more embodiments, a system includes: a confinement system for arranging an array of atoms in a first array state, the confinement system including: a crystal; an adjustable acoustic tone frequency application source configured to selectively apply a plurality of discrete adjustable acoustic tone frequencies to the crystal; and a laser source arranged to pass light through the crystal to form a plurality of confinement regions, wherein each acoustic tone frequency corresponds to a separate confinement region; a source of an atomic cloud capable of being positioned to at least partially overlap the plurality of confinement regions; an excitation source for causing at least some of the plurality of atoms in the first array state to evolve into a plurality of atoms in the second array state, the excitation source including at least one photon energy source; and an observation system for observing the plurality of atoms in the second array state.

[0028] In one or more embodiments, the excitation source is configured to excite at least some of the plurality of atoms in the first array state into a Rydberg state.

[0029] In one or more embodiments, the plurality of atoms in the first array state includes 5 to 51 atoms.

[0030] In one or more embodiments, the excitation source is configured to excite at least some of the plurality of atoms in the first array state to a Zeeman sublevel of a ground state prior to subjecting at least some of the atoms to photon energy.

[0031] In one or more embodiments, the excitation source further includes an optical pumping system and a magnetic field generator.

[0032] In one or more embodiments, the at least one photon energy source includes a light source having a first wavelength and a second wavelength for producing a two-photon transition of the at least some of the plurality of atoms in the first array state.

[0033] In one or more embodiments, the two different wavelengths are approximately 420 nm and approximately 1013 nm.

[0034] In one or more embodiments, the at least one photon energy source includes a source having a third wavelength for applying a phase gate.

[0035] In one or more embodiments, the third wavelength is approximately 809 nm.

[0036] In one or more embodiments, the excitation source is configured to apply two half-π pulses.

[0037] In one or more embodiments, the excitation source is configured to apply a π pulse between the two half π pulses.

[0038] In one or more embodiments, the confinement system is a first controlled acousto-optic deflector (AOD), and wherein the system further comprises a holding well array having at least three wells spaced apart in two dimensions, the holding well array being generated by a holding well source.

[0039] In one or more embodiments, the holding well source includes at least one of at least one holding AOD, a spatial light modulator (SLM), and an optical lattice.

[0040] In one or more embodiments, the system also includes a second control AOD in a cross relationship with the first control AOD, wherein the first control AOD controls the deflection of the light beam in a first direction, and the second control AOD controls the deflection of the light beam from the first control AOD in a second direction different from the first direction.

[0041] In one or more embodiments, the confinement system is a first controlled acousto-optic deflector (AOD), and wherein the system further comprises a second controlled AOD in a stacked relationship with the first controlled AOD, wherein the first controlled AOD is configured to produce a plurality of confinement regions in a first array having a first direction, and the second controlled AOD is configured to produce a plurality of confinement regions in a second array substantially parallel to the first direction.

[0042] In one or more embodiments, a system for controlling an array of trapped atoms includes: a laser source, the laser source being used to generate a laser output; a laser source controller, the laser source controller controlling the laser source; a Pound-Dreif-Hall (PDH) lock, the PDH lock being optically coupled to the laser source, the PDH lock receiving at least some of the laser output and providing a feedback signal to the laser source controller to stabilize the laser output; a reference optical cavity, the reference optical cavity being optically coupled to the laser source, the reference optical cavity being configured to receive at least some of the laser output and to transmit a reference optical cavity output, the reference optical cavity output corresponding to a portion of the at least some of the laser output that falls within a reference optical cavity transmission window; and an optical isolator, the optical isolator being optically coupled to the reference optical cavity, the optical isolator being configured to split the reference optical cavity output and provide at least a portion of the split reference optical cavity output to a Fabry-Perot laser diode to injection lock the reference optical cavity output, wherein the optical isolator provides injection locked light to at least some of the trapped atoms.

[0043] In one or more embodiments, the PDH further includes a photodetector that receives at least some of the laser output and outputs a photodetector signal to the laser controller.

[0044] In one or more embodiments, the system further includes a second laser source for providing a second laser output having a different wavelength than the first laser output.

[0045] In one or more embodiments, the system further includes: a second laser source controller, the second laser source controller controlling the second laser source; a second Pound-Dreif Hall (PDH) lock, the second PDH lock optically coupled to the second laser source, the second PDH lock receiving at least some of the second laser output and providing a second feedback signal to the second laser source controller to stabilize the second laser output; a second reference optical cavity, the second reference optical cavity optically coupled to the second laser source, the second reference optical cavity configured to receive at least some of the second laser output and transmit a second reference optical cavity output, the second reference optical cavity output corresponding to a portion of the at least some of the second laser output that falls within a second reference optical cavity transmission window; and a second optical isolator, the second optical isolator optically coupled to the second reference optical cavity, the second optical isolator configured to split the second reference optical cavity output and provide at least a portion of the split second reference optical cavity output to a second Fabry-Perot laser diode to injection lock the second reference optical cavity output, wherein the second optical isolator provides second injection locked light to at least some of the trapped atoms.

[0046] In one or more embodiments, the second laser source generates light at approximately 1013 nm.

[0047] In one or more embodiments, the second reference optical cavity and the first reference optical cavity are the same element.

[0048] In one or more embodiments, the first injection-locked light and the second injection-locked light are provided to the at least some of the trapped atoms in a counter-propagating configuration.

[0049] In one or more embodiments, the first laser source generates light at approximately 420 nm.

[0050] In one or more embodiments, the system further includes an optical device optically positioned between the optical isolator and the array of trapped atoms, the optical device configured to focus the injection-locked light onto at least some of the trapped atoms.

[0051] In one or more embodiments, the system further comprises a spatially resolved imaging device configured to pick up at least a portion of the injection-locked light so as to align the injection-locked light.

[0052] The present invention also includes the following items:

[0053] 1. A method comprising:

[0054] forming an array of atoms in a first array state, wherein the forming comprises:

[0055] The crystal is excited by multiple discrete adjustable acoustic tone frequencies.

[0056] Passing laser light through the crystal to create multiple confinement regions, wherein each acoustic tone frequency corresponds to a separate confinement region of a single atom,

[0057] trapping at least two atoms in at least two of the plurality of confinement regions,

[0058] associating with said discrete adjustable acoustic tone frequency to identify said confinement region containing said trapped atoms, and

[0059] adjusting a spacing between at least two of the trapped atoms by sweeping at least one associated adjustable acoustic tone frequency;

[0060] evolving the plurality of atoms in the first array state to a plurality of atoms in a second array state by subjecting at least some of the trapped atoms to photon energy to transform at least some of the trapped atoms into an excited state; and

[0061] The plurality of atoms are observed in the second array state.

[0062] 2. The method of item 1, wherein the excited state is a Rydberg state.

[0063] 3. The method of claim 1, wherein the plurality of atoms in the first array state comprises 7 to 51 atoms.

[0064] 4. A method as described in item 1, wherein evolving the plurality of atoms includes preparing at least some of the atoms in the first array state to a ground state Zeeman sublevel before subjecting at least some of the atoms to photon energy.

[0065] 5. The method of item 4, wherein preparing the atoms in the first array state to a Zeeman sublevel of the ground state comprises optical pumping in a magnetic field.

[0066] 6. A method as described in item 1, wherein subjecting at least some of the atoms to photon energy includes applying light having two different wavelengths, and wherein converting at least some of the atoms to an excited state includes a two-photon transition.

[0067] 7. The method of item 6, wherein the two different wavelengths are approximately 420 nm and approximately 1013 nm.

[0068] 8. The method of item 6, further comprising applying a phase gate having a third wavelength.

[0069] 9. The method of item 8, wherein the third wavelength is approximately 809 nm.

[0070] 10. The method of claim 1, wherein subjecting at least some of the atoms to photon energy comprises applying two half-π pulses.

[0071] 11. The method of item 10, wherein subjecting said at least some of said atoms to photon energy further comprises applying a π pulse between said two half-π pulses.

[0072] 12. A method as described in item 1, wherein capturing the at least two atoms includes capturing at least two atoms from an atomic cloud and dispersing atoms from the atomic cloud that are not captured in one of the multiple confinement regions.

[0073] 13. A method as described in item 1, wherein the crystal and the laser constitute a first controlled acousto-optic deflector (AOD), and wherein capturing the at least two atoms includes capturing atoms from a holding well array, the holding well array having at least three wells spaced apart in two dimensions.

[0074] 14. A method as described in item 13, wherein the holding well array is generated by at least one of at least one holding AOD, a spatial light modulator (SLM) and an optical lattice.

[0075] 15. The method of item 13, further comprising: a second control AOD, the second control AOD being configured in a crossed relationship with the first control AOD, and wherein:

[0076] associating with the discrete adjustable acoustic tone frequencies to identify the confinement region containing the trapped atoms comprises associating with discrete adjustable acoustic tone frequencies of the first control AOD and the second control AOD, and

[0077] Adjusting the separation between the at least two of the trapped atoms includes sweeping at least one associated adjustable acoustic tone frequency of the first control AOD or the second control AOD.

[0078] 16. The method of clause 15, wherein adjusting the spacing between the at least two of the trapped atoms further comprises adjusting the positions of a plurality of atoms in a row.

[0079] 17. The method according to item 1, further comprising:

[0080] forming a second atomic array adjacent to the first atomic array in a third array state, wherein the forming comprises:

[0081] exciting the second crystal with a second plurality of discrete adjustable acoustic tone frequencies,

[0082] passing a second laser through the second crystal to form a plurality of second confinement regions, wherein each second acoustic tone frequency corresponds to a separate second confinement region of a single atom,

[0083] trapping at least two second atoms in at least two of the plurality of second confinement regions,

[0084] associating with said second discrete adjustable acoustic tone frequency to identify said second confinement region containing said trapped atoms, and

[0085] adjusting a spacing between at least two of the trapped second atoms by sweeping at least one second associated adjustable acoustic tone frequency;

[0086] wherein evolving the plurality of atoms in the first array state into a plurality of atoms in a second array state by subjecting at least some of the trapped atoms to photon energy so as to transform the at least some of the trapped atoms into the excited state further comprises evolving the plurality of second atoms in the third array state into a plurality of second atoms in a fourth array state by subjecting at least some of the second trapped atoms to photon energy so as to transform the at least some of the second trapped atoms into the excited state; and

[0087] Wherein observing the plurality of atoms in the second array state further comprises observing the plurality of second atoms in the fourth array state.

[0088] 18. The method of item 1, wherein:

[0089] Adjusting said spacing between at least two of said trapped atoms by sweeping at least one associated adjustable acoustic tone frequency comprises encoding a quantum computing problem;

[0090] evolving the plurality of atoms in the first array state to the plurality of atoms in the second array state produces a solution to the quantum computing problem; and

[0091] Observing the plurality of atoms in the second array state includes reading out the solution to the quantum computing problem.

[0092] 19. A method as described in item 18, wherein the quantum computing problem includes at least one of an Ising problem and a maximum independent set (MIS) optimization problem.

[0093] 20. A system comprising:

[0094] A confinement system for arranging an array of atoms in a first array state, the confinement system comprising:

[0095] Crystals,

[0096] an adjustable acoustic tone frequency application source configured to selectively apply a plurality of discrete adjustable acoustic tone frequencies to the crystal, and

[0097] a laser source arranged to pass light through the crystal to form a plurality of confinement regions, wherein each acoustic tone frequency corresponds to a separate confinement region,

[0098] a source of a cloud of atoms positionable to at least partially overlap the plurality of confinement regions;

[0099] an excitation source, the excitation source being used to cause at least some of the plurality of atoms in the first array state to evolve into a plurality of atoms in a second array state, the excitation source comprising at least one photon energy source;

[0100] An observation system is used to observe the plurality of atoms in the second array state.

[0101] 21. The system of item 20, wherein the excitation source is configured to excite at least some of the plurality of atoms in the first array state into Rydberg states.

[0102] 22. The system of claim 20, wherein the plurality of atoms in the first array state comprises 5 to 51 atoms.

[0103] 23. The system of claim 20, wherein the excitation source is configured to excite at least some of the plurality of atoms in the first array state to a Zeeman sublevel of a ground state prior to subjecting at least some of the atoms to photon energy.

[0104] 24. The system of item 23, wherein the excitation source further comprises an optical pumping system and a magnetic field generator.

[0105] 25. A system as described in item 20, wherein the at least one photon energy source includes a light source having a first wavelength and a second wavelength for producing two-photon transitions of at least some of the plurality of atoms in the first array state.

[0106] 26. The system of item 25, wherein the two different wavelengths are approximately 420 nm and approximately 1013 nm.

[0107] 27. A system as described in item 25, wherein the at least one photon energy source includes a source having a third wavelength for applying a phase gate.

[0108] 28. The system of item 27, wherein the third wavelength is approximately 809 nm.

[0109] 29. The system of item 28, wherein the excitation source is configured to apply two half-π pulses.

[0110] 30. The system of item 29, wherein the excitation source is configured to apply a π pulse between the two half π pulses.

[0111] 31. A system as described in item 20, wherein the confinement system is a first controlled acousto-optic deflector (AOD), and wherein the system also includes a holding well array, the holding well array having at least three wells spaced apart in two dimensions, the holding well array being generated by a holding well source.

[0112] 32. A system as described in item 31, wherein the holding trap source includes at least one of at least one holding AOD, a spatial light modulator (SLM), and an optical lattice.

[0113] 33. A method as described in Item 31, which also includes a second control AOD in a cross relationship with the first control AOD, wherein the first control AOD controls the deflection of the light beam in a first direction, and the second control AOD controls the deflection of the light beam from the first control AOD in a second direction different from the first direction.

[0114] 34. A method as described in item 20, wherein the restriction system is a first controlled acousto-optic deflector (AOD), and wherein the system also includes a second controlled AOD in a stacked relationship with the first controlled AOD, wherein the first controlled AOD is configured to produce a plurality of restricted areas in a first array having a first direction, and the second controlled AOD control is configured to produce a plurality of restricted areas in a second array substantially parallel to the first direction.

[0115] 35. A system for controlling an array of trapped atoms, comprising:

[0116] A laser source, the laser source is used to generate a laser output;

[0117] A laser source controller, the laser source controller controls the laser source;

[0118] a Pound-Driv Hall (PDH) lock optically coupled to the laser source, the PDH lock receiving at least some of the laser output and providing a feedback signal to the laser source controller to stabilize the laser output;

[0119] a reference optical cavity optically coupled to the laser source, the reference optical cavity configured to receive at least some of the laser output and to transmit a reference optical cavity output corresponding to a portion of the at least some of the laser output that falls within a reference optical cavity transmission window; and

[0120] an optical isolator optically coupled to the reference optical cavity, the optical isolator configured to split the reference optical cavity output and provide at least a portion of the split reference optical cavity output to a Fabry-Perot laser diode to injection lock the reference optical cavity output, wherein the optical isolator provides injection-locked light to at least some of the trapped atoms.

[0121] 36. The system of item 35, wherein the PDH further comprises a photodetector that receives at least some of the laser output and outputs a photodetector signal to the laser controller.

[0122] 37. The system of clause 35, further comprising a second laser source for providing a second laser output having a different wavelength than the first laser output.

[0123] 38. The system of item 37, further comprising:

[0124] a second laser source controller, the second laser source controller controlling the second laser source;

[0125] a second Pound-Driv Hall (PDH) lock optically coupled to the second laser source, the second PDH lock receiving at least some of the second laser output and providing a second feedback signal to the second laser source controller to stabilize the second laser output;

[0126] a second reference optical cavity optically coupled to the second laser source, the second reference optical cavity configured to receive at least some of the second laser output and to transmit a second reference optical cavity output corresponding to a portion of the at least some of the second laser output that falls within a second reference optical cavity transmission window; and

[0127] a second optical isolator optically coupled to the second reference optical cavity, the second optical isolator configured to split the second reference optical cavity output and provide at least a portion of the split second reference optical cavity output to a second Fabry-Perot laser diode to injection lock the second reference optical cavity output, wherein the second optical isolator provides second injection-locked light to at least some of the trapped atoms.

[0128] 39. The system of item 38, wherein the second laser source generates light at approximately 1013 nm.

[0129] 40. The system of clause 38, wherein the second reference optical cavity and the first reference optical cavity are the same element.

[0130] 41. The system of item 38, wherein the first injection-locked light and the second injection-locked light are provided to the at least some of the trapped atoms in a counter-propagating configuration.

[0131] 42. The system of item 35, wherein the first laser source generates light at approximately 420 nm.

[0132] 43. The system of item 35, further comprising an optical device optically positioned between the optical isolator and the array of trapped atoms, the optical device configured to focus the injection-locked light onto at least some of the trapped atoms.

[0133] 44. The system of item 35, further comprising a spatially resolved imaging device configured to pick up at least a portion of the injection-locked light so as to align the injection-locked light.

[0134] These and other capabilities of the disclosed subject matter will be more fully understood after reviewing the following drawings, detailed description and claims.It is to be understood that the phraseology and terminology employed herein are for the purpose of description and should not be regarded as limiting. BRIEF DESCRIPTION OF THE DRAWINGS

[0135] Various objects, features, and advantages of the disclosed subject matter may be more fully understood with reference to the following detailed description of the disclosed subject matter when considered in conjunction with the accompanying drawings in which like reference numerals identify like elements.

[0136] Figures 1A to 1F Aspects of systems and methods for preparing atomic arrays according to some embodiments are shown.

[0137] FIG. 2A to FIG. 2B Phase diagrams and creation of crystalline phases according to some embodiments are shown.

[0138] FIG. 3A to FIG. 3C A comparison of the methods described in this disclosure according to some embodiments with simulations is shown.

[0139] FIG. 4A to FIG. 4B Scaling behavior according to some embodiments is shown.

[0140] FIG. 5A to FIG. 5D An array of atoms and their properties are shown before and after adiabatic evolution, according to some embodiments.

[0141] FIG. 6A to FIG. 6D Shown is a graphical representation of oscillations in multibody dynamics, according to some embodiments.

[0142] FIG. 7A to FIG. 7D Shown are systems for single atom placement and manipulation and graphs characterizing control of the single atom placement and manipulation according to some embodiments.

[0143] FIG. 8A to FIG. 8C Graphs characterizing single-atom coherence and phase control are shown according to some embodiments.

[0144] 9A to 9C A graph characterizing the generation of entanglement with two atoms is shown.

[0145] Fig.10 A graph showing the extension of entangled state lifetime via dynamical decoupling according to one embodiment is shown.

[0146] FIG. 11A to FIG. 11B Examples of independent sets including maximal independent sets are shown according to some embodiments.

[0147] Fig.12 An example of a unit disk graph is shown according to one embodiment.

[0148] FIG. 13A to FIG. 13B An example of a unit disk plot indicating a maximum independent set, a probability distribution for finding an independent set, is shown according to some embodiments.

[0149] Fig.14A system for two-dimensional ordering of atoms is shown according to one embodiment.

[0150] FIG. 15A to FIG. 15H Methods for two-dimensional ordering of atoms according to some embodiments are shown.

[0151] Fig.16 A system for two-dimensional ordering of atoms is shown according to one embodiment.

[0152] FIG. 17A to FIG. 17E Methods for two-dimensional ordering of atoms according to some embodiments are shown.

[0153] 18A to 18H Methods for two-dimensional ordering of atoms according to some embodiments are shown.

[0154] FIG. 19A to FIG. 19B A system for two-dimensional ordering of atoms is shown according to one embodiment.

[0155] FIG. 20A to FIG. 20C Methods for two-dimensional ordering of atoms according to some embodiments are shown.

[0156] FIG. 21A to FIG. 21B Methods for two-dimensional ordering of atoms according to some embodiments are shown.

[0157] FIG. 22A to FIG. 22P A system for two-dimensional ordering of atoms is shown, according to some embodiments.

[0158] Fig.23 A system for two-dimensional ordering of atoms is shown according to one embodiment.

[0159] FIG. 24A to FIG. 24E Methods for two-dimensional ordering of atoms according to some embodiments are shown.

[0160] FIG. 25A to FIG. 25H Methods for two-dimensional ordering of atoms according to some embodiments are shown.

[0161] Fig.26 An image of an array of wells produced using an SLM, according to one embodiment, is shown.

[0162] Fig. 27 A spatial phase diagram according to one embodiment is shown. DETAILED DESCRIPTION

[0163] As quantum simulators, fully controlled coherent many-body quantum systems can provide unique insights into closely related quantum systems and the role of quantum entanglement, and enable the realization and study of new states of matter even far from equilibrium. These systems also form the basis for the realization of quantum information processors. Although the basic building blocks of such processors have been demonstrated in systems of several coupled qubits, it is a challenge to increase the number of coherently coupled qubits to perform tasks that are not possible with modern classical machines. In addition, current systems lack the coherence and / or quantum nonlinearities required to realize full quantum dynamics.

[0164] Neutral atoms can be used as building blocks for large-scale quantum systems. They can be well isolated from the environment, enabling long-term quantum storage. Initialization, control, and readout of the internal and motional states of neutral atoms are accomplished through resonance methods developed over the past four decades. Arrays with a large number of identical atoms can be assembled quickly while maintaining single-atom optical control. These bottom-up methods are complementary to methods involving optical lattices loaded with ultracold atoms prepared via evaporative cooling, and typically result in atomic separations of a few microns. Controllable interactions between atoms can be introduced to use these arrays for quantum simulations and quantum information processing. This can be achieved by coherent coupling to highly excited Rydberg states, which exhibit strong long-range interactions. This approach provides a powerful platform for many applications, including fast multi-qubit quantum gates, quantum simulations of Ising-type spin models with up to 250 spins, and the study of collective behavior in mesoscopic ensembles. The short coherence times and relatively low gate fidelity associated with such Rydberg excitations are challenging. This imperfect coherence can limit the quality of quantum simulations and may dim the prospects for neutral atom quantum information processing. Even at the level of single isolated atomic qubits, limited coherence becomes apparent.

[0165] The present disclosure describes embodiments related to quantum computing. According to some embodiments, methods and systems for quantum computing involve first capturing individual atoms and arranging them into specific geometric configurations of multiple atoms, for example using the disclosed acousto-optic deflector systems and techniques. Systems and methods that allow precise placement of individual atoms help encode quantum computing problems. Next, one or more of the arranged atoms can be excited into Rydberg states, as described below, which produce specific interactions between atoms in the array. Next, the system can evolve. Finally, the state of the atoms can be read out to observe the solution to the encoded problem. According to some embodiments, the systems and methods described herein involve: (1) arranging atoms into 1D and / or 2D arrays (see Section 1); (2) using, for example, laser manipulation techniques and high-fidelity laser systems described herein to excite atoms into Rydberg states and evolve atomic arrays (see Section 2); and (3) observing the resulting final state (see Section 3). In addition, additional improvements to the systems and methods described herein are described in Section 4, such as to provide high-fidelity and coherent control of assembled atomic arrays. Additionally, in Section 5, exemplary problems that can be solved using the systems and methods described herein are discussed.

[0166] Chapter 1: Arranging atoms into 1D and / or 2D arrays

[0167] According to some embodiments, the state and position of neutral atoms can be strictly controlled in a vacuum using a focused laser so that the atoms are arranged into arrays for encoding problems that can be solved by quantum computing. The system and method described in the present disclosure provide control of a larger number of atoms than previously reported, which greatly expands the range of problems that such quantum systems can solve. These atoms can be prepared in 1D or 2D arrays using, for example, acoustically excited crystals and lasers. The system and method described in the present disclosure allow for fine control of the position of individual atoms in these 1D or 2D arrays.

[0168] An initial state of atoms can be prepared according to the techniques described in the present disclosure, and the system can then be adiabatically evolved to produce a solution. Thus, the system can be prepared in a minimum energy state for a given initial parameter, and then slowly evolved to its final value so that the system remains in the instantaneous minimum energy state. In addition, according to some embodiments, such as in conjunction with FIG. 7 to FIG. Fig.10 As described, a specific laser control technique allows high-fidelity and coherent control of individual atoms after they are arranged in a 1D or 2D array. This technique allows for better control of quantum mechanical systems, thus producing more accurate solutions to encoded problems.

[0169] Section 1.A: Acousto-optic deflectors for arranging atoms into 1D arrays

[0170] According to one embodiment, a positioning system and method provide for preparing a large number of atoms, such as 51 or more, to be encoded by an initial question. The laser can pass through a crystal, which can be controlled with one or more discrete acoustic tones (tone frequencies that vibrate the crystal) to form discrete deflections of the laser path corresponding to the applied tone frequencies. The number of deflections can be controlled by the number of tone frequencies provided to the crystal. The tone frequency is an electronic radio frequency signal in a VHF band selected, for example, between 75MHz and 125MHz. The tone frequency can include a sound wave with a narrower frequency. Multiple tone frequencies can be superimposed to produce a signal including multiple tones. These tone frequencies can then be applied to the crystal to cause compression or vibration of the crystal. According to some embodiments, these tone frequencies can be provided to the crystal using, for example, one or more piezoelectric transducers that can be coupled to the crystal. According to some embodiments, the tone frequencies can be selected based on the acoustic and / or other properties of the crystal. In addition, adjusting the frequency of each individual tone can adjust the deflection amount of each discrete deflection, thereby forming a controllable spacing between each deflected beam. The tone frequencies may be converted to analog signals by software defined radio (SDR) from digital waveforms generated, for example, by a computer controller, or an arbitrary waveform generator may be implemented that synthesizes a superposition of all desired tone frequencies in the time domain. The frequency may be adjusted by updating the digital waveform output by the computer controller.

[0171] The frequency of the tone used depends on many factors, such as, but not limited to, the desired number of deflections, the specific crystal used, the wavelength of the light applied to the crystal, and the desired spacing of the confined area / well. The frequency range of the sound waves applied to the crystal can have limitations based on the speed of sound in the crystal material, and can be, for example, between 50 MHz and 500 MHz. According to some embodiments, a set of discrete frequencies in the range of about 75 MHZ to 125 MHZ can be used. According to some embodiments, 100 wells can be generated by spacing the individual tone frequencies about 0.5 MHz apart. According to some embodiments, when the individual tone frequencies are adjusted to the spacing of the wells, the angle that varies with the applied acoustic frequency can be about .1 mrad / MHz to 10 mrad / MHz. An exemplary value can be about 1.2 mrad / MHz. However, this value is exemplary only and varies significantly based on the specific crystal and optical device used.

[0172] The output laser array beam from the crystal can be focused into the cooling atom cloud. The atom can be cooled by the radiation pressure of the additional counter-propagating laser beam, wherein the quadrupole magnetic field causes a spatially dependent radiation pressure, which counteracts the atomic motion in all directions and generates a restoring force toward the center of the quadrupole field. The output laser array can be focused so that each laser array beam can only capture a single atom. The cloud can then disperse, leaving only the trapped atoms. Then the image of the atoms in the trap can be obtained, for example, based on the light scattering of the atoms. In the measurement and feedback process, the trapped atoms can then be associated with the individual tone frequencies provided to the crystal, for example, by averaging a large number of images of the atoms carried in the trap. The established position can be recorded and assigned to a separate tone frequency. When the atoms are subsequently loaded into the trap, the tone frequencies of the trapped atoms without the associated trapping can be turned off after the image is obtained and the trap position without atomic fluorescence is located. Each remaining tone frequency (for example, those tone frequencies that have not been turned off) can then be swept by adjusting each tone frequency to locate the remaining trapped atoms. Thus, according to some embodiments, even if each confinement region does not capture atoms, the system can be adjusted to reposition the confinement regions to form the desired array spacing. This technique and system can significantly increase the number of atoms that can be reliably captured in an array, such as 51 atoms or more, and allow precise control of the atoms spaced apart. After arranging the atoms in an array, the system can be excited and allowed to evolve, as discussed in more detail below, and the resulting changes observed to read out a solution to the encoded problem.

[0173] Figure 1C to Figure 1E An experimental protocol and setup for arranging and evolving an array of atoms according to one embodiment is shown. Figure 1E Schematic diagram of a system for arranging and controlling an array of atoms according to some embodiments is shown. Figure 1E As shown, the system may include a crystal 102, an optical tweezers laser source 106, an acoustic tone generator 104, and steering laser sources 108A, 108B. The acoustic tone generator generates one or more (n) tone frequencies, which are applied to the crystal 102. The optical tweezers laser source 106 supplies light to the crystal 102, which is then deflected into n individual optical tweezers beams forming an optical tweezers array 107, each optical tweezers beam being associated with one of the one or more tone frequencies. The frequency of each individual tone frequency determines the deflection of the corresponding optical tweezers beam. The optical tweezers beam can be used to capture atoms 190. The frequency of the individual tone frequencies can be adjusted to adjust the spacing of atoms 190. The atoms 190 can then be manipulated by steering laser sources 108A and 108B to evolve the system.

[0174] First, if Figure 1DAs shown, in step 110, atoms are loaded from a magneto-optical trap (not shown) into an optical tweezers array 107 formed by an acousto-optic deflector (AOD), which includes, for example, a crystal 102 and an acoustic tone generator 104. For example, as described above, an atomic cloud can be prepared, for example, in a magneto-optical trap. The AOD can provide multiple tone frequencies to the crystal 102 via the acoustic tone generator 104 to form a 1D optical tweezers array 107. Each laser optical tweezer is associated with a discrete tone frequency applied to the crystal. The atoms can then be brought to the optical tweezers array 107 (or vice versa) so that individual atoms 190 are trapped in the optical tweezers associated with the tone frequency applied to the crystal 102. As Figure 1C As shown, the spacing of the loaded atoms can be arbitrary due to at least two reasons. First, it is not guaranteed that every tone frequency can capture atoms from the cloud. Therefore, some tone frequencies may not be associated with atoms. Secondly, it is not necessary to prepare the tone frequencies in a specific ordered state in order to capture atoms. Therefore, atoms 190 can be spaced at arbitrary frequencies (and therefore at arbitrary relative spacing) before and during the loading process. After the atoms are captured in the AOD optical tweezers, the atomic cloud can be dispersed. The measurement and feedback process discussed above can be used to eliminate the entropy associated with probability well loading and lead to the rapid generation of defect-free arrays with more than 50 laser-cooled atoms.

[0175] Next, at step 120, the trapped atoms 190 may be prepared in a pre-programmed spatial configuration with a well-defined internal ground state g. As described above, each atom in the array 190 may be associated with one of the tone frequencies applied to the crystal 102, but not all final wells may be occupied by trapped atoms. The trapped atoms may be imaged and the positions of the atoms may be correlated to the tone frequencies. The occupied tone frequencies may be recorded and maintained while the unoccupied tone frequencies may be segmented out from the acoustic signal from the acoustic tone generator 104 applied to the crystal 102. Next, the occupied tone frequencies may be adjusted to rearrange the atoms 190 into, for example, Figure 1C Since the relative spacing of each deflected laser beam in the optical tweezers array 107 (and therefore each optical tweezer) depends on the specific tone that causes the deflection, the relative spacing of the optical tweezers in the optical tweezers array 107 can be adjusted by adjusting the individual tone frequencies supplied by the acoustic tone generator 104. Therefore, each trapped atom 190 can be repositioned in 1D by adjusting its associated tone frequency. Figure 1C As shown, by using this technique, arbitrarily spaced atoms at loading locations can be arranged in a specific pattern, such as in evenly spaced groups of seven atoms.

[0176] In step 114, the optical trap or optical tweezers can then be turned off, for example by turning off the optical tweezer laser source 106, so that the system evolves under the unitary time evolution U(Ω,Δ,t) for a sufficiently short time so as to neglect the atomic motion while avoiding the detrimental effects of the trapping potential on the unitary evolution U(t). Figure 1A And as shown in step 116, the atoms can be coupled into the Rydberg state |r>=|70S by laser along the array axis. 1 / 2 > to realize the evolution U(t). Figure 1E As shown, laser light from one or more steering lasers 108A, 108B may be applied to the atomic array 190 during the evolution. Figure 1C The application of the function U(t) is shown in FIG. Fig.10 Various control techniques are discussed. In step 118, the final state of the individual atoms is detected by reopening the trap and imaging the re-trapped ground state atoms via atomic fluorescence using, for example, a camera while ejecting the anti-trapped Rydberg atoms. Figure 1C The detection line is shown in , where the dots represent the atomic fluorescence associated with the current ground state atoms, and the circles mark the positions where atoms are lost due to Rydberg excitation.

[0177] Section 1.B: Two-dimensional atomic ordering for encoding more complex problems

[0178] According to some embodiments, the techniques described in the present disclosure can be applied and adapted to prepare arrays of hundreds of individual atoms arranged in 2D geometric structures. Compared with 1D arrays, 2D atomic arrays can be used to solve a wider range of problems. Although some techniques and systems described in the present disclosure refer to 1D arrays, they can be applied to 2D arrays using the 2D array systems and methods described below. In combination with the laser control systems and methods described below for controlling and manipulating atoms into Rydberg and other quantum states, quantum optimization algorithms can be implemented to solve real-world problems, such as but not limited to the maximum independent set optimization problem described in the present disclosure.

[0179] According to some embodiments, the systems and methods described in the present disclosure provide for forming a large number of wells in 2D. When atoms are loaded into these wells, they are loaded into each well with a finite probability of about 0.5. After identifying the positions of the atoms, the process performed in 1D can then sort the atoms. According to some additional embodiments, this sorting (or "rearrangement") process can be applied to a 2D array of wells.

[0180] Crossed AODs: According to some embodiments, an acousto-optic deflector (AOD) can be used to form multiple copies of a single beam in one direction to generate a 1D pattern (see above for a description of the operation of the AODs). A second AOD can then be used to form copies of the one-dimensional system in a different (e.g., perpendicular) direction to generate a 2D pattern. Other orientations can be considered. Next, atoms can be loaded into this pattern and defects can be eliminated by continuously turning off the frequency of the row or column in either AOD corresponding to the maximum number of empty wells. Next, the frequency can be modified on both AODs to transform the fully loaded pattern into a target 2D array.

[0181] like Fig.14 As shown, according to some embodiments, two AODs 1410 and 1420 can be placed close to each other and used to generate a set of 2D traps. Each AOD 1410 and 1420 can have an acoustic driver driven by RF signals 1450A and 1450B, respectively, to separate the incident light 1440 according to the specific tone frequency applied. When the 1D light array output from AOD 1410 is incident on AOD 1420, AOD 1420 can further separate each light beam in the 1D array into additional light beams in another direction. Fig.14 As shown, AOD 1410 and 1420 can be held at a relative angle (e.g., 90 degrees) to each other. The relative angle between the two AODs 1410 and 1420 determines the relative orientation of the "rows" and "columns" of the 2D capture pattern. For example, two AODs 1410 and 1420 perpendicular to each other can form a square pattern 1430; two AODs with a relative angle of 60 degrees can form a triangular pattern (not shown). This process can form any geometric shape, in which all "rows" are identical to each other and all "columns" are identical to each other. Atoms can be loaded into a collection of traps generated using this technology. The frequencies associated with individual "rows" (AOD1) and "columns" (AOD2) can be turned off to ensure that the remaining traps contain atoms of the desired pattern. The frequency set in each AOD can be modified to transmit atoms to the final configuration of any distance between "columns" and "rows".

[0182] FIG. 15A to FIG. 15D According to some embodiments, a method for using Fig.14 The system shown captures the process of multiple atoms in a 2D array. Specifically, FIG. 15A to FIG. 15D The process of forming a 3x3 array of wells is shown, all filled with atoms. It will be appreciated that similar techniques can be used to form other configurations, such as somewhere where some spaces are left empty. Fig.15AThe outputs of AODs 1410 and 1420 are shown at right angles to produce a square well configuration 1510a having 6 rows and 6 columns. The positions of the wells are indicated by the intersections of lines 1520 in the square pattern 1510, and atoms are indicated by solid circles 1530. Fig. 15B Array 1510B is shown having rows and columns 1540 marked with "Xs" that may be removed while retaining three trapped atoms per remaining row and column. Fig. 15C An array 1510C is shown without rows and columns 1540, which may be dropped by dropping the associated tone frequency applied to one of the AODs 1410 and 1420, as described herein. The frequencies of the remaining rows and columns may be adjusted to induce motion in direction 1550 to form a uniform spacing of the 3×3 atomic array 1510D, as shown. Fig.15D As shown. It should be understood that these techniques can be scaled up to larger arrays and can be used to form configurations where not all intersections of lines (i.e., wells) are filled. According to some embodiments, rows and columns are paired to control the position of exactly one atom at the intersection. Atoms can be arranged in a desired pattern by adjusting the frequency of the associated rows / columns.

[0183] Fig.15E An example of a randomly loaded 8x8 array is shown. The fluorescence of the atoms is shown as black dots, the underlying pattern of which can be inferred from the positions of the atoms present. FIG. 15F to FIG. 15H A randomly loaded 2 x 40 array is shown, with the fluorescence of trapped atoms shown by the dark areas. Figure 15G The crosses in the box indicate all closed columns. Fig.15H The final fluorescent image in shows atoms as dark areas in their new positions after rearrangement into a 2×18 fully loaded array.

[0184] Row-by-row rearrangement in two pairs of crossed AODs: According to some embodiments, two sets of AOD pairs as described above can be used to generate a two-dimensional array of optical tweezers to load a 2D array. Fig.16As shown, two AOD pairs 1600A and 1600B can be used. AOD pair 1600A can include AODs 1610A and 1620A close to each other for generating a 2D set of traps 1630A. The relative angle between the two AODs 1610A and 1620A determines the relative orientation of the "rows" and "columns" of the 2D capture pattern 1630, as described above. Atoms can be loaded into the set of traps generated in this way, as discussed above with reference to AODs 1410 and 1420. Another pair of AODS 1600B (pair 2) is used to generate different sets of traps 1630B using AODs 1610B and 1620B. These traps can overlap with the traps generated with pair 1. This can be accomplished, for example, by using optical elements 1660, such as, but not limited to, semi-reflective surfaces (e.g., non-polarizing beam splitters), surfaces that transmit one polarization of light and reflect an orthogonal one (e.g., polarizing beam splitters), optical elements that transmit light at some wavelengths and reflect at other wavelengths (e.g., dichroics), or by having two sets of traps come from different directions and meet at their focal points. The set of traps used for rearrangement (pair 2) can be adjusted to form stronger confinement than those used for loading (pair 1), such as by having each beam have greater optical power, have a wavelength closer to the atomic resonance, or have a smaller focal point (not an extensive list, and combinations of these effects can be used). Fig.17A An array of traps formed via AOD pair 1600A is shown, where a trap is a line intersection and a dot represents a trapped atom. Fig. 17B Overlapping a "row" or "column" of traps generated by AOD pair 1600A) on top of a "row" or "column" of traps generated by AOD pair 1600A can allow all atoms in that "row" or "column" to be dominated by the traps generated by AOD pair 1600B. In addition, according to some embodiments, once the traps from AOD pair 1600B have overlapped a "row" or "column", it is possible to turn off that particular "row" or "column" in AOD pair 1600A. The traps generated by AOD pair 1600B can now be rearranged using the process described for the 1D case (see Fig. 17C ). After rearrangement, if a particular "row" or "column" from AOD pair 1600A had been closed, then row 1 of AOD pair 1600A may be reopened. Closing the trap created with AOD pair 1600B may now allow the atoms to be held in their new positions by the underlying trap created with AOD pair 1600A (see Fig.17D). This process can now be repeated for different "rows" or "columns" of traps generated with AOD pair 1600A by changing the frequency associated with the AOD in AOD pair 1600B, so that a new set of traps can be generated with AOD pair 1600B that overlap with the new target "rows" or "columns" of traps generated with AOD pair 1600B. Fig.17E The array is shown after all rows are shifted to the left using this process. It should be understood that in 1D, where only AOD pair 1600B needs to be manipulated, AOD pair 1600B can be replaced by a single AOD. This AOD (or AOD pair) can be referred to as a "control AOD", while the AOD pair 1600A used to maintain the well can be referred to as a "holding AOD". According to some embodiments, the holding AOD can be viewed as a set of pixels filled by controlling the control AOD so as to form any desired pattern with a certain resolution based on the holding AOD. According to some embodiments, the trapped atoms are moved based on other configurations of the control AOD that do not correspond to entire rows or columns (e.g., squares with atoms at each edge).

[0185] about 18A to 18H A similar process is shown, but in which a single atom is moved via control of the AOD. Fig.18A As shown, the AOD is maintained with the initial set of atoms trapped therein. Fig.18B As shown, individual atoms are selected by controlling the AOD (shown as circles). Fig. 18C Shows the movement of individual atoms by controlling the AOD and then by shutting down Fig.18D The control AOD in the Fig.18E , select another atom and Fig.18F and then released in 18G. Fig.18H The array is shown trapped in a holding AOD after alignment. The resulting atomic array can have a higher yield.

[0186] According to some embodiments, the above method can also be performed without closing the row or column of the holding AOD. Instead, the control AOD can be formed to have a deeper (stronger) trap than the holding AOD. Therefore, when the trap of the control AOD overlapping the holding AOD trap moves, the atom will follow the deeper trap of the control AOD, as if the atom is not captured by the holding AOD. However, if it moves to another holding AOD trap and then closes the control AOD trap, the atom may still be captured by the holding AOD.

[0187] Stacked AODs: According to some embodiments, multiple (N) AODs can be stacked on top of each other. Multiple AODs can be used in parallel to generate a ID collection of independent wells in which atoms are trapped and rearranged. Fig.19A and Fig.19B An exemplary system for combining different 1D sets of wells into a 2D pattern is shown. Fig.19A As shown, a set of five AODs 1910 viewed from a top angle are aligned to send an array of light to a faceted reflective surface 1920. The light is then reflected from the faceted reflective surface 1920 to a lens 1930 which focuses the light beam. Fig.19B The same elements are shown in side view. The multi-faceted reflective surface 1920 can be, for example, a polished reflective surface, a combination of several reflective surfaces, the first two but with a non-reflective substrate coated with a reflective material, a deformable mirror or a mirror array (the list is not extensive. Each 1D set of traps from the AOD 1910 can be redirected via the multi-faceted reflective surface 1920. A transmissive multi-faceted surface with a controllable refractive index can also be used, such as a dielectric of variable density or width, or a patterned dielectric with holes or alternating sections with different refractive indices, or other suitable surfaces. This structure can be used to form a wavefront equivalent to multiple beams overlapping in a well-defined volume. A lens 1930 can be used to focus all beams onto a desired plane. By using N independent AODs 1910 (each AOD forming an independent 1D pattern of traps), atoms can be loaded into such traps and each independent 1D set of traps can be rearranged in a desired manner to position the atoms in a desired pattern. The rearrangement of all 1D sets of traps can occur simultaneously, or in any desired order.

[0188] FIG. 20A to FIG. 20C illustrates operation according to some embodiments FIG. 19A to FIG. 19B A systematic approach. Fig. 20A As shown, a 6×6 array of wells is produced. The N rows correspond to 1D AODs 2010A to 2010N (6 in this case). Independent 1D sets of wells from AODs 2010A to 2010N are represented by small open circles connected by straight lines, and atoms are represented by solid circles. Fig. 20B As shown, a square 6×6 array of wells is randomly loaded. Fig. 20C As shown, the wells are rearranged to collect all atoms on the left side of each 1D set of wells. The rearrangement may occur by adjusting the tone frequency associated with the well for each AOD 2010A to 2010N. It should be understood that these individual AOD arrays may be implemented as control and / or retention AODs as described above.

[0189] Trap generation in an SLM and rearrangement with crossed AODs: According to some embodiments, a spatial light modulator (SLM) can be used to generate a 2D array of traps. The SLM can be used to modify the wavefront of a light beam to generate an arbitrary spatial pattern of light intensity, which can be used instead of the holding AODs as described above to hold the trap array. There are different types of SLMs that affect the local intensity (example: digital micromirror device, DMD), phase (liquid crystal on silicon, LCOS) or both of the transmitted or reflected light field in order to change its wavefront in a programmable manner. Using such a device, a 2D pattern of traps with arbitrary geometry can be generated, and atoms can be loaded into a collection of traps generated in this way, such as via one or more controlled AODs as described above. Fig.21A An embodiment is shown in which a reflective SLM 2110A is used as a holding SLM and a pair of AODs 2120A are used to manipulate the positions of atoms in an array 2130A. Fig. 22B An embodiment is shown where a transmissive SLM 2110B is used as a holding SLM and a pair of AODs 2120B are used to manipulate the positions of atoms in an array 2130B. According to some embodiments, a pair of AODs 2120A or 2120B may be used to generate different sets of wells loaded with atoms. Fig.22A and Fig. 22BThe traps 2130A or 2130B shown in the can overlap with the traps generated with the SLM. This can be achieved, for example, by element 2140, or by having two sets of traps come from different directions and meet at their focal points, such as optical elements such as semi-reflective surfaces (example: non-polarizing beam splitters), surfaces that transmit one polarization of light and reflect an orthogonal polarization of light (example: polarizing beam splitters), optical elements that transmit light at some wavelengths and reflect at other wavelengths (e.g., dichroics). The set of traps generated with the AOD 2120A or 2120B for rearrangement can be adjusted to form stronger confinement than those generated with the SLM 2110A or 2110B for loading, for example by having each beam have greater optical power, have a wavelength closer to the resonance of the atoms, and / or have a smaller focal point (not an extensive list, and combinations of these effects can be used). Overlapping a row of wells generated by AOD pair 2120A or 2120B on top of a subset of wells generated by SLM 2110A or 2110B can allow all atoms within the subset to be primarily controlled by the wells generated by the AOD pair. Wells generated by AOD pair 2120A or 2120B can therefore be rearranged within the 2D plane of the wells so that they overlap simultaneously or sequentially with other wells generated by SLM 2110A or 2110B. Closing specific wells generated by AOD pair 2120A or 2120B when they overlap with other wells generated by SLM 2110A or 2110B can allow atoms to be held in their new positions by the well below generated by SLM 2110A or 2110B. This process (which simulates if SLM 2110A or 2110B is regular in spacing between wells) can be used to generate a large number of wells. FIG. 17A to FIG. 18D or 18A to 18H The process shown in ) can be repeated for different subsets of traps generated by SLM 2110A or 2110B by changing the frequency associated with the AOD pair, so that a new set of traps can be generated with AOD pair 2120A or 2120B, which overlaps with a new target subset of traps generated with SLM 2110A or 2110B.

[0190] According to some embodiments, the locations of the wells in SLM 2110A or 2110B include separate arbitrary and regular locations. Fig.22A The pattern generated by an arbitrary portion of SLM 2110A or 2110B is shown, the arbitrary portion being indicated by the small open circle near the top. Also shown below is an array of atoms trapped in a regular portion of SLM 2110A or 2110B (the distance between the two may or may not be to scale). The positions of the traps in the regular array are indicated by the intersection of the lines, and the atoms are indicated by the solid circles. According to some embodiments, the control AOD (shown as a large open circle) can be used to reposition the atoms in an arbitrary array, such as FIG. 22A to FIG. 22D As shown, and atoms are taken from the regular array and moved to the arbitrary array for repositioning, as FIG. 22E to FIG. 22P It should be noted that multiple atoms can be moved from the regular part to be placed in the irregular part at the same time, as shown in FIG. 22E to FIG. 22F and Figures 22L to 22M After moving to the irregular part, the horizontal spacing between the controlled atoms can be adjusted first, as shown in Fig.22F and Figures 22M to 22N shown. Fig.26 An illustrative example of a 30x50 regular array of wells generated with a reflective LCOS-SLM is shown, where a corresponding phase pattern is imprinted on the wavefront of the reflected light beam by the LCOS-SLM. Fig. 27 The corresponding spatial phase pattern added to the laser field to convert a single input beam into a 30×50 array of wells is shown.

[0191] Generating an optical lattice and using traps to rearrange atoms therein: According to some embodiments, a 2D trap array can be generated alternatively by using an optical lattice with a large lattice spacing. The interference from two light sources can form a light intensity pattern that can be used to trap cold neutral atoms, which is called an optical lattice. These traps can be used as a holding trap array instead of holding AODs. Thus, an optical lattice can be used to hold trapped atoms and combined with a control AOD, such as described above, to, for example, utilize FIG. 17A to FIG. 17E (Similarly in FIG. 24A to FIG. 24E ) and 18A to 18H (Similarly in FIG. 25A to FIG. 25H The method and system described in FIG. 2 are used to rearrange atoms in an optical lattice. Fig.23As shown, the light source 2320 can generate an interference form 2330 that forms a holding trap array. AODs 2310A, 2310B can be used to manipulate the trapped atoms of the interference form 2330. This pair of AODS is used to generate a set of traps. These traps can overlap with the capture area of ​​the optical lattice. This can be achieved, for example, by using a semi-reflective surface (example: non-polarizing beam splitter), a surface that transmits one light polarization and reflects a perpendicular light polarization (example: polarizing beam splitter), an optical element that transmits light at certain wavelengths and reflects at other wavelengths (for example, dichroic), or by making two sets of traps come from different directions and meet at their focus. The set of traps used for rearrangement can be adjusted to form a stronger constraint than the trap provided by the optical lattice used for loading, for example by making each beam have a larger optical power, a wavelength closer to the atomic resonance, or a smaller focus (non-extensive list, and a combination of these effects can be used). Overlapping a row of traps generated by AOD pairs on top of a "row" or "column" of an optical lattice can allow all atoms in that "row" or "column" to be primarily controlled by the traps generated by the AOD pairs. The traps generated by the AOD pairs can be rearranged using the process described for the 1D case. Turning off the traps generated by the AOD pairs can allow the atoms to remain in their new positions by the underlying optical lattice. This process can be repeated for different "rows" or "columns" of the optical lattice by changing the frequency associated with the AODs, so that a new set of traps can be generated with the AOD pairs that overlap with a new target "row" or "column" of the optical lattice.

[0192] Chapter 2. Inspiration and Evolution

[0193] According to one embodiment, the arranged atomic array can then be excited and evolved to calculate the answer to the encoded problem. A laser with a photon energy approximately equal to the transition energy of the outermost electron of the atom can be used to excite the outermost electron in the atom, thereby transforming the atom into an excited state. Specific laser control and application techniques are described in more detail in the present disclosure. The interaction between atoms may be so strong that only some atoms and especially only some atoms in a specific area may be transformed into an excited state. For example, approaching another excited atom can increase the excitation energy of nearby non-excited atoms, making the transition of nearby atoms impossible. The possibility of atomic transitions can initially be controlled by the distance between individual atoms. According to one embodiment, the atoms leaving can be diffused away from the well, and the remaining atoms can be imaged to determine which atoms are not excited. This final result can produce a solution to the encoded problem.

[0194] According to one embodiment, atom-by-atom assembly can be used to deterministically prepare individually trapped cold neutral atoms in optical tweezers. 87 An array of Rb atoms. Figure 1AAs shown, a controlled coherent interaction between atoms 190 can be introduced by coupling the atoms to Rydberg states. This results in a repulsive van der Waals interaction (Vi = C / R 6 , C > 0) between pairs of Rydberg atoms at a distance Rij. The quantum dynamics of such a system is described by the following Hamiltonian equation (1):

[0195]

[0196] where Ωi is the Rabi frequency associated with an individual atom, Δi is the detuning of the driving laser from the Rydberg state (see Figure 1B ), σi = |gi><ri| + |ri><gi| describes the coupling between the ground state |g> and the Rydberg state |r> of the atom at position i, and ni = |ri><ri|. Typically, within this platform, the control parameters Ωi, Δi can be programmed by changing the laser intensity and detuning in a timely manner. According to one embodiment, a uniform coherent coupling (|Ωi| = Ω, Δi = Δ) can be used. The interaction strength Vij can be tuned by changing the distance between the atoms or by carefully selecting the desired Rydberg state.

[0197] The ground state |g> and the Rydberg state |r> can be used as qubit states to encode quantum information. The coherent coupling between these states is provided by the laser and allows manipulation of the qubits. In addition, the Rydberg states of multiple atoms interact strongly with each other, enabling engineered coherent interactions. These strong coherent interactions between Rydberg atoms can provide effective confinement, which prevents nearby atoms from being simultaneously excited to the Rydberg state. Figure 1F This effect, sometimes referred to as Rydberg blockade, is shown. When two atoms are close enough such that their Rydberg-Rydberg interaction Vij exceeds the effective Rabi frequency Ω, multiple Rydberg excitations can be suppressed. This provides a Rydberg blockade radius Rb, where Vij = Ω (as used herein, for |r> = |70S>, Rb = 9 μm, and Ω = 2π x 2 MHz). In the case of resonantly driving atoms separated by a distance a = 24 μm, we observe Rabi oscillations associated with non-interacting atoms, as shown in the top curve of Figure 1F . However, when we bring multiple atoms closer to each other (a = 2.95 μm < Rb), the dynamics change significantly. In this case, Rabi oscillations between the ground state and a collective W state can be observed, where exactly one excitation ~ ∑ i Ω i |g1…r i …g N > has a characteristic N with a collective Rabi frequency 1 / 2 scaling. These observations allow the coherent properties of the system to be quantified. Specifically, Figure 1FThe contrast of the Rabi oscillations in is mainly limited by the state detection fidelity (93% for r and 98% for g). The individual Rabi frequencies and detuning are controlled to better than 3% over the entire array, while the coherence time is ultimately limited by the probability of spontaneous emission from the state |e> during the laser pulse (scattering rate 0.022 / μs).

[0198] like Figure 1A As shown, optical tweezers are used to capture individual 87 Rb atoms and arrange them into a defect-free array. The coherent interaction Vij between atoms is achieved by exciting the atoms into Rydberg states with intensity Ω and detuning Δ.

[0199] Figure 1B It is shown that a two-photon process can be used to convert the ground state |g>=|5S using circularly polarized 420 nm and 1013 nm lasers. 1 / 2 , F = 2, mF = -2> via the intermediate state |e> = |6P 3 / 2 , F=3,m F =-3> coupled to the Rydberg state |r> = |71S 1 / 2 , J = 1 / 2, m J =-1 / 2>(usually δ~2πx 560MHz>>Ω B ,Ω R ~2πx 60,36MHz).

[0200] like Figure 1C As shown, the experimental protocol may include loading atoms into an optical tweezers array (1) and rearranging them into a pre-programmed configuration (2). Thereafter, the system may be evolved under U(t) with tunable parameters Δ(t), Ω(t), Vij. This may be achieved in parallel on several non-interacting subsystems (3). The final state may be detected by appropriate techniques, such as using fluorescence imaging (4).

[0201] like Figure 1F As shown, for resonant driving (Δ = 0), isolated atoms (top points) show Rabi oscillations between |g> and |r>. Arranging the atoms into fully blocked clusters of N = 2 (as shown in the middle graph) and N = 3 (as shown in the bottom graph) atoms results in only one excitation being shared between the atoms in the cluster, and the Rabi frequency is enhanced by N 1 / 2 Multiple excitations (middle and bottom points) were strongly suppressed. Error bars indicate 68% confidence intervals (CI) and are smaller than the marker size.

[0202] Section 2.A: Exemplary Control and Evolution of 1D Atomic Arrays

[0203] Finding ways to engineer and control large quantum systems is a major challenge in quantum computing. The control and evolution examples discussed below allow arrays of up to 51 atoms or more that can be used as qubits, with a great deal of controllability and engineered coherent interactions between them. In addition, as discussed in the present disclosure, such systems are very suitable for scaling up to a larger number of atoms and controllability at the level of individual atoms. The techniques and experiments described in the present disclosure show that it is possible to engineer and control such a large number of subsystems. Such control is required to perform quantum simulations. Such quantum simulations can be used to solve other real-world problems, such as finding new materials (a famous example is high-temperature superconductivity), understanding complex molecular structures, and making new designs. Other applications include optimization problems, such as the maximum independent set problem discussed in more detail below. These optimization problems map directly to real-world problems.

[0204] According to some embodiments, controlled arrays of Rydberg crystals or Rydberg atoms can be formed, as discussed in more detail below. These Rydberg crystals provide a good test bed for large quantum systems produced using the methods and systems discussed herein. Generally, since complexity increases exponentially with the increase in system size, it is difficult to characterize large quantum systems. Due to the known solution to the ordered state of Rydberg crystals, the formation and characterization of Rydberg crystals provide a benchmark for the systems and techniques for forming and manipulating Rydberg crystals. As discussed below, these systems and methods demonstrate coherent control, and this large quantum system (Rydberg crystal) shows a high degree of coherence. In addition, it is demonstrated that the Rydberg crystals produced and controlled using the systems and methods described herein have special quantum states, and when they are out of equilibrium, the quantum states show surprisingly robust dynamics. This unique attribute is discussed in more detail below.

[0205] According to some embodiments, arranged 1D atomic arrays can be excited and evolved to produce solutions to quantum computing problems and can be used as quantum simulators. The following describes techniques for exciting and controlling 1D atomic arrays and characterizing the interactions between atoms. In the case of uniform coherent coupling, the Hamiltonian equation (1) is very similar to the paradigmatic Ising model of effective spin 1 / 2 particles with a variable interaction range. Its ground state exhibits a variety of many-body phases that break different spatial symmetries, such as Figure 2A For example, at large negative values ​​of Δ / Ω, its ground state corresponds to all atoms in state g, thus corresponding to a paramagnetic or disordered phase. As Δ / Ω increases towards larger positive values, the number of atoms in r increases and the interactions between them become significant. This produces a spatially ordered phase in which the Rydberg atoms are regularly arranged throughout the array, giving rise to a "Rydberg crystal" with a different spatial symmetry, such as Figure 2AThe origin of these related states can be explained by first considering that when V i,i+1 >>Δ>>Ω>>V i,i+2 In this case, Rydberg blockade reduces the nearest neighbor excitations while long-range interactions are negligible, resulting in a Rydberg crystal that breaks the Z2 translational symmetry, which is similar to the antiferromagnetic order in magnetic systems. In addition, by tuning the parameters V i,i+1 、V i,i+2 >>Δ>>Ω>>V i,i+3 And V i,i+1 、V i,i+2 、V i,i+3 >>Δ>>Ω>>V i,i+4 , we can obtain arrays with broken Z3 and Z4 symmetries, respectively, as Figure 2A shown. Figure 2A The boxed area 220 in indicates a potential incommensurate phase.

[0206] To prepare the system in these phases, the detuning Δ(t) of the driving laser can be dynamically controlled to adiabatically convert the ground state of the Hamiltonian from the product state of all atoms in g to the crystalline Rydberg state. First, the ground state of the Hamiltonian can be adiabatically converted from the product state of all atoms in g to the crystalline Rydberg state. S1 / 2 ,F=2,m F =2> to prepare all atoms. Then you can use Figure 3A The functional form shown turns on the laser field and sweeps the detuning from negative values ​​to positive values ​​of the two photons.

[0207] like Figure 2B As shown, single-atom trajectories are obtained in groups of 13 atoms for three different interaction strengths as the detuning Δ is varied. In each of these cases, it is observed that the initial state |g1,...,g 13 >Clear transitions to Rydberg crystals of different symmetries. The distance between atoms determines the strength of the interaction, which leads to different crystallographic orders for a given final detuning. For example, to achieve Z2 order, we arrange the atoms with a spacing of 5.9 μm, which leads to V i,i+1 =2πx 24MHz>Ω=2πx2MHz nearest neighbor interaction, while the next nearest interaction is small (2πx0.38MHz). This leads to the accumulation of antiferromagnetic crystals where every other well site is occupied by Rydberg atoms (Z2 order). By reducing the spacing between atoms to 3.67μm and 2.95μm, respectively, Z 3- Sequence and Z 4- Sequence, such as Figure 2B shown.

[0208] More specifically, Figure 2B shows the accumulation of a Rydberg crystal on an array of 13 atoms observed by slowly varying the laser parameters, as indicated by the red arrow in a (see also Figure 3A ). The bottom panel shows a configuration where the atomic spacing a = 5.9 μm, which results in V i,i+1 = 2π × 24 MHz nearest neighbor interaction and leads to a Z2 order where every other atom is excited into a Rydberg state |r>. The bar graph on the right shows the final position-dependent Rydberg probabilities (error bars indicate 68% CI). The configuration in the middle panel (a = 3.67 μm, V i,i+1 = 2π x 414.3MHz) leads to Z3 order, and the top panel (a = 2.95μm, V i,i+1 =2π x 1536 MHz) leads to the Z4 ordered phase. For each configuration, single emission fluorescence images before (left) and after (right) the pulse are shown. The open circles highlight the missing atoms attributed to Rydberg excitation.

[0209] The performance of the quantum simulator can be compared with the measured Z constructed using theoretical predictions for an N = 7 atom system obtained via accurate numerical simulations. 2 Compare in order. FIG. 3A to FIG. 3C A comparison with a fully coherent simulation is shown. FIG. 3A to FIG. 3C As shown, such fully coherent simulations with no free parameters show excellent agreement with the observed data when considering limited detection fidelity. Figure 3A As shown, the laser drive comprises square pulses Ω(t) with a detuning Δ(t) from a negative value chirp to a positive value. Figure 3B The time evolution of the Rydberg excitation probability for each atom in a cluster of N = 7 atoms (colored dots) is shown, which can be obtained by varying the duration of the laser excitation pulse Ω(t). The corresponding curves are theoretical single-atom trajectories obtained by accurately simulating quantum dynamics with (1) (where the functional form of Δ(t) and Ω(t) is used as discussed in this example) and with limited detection fidelity. Figure 3C The evolution of the seven most likely many-body states is shown. Figure 3C The evolution of the many-body state in shows that the perfect antiferromagnetic target state can be measured with a probability of 54(4)%. When corrected for the known detection fidelity, the desired many-body state is reached with a probability of p=77(6). The error bars represent the 68% CI.

[0210] As shown in Figure 4, by performing an adiabatic sweep over arrays of various sizes, the preparation fidelity depends on the size of the system. Figure 4AThe fidelity of preparation of the crystal ground state as a function of cluster size is shown. The open points are measured values, and the solid points are corrected for limited detection fidelity. The error bars represent 68% CI. Figure 4B A histogram of the number of observed occurrences of microstates without correction for detection errors is shown for each of the 51 atomic clusters for 18439 experimental realizations. Most of the occurring microstates are the ground state of the many-body Hamiltonian. As the system size increases, the probability of finding the system in the many-body ground state at the end of the sweep decreases. However, even at system sizes as large as 51 atoms or more, perfectly ordered crystalline many-body states are obtained with p = 0.11(2)% (p = 0.9(2)% when correcting for detection fidelity), which is remarkable considering the exponentially large 251-dimensional Hilbert space of the system. Moreover, as Figure 4B As shown, this state with perfect Z2 order is by far the most commonly used preparation state.

[0211] Chapter 3: Observe the resulting final state

[0212] After the quantum computation, the state of the atoms can be detected by fluorescence imaging. This can be done by state-dependent atomic loss and subsequent imaging to expose the remaining atoms. In the example described in this article, the optical tweezers potential can be restored after the laser pulse. The atoms in the ground state are recaptured by the optical tweezers, while those atoms remaining in the Rydberg state are pushed away by the optical tweezers beam. Subsequent fluorescence detection can expose the state of each atom. An extension of this detection can be to first map the Rydberg state to a second hyperfine state, and then state-selective fluorescence can be used to image the atomic groups in each state. This provides the additional advantage that atoms are not lost at the end of the calculation.

[0213] Chapter 4. Laser control systems for manipulating atomic arrays

[0214] Individual neutral atoms excited to Rydberg states are a promising platform for quantum simulation and quantum information processing. However, experimental progress to date has been limited by the short coherence times and relatively low gate fidelity associated with such Rydberg excitations. Therefore, even in the case where the above methods can be used to assemble large atomic arrays for quantum computing, it is still necessary to have high fidelity and coherent control over the assembled atomic arrays in order to evolve the arranged atomic arrays to solve specific problems. Therefore, according to one embodiment, one can use, for example, FIG. 7A to FIG. 7DThe described systems and methods and / or the above methods enable high-fidelity (low error, such as close to 0% error) quantum control of Rydberg atom qubits. This approach produces significant improvements in the coherence properties of individual qubits by reducing laser phase noise. This high-fidelity control is extended to multi-particle situations by preparing diatomic entangled states with fidelity exceeding 0.97(3) (i.e., an error rate of only 3 / 100) and using a diatomic dynamic decoupling protocol to extend their lifetime. These advances provide scalable quantum simulation and quantum computing for neutral atoms that can more accurately and consistently manipulate atoms in ordered arrays.

[0215] According to some embodiments, high-fidelity quantum control of Rydberg atom qubits can be achieved by reducing laser phase noise, resulting in significant improvements in the coherence properties of single qubits. The extension of this high-fidelity control to the multi-particle case is demonstrated by experimental results of preparing two-atom entangled states with fidelities exceeding 0.97(3). According to some embodiments, the lifetime of the prepared Bell states can also be extended using a novel two-atom dynamical decoupling protocol.

[0216] FIG. 7A to FIG. 7D A control system for single-atom Rabi oscillations of individual cold Rubidium-87 atoms in optical tweezers at programmable positions in one dimension is shown and characterized according to one embodiment. Fig. 7AAs shown, a laser such as an external cavity diode laser (ECDL) 710A can provide light to a reference optical cavity (REF) 720. The light from the ECDL 710A can be locked to the REF 720 using, for example, the Pound-Driv Hall technique. To do this, the light from the ECDL can be phase modulated and the reflection of the cavity 720 is detected on the photodiode PD 770. This signal is demodulated to produce an error signal for locking the laser. This type of locking can produce a very narrow linewidth. However, it can also produce high-frequency phase noise at the locked bandwidth. To mitigate this noise, the cavity Ref720 can be used as a filtering cavity, the narrow transmission window of which (the shaded area in the inset) suppresses the high-frequency phase noise. To enhance the output power of REF 720, the transmitted light can be used to injection lock a Fabry-Perot (FP) laser diode 740. This can be achieved by projecting the light from the REF 720 to the FP 740 through an optical isolator (ISO) 730. By using this technique, the light from FP 740 will inherit the spectral characteristics of the light from REF 720, but with higher power. The light from FP 740 can be provided to the atomic array 790 by ISO 730 and focusing optics 750A. The second laser source 710B can provide lasers of different frequencies, which can be focused on the atomic array 790 by optics 750B. In certain embodiments, the light from optics 750A and 750B can be provided in a counter-propagation configuration to minimize Doppler transition sensitivity. According to some embodiments, it is desirable to ensure good alignment of the excitation beam relative to the atomic array 790. To achieve this, an active feedback scheme can be adopted, which controls the alignment of the incident beam. A small amount of light from the optics 750A can be picked up and provided to a spatially resolved imaging device (such as CCD 760) for alignment. This narrow and precise control system for lasers 710A and 710B allows control of the internal states of atoms |g> and |r>. Additional components may be added, such as cooling of the laser and magnetic field generating structures (such as field coils).

[0217] According to some embodiments, atom 790 is optically pumped in a 1.5 G magnetic field at the ground state Zeeman sublevel |g>=|Sg>=|S 1 / 2 , F=2 for m F =-2>m F =-2>initialize. Then the optical tweezer potential is quickly turned off, and the laser fields from the two lasers 710A and 710B are applied to couple the atom 790 to the Rydberg state |r>r>=|70S,J=1 / 2,m J =-1 / 2>70S,J=1 / 2,m J=-1 / 2>. After a laser pulse of, for example, 3 to 8 μs in duration, the optical tweezers potential is restored. The atoms 790 in the ground state are recaptured by the optical tweezers, while those atoms remaining in the Rydberg state are pushed away by the optical tweezers beam. According to some experimental embodiments, this detection method has a Rydberg state detection fidelity f r f r =0.96(1) and the range is f g f g Ground state detection fidelity of 0.955(5) to 0.990(2), depending on capture time.

[0218] According to some embodiments, the Rydberg states are excited via a two-photon transition. ECDL710A may be a 420 nm laser, ΔΔ detuned over |g>g> to |e>e>=|6P 3 / 2 6P 3 / 2 ,F=3,m F m F = -3 transition frequency. The second laser source 710B provides a laser field of, for example, 1013 nm, which couples |e>e> to |r>r>. The two lasers 710A and 710B are polarized to drive σ - σ - and σ + σ + transitions, so that only a single intermediate sublevel can be coupled to the Rydberg state, thus avoiding the addition of additional energy levels and the associated dephasing. Figure 7B Shown in.

[0219] The two lasers 710A, 710B can be any known lasers, such as external cavity diode lasers sold by MOG Laboratories Pty Ltd. Laser 710A can be frequency stabilized by a Pound-Driv Hall (PDH) 770 (such as the PDH provided by Stable Laser Systems), locked to an ultra-low expansion reference cavity. Laser 710B can also be stabilized by a similar PDH (not shown). PDH770 locking strongly suppresses laser noise at frequencies below the effective bandwidth of locking, resulting in a narrow linewidth of <1kHz as estimated from the in-loop noise. However, noise above the locking bandwidth cannot be suppressed and can be amplified with a high locking gain. This results in a broad peak in phase noise of approximately ~2π×1MHz~2π×1MHz (see Fig. 7A). This high frequency phase noise exhibits coherence limitations in Rydberg experiments and experiments using trapped ions. To suppress this phase noise, the reference cavity 730 can be used as a spectral filter. Specifically, the transmission function of the cavity can be a Lorentzian function with a full width at half maximum of ΓΓ~2π×~2π×500kHz (corresponding to a finesse of F~~3000). Other functions can be used for cavities with different functions. The smaller the linewidth, the better the filtering. When the laser is locked, its narrow linewidth carrier component is transmitted through the cavity, while the high frequency noise at 2π×1MHz2π×1MHz is suppressed 4 times. Although only Fig. 7A , but to amplify this light at 420nm and 1013nm, the two colors can be separated and each beam used to injection lock a separate laser diode that inherits the same spectral characteristics. This amplifies the pure spectral transmission light to 5mW at 420nm and 50mW at 1013nm. Although the 420nm power is sufficient to drive the blue transition directly, in some embodiments, the 1013nm can be further amplified by a tapered amplifier (not shown).

[0220] Lasers 710A, 710B may be provided to the atomic array 790 in a counter-propagating configuration to minimize the Doppler shift due to finite atomic temperature. Lasers 710A, 710B may be focused to a waist of 20 or 30 μm, respectively. According to an experimental implementation, this may be achieved The single-photon Rabi frequency. This results in a two-photon Rabi frequency of Each beam power is stabilized to <1% by an acousto-optic modulator, which is also used for fast (~~20ns) switching. To minimize sensitivity to pointing fluctuations, a reference camera 760 and an automated beam alignment process can be used to ensure good centered alignment onto the atoms, where the beam position is stabilized at a fixed position on the reference camera using a steering mirror mount with a piezoelectric actuator. The optimal position can be established by measuring the Rabi frequency on the atom for different beam positions on the camera and choosing the position that maximizes coupling to the Rydberg state.

[0221] Fig.7D According to some embodiments, Fig. 7A A similar system with the addition of further control over laser 710B. Fig.7DTwo lasers 710A and 710B at 420nm and 1013nm are shown. Each laser 710A, 710B can be, for example, an external cavity diode laser. Lasers 710A, 710B are used to coherently excite atoms 790 to Rydberg states. Both lasers are stabilized to a reference optical cavity 720 by Pound-Driv Hall (PDH) locking. The mirror coating of REF720 can be selected so that the cavity is suitable for a variety of wavelengths and can be used by both 710A and 710B. To this end, each laser is phase modulated using an electro-optical modulator (EOM) placed between each laser source and the reference cavity. (not shown), and the light reflected from the optical cavity is measured on photodetectors (PD) 770A, 770B, respectively, and the light is used to feedback the laser by tuning the current through the laser diode. This can stabilize each laser to a narrow linewidth of <1kHz, as measured by the noise on the error signal locked by PDH. However, due to the limited bandwidth of the laser diode, the high frequency noise from the laser diode cannot be suppressed and can instead be amplified by the PDH lock-in. This results in a broad peak in the noise at approximately + / -1 MHz relative to the central narrow carrier linewidth (shown in the power spectrum inset).

[0222] When lasers 710A, 710B are locked to a reference cavity, light is primarily transmitted through cavity 720. However, cavity 720 acts as a low-pass filter with a bandwidth of about 500 kHz, and so its transmission suppresses noise outside this "transmission window" (schematically shown as the boxed area with dashed lines in the power spectrum inset). Thus, the transmitted light through the cavity has a linewidth, but also suppresses high frequency noise.

[0223] Since high power is beneficial for controlling atoms 790, the transmitted light of each laser 710A, 710B is separated and injected and locked into independent Fabry-Perot (FP) laser diodes 740A, 740B respectively through optical isolators (ISO) 730A, 730B. These laser diodes 740A, 740B inherit the same high-quality spectral characteristics of the light used to seed them, and effectively amplify this seed light to about 5mW light at 420nm and about 50mW light at 1013nm. The 1013nm power can be additionally amplified by a tapered amplifier (not shown) located after the laser diode 740B. The two lasers 710A, 710B can then be focused on the atomic array 790 in a counter-propagating configuration by lenses 750A, 750B to minimize the Doppler sensitivity of the transition.

[0224] To date, this laser stabilization and filtering scheme has achieved high-fidelity coherent control of Rydberg atom qubits. The scheme can be further improved by using commercially available higher-fineness optical cavities with narrower linewidths caused by higher reflectivity mirrors and therefore higher noise suppression factors. According to some embodiments, intrinsically lower-noise laser sources (such as titanium-sapphire lasers or dye lasers) can be used to drive this transition without the need to spectrally filter the high-frequency laser phase noise.

[0225] Section 4.A: Experimental results from improved laser control

[0226] According to some embodiments, various control methods and systems disclosed herein can be implemented to extend the coherence time and improve the control of atoms. FIG. 7A to FIG. 7D The experimental implementation of the control system shown in ) measured the e1 / ee lifetime of the long-term Rabi oscillation to be ττ = 27(4)μs, which is consistent with the typical 7μs lifetime. Figure 7C , which shows that resonant two-photon coupling induces Rabi oscillations between |g>=|g>= and |r>r>. Each graph shows the Rydberg probability as a function of time (in microseconds). The upper graph is a measured value for a setup used in the art. The lower graph shows a typical result for the above setup, where the fitted coherence time is 27(4)μμs. Each data point is calculated from 50 to 100 repeated measurements and averaged over two identically coupled atoms separated by 23μμm so that they do not interact. The error bars are 68% confidence intervals. The solid line fits the experimental data, while the dashed line indicates the expected comparison with the numerical model. As shown Fig.7D As shown, there is excellent agreement between these new measurements and simple numerical models of single-atom systems, such as Fig.7D The implemented numerical model has no free parameters and only considers the effects of random Doppler shifts at finite atomic temperature, non-resonant scattering of the intermediate states |e>e>, and the finite lifetime of the Rydberg states |r>r>. The results from the numerical model are additionally scaled to take into account the detection fidelity.

[0227] According to another experimental embodiment, the coherence and single-qubit control of a single atom can be characterized. First, the lifetime of the Rydberg state is measured, e.g. Fig. 8A In order to demonstrate and determine the time scale over which quantum control can be performed. Fig. 8AAs shown, the lifetime of |r>r> can be characterized by exciting from |g>|g> to |r>r> (with a ππ-pulse) and then de-exciting after a variable delay. The probability of terminating in |g>|g> (denoted as PgPg) decays within the extraction lifetime of TT1=51(6)μμs. The measured T1=T r→g T1=T r→g = 51(6)μμs is consistent with a Rydberg lifetime of 146μs, which is the case when combined with the ~~80μs time scale of non-resonant scattering of the 1013 nm laser (from |e>e>). The Ramsey experiment shows a Gaussian decay that is well explained by the thermal Doppler shift 810 (see Figure 8B ).like Figure 8B As shown, the Ramsey experiment plotted as line 810 shows that the 1 / e lifetime is Gaussian decay, which is limited by the thermal Doppler shift. Inserting an additional ππ-pulse 830 between the π / π / 2 pulses 832, 834 eliminates the effect of the Doppler shift and results in a substantially longer coherent lifetime of TT2 = 32(6)μμs (fit to an exponential decay down to 0.5). At 10μμK, the random atomic velocity of each shot of the experiment exhibits a random detuning δ D δ D (from a Gaussian distribution of width 2π×π×43.5kHz), resulting in a phase shift of However, since the random Doppler shift is constant over the duration of each pulse sequence, its effect can be eliminated by a spin echo sequence (see Figure 8B The spin echo measurements show some small deviations from the numerical simulations (dashed lines), indicating the presence of an additional dephasing channel. Assuming exponential decay, the fitted T2T2 = 32(6) μμs, and the pure dephasing time T φ =(1 / T2-1 / (2T r→g )T φ =(1 / T2-1 / (2T r→g )) –1 = 47(13)μμs. This phase shift may be caused by residual laser phase noise. In addition to resonantly manipulating atoms between their states |g> and |r>, it is also desirable to be able to manipulate the phase between these states, which can be called a phase gate. Figure 8CSuch a single atom phase gate is shown as being achieved by applying an independent 809 nm laser 840. According to some embodiments, other wavelengths can be used that are far away from the atomic transition but close enough to induce a light shift on the ground state that is different from the Rydberg state. This induces a light shift on the ground state of δ=2π×5δ=2π×5 MHz (in time t), resulting in an accumulated dynamic phase φ=δtφ=δt. The gate can be embedded in a spin echo sequence to eliminate Doppler shifts. In each measurement shown here, the 1013 nm laser remained on throughout the pulse sequence, while the 420 nm laser was pulsed according to the sequence shown above each figure. Each data point is calculated based on 200 to 500 repeated measurements of a single atom, with error bars representing a 68% confidence interval. Figure 8B and Figure 8C In Figure 2, the solid line fits the experimental data. The dashed line shows the contrast expected from the numerical model, including finite detection fidelity. Such long coherence times and single qubit control gates can be used for quantum computing.

[0228] According to some embodiments, a single-atom phase gate can be achieved by applying an independent focused laser that shifts the energy of the ground state |g>|g> by 5 MHz. By controlling the duration of the applied laser pulse, a controlled dynamic phase can be imposed on |g>|g>, which is relative to |r>r>. The contrast of the resulting phase gate (embedded in a spin echo sequence) approaches the limits imposed by the detection and spin echo fidelity.

[0229] According to some embodiments, two atoms can be controlled. It should be understood that such techniques and systems can be applied to more than two atoms. To this end, two atoms can be positioned at a separation of 5.7 μm, where the Rydberg-Rydberg interaction is In this mechanism, which can be called the Rydberg blockade mechanism, the laser field globally couples the two atoms from |gg>|gg> to the symmetric state This is in The enhanced Rabi frequency is carried out (see Fig. 9A ). Fig. 9A shows the energy level structure of two nearby atoms, which is characterized by a double excited state |rr>|rr>, which is excited by the interaction energy In this Rydberg blockade mechanism, the laser field only couples |gg>|gg> to |W>W>. The symmetric and antisymmetric states |W>W>, The coupling may be via a local phase gate on one atom (indicated via arrow 910).

[0230] The probabilities can be measured for the states |gg>|gg>, |gr>|gr>, |rg>|rg>, and |rr>|rr> (denoted by P gg , P gr , P rg and P rr ), and indicates that no cluster enters the doubly excited state, see 9B top (P rr <0.02, which is consistent only with detection error). In contrast, the manifold with zero excitation ( Fig. 9B The bottom of the manifold) and the first excitation manifold ( Fig. 9B There are oscillations between the centers of (See Fig. 9B ). After resonantly driving the two atoms for a variable time, the probability of the resulting diatomic state can be measured. The cluster oscillates from |gg>|gg> to |W>W> which is at the enhanced Rabi frequency Next. This demonstrates high-fidelity two-qubit control.

[0231] These collective Rabi oscillations can be used to directly prepare the maximally entangled Bell state |W>W> by applying ππ pulses (at an enhanced Rabi frequency (given by To determine the fidelity of the entangled state prepared in this experiment, Given , it can be expressed in terms of the diagonal and off-diagonal matrix elements of the density operator ρρ:

[0232]

[0233] where ρ αβ,γδ = <αβ|ρ|γδ><αβ|ρ|γδ> for αα, ββ, γγ, δδ∈∈{g,r}. The diagonal elements can be measured directly by applying ππ pulses and then measuring the cluster. After accounting for state detection errors, the results closely match those for the perfect |W> state, where ρ gr，gr +ρ rg，rg ρ gr，gr +ρ rg，rg =0.94(1), relative to the maximum possible value of 0.95(1).

[0234] To measure off-diagonal elements of the density matrix, a single-atom phase gate can be used 920, such as Fig. 9C As shown, it introduces a variable phase on one atom. For example, the local beam adds a light shift δδ to |gr>|gr> but not to |rg>|rg>, so that This phase accumulation rotates |W>|W> into the orthogonal dark state according to:

[0235] |W→cos(δt / 2)|W>+i sin(δt / 2)|W>→cos(δt / 2)|W>+i sin(δt / 2)|D>|D>

[0236] Since |D>|D> is decoupled by the laser field, the subsequent ππ pulse only maps the cluster of |W>|W> back to |gg>|gg>. Therefore, the probability that the system ends up in |gg>|gg> depends on the phase accumulation time, since P gg (t)P gg (t) = A cos 2 (δt / 2)(δt / 2). Here, the amplitude of the oscillation A is precisely measured by the off-diagonal matrix elements (ρ gr，rg =ρ rg，gr (ρ gr，rg =ρ rg，gr To mitigate sensitivity to random Doppler shifts, the entire sequence can be embedded in a spin-echo protocol (see Fig. 9C ). Fig. 9C Shown is the measurement of the entanglement fidelity of two atoms following a resonant π-pulse in the blockade regime. Local phase gate |W>W> is rotated into |D>D> which is detected by a subsequent ππ-pulse. The contrast of the fit is 0.88(2) measuring the off-diagonal density matrix elements. The phase gate is implemented by a non-resonant laser focused on one atom with crosstalk <2%. The measurements are embedded in a spin echo sequence to remove the phase shift caused by the thermal Doppler shift. As mentioned above, the phase gate is a single-qubit gate that can be used in quantum computing. Here, it can be used to characterize the entanglement between two atoms. The echo sequence used here helps to remove the noise from the Doppler shift and thus increase the coherence of the system.

[0237] The obtained contrast is A = 0.88(2) = 2ρ gr，rg =2ρ rg，gr 2ρ gr，rg =2ρ rg，gr Combining these values ​​with the diagonal matrix elements, the entanglement fidelity is measured to be FF = 0.91(2). Given that the maximum measurable fidelity of the experimental state detection error rate would be 0.94(2), after correcting for the imperfect detection, an entangled Bell state is found to have formed with a fidelity of FF = 0.97(3). This fidelity includes errors introduced during the pulses following the initial ππ-pulse and therefore constitutes a lower limit on the true fidelity.

[0238] Entanglement is a useful resource in quantum computing. However, entangled states can be very fragile and subject to rapid phase shifts. The methods discussed herein can be used to protect entangled states from certain noise sources. According to some embodiments, the lifetime of an entangled state can be explored with a ππ-pulse by exciting |W>|W> and then de-exciting after a variable delay (see Fig.10 ). Fig.10 The lifetime of entangled states is extended via dynamic decoupling. The lifetime of |W>W> can be measured by exciting |gg>> to |W>W> and then de-exciting after a variable time, as shown in graph 810. The lifetime is limited by the phase shift caused by the random Doppler shift. Inserting an additional 2ππ-pulse 1030 in the blockade mechanism exchanges the cluster of |gr>> with |rg>> to refocus the random phase accumulation, thereby extending the lifetime to ~36μμs, as shown in graph 1020 (fit to exponential decay, shown as a solid line). The initial offset in each curve 1010, 1020 is set by the ground state detection fidelity associated with a given capture time. All data points are calculated from 30 to 100 repeated measurements, averaging nine independent identically coupled atomic pairs, with error bars indicating a 68% confidence interval. The dashed and fitted lines near the graph show predictions from the numerical model, including detection errors. The contrast attenuation is in good agreement with the numerical prediction associated with the mechanical Doppler shift. Specifically, the phase shift of the two components |gr>> and |rg>> (of the |W>W> state) is in is the two-photon Doppler shift on atom ii.

[0239] According to some embodiments, the lifetime of a diatomic entangled state can be extended using an echo sequence acting on multiple qubits. This allows for longer control periods. After the |W>W> state has evolved for time T, a 2ππ-pulse can be applied to the diatomic system. In the Rydberg blockade mechanism, this pulse exchanges clusters of |gr>> and |rg>>. After evolving again for time T, the total accumulated Doppler shift is the same for each part of the diatomic wave function, and therefore does not affect the final |W>W> state fidelity. Fig.10 It shows that its lifetime is extended far beyond the Doppler-limited attenuation. As in the single-atom case, the pure dephasing time scale is extracted

[0240] Bell state phase shift time of two atoms Significantly longer than the single-atom dephasing time This can be understood by noting that the states |gr>> and |rg>> form a dephasing-free subspace that is insensitive to global perturbations, such as laser phase and intensity fluctuations that are also coupled to both atoms. The individual atoms are sensitive to both the laser phase and the laser intensity. This decoherence-free subspace can be used to protect quantum information from certain noise sources. These measurements further show that even though the laser noise was greatly reduced in these experiments, it was still not completely eliminated in our experiments. A higher finesse cavity, REF 720, can be used to filter out even more laser noise and achieve even longer coherence times. Furthermore, these coherence manipulation techniques between the ground state and the Rydberg state are still much better than those reported previously.

[0241] These measurements establish Rydberg atom qubits as a platform for high-fidelity quantum simulation and computing. The technology demonstrated in the present disclosure shows a method for controlling neutral atom arrays. The fidelity demonstrated by these technologies can be further improved in the following ways: by increasing the laser intensity and operating under a larger detuning with the intermediate state to reduce the harmful effects of non-resonant scattering, or by using direct single-photon transitions. In addition, the sideband cooling of atoms in optical tweezers can significantly reduce the amplitude of the Doppler shift, and low-noise laser sources (such as titanium-sapphire lasers or diode lasers filtered by higher finesse cavities) can further eliminate errors caused by phase noise. Advanced control techniques (such as laser pulse shaping) can also be used to achieve higher fidelity. Finally, state detection fidelity, as the main defect source in current work, can be improved by field ionization of Rydberg atoms or by mapping Rydberg states to a separate ground state level.

[0242] Section 5.A: Example - Quantum Dynamics on Phase Transitions

[0243] The systems and methods described herein provide identification of a solution to the Ising problem, as described below. The techniques applied herein can also be transferred to other models, such as the maximum independent set problem described below.

[0244] Once atoms can be arranged into large arrays, such as 1D arrays of up to 51 atoms or more, phase transitions can be observed as the atoms alternate between the Rydberg state and the ground state. These transitions are discussed in more detail below. FIG. 5A to FIG. 5D The characteristics of the transition to the Z2 phase in an array of 51 atoms according to one embodiment are shown. Long ordered chains of atoms alternating between the Rydberg state and the ground state may appear. Figure 5A As shown, these ordered domains can be separated by domain walls consisting of two neighboring atoms in the same electronic state.

[0245] Figure 5Ashows the heat transfer characteristics of the heat transfer device according to an exemplary embodiment before applying the adiabatic pulse (top row 501, i.e., about Figure 1D The evolution step 116 in question) and after the pulse (the bottom three rows 502, 503, 504, corresponding to three separate instances, i.e., Figure 1D Single shot fluorescence image of an array of 51 atoms (detection step 118 in FIG. 5 ). Small circles 505 mark missing atoms, which may be attributed to Rydberg excitations. Domain walls are defects in a perfectly ordered Rydberg crystal. Domain walls allow characterization of the extent to which the system reaches the ground state at the end of an adiabatic sweep. Observing these domain walls themselves is also useful. For example, the system can be better characterized by how the sweep speed affects the number of domain walls or whether there is a correlation between domain walls. Domain walls 506 (circled points) are identified as two adjacent atoms or ground state atoms in the same state at the edge of the array and are indicated by ellipses. Ordered chains of length Z2 can be observed between domain walls.

[0246] Figure 5B The domain wall density as a function of detuning during a frequency sweep according to an exemplary embodiment is shown. The points on curve 560 are the average values ​​of the domain wall density as a function of detuning during the sweep. The error bars are the standard error of the mean and are smaller than the marker size. The points on curve 570 are the corresponding variances, where the shaded area represents the jackknife error estimate. The decrease in domain wall density and the peak in the variance witness the onset of the phase transition. Each point is obtained from 1000 realizations. The solid line 560 is a fully coherent MPS simulation with no free parameters (bond size D=256), thereby taking into account the measurement fidelity.

[0247] The domain wall density can be used to quantify the transition from the disordered phase to the ordered Z2 phase as a function of detuning Δ and serves as an order parameter. As the system enters the Z2 phase, the size of the ordered domains increases, resulting in a significant decrease in the domain wall density ( Figure 5B points on curve 560 in ). Consistent with expectations for an Ising-type second-order quantum phase transition, domains of fluctuating length close to the transition point between the two phases are observed, as reflected by a clear peak in the variance of the domain wall density. This peak is shifted toward positive values ​​of Δ≈0.5Ω, consistent with predictions from finite-size scaling analysis. The observed domain wall density is in good agreement with fully coherent simulations of quantum dynamics based on a 51-atom matrix product state (line 560); however, these simulations underestimate the variance at the phase transition.

[0248] At the end of the sweep, deep in the Z2 phase (Δ / Ω>>1), Ω can be neglected, making the Hamiltonian (1) classical in nature. In this case, the measured domain wall number distribution allows us to directly infer the statistics of the excitations generated when crossing the phase transition. Figure 5CA histogram of the normalized number of domain walls that occur during 18439 individual experimental realizations for a 51-atom array is shown. The distribution is depicted with an average of 9.01 (2) domain walls. This distribution is affected by the detection fidelity of the state |g> and the state |r>, which results in a higher number of domain walls. In other words, a system with perfect detection fidelity will produce a different distribution of domain walls than a system in which imperfect detection fidelity introduces additional domain walls. Therefore, in order to determine the actual number of domain walls, the effect of detection fidelity can be modeled on the domain wall distribution to determine the number of domain walls that would be formed in the absence of the effect of lower detection fidelity. This allows obtaining a maximum likelihood estimate of the distribution corrected for detection fidelity, which corresponds to a state with an average of 5.4 domain walls. These remaining domain walls (those not caused by lower detection fidelity) are most likely due to non-adiabatic transitions from the ground state at the phase transition across which the energy gap becomes smallest. Furthermore, preparation fidelity is limited by spontaneous emission during the laser pulse (scattering average of 1.1 photons per μs for the entire array).

[0249] Figure 5C The distribution of the number of domain walls at Δ=14 MHz obtained from 18439 experimental realizations (top graph) according to an exemplary embodiment is shown. The error bars represent the 68% CI. Due to boundary conditions, only an even number of domain walls can appear. The bar on the right side of each coupled pair in the bottom graph shows the distribution obtained by correcting the limited detection fidelity using the maximum likelihood method, which results in an average of 5.4 domain walls. The bar on the left side of each coupled pair shows the distribution of thermal states with the same average domain wall density. Figure 5D The correlation function of the measurement in the Z2 phase is shown.

[0250] To further characterize the formed Z2-ordered state, the correlation function can be evaluated:

[0251]

[0252] where 〈...〉 is averaged over experimental replicates. We found that the association decays exponentially with distance, with a decay length of ξ = 3.03(6) sites (see Fig. 5d and SI).

[0253] FIG. 6 shows a graph demonstrating that the method described within the present disclosure also enables the study of coherent dynamics of multi-body systems far from equilibrium. Fig. 6AA schematic sequence is shown (top, showing Δ(t)) involving adiabatic preparation and then sudden quenching to a single-atom resonance. The heat map shows the single-atom trajectories of a 9-atom cluster. The initial (left inset) crystal is observed to have Rydberg excitations when each odd-numbered well site collapses after quenching, and a crystal with excitations is observed when each even-numbered site is built (middle inset). At a later time, the initial crystal recovers (right inset). The error bars represent 68% CI. Figure 6B The density of domain walls after quenching is shown. The dynamics decay slowly on a time scale of 0.88 μs. The shaded area indicates the standard error of the mean. The solid line in the top panel is a fully coherent MPS simulation with a bond size of D = 256, thus accounting for measurement fidelity. Figure 6C A toy model of a non-interacting dimer is shown. Fig.6D Numerical calculations of the dynamics obtained by exact diagonalization after quenching starting from an ideal 25-atom crystal are shown. The domain wall density as a function of time 610 and the growth of the entanglement entropy 620 of the half chain (13 atoms). The dashed line considers only the nearest neighbor blockade constraints. The solid line corresponds to the full 1 / r 6 Interaction potential.

[0254] like Fig. 6A As shown, focusing on the quenching dynamics of a Rydberg crystal initially prepared deep in the Z2 ordered phase, the detuning Δ(t) suddenly changes to a single-atom resonance Δ = 0. After this quenching, oscillations of the many-body state appear between the initial crystal and the complementary one, where each internal atomic state is inverted. These oscillations are very robust, persisting for several cycles, where the frequency is largely independent of the system size for large arrays. This is determined by measuring the dynamics of the domain wall density, thereby signaling the appearance and disappearance of crystalline states, as Figure 6B Shown for arrays of 9 (solid dots) and 51 atoms (open dots). The repetitive recovery period of the initial crystal is 1.4 times slower than the Rabi oscillation period of independent non-interacting atoms.

[0255] According to one embodiment, several important features result from the techniques described in this disclosure. First, the Z2 ordered state cannot be characterized by a simple thermal ensemble. More specifically, if the effective temperature is estimated based on the measured domain wall density, the corresponding thermal ensemble predicts a correlation length ξth = 4.48 (3), which is significantly longer than the measured value ξ = 3.03 (6). This difference is also reflected in a different probability distribution for the number of domain walls (see FIG5c ). These observations suggest that the system does not thermalize within the time scale of the Z2 state preparation.

[0256] Second, the coherent and persistent oscillations of the crystalline order after quantum quenching. For the quenched Hamiltonian (Δ = 0), the energy density of the Z2 ordered state corresponds to the energy density of the infinite temperature ensemble within the manifold constrained by the Rydberg blockade. Again, the Hamiltonian does not have any explicit conserved quantities other than the total energy. Nevertheless, the oscillations still far exceed the natural time scale of local relaxation ~1 / Ω and the fastest time scale 1 / V i,i+1 .

[0257] To understand these observations, we consider a simplified model in which the effects of long-range interactions are neglected and the nearest-neighbor interactions are replaced by hard constraints on neighboring excitations of the Rydberg state. In this limit, the qualitative behavior of the quenching dynamics can be understood with respect to the dimerization spin (e.g. Figure 6C ); due to the constraints, each dimer forms an effective spin-1 system with three states |r,g>, |gg> and |gr>, where the resonant actuator "rotates" these three states with a period 2 (2π / Ω) close to the experimentally observed. Although this qualitative picture does not take into account the strong interactions (constraints) between neighboring dimers, it can be extended by considering a minimum variational postulate of the many-body wave function based on a matrix product state (MPS) that obeys all Rydberg blockade constraints. By using the time-dependent variational principle, the analytical equations of motion can be derived and crystalline order oscillations with a frequency of Ω / 1.51 can be obtained, which is within 10% of the experimental observations. These considerations can be supported by various numerical simulations. For example, MPS simulations with large bond sizes predict that simplified models exhibit crystal oscillations for long times, while the entanglement entropy grows at a rate much smaller than Ω, indicating that the oscillations persist for many periods ( Fig.6D ). However, adding long-range interactions causes the oscillations to decay more rapidly, with the time scale being reduced by 1 / V i,i+2 Sure enough, this is consistent with experimental observations, and the entanglement entropy also grows on this time scale.

[0258] The decay of the crystal oscillations is therefore limited by the influence of the weaker next-nearest-neighbor interaction. This slow thermalization is quite unexpected, since our Hamiltonian (with or without long-range correction) is very far from any known integrable system and features no strong disorder and no clearly conserved quantities. Instead, the observations may be associated with constrained dynamics due to Rydberg blockade, leading to timescales V i,i+1 >>ΩV i,i+2 These give rise to the so-called constrained dimer model, in which the Hilbert space dimension is determined by the golden ratio (1+5) N / 2 N and non-trivial dynamics determination.

[0259] According to one embodiment, the initial spacing between atoms trapped in the array can be used to encode a problem (such as one governed by quantum mechanics). After the adiabatic evolution of the system, the atoms can then be observed to determine the solution to the problem. The state of the atoms after the evolution can indicate the solution to the problem.

[0260] Section 5.B: Example - Using quantum computers to solve the maximum independent set optimization problem

[0261] The method and system for arranging and controlling the neutral atoms and Rydberg interactions of individual captures described above can be used to solve various different types of problems. For example, as described below, according to some embodiments, the above-mentioned system and method can be used to solve the maximum independent set (MIS) optimization problem based on the quantum adiabatic principle. It is challenging to solve the MIS optimization problem using numerical techniques alone, but it can be easily solved using quantum computing techniques. Therefore, the above-mentioned system and method for quantum computing are very suitable for finding solutions to the following MIS optimization problems.

[0262] Adiabatic quantum computing is a new general approach to solving combinatorial optimization problems. It consists of constructing a set of qubits and designing a time-dependent Hamiltonian H(t) whose starting point H(0) has a ground state that can be easily prepared and whose end point H(T) has such a form that its eigenstates encode the solution to the optimization problem. The name "adiabatic" comes from the fact that if H(t) varies slowly enough, the system will be mostly in the ground state of the instantaneous Hamiltonian H(t) at all times t, so that at the last time t=T, the system is found to be in the ground state of H(T), from which the solution to the optimization problem can be found. According to some embodiments, if the Hamiltonian does not vary slowly enough for a completely adiabatic evolution, the dynamics caused by the time-dependent Hamiltonian can inject finite energy into the system. As long as the cost function of the optimization problem is correctly encoded in the final Hamiltonian, and the evolution is slow enough so that the injected energy is low, then measuring the final state of the system provides a good approximate solution to the optimization problem. The working principle of this quantum adiabatic optimization (or approximation) is fundamentally different from the known classical algorithms for finding (or approximating) solutions to optimization problems and may therefore lead to quantum speedups (i.e., performing calculations faster using a quantum computer).

[0263] The maximum independent set problem (discussed in more detail below) is a classic combinatorial optimization problem in graph theory. The task is to select a subset of vertices from the graph so that none of them are adjacent. In some embodiments, the challenge is to find such a subset with the maximum number of vertices. In complexity theory, this is a well-studied problem and is known to be NP-hard to approach. When formulated as a decision problem, it is NP-complete (i.e., both an NP (nondeterministic polynomial time) problem and an NP-hard problem (a problem at least as difficult as an NP problem) problem).

[0264] According to some embodiments, there are different variations of the maximum independent set problem. The embodiments discussed herein focus on the problem of disk graphs in which the class of graphs can be restricted to the optimization of the maximum independent set problem. The unit disk graph is a special case of a geometric intersection graph (i.e., a graph representing the intersection pattern of a series of sets), in which only vertices located within a certain distance are considered adjacent. The optimization problem of this intersection graph plays an important role in various applications, including but not limited to broadcast network design, map marking, and problems caused by determining the optimal location of facilities. The maximum independent set problem on the unit circle graph is NP-complete. Although there are polynomial time approximation algorithms, fast algorithms that achieve good approximation ratios have not yet been implemented.

[0265] As discussed in more detail below, a setup including individually trapped atoms can be used to implement a quantum algorithm to solve the maximum independent set problem on a unit disk graph.

[0266] As described herein, the maximum independent set problem may include an undirected graph G = (V, E) with a set of vertices V and edges E. An independent set is a subset of vertices So that no two vertices in S are connected by an edge. For examples of independent sets, see FIG. 11A to FIG. 11B . Fig.11A Two examples of graphs with different independent sets marked as combinations of empty circles and black circles are shown. The maximum independent set in each case is depicted on the right. In other words, the maximum independent set is the largest independent set, i.e., the independent set with the largest number of vertices. It should be noted that there is also the concept of a maximum independent set, i.e., an independent set to which one cannot add any other vertices without making it non-independent. The maximum independent set is the largest of the maximum independent sets.

[0267] A generalization of the maximum independent set problem is the maximum weight independent set problem. According to some embodiments, an undirected weighted graph G = (V, W, E) can be considered using a set of vertices V with associated weights W and edges E. The maximum weight independent set is the independent set with the largest weight. The maximum independent set problem can be formulated as a decision problem: "Given a graph G, are there independent sets of size k?" This decision problem is NP-complete. It can also be formulated as an optimization problem: "Given a graph G, find independent sets of maximum size k". Even approximate optimization is NP-hard (an approximation of k, within a constant factor). Finding the maximum independent set is equivalent to finding the minimum vertex cover: this is a dual problem. These principles extend to the maximum weight independent set problem.

[0268] If only vertices within a unit distance R in Euclidean space are connected by edges, then the graph can be called a unit disk graph. Fig.12 An example of a unit disk graph is shown. As shown in FIG2 , each pair of vertices 1210 that are closer than distance R are connected by an edge 1220. This is equivalent to drawing a disk 1230 of radius R / 2 around each vertex and connecting two disks if they overlap. It should be noted that finding the maximum independent set of the unit disk graph is still NP-complete.

[0269] According to some embodiments, the atomic array can be arranged to solve problems such as those presented by the unit disk diagram. According to some embodiments, Rydberg atoms can be used, where the Rydberg implementation R plays the role of the blocking radius corresponding to the disk 1230. Such implementations are discussed in more detail below.

[0270] Given a graph, we can find the maximum independent set from the ground state of the classical Hamiltonian. To do this, we can assign a classical Ising variable to each vertex n v ∈{0, 1}. The Hamiltonian is as follows

[0271]

[0272] Among them U u,w >Δ>0. The configuration that minimizes H {n v} Encode the maximum independent set: in the ground state, it has n v = 1 forms a maximum independent set. It should be noted that U n,m The value of is not important as long as it is greater than Δ.

[0273] In addition, it should be noted that (for U n,m >>>>Δ) The lowest energy level state encodes different independent sets, and further note that the corresponding energy is directly related to the size of the independent set k, such as E=-kΔE=-kΔ.

[0274] To design a quantum adiabatic algorithm for MIS, the Hamiltonian can be lifted to the operator level, including adding terms that couple different configurations of the Ising spin. For example, the Hamiltonian can be written as

[0275]

[0276] Instead of the classical Ising spin, there are now qubits with states |0>|0> and |1>|1>, such that , (x∈{0,1})(x∈{0,1}) and σ x =|0><1|+|1><0|σ x =|0><1|+|1><0|. Thus, the adiabatic algorithm can be obtained by initializing all qubits at time t=0 in |0> and then evolving the system for time T under the time-dependent Hamiltonian H(t), where the parameters are chosen such that Δ(0)<0, Δ(T)>0, Ω(0)=Ω(T)=0Δ(0)<0, Δ(T)>0, Ω(0)=Ω(T)=0, and Ω(0<t<T)>0 Ω(0<t<T)>0 As a specific example, consider ,in Note that by making the parameter Δ(t) different for each vertex, this immediately generalizes to the maximum weighted independent set problem.

[0277] According to some embodiments, the systems and methods for arranging and manipulating individual atoms described in more detail above can be used to encode and evolve such problems. For example, a set of individually positioned optical tweezers, as discussed in more detail above, can be used to capture individual atoms each having a ground state |0>|0> and a Rydberg state |1>|1>. The atoms can be coherently driven with a Rabi frequency Ω(t)Ω(t) that couples the ground state to the Rydberg state. The frequency of the driving field can vary in a time-dependent manner, causing a time-dependent detuning Δ(t). This drive can be global, or alternatively each atom can be driven individually with a specific field at a specific time. If two atoms u and v are in a Rydberg state, they interact so that the energy shift amount W of this configuration is u,v , which depends on the geometric distance between the two well locations For example Therefore, the Hamiltonian describing the dynamics of this array of trapped atoms is:

[0278]

[0279] For two atoms trapped in close proximity, being in the Rydberg state at the same time is energetically extremely expensive.

[0280] Since the unit disk graph has a geometric interpretation, the wells can be arranged according to the arrangement of the vertices in the unit disk graph. The length unit is chosen so that the Rydberg blockade radius corresponds to the unit distance in the graph, that is, so that

[0281] W u,v >Δ(T), provided that d u,v <R (8)

[0282] W u,v <Δ(T), provided that d u,v >R (9)

[0283] Experimentally, a quantum optimization algorithm can be implemented by slowly varying the parameters Ω(t)Ω(t) and Δ(t)Δ(t) and measuring which atoms are in Rydberg states at the end. If the evolution is slow enough, this will be the maximum independent set. If the evolution is not completely adiabatic, but the injected energy is low, the final state will usually be a superposition of "independent set states", that is, a configuration of atoms that would be in Rydberg states if they were not within the blockade radius. The larger the time T, the better the approximation ratio that the protocol can achieve.

[0284] The above method of encoding the MIS problem ignores interactions smaller than the blocking interactions. If long-tail interactions are included, the geometric arrangement of the wells can be chosen so that all wells v are defined as in in is the maximum possible energy transfer that can occur for an atom in a Rydberg state at vertex v due to interactions outside the blockade radius. Therefore, as long as δ v δ v If is very small (i.e., the interactions between atoms outside the blockade radius can be neglected), the quantum algorithm gives (or approximates) a solution to the maximum independent set problem.

[0285] Fig.13A An example of a unit disk diagram is shown and the maximum independent set is indicated. Fig.13A As shown, the unit disk graph has 25 vertices (small circle 1310) and a vertex density of 2.7. The vertex 1320 at the center of the larger circle 1330 consists of the largest independent set (there are more than one). The larger circle 1330 indicates the blocked radius. Fig. 13B The probability distribution of finding an independent set of size k when the quantum algorithm runs for time T is shown. The longer the time T, the higher the probability that the algorithm will discover a larger (or even the largest) independent set. The probability distribution of the size of the independent set is found by the adiabatic algorithm after the evolution with Hamiltonian 2 and time T. Here, the units are such that and For time T~5, the probability of finding the global optimal value is very high.

[0286] The technology described in the present disclosure may also include many variations or applications. For example, the coherence properties of atoms can be improved by increasing the intermediate state detuning to further suppress spontaneous emission and cooling the atomic motion to the ground state to eliminate the residual Doppler shift by Raman sidebands. The light offset associated with the captured light can be used to realize the rotation of a single qubit around the z-axis, and the second AOD can be used for the separate control of the coherent rotation around other directions. Coherence and controllability can be further improved by encoding the qubit into the hyperfine sub-level of the electronic ground state and using state-selective Rydberg excitation. It is possible to implement two-dimensional (2d) to form thousands of wells. Such 2d configurations can be realized by directly using 2d-AOD or by forming a static 2d lattice of a well and using an independent AOD to sort atoms.

[0287] It should be understood that the application of the disclosed subject matter is not limited to the construction details and component arrangements set forth in the following description or shown in the accompanying drawings. The disclosed subject matter can have other embodiments and can be practiced and implemented in various ways. In addition, it should be understood that the wording and terminology used herein are for descriptive purposes and should not be considered as limiting.

[0288] Therefore, it will be appreciated by those skilled in the art that the concepts upon which the present disclosure is based can be readily used as a basis for designing other structures, methods and systems for implementing the multiple purposes of the disclosed subject matter. It is therefore important that the claims be deemed to include such equivalent constructions without departing from the spirit and scope of the disclosed subject matter.

[0289] While the disclosed subject matter has been described and illustrated in the foregoing exemplary embodiments, it should be understood that the present disclosure has been implemented by way of example only and that many changes may be made in the details of the implementation of the disclosed subject matter without departing from the spirit and scope of the disclosed subject matter, which is limited only by the appended claims.

[0290] Techniques and systems disclosed herein, such as a particular AOD or laser system, may be controlled, for example, by using a computer program product for use with a network, a computer system, or a computerized electronic device. Such an implementation may include a series of computer instructions or logic fixed on a tangible medium, such as a computer readable medium (e.g., a disk, CD-ROM, ROM, flash memory, or other memory or fixed disk), or transmittable to a network, computer system, or device via a modem or other interface device (such as a communication adapter connected to the network through the medium).

[0291] The medium may be a tangible medium (e.g., optical or analog communication lines) or a medium implemented with wireless technology (e.g., Wi-Fi, cellular, microwave, infrared or other transmission technology). The series of computer instructions embody at least a portion of the functionality described herein with respect to the system. Those skilled in the art will appreciate that such computer instructions may be written in a plurality of programming languages ​​for use with many computer architectures or operating systems.

[0292] Furthermore, such instructions may be stored in any tangible memory device such as semiconductor, magnetic, optical or other memory devices and may be transmitted using any communication technology such as optical, infrared, microwave or other transmission technology.

[0293] It is desirable that such a computer program product can be distributed as a removable medium with a printed document or electronic document (e.g., compressed packaged software), pre-installed by a computer system (e.g., pre-installed on a system ROM or fixed disk), or distributed from a server or electronic bulletin board over a network (e.g., the Internet or the World Wide Web). Of course, some embodiments of the present invention can be implemented as a combination of both software (e.g., a computer program product) and hardware. Other embodiments of the present invention are implemented as complete hardware or complete software (e.g., a computer program product).

[0294] In the foregoing description, certain steps or processes may be performed on a particular server or as part of a particular engine. These descriptions are illustrative only, as the particular steps may be performed on a variety of hardware devices, including but not limited to server systems and / or mobile devices. Alternatively or in addition, any or all of the steps described herein may be performed on a virtual machine running on the physical server itself. Similarly, the area in which a particular step is performed may vary, and it should be understood that any area or different areas are not within the scope of the present invention. In addition, the use of "modules" and / or other terms used to describe computer system processing is intended to be interchangeable and represent logic or circuits in which a function may be performed.

Claims

1. A method comprising: selectively arranging a plurality of atomic qubits into a spatial structure to encode a quantum computing problem, the spatial structure comprising a plurality of vertex qubits, wherein each vertex qubit corresponds to a vertex in the quantum computing problem, and wherein spatial proximity of the plurality of atomic qubits represents an edge in the quantum computing problem; Initializing the plurality of atomic qubits to an initial state; driving the plurality of atomic qubits to a final state by applying a sequence of resonant light pulses to at least some of the plurality of atomic qubits, wherein the final state comprises a solution to the quantum computing problem; and At least some of the plurality of atomic qubits in the final state are measured.

2. The method of claim 1, wherein the resonant light pulses have a variable detuning. The method of claim 1 , wherein the resonant light pulses have a variable Rabi frequency.

4. The method of claim 1, wherein the spatial structure comprises a one-dimensional or two-dimensional array of qubits.

5. The method of claim 1, wherein the encoded quantum computing problem comprises one or more of an unweighted maximum independent set problem, a maximum weighted independent set problem, a maximum clique problem, and a minimum vertex cover problem.

6. The method of claim 5, wherein the weights in the maximum weight independent set problem are encoded by applying a different detuning to each vertex qubit.

7. The method of claim 5, wherein the final state of the plurality of atomic qubits comprises one or more of a solution to an encoded unweighted maximum independent set problem, a solution to an encoded maximum weighted independent set problem, a solution to an encoded maximum clique problem, and a solution to an encoded minimum vertex cover.

8. The method of claim 1, wherein the solution to the quantum computing problem comprises an approximate solution to the quantum computing problem.

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