Dynamically Reconfigurable Architectures for Quantum Information and Simulation

Abstract

A dynamically reconfigurable architecture for quantum information and simulation is provided. A plurality of neutral atoms are provided. Each neutral atom is disposed in a corresponding optical trap. Each of the plurality of neutral atoms is m F =0 clock state. Pairs of neutral atoms of a plurality of neutral atoms are entangled by directing a laser pulse thereon. The laser pulse is configured to transition the pair of neutral atoms through a Rydberg state. An optical trap corresponding to at least one neutral atom of the pair is moved adiabatically, thereby moving one atom of the pair relative to the other atom of the pair without destroying the entanglement of the pair.

Description

[Technical field]

[0001] REFERENCE TO RELATED APPLICATIONS This application claims the benefit of U.S. Provisional Application No. 63 / 228,940, filed Aug. 3, 2021, which is incorporated by reference in its entirety.

[0002] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT This invention was made with government support under awards 1745303, 1734011, 2012023 from the National Science Foundation, and W911NF2010021 and W911NF2010082 awarded by the U.S. Army Research Office, and N00014-15-1-2846 and N00014-15-1-2761 awarded by the U.S. Naval Research Laboratory, and DE-SC0021013 awarded by the U.S. Department of Energy. The Government has certain rights in this invention. [Background technology]

[0003] background Aspects of the present disclosure relate to dynamically reconfigurable architectures for quantum computing, and more specifically, quantum information and simulation. Summary of the Invention

[0004] Quick Overview According to an aspect of the present disclosure, a method of quantum computing is provided. A plurality of neutral atoms is provided. Each of the plurality of neutral atoms is disposed in a corresponding optical trap. Each of the plurality of neutral atoms is a F =0 clock state. A pair of neutral atoms of a plurality of neutral atoms is entangled by directing a laser pulse thereto. The laser pulse is configured to transition the pair of neutral atoms through a Rydberg state. An optical trap corresponding to at least one neutral atom of the pair is moved adiabatically, and a Raman pulse is applied to at least one neutral atom during the movement, thereby moving the neutral atoms of the pair relative to each other without destroying the entanglement of the pair.

[0005] In various embodiments, the Raman pulse is applied at a midpoint of the motion. In various embodiments, the adiabatic motion has a constant jerk. In various embodiments, the adiabatic motion has an average velocity of less than 0.55 μm / μs.

[0006] In various embodiments, an optical trap corresponding to at least one neutral atom is moved within a blockade radius of a target neutral atom of the plurality of neutral atoms. In various embodiments, the at least one neutral atom is entangled with the target neutral atom. In various embodiments, a gate is applied to the at least one neutral atom and the target neutral atom.

[0007] In various embodiments, the plurality of neutral atoms forms a two-dimensional array. In various embodiments, the at least one neutral atom and the target neutral atom are non-adjacent in the two-dimensional array prior to said movement.

[0008] In various embodiments, the optical trap corresponding to the at least one neutral atom is generated by directing a beam of light to at least one acousto-optical deflector (AOD), where adiabatically moving the optical trap corresponding to the at least one neutral atom comprises varying a drive frequency of the at least one AOD. In various embodiments, at least a first subset of the optical traps corresponding to the plurality of neutral atoms are generated by directing a beam of light to a spatial light modulator (SLM).

[0009] According to an aspect of the present disclosure, a method of quantum computing is provided. A plurality of neutral atoms is provided. Each of the plurality of neutral atoms is disposed in a corresponding optical trap. Each of the plurality of neutral atoms is a F=0 clock state. A pair of neutral atoms of a plurality of neutral atoms is entangled by directing a laser pulse thereto, the laser pulse configured to transition the pair of neutral atoms through a Rydberg state. An optical trap corresponding to at least one neutral atom of the pair is moved adiabatically, thereby moving the neutral atoms of the pair relative to each other without destroying the entanglement of the pair. A first region is illuminated, the first region containing the first atom of the pair therein, thereby applying a rotation to the first atom of the pair. The optical trap corresponding to the first atom of the pair is moved adiabatically outside the first region. The optical trap corresponding to the second atom of the pair is moved adiabatically into the first region. The first region is illuminated, thereby applying a rotation to the second atom of the pair.

[0010] In various embodiments, a Raman pulse is applied to at least one neutral atom during said motion, hi various embodiments, a Raman pulse is applied at a midpoint of said motion.

[0011] In various embodiments, the adiabatic motion has a constant jerk. In various embodiments, the adiabatic motion has an average velocity of less than 0.55 μm / μs.

[0012] In various embodiments, the plurality of neutral atoms forms a two-dimensional array.

[0013] In various embodiments, the optical trap corresponding to the at least one neutral atom is generated by directing a beam of light to at least one acousto-optical deflector (AOD), where adiabatically moving the optical trap corresponding to the at least one neutral atom comprises varying a drive frequency of the at least one AOD. In various embodiments, at least a first subset of the optical traps corresponding to the plurality of neutral atoms are generated by directing a beam of light to a spatial light modulator (SLM).

[0014] According to an aspect of the present disclosure, a method of quantum computing is provided. A plurality of neutral atoms is provided. Each of the plurality of neutral atoms is disposed in a corresponding optical trap. The plurality of neutral atoms includes a first subset and a second subset. Each neutral atom of the first subset is disposed within a blockade radius of a first corresponding neutral atom of the second subset, thereby forming a first plurality of pairs. Each of the plurality of neutral atoms is disposed within a blockade radius of a first corresponding neutral atom of the second subset, thereby forming a first plurality of pairs. F =0 clock state. A first gate is applied to each of the first plurality of pairs. An optical trap corresponding to the first subset is moved adiabatically such that each neutral atom of the first subset is within a blockade radius of a second corresponding neutral atom of the second subset, thereby forming a second plurality of pairs. A Raman pulse is applied to the first subset during said movement. A second gate is applied to each of the second plurality of pairs.

[0015] In various embodiments, the first and / or second gate is a CZ gate.

[0016] In various embodiments, an optical trap corresponding to the first subset is adiabatically moved to an imaging region that does not include the second subset. The imaging region is illuminated to measure the state of the first subset.

[0017] In various embodiments, the optical traps corresponding to the first subset are moved simultaneously.

[0018] In various embodiments, a Raman pulse is applied at a midpoint of the motion.

[0019] In various embodiments, the adiabatic motion has a constant jerk. In various embodiments, the adiabatic motion has an average velocity of less than 0.55 μm / μs.

[0020] In various embodiments, the plurality of neutral atoms forms a two-dimensional array.

[0021] In various embodiments, the optical trap corresponding to the at least one neutral atom is generated by directing a beam of light to at least one acousto-optical deflector (AOD), where adiabatically moving the optical trap corresponding to the at least one neutral atom comprises varying a drive frequency of the at least one AOD. In various embodiments, at least a first subset of the optical traps corresponding to the plurality of neutral atoms are generated by directing a beam of light to a spatial light modulator (SLM).

[0022] According to an aspect of the present disclosure, a method of quantum computing is provided. A plurality of neutral atoms is provided. Each of the plurality of neutral atoms is disposed in a corresponding optical trap. Each of the plurality of neutral atoms is a F =0 clock state. The plurality of neutral atoms are moved adiabatically between a first configuration and a second configuration different from the first configuration. The first array configuration includes at least one pair of neutral atoms within a blockade radius of each other. A gate is applied to the at least one pair of neutral atoms when in the first configuration. The plurality of neutral atoms are advanced according to a first Hamiltonian when in the second configuration.

[0023] In various embodiments, a Raman pulse is applied to at least one neutral atom during said motion, hi various embodiments, a Raman pulse is applied at a midpoint of said motion.

[0024] In various embodiments, the adiabatic motion has a constant jerk. In various embodiments, the adiabatic motion has an average velocity of less than 0.55 μm / μs.

[0025] In various embodiments, the plurality of neutral atoms forms a two-dimensional array.

[0026] In various embodiments, the optical trap corresponding to the at least one neutral atom is generated by directing a beam of light to at least one acousto-optical deflector (AOD), where adiabatically moving the optical trap corresponding to the at least one neutral atom comprises varying a drive frequency of the at least one AOD. In various embodiments, at least a first subset of the optical traps corresponding to the plurality of neutral atoms are generated by directing a beam of light to a spatial light modulator (SLM).

[0027] According to various embodiments, a quantum computer is provided that includes a plurality of optical traps, a source of a plurality of neutral atoms, where each of the plurality of neutral atoms is disposable in a corresponding one of the plurality of optical traps, and at least one laser, the quantum computer configured to perform any of the aforementioned methods. [Brief description of the drawings]

[0028] Brief description of some figures of the drawing [Figure 1A] FIG. 1A is a schematic diagram of a quantum information architecture according to an embodiment of the present disclosure. [Figure 1B] FIG. 1B is a pair of images of a neutral atom before and after motion, according to an embodiment of the present disclosure. [Figure 1C] FIG. 1C is a graph of the parity oscillations of stationary and transported atoms according to an embodiment of the present disclosure. [Figure 1D] FIG. 1D is a graph of measured Bell state fidelity as a function of separation rate, according to an embodiment of the present disclosure. [Figure 2A] FIG. 2A is a series of images of neutral atoms illustrating the generation of a 12-atom 1D cluster state graph according to an embodiment of the present disclosure. [Figure 2B] FIG. 2B is a quantum circuit representation of 1D cluster state preparation and measurement according to an embodiment of the present disclosure. [Figure 2C] FIG. 2C is a graph of the raw measured stabilizer of the resulting 1D cluster state, according to an embodiment of the present disclosure. [Figure 2D] FIG. 2D is a graph state representation of a 7-qubit Steane code, according to an embodiment of the present disclosure. [Figure 2E] FIG. 2E is a circuit for preparing Steane code logic states according to an embodiment of the disclosure. [Figure 2F] FIG. 2F is a graph pair of measured stabilizers and logical operators according to an embodiment of the present disclosure. [Figure 3A] FIG. 3A illustrates a graphical representation of a surface code implementation according to an embodiment of the present disclosure. [Figure 3B] FIG. 3B is a graph of the measured X-plaquette and Z-star stabilizer of the resulting surface code according to an embodiment of the present disclosure. [Figure 3C] FIG. 3C is a schematic diagram of trick code execution according to an embodiment of the present disclosure. [Figure 3D] FIG. 3D shows the measured X-plaquette and Z-star stabilizers along with logical operators for two logical qubits with and without error detection according to an embodiment of the present disclosure. [Figure 4A] FIG. 4A illustrates a hybrid quantum circuit that combines coherent atomic transport with analog Hamiltonian evolution and digital quantum gates, according to an embodiment of the present disclosure. [Figure 4B] FIG. 4B includes two atomic images illustrating the measurement of entanglement entropy in a many-body Rydberg system via two-copy interferometry, according to an embodiment of the present disclosure. [Figure 4C] FIG. 4C is a graph of the measured half-chain Renyi entanglement entropy after many-body dynamics, according to an embodiment of the present disclosure. [Figure 4D] FIG. 4D is a graph of mutual information for various system sizes, according to an embodiment of the disclosure. [Figure 4E] FIG. 4E is a graph of single-site Renyi entropy according to an embodiment of the present disclosure. [Figure 5A] FIG. 5A is a diagram of a CZ gate according to an embodiment of the present disclosure. [Figure 5B] FIG. 5B is a level diagram showing important 87Rb atomic levels, according to an embodiment of the present disclosure. [Figure 5C] FIG. 5C is a schematic diagram of an exemplary pulse sequence for running a quantum circuit according to an embodiment of the present disclosure. [Figure 6A] 6A-6D are graphs of atom loss and atom retention according to embodiments of the present disclosure. [Figure 6B] 6A-6D are graphs of atom loss and atom retention according to embodiments of the present disclosure. [Figure 6C] 6A-6D are graphs of atom loss and atom retention according to embodiments of the present disclosure. [Figure 6D] 6A-6D are graphs of atom loss and atom retention according to embodiments of the present disclosure. [Figure 7A] 7A-7C are graphs of pulse fidelity, coherence and population differences according to embodiments of the present disclosure. [Figure 7B] 7A-7C are graphs of pulse fidelity, coherence and population differences according to embodiments of the present disclosure. [Figure 7C] 7A-7C are graphs of pulse fidelity, coherence and population differences according to embodiments of the present disclosure. [Figure 8A] FIG. 8A is a schematic diagram of an exemplary pulse sequence according to an embodiment of the present disclosure. [Figure 8B] FIG. 8B is a graph of a hyperfine coherence sequence according to an embodiment of the present disclosure. [Figure 8C] FIG. 8C is a graph of vibration frequency according to an embodiment of the present disclosure. [Figure 9A] 9A-9D are schematic diagrams of the generation of 1D cluster states, Steane codes, surface codes and trick codes according to embodiments of the present disclosure. [Figure 9B] 9A-9D are schematic diagrams of the generation of 1D cluster states, Steane codes, surface codes and trick codes according to embodiments of the present disclosure. [Figure 9C]9A-9D are schematic diagrams of the generation of 1D cluster states, Steane codes, surface codes and trick codes according to embodiments of the present disclosure. [Figure 9D] 9A-9D are schematic diagrams of the generation of 1D cluster states, Steane codes, surface codes and trick codes according to embodiments of the present disclosure. [Figure 10A] 10A-10B are graphs of error estimation according to an embodiment of the present disclosure. [Figure 10B] 10A-10B are graphs of error estimation according to an embodiment of the present disclosure. [Figure 10C] FIG. 10C is a table of single qubit (SQ) and two qubit (TQ) gate errors according to an embodiment of the present disclosure. [Figure 11A] 11A-11C are graphs of error probability and expectation according to an embodiment of the present disclosure. [Figure 11B] 11A-11C are graphs of error probability and expectation according to an embodiment of the present disclosure. [Figure 11C] 11A-11C are graphs of error probability and expectation according to an embodiment of the present disclosure. [Figure 12A] 12A-12B are graphs testing interferometry measurements on a benchmark problem, according to an embodiment of the present disclosure. [Figure 12B] 12A-12B are graphs testing interferometry measurements on a benchmark problem, according to an embodiment of the present disclosure. [Figure 13A] 13A-13C are graphs of raw multi-body data and numerical modeling of errors according to embodiments of the present disclosure. [Figure 13B] 13A-13C are graphs of raw multi-body data and numerical modeling of errors according to embodiments of the present disclosure. [Figure 13C] 13A-13C are graphs of raw multi-body data and numerical modeling of errors according to embodiments of the present disclosure. [Figure 14A] 14A-14C are graphs of local observables and entanglement entropy for a quantum many-body scar, according to an embodiment of the present disclosure. [Figure 14B] 14A-14C are graphs of local observables and entanglement entropy for a quantum many-body scar, according to an embodiment of the present disclosure. [Figure 14C] 14A-14C are graphs of local observables and entanglement entropy for a quantum many-body scar, according to an embodiment of the present disclosure. [Figure 14D] FIG. 14D is a diagram of a bounded Hilbert space, according to an embodiment of the present disclosure. [Figure 15] FIG. 15 is a schematic diagram of an apparatus for quantum computing according to an embodiment of the present disclosure. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0029] Detailed Description The ability to engineer parallel programmable operations between desired qubits in a quantum processor is central to building scalable quantum information systems. In state-of-the-art approaches, qubits interact locally and are constrained by connectivity associated with their fixed spatial layout. The present disclosure provides a quantum processor with dynamic, nonlocal connectivity in which entangled qubits are coherently transported in a highly parallel manner across two spatial dimensions between layers of single- and two-qubit operations. This approach utilizes neutral atomic arrays that are trapped and transported by optical tweezers; hyperfine states are used for robust quantum information storage, and excitation to Rydberg states is used for entanglement generation.

[0030] In various examples, the architecture is used to realize programmable generation of entangled graph states such as cluster states and 7-qubit Steane code states. Additionally, entangled ancillary arrays are shuttled to realize surface code states with 13 data and 6 ancillary qubits and toric code states on a torus with 16 data and 8 ancillary qubits. The architecture is also used to realize hybrid analog-digital extension and to use it to measure entanglement entropy in quantum simulations, experimentally observing nonmonotonic entanglement dynamics associated with quantum many-body scars. By achieving a long-standing goal, these results pave the way toward scalable quantum processing and enable novel applications ranging from simulation to metrology.

[0031] A quantum bit (qubit) is the basic building block for a quantum computer. By analogy with classical bits (each bit is either 0 or 1) used to store information in traditional computers, a qubit can occupy two distinct states, labeled |0> and |1>, or any quantum superposition of two states. In various applications, multiple qubits are entangled to construct multi-qubit quantum gates.

[0032] Bits and qubits are each encoded in the state of a real physical system: for example, a classical bit (0 or 1) can be encoded in whether a capacitor is charged or discharged or whether a switch is "on" or "off."

[0033] The term qudit (quantum digit) denotes a unit of quantum information that can be realized in a suitable d-level quantum system. A collection of qubits that can be measured to N states can implement an N-level qudit.

[0034] A qubit is encoded in a quantum system that has two (or more) distinct quantum states. There are many physical realizations that can be used. One example is based on individual particles such as atoms, ions or molecules that are isolated in a vacuum. These isolated atoms, ions and molecules have many distinct quantum states that correspond to different orientations of electron spin, nuclear spin, electron orbitals and molecular rotation / vibration.

[0035] In principle, qubits can be encoded into any pair of atomic / ionic / molecular quantum states. In practice, an important parameter of qubits is described by their quantum coherence property. Coherence measures the lifetime of a qubit before its information disappears. This has a close analogy with classical bits: if one prepares a classical bit in the 0 state, due to environmental noise, it can be randomly flipped to 1 after some time. Quantum mechanically, the same error can occur: |0> can be randomly flipped to |1> after some characteristic timescale. However, qubits can suffer further errors: for example, a superposition state

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[0036] A quantum computer may generally contain many qubits, each encoded into its own atom / molecule / ion / etc. Besides simply containing qubits, a quantum computer should be able to (1) initialize the qubits, (2) manipulate the state of the qubits in a controlled way, and (3) read out the final state of the qubits. When it comes to qubit manipulation, this is usually decomposed into two types: One type of qubit manipulation is the so-called single-qubit gate, which means an operation applied to a qubit individually. For example, this can flip the state of a qubit from |0> to |1> or flip |0> into a superposition state

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[0037] In various embodiments of quantum computers, qubits are encoded in energy levels close to the two ground states of an atom, ion or molecule. An example of this is the hyperfine qubit. Such qubits are encoded in two electrical ground states that differ by the relative orientation of the nuclear spin with respect to the outer electron spin. Such pairs of states can be chosen so that they are particularly robust / insensitive to environmental perturbations, resulting in long coherence times. These states are split in energy by the hyperfine interaction energy of the atom / ion / molecule, which is the interaction energy between the nuclear spin and the electron spin. The robustness of a qubit can be understood as the energy division between two states that are particularly stable. For this reason, such states are referred to as clock states, since a stable energy division can form an excellent frequency reference and thus a basis for atomic clocks. Typical hyperfine splitting between these qubit states is in the 1-13 GHz frequency range.

[0038] To perform single-qubit gates on such hyperfine qubits, it is possible to apply coherent microwave radiation at the exact frequency of the energy division between the states. However, this approach has two drawbacks. First, microwaves cannot be applied to just one qubit without affecting neighboring qubits. This is because qubits are typically encoded in particles that are just a few microns apart from each other, and because of their large wavelengths, microwaves cannot be focused on such small scales. Second, microwave strength is quite limited, and therefore the maximum speed of single-qubit gates is correspondingly limited.

[0039] An alternative approach is based on stimulated Raman transitions. In this case, a laser field is applied to the atom / ion / molecule. The laser field is nearly (but not exactly) resonant with an optical transition from one of the ground states to an optically excited state. The laser contains multiple frequency components that are separated in frequency by an amount exactly equal to the hyperfine splitting of the qubit. The atom / ion / molecule can absorb a photon from one frequency component and coherently emit a different frequency component, and in doing so it changes its state. This approach benefits from the ability in quantum computers to focus the laser field on individual particles or subsets of particles. The laser field can also be applied at high intensities, allowing for fairly fast gate operations.

[0040] Neutral atom quantum computers encode qubits in individual neutral atoms. Neutral atoms are trapped in a vacuum chamber and levitated by a trapping laser. Most commonly, the trapping laser is an individual optical tweezers, which is an individual tightly focused laser beam that traps individual atoms at the focal point. Alternatively, individual atoms can be trapped in an optical lattice formed by a standing wave of laser light that produces a periodic structure of nodes / antinodes.

[0041] A typical approach for encoding qubits in neutral atoms is the hyperfine qubit approach, where the two ground states are separated from the qubit by a few GHz. Multi-qubit gates in neutral atom quantum computers are realized using a third atomic state that is a highly excited Rydberg state. If one atom is excited to a Rydberg state, a neighboring atom is prevented from exciting to a Rydberg state. This conditional behavior forms the basis for multi-qubit gates such as the controlled-NOT gate. The Rydberg state is used temporarily to mediate the multi-qubit gate, and then the atoms are driven back from the Rydberg state to the ground state level to preserve their coherence.

[0042] Trapped ion quantum computers use atomic species that are ionized, meaning that they have a net charge. In most cases, many ions are trapped in one large trapping potential formed by electrodes in a vacuum chamber. The ions are attracted to the smallest trapping potential, but Coulomb repulsion between the ions causes them to form crystalline structures centered in the middle of the trapping potential. Most commonly, the ions align into linear chains. Other methods for trapping ions are also possible, such as using optical tweezers or trapping ions individually by local electric fields with more complex on-chip electrode structures.

[0043] Qubits can be encoded in trapped ions in a number of ways. One common approach is to use the ground state hyperfine level, as described for neutral atoms. As with neutral atoms, in trapped ions with hyperfine qubit encoding, single qubit gates can use microwave irradiation or stimulated Raman transitions.

[0044] Unlike in neutral atoms, trapped ion hyperfine qubits rely heavily on stimulated Raman transitions to perform multi-qubit gates. Stimulated Raman transitions can be used both to change the ion's state of motion (i.e., adding momentum) as well as to control the ion's hyperfine state. This can be understood as absorbing a photon moving in one direction and emitting a photon in a different direction, such that the difference in the momentum of the photons is absorbed by the ion. Often many ions are trapped in one collective trapping potential, each repelling each other, so that changing the state of motion of one ion affects the other ions in the system, and this mechanism forms the basis for multi-qubit gates.

[0045] According to various embodiments of the quantum computer, individual particles (atoms / ions / molecules) can first be trapped in an array and aligned in a specific configuration. Next, one or more particles are prepared in a desired quantum state. A quantum circuit can then be performed by a sequence of qubit operations acting on individual qubits (single-qubit gates) or groups of two or more qubits (multi-qubit gates). Finally, the state of the particles can be read out to observe the results of the quantum circuit. Readout can be achieved using an observation system that typically includes an electron-multiplying CCD (EMCCD) camera image to detect the loaded position of the particles and a second camera image to read out the final state of the particles, for example by detecting the fluorescence emitted by the particles in their final state.

[0046] Quantum information platforms rely on interactions between qubits either to perform quantum gates or to perform analog many-body simulations. However, qubits often interact in a localized manner, which limits the connectivity of circuits or analog simulations and hinders possible computations. Some platforms can communicate in a non-local manner through the use of shared buses (e.g., trapped ions), but these shared bus approaches are limited to small systems and therefore still require a way to dynamically move qubits around to truly scale up the platform.

[0047] This disclosure shows that neutral atomic arrays can be dynamically reshaped while storing quantum information in hyperfine states and preserving quantum coherence and entanglement between qubits by shuttling atoms in optical tweezers. This approach provides a scalable way to realize quantum information systems with many qubits and arbitrary programmability--where any qubit can perform an entanglement gate with any other qubit in the array. Various quantum information circuits are described herein that use high-fidelity two-qubit Rydberg gates to enhance the programmability and nonlocal coupling achievable with these approaches. Examples of high-fidelity Rydberg gates are described in Levine, et al., Parallel Implementation of High-Fidelity Multiqubit Gates with Neutral Atoms, Phys. Rev. Lett., vol. 123, issue 17, https: / / link.aps.org / doi / 10.1103 / PhysRevLett.123.170503, which is incorporated herein by reference.

[0048] The approach described herein is naturally adapted to create graph states, an important class of quantum information states defined by stabilizer states or graphs, some of which have nonlocal connectivity. In particular, this disclosure demonstrates the preparation of 1D cluster states, 7-qubit Steane code quantum error correcting codes, and surface code quantum error correcting codes with high fidelity.

[0049] To further demonstrate the true nonlocal capabilities of these approaches, this disclosure demonstrates entanglement of qubits on opposite ends of the array to implement periodic boundary conditions with 24 qubits and realize a Trick code on a torus, a canonical topological error-correcting code whose physical realization is impractical in other systems due to the required nonlocal connectivity, highlighting the unique power of this approach.

[0050] The approaches provided herein also provide various novel tools for analog quantum simulation with Rydberg atoms. As an example of this, the present disclosure demonstrates a quantum many-body quench on two identical many-body copies, then interferes the two systems with a gate-based protocol, yielding the entanglement entropy of the system - a significant quantity not previously experimentally measured in Rydberg atomic systems.

[0051] It will be appreciated that the approaches described herein have various advantages, such as the ability to preserve qubit coherence during motion and the ability to avoid breaking entanglement during motion.

[0052] As described in more detail below, the methods provided herein enable various computational scenarios. In some scenarios, multiple neutral atoms are moved in parallel between multiple regions in space. For example, a source of illumination can be directed to a first region, and atoms are moved inside or outside the region between the application of pulses by the source of illumination. Similarly, a camera can be directed to an imaging region, and atoms are moved inside and outside the imaging region for imaging. Similarly, atoms can be moved inside and outside the blockade radius of other atoms, thereby enabling the application of gates to different groups of atoms at different stages of an algorithm or layers of a quantum circuit.

[0053] It will be appreciated that various stabilizer codes entail readout of the ancilla qubits, and the present disclosure allows for physical relocation of the ancilla qubits to the imaging region separate from the data qubits. In this manner, readout of the ancilla qubits can be provided without destruction of the data qubits.

[0054] More generally, an array of atoms can be moved between multiple configurations to facilitate both digital gates between different selections of atoms and analog evolution of the array as a whole. As used herein, an array of atoms or a configuration of multiple atoms refers to the positioning of these atoms relative to each other. It is understood that a particular configuration provides connectivity between qubits that allows for a particular gate or analog evolution according to a particular Hamiltonian. One advantage of the methods provided herein is that atoms can be moved to the vicinity of atoms that were not adjacent in the array. Non-adjacent atoms are those that are not within the unit cell in a regular lattice or are not nearest neighbors in an irregular array. For example, in a rectangular lattice, each atom has eight atoms that are within its unit cell and therefore eight adjacent atoms (regardless of edges).

[0055] As further defined below, to preserve entanglement, the atoms are moved adiabatically. As used herein, the term adiabatic motion refers to motion that avoids transitions of the subject atoms within their traps. For example, motion is considered adiabatic if the first derivative of the acceleration of the subject atoms is not greater than a predefined value. Typically, jerk<(atom size)×(trap frequency) 3 Adiabatic motion occurs when In physics, a jerk or jolt is the term given to the rate at which an object's acceleration changes with respect to time.

[0056] In addition to adiabatic motion, in some embodiments, dynamic decoupling is applied during motion. As described further below, a π-pulse during motion counteracts the dephasing induced by the trap differential light shift. The trap differential light shift changes when the atom is moving (depending on its acceleration) because the atom moves in the trap and samples different portions of the light intensity and therefore has a different differential light shift.

[0057] In general, the more pulses applied, the greater the decoupling from fluctuations, which may arise, for example, from laser intensity fluctuations at different transition positions of atoms or different magnetic fields in space.

[0058] In embodiments where acceleration and deceleration are symmetric, both change the differential light shift in the same way. Therefore, in such embodiments, it is advantageous to apply a π pulse at the midpoint of the motion. In this way, the changes in differential light shift induced by acceleration and deceleration cancel each other out.

[0059] As is known in the art, analog evolution of a system of neutral atoms under a Hamiltonian can be used to perform quantum simulations and related problems. As described below, the methods provided herein can be used to move atoms into configurations suitable for analog Hamiltonian evolution according to a given Hamiltonian. The atoms can further be moved backwards and forwards between such configurations and configurations suitable for the application of digital quantum gates.

[0060] In the following example, such an approach is described to measure entanglement entropy in a many-body system. However, it is understood that the approach can be used for a variety of additional problems. For example, moving atoms between multiple configurations and performing multiple rounds of analog evolution allows the formation of a maximum independent set problem on a graph with non-local connectivity. Using digital gates and multiple copies, error mitigation can be performed on an analog quantum simulator. More generally, applying gates in this manner allows for more precise control of analog evolution (such as spin liquids). This control can also be used to perform shadow tomography in complex systems as a method for investigating many-body physics.

[0061] Referring to FIG. 1, a quantum information architecture enabled by coherent transport of neutral atoms is illustrated. Qubits are transported to perform entanglement gates with distant qubits, allowing programmable non-local connectivity. Shuttling of atoms is performed using optical tweezers, allowing selective manipulation with high parallelism between multiple domains in two dimensions. The inset shows the atomic levels used: |0>, |1> qubit states are 87 Rbm F =0 clock state, and |r> is the Rydberg state used to generate entanglement between the qubits (Figure 5B). Figure 1B shows atomic images illustrating the coherent transport of entangled qubits. Using a sequence of single-qubit and two-qubit gates, the atomic pairs are transported through |Φ + >The qubits are prepared in the Bell state and then separated by 110 μm over a 300 μs interval. Figure 1C is a graph showing parity oscillations indicating that motion does not observably affect entanglement or coherence. For both motion and stationary measurements, qubit coherence is preserved by using the XY8 dynamic decoupling sequence for 300 μs. Figure 1D is a graph of measured Bell state fidelity as a function of separation speed over 110 μm, showing that fidelity is unaffected for motions slower than 200 μs (average separation speed of 0.55 μm / μs). Inset: normalization by atom loss during motion yields constant fidelity, indicating that atom loss is the dominant error mechanism.

[0062] Quantum information systems derive their power from controllable interactions that give rise to quantum entanglement. However, the natural local character of the interactions limits the connectivity of quantum circuits and simulations. Nonlocal connectivity can be engineered via global shared quantum data buses, but these approaches are limited in either control or size.

[0063] According to various aspects of the present disclosure, this long-standing challenge is addressed by a dynamically reconfigurable array of entangled neutral atoms that are shuttled by optical tweezers in two spatial dimensions (FIG. 1A). Hyperfine states are used to store and transport quantum information between quantum operations, and excitation to Rydberg states is used to generate entanglement. Highly parallel operations are enabled by selective qubit manipulation in distinct regions where qubits are dynamically shuttled. Together, these components enable powerful quantum information architectures that are used to realize applications such as the generation of entangled states, the creation of topological surfaces and trick code states, and hybrid analog-digital quantum simulation.

[0064] Entanglement transport in atomic arrays In various embodiments, the two-dimensional atomic array systems described below are used to implement multiple layers of coherent transport and single-qubit and two-qubit gates. Quantum information is 87 The Rb atoms are stored in a magnetically insensitive clock state within the ground-state hyperfine manifold. Robust single-qubit Raman rotation (scattering error per π pulse, approx. 7×10 -5 ) is realized by a composite pulse that is robust against pulse errors (Fig. 7A-B). A high-fidelity controlled-Z (CZ) entanglement gate (Fig. 1A) in the hyperfine basis {|0>, |1>} is

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[0065] 1A-D show the ability to transport qubits over large spacings while preserving entanglement and coherence. The pairs are initialized with an atom-atom spacing of 3 μm (FIG. 1B) and then transported into the Bell state

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[0066] Programmable circuits and graph states To illustrate the ability to generate nonlocal connectivity among qubit arrays in parallel, entangled graph states are prepared: a large class of useful quantum information states with examples ranging from GHZ states and cluster states to quantum error-correcting codes.

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[0067] Referring to Fig. 2, 1D and 2D graph states using dynamical entanglement transport are illustrated. In Fig. 2A, the generation of a 12-atom 1D cluster state graph is illustrated, which is generated by initializing all qubits (vertices) at |+> and applying controlled-Z gates on the connections (edges) between the qubits. Atomic images show the configuration for the first and second gate layers. Fig. 2B shows a quantum circuit representation of the 1D cluster state preparation and measurement. Dynamical decoupling is applied throughout the entire quantum circuit (see Methods). Fig. 2C shows the quantum circuit representation of the S i =Z i-1 X i Z i+1 (For edge qubits, X1Z2 and Z 11 X 12) for the resulting 1D cluster state. Figure 2D shows a graph state representation of the 7-qubit Steane code (shading indicates the stabilizer plaquette). Figure 2E shows the Steane code logic implemented in four parallel gate layers. L The circuit for adjusting the state is shown in FIG. 2F. L Figure 1 shows the measured stabilizers and logical operators after adjusting for . Error detection is done by postselecting measurements where all stabilizers are +1. For both the 1D cluster state and the Steane code, the stabilizers and logical operators are measured with two measurement settings. Error bars represent 68% confidence intervals.

[0068] Figure 2A shows the preparation of a 1D cluster state, a graph state, defined by a linear chain of qubits. To achieve this state, one comprehensive parallel layer of CZ gates is performed on adjacent atomic pairs, half the atoms are moved to form a new pair, and then another parallel layer of CZ gates is performed (Figure 2A,B). To probe the resulting 12-qubit cluster state, we first measure the stabiliser set {S i}={Z i-1 X i Z i+1 Local rotation is achieved by moving one sublattice of qubits to another region and then performing a rotation on the stationary qubit with a uniform beam illuminating the experimental region (Fig. 1A, Methods). <s1>is measured by analyzing the resulting bit-string output and plotting the resulting raw stabilizer measurements (Figure 2C). Across all 12 stabilizers, the average i >=0.87(1) was observed (Figure 2C) (state preparation and measurement SPAM errors were <s1>= 0.91(1)), two biseparable entanglements are demonstrated in the cluster state (all <s1>>0.5). The measured fidelity corresponds to a few percent of error per operation for the measurement system quantum computer calculation.

[0069] An important class of graph states are quantum error correction (QEC) codes, where the graph state stabilizers, denoted as QEC code stabilizers, can be measured to correct errors on the encoded logical qubits. In fact, all stabilizer QEC states are equivalent to some graph state up to a single-qubit Clifford rotation, so the ability to generate arbitrary graph states allows a wide range of QEC states to be easily prepared. As an example, the seven-qubit Steane code, a topological color code depicted by the graph in Figure 2D, can be written in the logical state |+> L To prepare this state, all qubits are initialized with |+> and CZ is applied on the connections between the qubits (in four parallel layers, see Fig. 9B). Then, one of the two sublattices is rotated to measure the stabilizers (Fig. 2E). After the sublattice rotation, six of the graph state stabilizers are i or Z i The seventh graph state stabilizer is given by the product of four fields: L and the logical qubit operator X L which has an eigenvalue of +1 for the graph state |G>. Thus, in Figure 2F, <X L >=0.71(2) and <Z L >=-0.02(3) and the logical qubit state |+> L In addition, error detection is performed by postselecting the measurement results, where all measured stabilizers yield +1 (having a 66(1)% chance of no errors being detected). Using this procedure, the corrected value

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[0070] Topological states with ancillary arrays Transportable ancillary qubit arrays are also used to mediate quantum operations between remote qubits. Due to the ability to rapidly move arrays of atoms throughout the system, the use of ancillary qubits naturally complements the motor capabilities provided herein. Specifically, ancillaries are used for state preparation by mediating entanglement between physical qubits that do not directly interact, followed by a form of measurement-based quantum computer calculation, projective measurement of the ancillary array (performed simultaneously with the measurement of the data qubits). In particular, topological surface code and trick code states are prepared, whose states are more difficult to construct by direct CZ gates between physical qubits (requiring a large number of layers). For these codes, the measured values ​​of the ancillary qubits are handled in-software for the actual QEC operations by simply redefining the stabilizers. Because the redefinition is applied in-software, without physical intervention, projective measurements on the ancillaries are interchangeable with all operations on the data qubits and can be performed at any time, so that all qubits are measured simultaneously.

[0071] Referring to Figure 3, topological surface code and trick code states are illustrated using movable ancilla qubit arrays. Figure 3A shows the graph states that realize the surface code. The circuit shows the formation of the graph state with movable ancilla qubits; each movement corresponds to the execution of a CZ gate with adjacent data qubits (illustrated in a box). Logic |+> L The states are generated upon projective measurement of the ancilla qubit in the X-reference. The schematic on the right shows the stabilizers and the logical operators of the code. Figure 3B shows the measured X-plaquette and Z-star stabilizers of the resulting surface code, together with the logical operators with and without error detection (performed in postselection). Figure 3C illustrates the execution of the trick code. (Top) Two logical qubit seed states of the trick code upon projective measurement of the ancilla qubit in the X-reference

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[0072] Figure 3A shows the surface code |+> L We show the preparation of a 19-qubit graph state that generates a logical state. The surface code is defined by the X-plaquette and Z-star stabilizers and is represented by the logical operator X L (Z L ) is defined as a string of X(Z) products over the height (width) of the graph. To prepare this state, the ancillars are moved to perform a CZ gate with each of their four neighbors, which are then measured, projecting the data qubits onto the surface code state. Here the graph state stabilizers are the X-plaquette, the Z-star (which has values ​​±1 for ±1 measurements of the central ancillar), and the logical X L Notably, this procedure produces topologically ordered states in a circuit of constant depth, where the measured ancillary values ​​can be used to redefine the stabilizers, which can be handled in software for the actual QEC operation.

[0073] FIG. 3B shows the measured expectation values ​​of the 12 obtained stabilizers and the logical operator expectation values ​​with and without fault detection. <X L Using the stabilizer measured for false detection, we see raw values ​​of >=0.64(3).

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[0074] Although surface codes can be prepared by other methods, the transport capabilities provided herein allow for periodic boundary conditions and realization of trick code states on the torus. To this end, the 24-qubit graph state shown in FIG. 3C is generated by executing five layers of parallel gates and moving the ancillaries to their separate regions for readout on a separate basis. The prepared state has 7 independent X-plaquette and 7 independent Z-star (due to the periodic boundary conditions). Furthermore, due to the topological properties of this graph, two independent logical qubits can be used to perform logical operators that wrap around the entire torus along two topologically distinct directions.

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[0075] The state preparation is verified in FIG. 3D by measuring the trick code stabilizer. For two encoded logical qubits,

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[0076] Hybrid Analog-Digital Circuit Referring to FIG. 4, dynamic reconfigurability for hybrid analog-digital quantum simulation is illustrated. FIG. 4A shows a hybrid quantum circuit combining coherent atomic transport with analog Hamiltonian evolution and digital quantum gates. FIG. 4B illustrates the measurement of entanglement entropy in a many-body Rydberg system via two-copy interferometry. FIG. 4C shows the measured half-chain Renyi entanglement entropy after many-body dynamics and subsequent quench for two eight-atom systems. Quench from |gggg...>(

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[0077] Atomic motion is further applicable to quantum simulation. In particular, the present disclosure provides a hybrid modular quantum circuit composed of analog Hamiltonian evolution, reshaping, and digital gates (FIG. 4A). Together, these tools open up a variety of novel possibilities in quantum simulation and many-body physics. As a specific example, the Renyi entanglement entropy is measured after a quantum quench by efficiently interfering two copies of a many-body system.

[0078] FIG. 4B illustrates the experimental procedure. After initializing both copies with all qubits at |1〉, each copy evolves over time t by the Rydberg Hamiltonian H Ryd The entanglement entropy is then measured by using a Bell measurement circuit to rearrange the system and interfere each qubit in the first copy with its identical paired counterpart in the second copy. Measuring the pairs in the Bell basis yields an antisymmetric singlet state

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[0079] The method is used to investigate the growth of entanglement entropy caused by many-body dynamics (see Methods for testing on further benchmark problems of the technique). Specifically, the evolution of two eight-atom copies under the Rydberg Hamiltonian, subject to nearest-neighbor blockade constraints, is examined.

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[0080] Such thermalization dynamics are generally predicted in strongly interacting many-body systems, but notably, it has been previously shown that for certain initial states, the system may avoid thermalization. Augmented by special non-thermal eigenstates, termed quantum many-body scars, these states have been theoretically predicted to feature dynamics associated with slow, non-monotonic entanglement growth. Figure 4 shows the initial state initialized by applying a local shift within one sublattice and by performing a global Rydberg π-pulse.

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[0081] These observations are in excellent agreement with accurate numerical simulations in the isolated system (Fig. 4C,E and lines plotted in Fig. 14). Furthermore, while the single-site purity approaches that of the fully-mixed state, the overall purity (16-body observables composed of a 3-level system) remains >100× that of the fully-mixed state (see Fig. 13), overall demonstrating the high precision and fidelity of this circuit-based technique. These results demonstrate that combining atomic motion, many-body Hamiltonian evolution and digital quantum circuits yields a powerful novel tool for simulating and probing the quantum physics of complex systems.

[0082] Observations and Perspectives The experiments described herein illustrate highly parallel coherent qubit transport and entanglement that enables powerful quantum information architectures. The technique can be expanded along several directions. Local Rydberg excitation on a subset of qubit pairs eliminates residual interactions from unintended atoms, allowing parallel independent operation on arrays with significantly higher qubit density. Two-qubit gate fidelity can be improved using higher Rydberg laser power or more efficient delivery methods as well as more advanced atomic cooling. These technological improvements should enable scaling to deep quantum circuits operating on thousands of neutral atomic qubits. These improvements can be further complemented by more sophisticated local single-qubit control using parallel Raman excitation, for example, via AOM arrays. Mid-circuit readout can be performed by moving the ancillary to another region and imaging, for example, using an avalanche photodiode array within hundreds of microseconds.

[0083] These methods have clear potential for realizing scalable quantum error correction. For example, the procedure shown in FIG. 3C can be used for syndrome extraction in a real QEC sequence, where ancillaries are entangled with their data qubit neighbors and then moved to another region for mid-circuit readout. The entire QEC round can be performed within milliseconds, much faster than the measured T2>1s, and the projected fidelity improvement theoretically exceeds the surface code threshold (Methods). Such mid-circuit readout is essential for realizing scalable fault-tolerant quantum computing. Furthermore, the ability to reshape and combine arrays allows for efficient parallel execution of transverse entanglement gates between many logical qubits. These techniques also enable the implementation of higher-order or nonlocal error correction codes with more favorable properties. Together, these building blocks can enable novel approaches for universal fault-tolerant quantum computing with thousands of physical qubits.

[0084] The dynamically reconfigurable architectures provided herein also open up many new opportunities for digital and analog quantum simulations. For example, the hybrid approach can be extended to explore the full entanglement spectrum, simulate wormhole creation, perform many-body refinements, and reshape novel non-equilibrium states. Entanglement transport can also empower metrological applications, such as the creation of distributed states to explore gravity gradients. Finally, these approaches can facilitate quantum network connections between isolated arrays, paving the way toward large-scale quantum information systems and distributed quantum metrology.

[0085] method Dynamic reshaping in 2D tweezers arrays These experiments used the same apparatus described below. Inside the vacuum vessel, 87 Rb atoms are loaded from the magneto-optical trap into a backbone array of programmable optical tweezers created by a spatial light modulator (SLM). The atoms are repositioned in parallel into defect-free target locations in this SLM backbone by further optical tweezers created from crossed 2D acousto-optical deflectors (AODs). After the repositioning procedure, selected atoms are transferred back from the stationary SLM trap into the mobile AOD trap, and then these mobile atoms are moved to their starting positions in the quantum circuit. During this entire process, the atoms are cooled by deflection gradient cooling. Before running the quantum circuit, a camera image is taken of the atoms in their initial starting positions. A final camera image is taken after the circuit to detect the qubit states |0> (atom presence) and |1> (atom loss after resonant push-out). Before running the circuit, all data is post-selected to find the complete repositioning of the AOD and SLM atoms. In all experiments here, each atom remains in a single stationary or single mobile trap throughout the duration of the quantum circuit.

[0086] The crossed AOD system consists of two independently controlled AODs (AA Opto Electronic DTSX-400) for x and y control of the beam position. Both AODs are driven by independent arbitrary waveforms generated by a dual channel arbitrary waveform generator (AWG) (M4i.6631-x8 by Spectrum Instrumentation) and then amplified by independent MW amplifiers (Minicircuits ZHL-5W-1). The time domain arbitrary waveforms consist of multiple frequency gradations corresponding to the x and y positions of the columns and rows, which are independently varied as a function of time to dynamically orient the AOD trapped atoms around; the full x and y waveforms are calculated by adding together the time domain profiles of all frequency components with the predefined amplitudes and phases for each component. To run the quantum circuit, we program the position of the AOD atoms at each gate position and then smoothly interpolate the AOD frequency (with a cubic profile) as a function of time between the gate positions. The cubic profile defines a constant jerk for the atoms, which allows for movement approximately 5-10x faster (without heating and loss) than when moving at a constant velocity (linear profile). In the movement protocol, tensions, compressions and deformations of the AOD trap array are applied: i.e., the rows and columns of the AODs do not cross each other to avoid atomic loss and heating associated with the two frequency components crossing each other.

[0087] To minimize the dephasing induced by the time-varying magnitude of the differential light shift, the AOD tweezer intensity is homogenized across all atomic trajectories. For this purpose, a reference camera is used in the image plane to measure the intensity of each AOD tweezer at each gate position and homogenize it by varying the amplitude of each frequency component; upon movement between the two positions, the amplitude of each individual frequency component is interpolated.

[0088] The SLM tweezer light (830 nm) and the AOD tweezer light (828 nm) are generated by two separate free-running Ti:sapphire lasers (M Squared, 18-W pump). When projected through a 0.5 NA objective, the SLM tweezers have a rough waist of about 900 nm (about 1000 nm for the AOD). When loading atoms, the trap depth is about 2π×16 MHz and the radial trapping frequency is about 2π×80 kHz; when running quantum circuits, the trap depth is about 2π×4 MHz and the radial trapping frequency is about 2π×40 kHz.

[0089] Raman Laser System Fast, high-fidelity single-qubit operations are a crucial component of the quantum circuits presented in this work. To this end, F We use a high-power 795 nm Raman laser system to drive the entire single qubit rotation between =0 clock states. This Raman laser system is based on dispersive optics. 795 nm light (Toptica TA pro, 1.8 W) is phase modulated by an electro-optic modulator (Qubig), which is driven by a 3.4 GHz microwave (Stanford Research Systems SRS SG384) that is doubled and amplified to 6.8 GHz. The laser phase modulation is converted to amplitude modulation to drive the Raman transition using a Chirped Bragg Grating (Optigrate). The IQ control of the SG384 is used to control the frequency and phase of the microwave, which is imprinted on the laser amplitude modulation, thus giving us direct frequency and phase control for the hyperfine qubit drive.

[0090] The Raman laser illuminates the atomic plane laterally in a circularly polarized elliptical beam with waists of 40 μm and 560 μm on the thin and tall axes, respectively, with a total average optical power of 150 mW at the atom. The large vertical spread ensures a non-uniformity of <1% across the atom, and shot-to-shot fluctuations in laser intensity are also <1%. For Figures 1-3, the Raman laser operates with a blue-detuned intermediate state detuning of 180 GHz, a two-photon Rabi frequency of 1 MHz, and a dc of 7 × 10 -5 of π For Figure 4, to shorten the duration of the coherent mapping pulse sequence, the Raman laser power is increased and a smaller blue-detuned intermediate state detuning of 63 GHz is used, with a corresponding two-photon Rabi frequency of 3.2 MHz, resulting in an estimated scattering error per π pulse of 2 × 10 -4 It is.

[0091] Robust single-qubit rotation For nearly all single-qubit rotations in this work (other than the XY8 / XY16 self-correcting sequences), robust single-qubit rotations are performed in the form of composite pulse sequences. These composite pulse sequences can be highly insensitive to pulse errors such as amplitude or detuning miscalibration. The dominant source of coherent single-qubit errors arises from amplitude drifts of ≤1% and non-uniformity across the array; therefore "BB1" (broadband 1) pulse sequences are primarily used, which are sequences of four pulses that perform arbitrary rotations on the Bloch sphere while being insensitive to amplitude errors up to the sixth order. The performance of these robust pulses is tested with a benchmark problem in Figure 7A. Furthermore, by applying a series of BB1 pulses, accumulated errors are seen that match the estimated scattering limit (not plotted here), indicating that the scattering limit is insensitive to the single-qubit rotation infidelity (approximately 3 × 10 per BB1 pulse due to the increased length of the composite pulse sequences). -4 It is suggested that this roughly represents the error of 1. Testing on randomized benchmark problems could be applied in future trials to further test single-qubit rotation fidelity.

[0092] Qubit coherence and dynamical decoupling In an 830 nm trap, the hyperfine qubit coherence is

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[0093] The transport sequence is accompanied by a dynamic decoupling sequence. The number of pulses used is a compromise between preserving qubit coherence while minimizing pulse errors. In various embodiments, there is an interchange between two types of dynamic decoupling sequences: the XY8 / XY16 sequence is a phase-shifted individual pulse that is self-correcting for amplitude and detuning errors. π The CPMG-type dynamic decoupling sequence is composed of a robust BB1 pulse, while the CPMG-type dynamic decoupling sequence is composed of a robust BB1 pulse. The CPMG-BB1 sequence is more robust against amplitude errors but suffers from more scattering errors. The sequences are compared with these different sequences and a variable number of decoupling sequences. π The decoupling sequence can be empirically optimized for any given experiment by choosing between pulses to optimize either single qubit coherence (including motion) or the final signal. Typically, the decoupling sequence consists of a total of 12 to 18 π It consists of pulses.

[0094] Kinetic effects on atomic heating and loss. Below we discuss the effect of motion on the loss and heating of atoms in the harmonic oscillator potential imposed by the tweezer trap. The motion of the trapping potential is equivalent to a non-inertial frame of reference in which the harmonic oscillator potential is at rest but the atoms experience an imaginary force given by F(t) = -ma(t), where m is the mass of the particle and a(t) is the acceleration of the trap as a function of time. The mean oscillation quantum number increase ΔN is given by

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[0095] Several relevant insights can be gleaned from this equation. First, this expression shows the ability to move large distances D with comparably small increments in time T. Furthermore, to maintain a constant ΔN, the movement time must be

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[0096] We now compare Equation 2 with experimental observations. In FIG. 1D, atom loss is observed with a movement of 55 μm in 200 μs under constant negative jerk. This velocity limit is consistent with the estimates above: ω0=2π×40 kHz and x zpf Using N = 38 nm, we predict that ΔN ≈ 6 for this motion, which corresponds to the onset of specific heating at this motion rate. More quantitatively, we assume a Poisson distribution with mean N and variance N, and estimate some critical N max The atom retention is then calculated by multiplying the total number of atoms by 100, which is the number of atoms that are trapped.

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[0097] Figure 6A and B show the relationship between the motion time T and the trap frequency.

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[0098] Further heating and loss during the circuit could also be caused by repeated short drops to implement a two-qubit gate, where the tweezers are briefly turned off to avoid anti-trapping of the Rydberg state and photo-shifting of the ground-Rydberg transition. However, the drop-recapture measurements in Figure 6C suggest that the 500 ns drops used experimentally have negligible effect up to several hundred drops per atom (corresponding to several hundred CZ gates). The atom loss and heating as a function of the number of drops is well described by a diffusion model, which then reduces the atomic temperature by a factor of 2× (the thermal rate

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[0099] Two-qubit CZ gate implementation Two-qubit gating and calibration can be performed using the techniques provided herein. Specifically, a two-qubit CZ gate is performed with two overall Rydberg pulses, each at a detuning Δ and length τ, with a phase jump ξ between the two pulses. The pulse parameters are selected such that qubit pairs adjacent to and below the Rydberg blockade constraint transition back from the Rydberg state to the hyperfine qubit manifold, with the phase depending on the state of the other qubit. Numerical values ​​for these pulse parameters are: Δ=-0.377371Ω ξ=-0.621089×(2π) τ=0.683201 / [Ω / (2π)] It is.

[0100] The experiments in Figures 1-3 were operated with a two-photon Rydberg-Rabi frequency of Ω / 2π = 3.6 MHz, giving a theoretical τ = 190 ns and a theoretical Δ / (2π) = -1.36 MHz. The negative detuning sign corresponds to a detuning of m by about 24 MHz under a field of 8.5 G. j = +1 / 2 is chosen to help minimize excitation to the Rydberg state (and the desired m j = -1 / 2 state). In this work, strong blockade between adjacent qubits is provided, and the Rydberg-Rydberg interaction V0 / 2π ranges from 200 MHz to 1 GHz. In Figure 4, Ω / 2π = 4.45 MHz for the two-qubit gate.

[0101] Managing spurious phase between CZ gates The two-qubit gate induces both an intrinsic single-qubit phase and a spurious phase induced primarily by the differential optical shift from the 420 nm laser. Under certain configurations, the differential optical shift induced by 420 nm on a hyperfine qubit can be extremely large (>8 MHz), resulting in phase accumulation on the hyperfine qubit of ≈6π. Thus, small percentage-level fluctuations in the 420 nm intensity can result in significant qubit detuning.

[0102] This 420 induced phase problem can be addressed by performing an echo sequence: after the CZ gate, the 1013 nm Rydberg laser is turned off, a Raman π pulse is applied, and then the 420 nm laser is pulsed again to cancel the 420 photoinduced phase during the CZ gate. This method echoes the 420 induced phase but at the cost of a 2x increase in the 420 induced scattering error, which is the dominant source of error in two-qubit CZ gates.

[0103] Echoes between CZ gates. To address these various issues, a Raman π-pulse is implemented between each CZ gate, echoing out of the spurious gate-induced phase on the hyperfine qubit (Figure 5). This approach has several advantages. Here, the 420-induced phase is cancelled by the pair of CZ gates without explicitly applying an additional 420 nm pulse to echo each individual CZ gate, thereby reducing the scattering error of the CZ gates in this work by a factor of about 2. This echo technique, which reduces the scattering error incurred between each gate, roughly compensates for the increased scattering rate incurred by spreading the optical dynamics over more space in 2D, thereby giving the two-qubit CZ gate fidelity an equivalent gate fidelity of ≥97.4(2)%. Furthermore, the echoes between the CZ gates also cancel out the intrinsic single-qubit phase of the CZ gate, eliminating errors in the calibration of this parameter, and cancel out any other gate-induced spurious single-qubit phases, such as the ≈0.01 rad phase induced by applying a pulse to trap off for 500 ns for a two-qubit gate (Figure 5). In examples where the number of CZ gates is odd, an echo for the last CZ gate is performed.

[0104] To further suppress the spurious 420-induced phase effects, the 420 nm laser was 3 / 2 The transition is operated to be red-detuned (by 2 GHz). For red-detuning, the optical shifts on the |0> and |1> states are of the same sign, minimizing the differential optical shift, while for blue-detuning <6.8 GHz, the optical shifts on the |0> and |1> states have opposite signs, amplifying the differential optical shift.

[0105] Sensitivity to axial trapped vibrations On typical Rydberg excitation timescales with optical tweezers, axial trap oscillation frequencies of a few kHz are insignificant. Now, with circuits running as long as 1.2 ms with Rydberg pulses throughout, axial trap oscillations can have significant effects. In particular, the axial oscillations cause the atoms to oscillate in and out of the Rydberg beam: with an estimated axial temperature of about 25 μK and an axial oscillation frequency of 6 kHz, the axial diffusion

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[0106] Bell state preparation and fidelity In Figure 1, |Φ + >A Bell state is prepared: after initializing a pair of qubits in |00>, a X / (π / 2) pulse-CZ gate-X / (π / 4) pulse is applied. This |Φ + The resulting raw fidelity of the >Bell state is averaged by the fitted amplitude of the parity oscillations (example in Fig. 1C), which measures off-diagonal coherence. In Fig. 1D, upon significant loss from motion, this fidelity estimate is skewed to account for the artificially large population measure in |11> (since state |1> is detected as a loss); thus, once the population difference between |11> and |00> becomes larger than 0.1 (an arbitrary cutoff where the effect of loss starts to become significant), |Φ + The > population is estimated as the 2× population of |00>. In Figure 1D, for movements slower than 300 μs, an average raw Bell state fidelity of 94.8(2)% is achieved after the movement. In the absence of movement or attempt to preserve coherence for 500 μs (i.e., measuring immediately after preparation of the Bell state), then a raw Bell state fidelity of 95.2(1)% is measured (not plotted here).

[0107] Analysis of error sources The following details some of the measurements and estimated sources of error for the entire sequence (especially the trick code preparation, the deepest example circuit). The total single qubit fidelity after running the entire sequence is approximately 96.5% for the trick code circuit, which is measured by embedding the entire experiment in a Ramsey sequence: a Raman π / 2 pulse is run, all motions and decouplings are run, and then a final π / 2 pulse where the fluctuating phase measures all contrasts. The single qubit fidelity is quantitatively illustrated in FIG. 10C as consisting of the known single qubit errors.

[0108] The estimated contributions to the two-qubit gate errors are summarized in FIG. 10C. These estimates arise from numerical simulations in QuTiP with the experimental parameters. The effects of intermediate state scattering and Rydberg damping are included in the Lindblad master equation solver via decay operators. Other error contributions include finite temperature random Doppler shifts and position fluctuations, as well as laser pulse-to-pulse fluctuations, all of which are simulated using classical Monte Carlo sampling of the experimental parameters. The experimental parameters used in the simulations are: blue and red Rabi frequencies (Ω b , Ω r )=2π×(160,90)MHz, 6P 3 / 2 Intermediate state detuning = 2GHz, intermediate state lifetime = 110ns, 70S 1 / 2 Rydberg state lifetime = 150 μs, Rydberg blockade energy = 500 ~ MHz, splitting into a second Rydberg state = 24 MHz, radial and axial trapping frequencies (ω r ,ω z )=2π×(40,6)kHz, and temperature T=20μK. This modeling can also be used to project future performance; by extrapolating a 10x increase in available 1013nm intensity and assuming the atoms are cooled to a 2uK temperature, a CZ gate fidelity of 99.7% above the surface code threshold is projected. Alkaline earth atoms may also offer another route to high fidelity operation for quantum error correction.

[0109] To understand how various single-qubit and two-qubit errors contribute to graph-state fidelity, stochastic simulations of the quantum circuit used for graph-state preparation are performed (Fig. 10A,B), exploiting the Clifford property of the circuit, which allows efficient numerical evaluation and random sampling of many possible error realizations. Simulations are performed under a realistic error model in which the ambient depolarization noise and atomic loss rates are measured in experiments (see Fig. 10C). The resulting stabilizer and logical qubit expectation values ​​are in good agreement with those measured experimentally.

[0110] Rydberg Beam Shaping and Uniformity The Rydberg beam is shaped into a tophat of variable size by wavefront control using a phase profile on a spatial light modulator (SLM). This capability allows the adaptation of the beam profile height to the experimental field size of any given experiment, thereby maximizing the 1013 nm light intensity and CZ gate fidelity. The Rydberg beam homogeneity is optimized until the peak-to-peak inhomogeneity is less than <1%. To this end, all aberrations are corrected up to the window of the vacuum chamber, which results in a few percent of atomic inhomogeneity that accounts for imperfections in the final window. To further optimize the homogeneity, the aberration correction is adjusted for the tophat via Zernike polynomial corrections up to the phase profile in the SLM plane (Fourier plane). With this procedure, the peak-to-peak inhomogeneity is reduced to <1% over a range of 40-50 μm in the atomic plane.

[0111] Graph layout generation and optimization The following outlines a description of how the graph layout is optimized for cluster state, Steane code, surface code and trick code preparation. The optimization in this example is recursive and other optimal circuits can be designed by atomic spatial arrangement and AOD trajectories. Figure 9 shows example graphs and the process for creating them. These are the results of optimization for several parameters: (1) Minimize the number of parallel two-qubit gate layers. (2) Minimize the total distance traveled for the moving atoms. (3) For all moving atoms in one sublattice (all graphs realized here are bipartite), facilitate a final local rotation of one sublattice. (4) Minimize the vertical extent of the array and the number of distinct rows (to maximize the intensity of the 1013 and minimize sensitivity to beam inhomogeneities between the rows). (5) When ordering gates, apply a two-qubit gate as early as possible in the circuit: if a layer of gates induces a bit flip (an X error), the error can propagate between subsequent gates (resulting in a Z error on the other qubit), so the gate should be in the earliest layer possible.

[0112] Local (sublattice) hyperfine rotation Local rotation is performed in the hyperfine basis using a horizontally propagating 420 nm beam, which imposes a differential light of several MHz on the hyperfine qubit and can therefore be used to realize fast Z rotation. To realize the local Y(π / 2) rotation used throughout this work, one sublattice of atoms is moved outside the 420 nm beam and then the following pulse is applied: [global Y(π / 4)]-[local Z(π)]-[global Y(π / 4)]. This realizes a Y(π / 2) rotation on one sublattice and a Z(π) rotation on the other sublattice (which is trivial if it is then interchangeable with an immediate measurement in the Z basis). To apply Y(π / 2) to the other sublattice of atoms, an additional global Z(π) is added between the two Y(π / 4) pulses (performed by jumping the Raman laser phase). An additional locally focused beam can be provided to perform local Raman control of the hyperfine qubit state. However, moving atoms (even >50 μm to move them outside the 420 nm beam) works very efficiently, and this approach is well suited to producing high-fidelity, homogeneous rotations of roughly half the qubits.

[0113] Local Rydberg Initialization Local Rydberg Control to Test the Dynamics of Multibody Scars

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[0114] A 50 MHz biased optical shift is significantly greater than the Rydberg-Rabi frequency Ω / 2π=4.45 MHz, resulting in a Rydberg population on undesired sites of <1%. The t=0 time point in Figure 14B was obtained using this approach.

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[0115] Rydberg Hamiltonian In FIG. 4, the dynamics under the many-body Rydberg Hamiltonian in Eq. 3 is considered.

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[0116] In Equation 3,

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[0117] Coherent Mapping Protocol A coherent mapping protocol is presented to transition a generic many-body state in {|1>,|r>} basis to a long-lived non-interacting {|0>,|1>} basis. To achieve this mapping, immediately after the Rydberg dynamics, a Raman π-pulse is applied to map |1> → |0>, and then a subsequent Rydberg π-pulse is applied to map |r> → |1>.

[0118] Even for perfect Raman and Rydberg π-pulses (on isolated atoms), there are three important sources of non-fidelity associated with this mapping process: (1) Any population in a blockade-violating state (i.e., two adjacent atoms are both at |r>) is strongly shifted off-resonant with respect to the final Rydberg π-pulse. Thus, this population of atoms remains in the Rydberg state and is lost. (2) Long-range interactions, e.g., from adjacent nearest neighbors, detune the final Rydberg π-pulse off resonance and therefore reduce the pulse fidelity. Because the long-range interactions are not the same for all many-body microstates, this effect cannot be mitigated by a simple shift in the detuning. (3) Dephasing of the state is Raman π Throughout the duration of the pulse, it arises from Doppler shifts preferentially between the ground states |0〉, |1〉 and the Rydberg state |r〉. These random on-site detunings are also present during many-body dynamics, but by turning off the Rydberg drive Ω, the system is free to accumulate phase, making us particularly susceptible to dephasing errors.

[0119] The above error mechanisms are mitigated as follows: To minimize the error from (1), the multibody dynamics is

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[0120] The overall Raman beam is π During the pulse, a photoinduced phase shift of ≈π on |0>, |1> for |r> is induced. Similarly, the overall 420 nm laser also π A light-shift-induced phase shift of ≈π between |0> and |1> during the pulse is induced. Because the measurements performed here are interferometric (i.e. the singlet state being measured is invariant under global rotation) they are unaffected by these global phase shifts, which can be measured and accounted for when relevant.

[0121] Measuring entanglement entropy The second order Renyi entanglement entropy is

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[0122] The Bell measurement circuit can be decomposed to apply an X / (π / 2) rotation to one atom of one of the pairs, then a CZ gate, then a global X / (π / 2) rotation. In the other measurement, the local X / (π / 2) is realized by doing a global X / (π / 4) rotation, then a local Z(π) rotation, and then a global X / (π / 4). However, for this singlet measurement circuit, the first X / (π / 4) overlaps if the singlet state is invariant under the global rotation, so for the local X / (π / 2), only a local Z(π) and then a second global X / (π / 4) are applied. This effectively realizes X / (π / 2) on one qubit up to Z(π) on the other qubit (not shown in the circuit diagram of Figure 4). Under this simplification, |Ψ - The Bell measurement circuit for mapping >→|00> can be roughly understood as the inverse of the Bell state preparation circuit, which is precisely how the parameters of the Bell measurement are calibrated.

[0123] Interferometry calibration and benchmark test. To validate the interferometry measurement (and check for proper calibration), it is tested in a benchmark separately from the many-body dynamics and coherent mapping protocols. The benchmark test is performed by preparing independent qubits (with a global Raman pulse of varying time) in identical, variable single-qubit superpositions and ensuring that the interferometry rarely produces |00> for all variable initial product states (Figure 12A). This is an important benchmark test step, since small miscalibrations of the Bell measurement can result in lower fidelity (i.e., higher entropy) for different initial product states, thereby resulting in additional spurious signals in the entanglement entropy measurement. The measurement is particularly sensitive to the single-qubit phase just before the final X / (π / 2) pulse (induced by the CZ gate and countered by the global Z(θ) pulse).

[0124] More multi-body data and details To test the method for measuring entanglement entropy in many-body systems on a benchmark problem, in Fig. 12B we test the entanglement dynamics after initializing two proximal atoms at |1〉 and exciting them resonantly to the Rydberg state for a variable time t. Under the condition of Rydberg blockade, this excitation can be transformed into the entangled state |11〉

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[0125] For the data shown in Figures 4C and 4E, the data is subtracted from the global classical entropy. This fixed time-independent offset is given by the entropy per particle, i.e., (global entropy at quench time t = 0) x (subsystem size) / (global system size). In Figure 13A, the raw entanglement entropy measurements are shown along with numerical values ​​to indicate the size of the global classical entropy contribution. In the plots, the Raman π To account for the fact that the pulse interrupts the final 10 ns of the Rydberg evolution, the theoretical curve is delayed by 10 ns, this is done to keep the coherent mapping gap as short as possible and minimize Doppler dephasing.Furthermore, in Fig. 13B, the measured overall purity is plotted and compared to a numerical simulation that incorporates the experimental error (Fig. 13C).

[0126] In Figure 14, further many-body data for an eight-atom chain system with the same parameters as those used in the main text are shown. The measured single-site entropy for each site is shown in Figure 14A.

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[0127] Referring to FIG. 5, the CZ gate echo, atomic level structure and a typical pulse sequence are illustrated. As shown in FIG. 5A, the two-qubit gate, in addition to applying a control Z operation between the two qubits, also induces a single qubit phase Z(ζ) to both qubits applying a pulse to trap off, consisting of the intrinsic phase of the CZ gate and an additional spurious phase from the 420 nm Rydberg laser. Since all gates are applied in parallel by the overall pulse of the Rydberg laser, if a qubit is not adjacent to another qubit, it does not perform a CZ gate but still acquires the same Z(ζ) (identical to that adjacent to another qubit in the dark state |0> with respect to the Rydberg laser). As shown in the figure, the additional unwanted Z(ζ) is cancelled by applying a π pulse between the pair of CZ gates. This echo procedure removes any need to calibrate the intrinsic phase from the CZ gate and makes us insensitive to any spurious changes in Z(ζ) slower than about 200 μs. An additional Y(π) propagates through the CZ gate in known fashion to multiply the particular stabilizer by a −1 sign, which simply redefines the signs of the stabilizers and logical qubits. 87 A level diagram showing the Rb atomic levels. The Rydberg excitation scheme from |1> to |r> consists of two-photon transitions driven by a 420 nm laser and a 1013 nm laser. A DC magnetic field of B=8.5 G is applied throughout this work. Figure 5C is a schematic of an exemplary pulse sequence for running the quantum circuit.

[0128] With reference to Figure 6, kinetic characterization and multiple drop recapture are illustrated. In Figure 6A, atom retention is given as a function of the average separation rate 2D / T with 0.7% background loss subtracted (as plotted in Figure 1D for separating Bell pairs). The inset in Figure 1D shows the (atom retention) (without subtracting background loss). 2 The dark curve is calculated using the experimental parameters and equation 2, N max Set = 26 and average

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[0129] Referring to Figure 7, robust single-qubit control and qubit coherence are illustrated. In Figure 7A, robust BB1 single-qubit rotation is compared against regular single-qubit rotation as a function of pulse area error. Any BB1(θ,φ) rotation on the Bloch sphere of angle θ around axis φ requires four pulses:

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[0130] Referring to FIG. 8, the effect of axial trap oscillations on the echo fidelity of a 420 nm Rydberg pulse is illustrated. FIG. 8A illustrates a noise correlation measurement of the 420 nm Rydberg laser pulse intensity. In the blue detuning configuration used only in this figure, the 420 nm laser induces an 8 MHz differential optical shift on the hyperfine qubit and consequently a phase accumulation of 32π during the 2 μs pulse (CZ gates are 400 ns total). Small fluctuations in the 420 nm laser intensity result in large fluctuations in the phase accumulation of the hyperfine qubit, resulting in significant dephasing. The echo sequence shown here probes the correlation of the accumulated phase between two 420 nm pulses separated by a fluctuating time τ, thus giving information on how far the 420 nm pulses can be separated in time while still adequately echoing out fluctuations in the 420 nm intensity. FIG. 8B is a graph of the hyperfine coherence (a proxy for echo fidelity) versus the gap time τ between the two 420 nm pulses. The echo fidelity initially decreases due to decorrelation of the 420 nm intensity, but then increases again, indicating that the correlation of the 420 nm intensity is non-monotonic.

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[0131] With reference to FIG. 9, an exemplary motion schematic is provided. The schematic shows the gate-by-gate generation of (FIG. 9A) 1D cluster states, (FIG. 9B) Steane codes, (FIG. 9C) surface codes, and (FIG. 9D) trick codes in a side-by-side comparison. These various graph states are all generated in the same way, and encoding the desired circuit is a matter of placing atoms at different starting positions and applying appropriate AOD waveforms. To achieve the desired circuit, we recursively optimize the atomic layout and trajectories in a manner described in the methods text. FIG. 9C also shows the definition of the surface code stabilizer.

[0132] Referring to FIG. 10, error simulations and single-qubit and two-qubit error estimates shown in a table are provided. The measured graph state fidelity is compared with that from stochastic Monte Carlo simulations for (FIG. 10A) surface code and (FIG. 10B) trick code. The simulated stabilizer matches well with the experimental data for this empirical depolarization noise model. Furthermore, for surface code (trick code) in the experiment, 35% (20%) measurements do not detect stabilizer errors, compared to 40% (26%) in the simulation. The two-qubit errors are described as 0.2% Y errors, 0.2% X errors, 0.5% Z errors and 0.5% loss per qubit per layer of parallel (4 layers for surface code and 5 layers for trick code), corresponding to 97.2% CZ gate fidelity. The surrounding single qubit errors are at rates of 0.1% Y error, 0.1% X error, 0.4% Z error, and 0.2% loss per qubit per layer in parallel as well as an initial 1% loss before the circuit starts (empirically including SPAM errors as a factor). Figure 10C provides a tabular entry of measured, estimated, and extrapolated single qubit (SQ) and two qubit (TQ) gate errors. The simulated TQ fidelity includes a 0.6% scattering error from the 420 nm echo pulse. The estimated TQ fidelity is given for the surface code and trick code experiments, but is an underestimate of the TQ fidelity for the cluster state and Steane code measurements, where the 1013 nm intensity is increased by 2x and the 420 nm intensity is reduced by 2x to increase the gate fidelity. The Bell state estimate of the CZ gate fidelity is performed similarly to the 2× higher 1013 intensity but includes a 420 nm echo pulse, resulting in gate fidelity similar to the surface and trichord estimates.

[0133] Referring to FIG. 11, the properties of the encoded logic states are illustrated. FIG. 11A shows this operation for both raw measurements and to perform error correction and detection in post-processing (all logic states |+> L We provide a summary of the logical error probabilities for various error correction graphs created in (see Figure 11B). Error correction for the Steane code is performed by the Steane code decoder and for the surface and trick codes using a minimum weighted perfect match algorithm. For the equi-spaced trick codes, no logical qubits are rejected if the correction is ambiguous, and thus the spacing d=2 logical qubits are unchanged under the correction procedure. The fidelity observed is comparable to similar displays in recent experiments with other platforms. Figure 11B shows the logical |+> for the surface code with correction and detection performed in post-processing similar to that of Figure 11A. L Indicates the life of the state. After state preparation, |+> L The state is held for a variable amount of time before the projection measurement, and two π A pulse is applied for dynamic decoupling (the lifetime is e.g. 128 as in FIG. 7B). π (This can be significantly further extended by applying a pulse.) Here some experimental parameters are slightly different compared to those in FIG. 11A, hence the higher error rate here at time point 0. FIG. 11C shows the logical qubit state |0> L We show a logical π / 2 rotation on the Steane code to prepare the lattice. The Steane code, the surface code and the trick code all have transverse single-qubit Clifford operations on the logical qubit (including in-software rotation of the lattice), and since the transverse rotation is performed in parallel with the overall Raman laser and the physical single-qubit fidelity is high, this is a high-fidelity operation in the system. Although a logical π / 2 rotation for the Steane code is shown here as an example, various reference states along the fundamental axes of the logical Bloch sphere can be realized for all of these codes.

[0134] Referring to Figure 12, testing of an interferometry benchmark problem is illustrated. To test the gated interferometry technique with the benchmark problem, a variable single particle pure state is prepared (by applying a resonant Raman pulse of variable length), then the system is reconfigured and an interferometry circuit is applied to one of the pairs. The interferometry circuit converts the symmetric triplet state to the other computational state while generating an antisymmetric singlet state |Ψ - > to the computer-calculated reference state |00>. The resulting pairwise output states are plotted in the left panel. The |00> state is observed rarely (1.95(2)% of measurements) and the measurement fidelity is independent of the initial state. This low probability P 00 is 2P 00 This corresponds to a high extracted single-particle purity of -1 = 0.961(3) (Figure 12A, right panel). This measurement is a useful benchmark, since interferometry miscalibration can produce significant state dependence of the observed purity that undermines the validity of multi-body entanglement entropy measurements. Benchmark problem test of entanglement entropy measurements with Bell state arrays. (Figure 12B, top) |11> under Rydberg pulses of variability duration and

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[0135] Referring to FIG. 13, the raw many-body data and numerical modeling of errors are provided. FIG. 13A shows the raw measured Renyi entropy without subtracting the long-range classical entropy as a function of subsystem size for quenches from |rgrgrgrg> and |gggggggg>. The Renyi entropy of the four-atom subsystem is the same underlying data as the half-chain entanglement entropy plotted in FIG. 4D. In the previous example, the data was subtracted a fixed offset given by the classical per-particle entropy, corresponding to the time=0 offset for each subsystem size. The long-range classical entropy offset is slightly larger for the |rgrgrgrg> quench due to the non-unitary fidelity of both preparing |r> and mapping |r>→|1>. FIG. 13B shows the raw overall purity after the |gggggggg> quench. The overall purity is a proxy for the sensitivity of the fidelity of the entire process. This 16-body observable, consisting of a three-level system, is a fully mixed state of eight qubits (1 / 2 8 ) leaves the predicted purity at >100× (see inset). For scale comparison, we also plot the single-particle purity to the power of 8 to show what the overall purity would be if the measurements for one of each pair were not corrected. Figure 13C shows the three-level system with various simulated error sources.

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[0136] Referring to FIG. 14, the local observables and entanglement entropy for a quantum many-body scar are illustrated. FIG. 14A shows the scarred state, such as classical entropy subtraction.

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[0137] Formation of arrays of particles using optical tweezers Optical trapping of neutral atoms is a powerful technique for isolating atoms in a vacuum. Atoms are polarized, and the oscillating electric field of a light beam induces an oscillating electric dipole moment in the atoms. The relevant energy shift in the atoms from the induced dipole, averaged over the period of the light oscillation, is called the AC Stark shift. Based on the AC Stark shift induced by light that is detuned (i.e., offset in wavelength) from the atomic resonance transition, atoms are attracted to light below the resonance frequency and thus trapped at a local intensity maximum (for detuned red, i.e., longer wavelength trapping light). The AC Stark shift is proportional to the intensity of the light. The shape of the intensity field is therefore the shape of the relevant atom trap. Optical tweezers exploit this principle by focusing a laser into a micron-scale constriction, where individual atoms are trapped at the focus. Two-dimensional (2D) arrays of optical tweezers can be generated, for example, by illuminating a spatial light modulator (SLM) that imparts a computer-generated hologram to the wavefront of the laser field. The 2D array of optical tweezers overlaps with a cloud of laser-cooled atoms in a magneto-optical trap (MOT). The tightly focused optical tweezers operate in the "collision blockade" regime where a single atom is loaded from the MOT and pairs of atoms are ejected for optically assisted collisions to ensure that at most one atom is loaded into the tweezers, but since the loading is probabilistic, there is about a 50-60% chance that the trap will have a single atom loaded.

[0138] To prepare deterministic atomic arrays, a real-time feedback procedure identifies randomly loaded atoms and rearranges them into preprogrammed geometric structures. Atomic rearrangement requires moving atoms in tweezers, which can be facilitated to minimize heating, for example by using an acousto-optical deflector (AOD) to deflect the laser beam by an adjustable angle controlled by the frequency of the acoustic waveform applied to the AOD crystal. Dynamic tuning of the acoustic frequency translates into smooth movement of the optical tweezers. Multi-frequency acoustic waves generate an array of laser deflections, which, after focusing through a microscope objective, form an array of optical tweezers with adjustable positions and amplitudes, both controlled by the acoustic waveform. The atoms are rearranged by using an additional set of dynamically moving tweezers overlaid on top of the SLM tweezers array.

[0139] Exemplary Hardware Optical tweezers arrays constitute a powerful and flexible way to build large systems composed of individual particles. Each optical tweezers traps a single particle, including but not limited to individual neutral atoms and molecules for applications in quantum technology. Loading individual particles into such tweezers arrays is a stochastic process, where each tweezers in the system is loaded with a single particle with a finite probability p<1, e.g., p~0.5, for many neutral atom tweezers runs. To compensate for this random loading, real-time feedback can be obtained by measuring which tweezers are loaded and then sorting the loaded particles into programmable geometric structures. This can be done by moving one particle at a time or in parallel.

[0140] Parallel sorting can be achieved by using two acousto-optical deflectors (AODs) to create multiple tweezers that can pick up particles from an existing particle trapping structure, move them simultaneously, and release them elsewhere. This can include moving particles around within a single trapping structure (e.g., a tweezer array) or transporting and sorting particles from one trapping system to another (e.g., between one tweezer array and another type of optical / magnetic trap). This sorting is flexible and allows for programmed placement of each particle. Each movable trap is formed by an AOD, and its position is dynamically controlled by the frequency components of a radio frequency (RF) driving field for the AOD. Because the RF driving of the AOD can be controlled in real time and can include any combination of frequency components, it is possible to create any grid of traps (such as a line of arbitrarily positioned traps), move rows or columns of the grid, and add or remove rows and columns of the grid by varying the number, magnitude, and distribution of frequency components in the RF driving field of the AOD.

[0141] In an exemplary embodiment, optical tweezers arrays are generated using liquid crystals on a silicon spatial light modulator (SLM) that can programmatically generate flexible configurations of tweezers. The tweezers are fixed in space for a given experimental sequence and stochastically loaded with individual atoms, so that each tweezers is loaded with a probability of p~0.5. Fluorescence images of the loaded atoms are taken to identify in real time which tweezers are loaded and which are empty.

[0142] After detecting which tweezers are loaded, the motile tweezers, overlapping the optical tweezers array, can dynamically reposition the atoms from their starting positions to fill the target configuration of the trap with near uniform packing. The motile tweezers are generated using a pair of crossed AODs. These AODs can be used to move one atom at a time to fill the target configuration or to generate a single motile trap that moves many atoms in parallel.

[0143] Referring to FIG. 15, a schematic diagram of an apparatus 1500 for quantum computing according to an embodiment of the present disclosure is provided. As shown in FIG. 15, using a beam generated by a light source 1502 (e.g., a coherent light source; in some exemplary embodiments—a monochromatic light source), an SLM 1504 forms an array of trapping beams (i.e., a tweezer array), which, in the exemplary embodiment shown in FIG. 15, is imaged onto a trapping surface 1508 in a vacuum chamber 1510 by an optical train including elements 1506a, 1506c, 1506d and a high numerical aperture (NA) objective lens 1506e. Other suitable optical trains may be used as would be readily understood by one of ordinary skill in the art. Using a beam generated by a light source 1512 (e.g., a coherent light source; in some exemplary embodiments—a monochromatic light source), a pair of AODs 1514 and 1516 with non-parallel (e.g., orthogonal) directions of acoustic wave propagation generate dynamically movable sorting beams. The sorting beam is overlapped with the trapping beam using an optical series such as that shown in Figure 15 (elements 1517, 1506b, 1506c, 1506d and 1506e). It will be appreciated that other optical series may be used to achieve the same result. For example, sources 1502 and 1512 may be a single source, with the trapping and sorting beams being generated by beam splitters.

[0144] Dynamic motion of the steering beam is achieved using two non-parallel AODs 1514, 1516 arranged in series. In the exemplary embodiment shown in FIG. 15, one AOD defines the "row" ("horizontal" - 'X' AOD) direction and the other defines the "column" ("vertical" - 'Y' AOD) direction. Each AOD is driven by an arbitrary RF waveform from an arbitrary waveform generator 1520, which is generated in real time by a computer 1522 that processes a feedback routine after analyzing the image of the position where the atoms are loaded. When each AOD is driven with a single frequency component, a single steering beam ("AOD trap") is generated in the same plane 1508 as the SLM trap array. The frequency of the X AOD drive determines the horizontal position of the AOD trap and the frequency of the Y AOD drive determines the vertical position; in this way, the single AOD trap can be stepped to overlap any SLM trap.

[0145] In FIG. 15, laser 1502 shines a beam of light onto SLM 1504. SLM 1504 can be controlled by computer 1522 to generate a pattern of beams (the "trapping beam" or "tweezer array"). The beam pattern is focused by lens 1506a, passes through mirror 1506b, and is collimated by lens 1506c on mirror 1506d. The reflected light passes through objective lens 1506e to focus the optical tweezer array in vacuum chamber 1510 on trapping plane 1508. The optical tweezer array laser light continues through objective lens 1524a, passes through dichroic mirror 1524b, and is detected by charge-coupled device (CCD) camera 1524c.

[0146] The vacuum chamber 1510 may be illuminated by an additional light source (not shown). Fluorescence from atoms trapped on the trapping surface also passes through the objective lens 1524a but is reflected by a dichroic mirror 1524b to an electronically multiplied charge-coupled device (EMCCD) camera 1524d. In this example, a laser 1512 directs a beam of light to the AODs 1514, 1516. The AODs 1514, 1516 are driven by an arbitrary wave generator (AWG) 1520, which is in turn controlled by a computer 1522. The crossed AODs 1514, 1516 emit one or more beams as described above, which are directed to a focusing lens 1517. The beams then enter the same optical train 1506b...1506e as described above for the optical tweezers array and are focused onto the trapping surface 1508.

[0147] It will be appreciated that alternative optical series components may be used to create an optical tweezers array suitable for use as described herein.

[0148] The description of various aspects of the present disclosure is given for illustrative purposes, but is not intended to be exhaustive or limited to the disclosed aspects. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described aspects. The terms used herein are selected to best explain the principles of the aspects, practical applications or technical improvements to the technology found in the market, or to enable other skilled in the art to understand the aspects disclosed herein. ​

Claims

1. A step of providing a plurality of neutral atoms, wherein each of the plurality of neutral atoms is disposed in a corresponding optical trap; The step of preparing each of a plurality of neutral atoms in the m F =0 clock state; A step of entangling pairs of neutral atoms among the plurality of neutral atoms by directing laser pulses at pairs of neutral atoms among the plurality of neutral atoms, wherein the laser pulses are configured to transition the pairs of neutral atoms through a Rydberg state; and A step of adiabatically moving in an optical trap corresponding to at least one neutral atom of the pair, applying a Raman pulse to at least one neutral atom during the movement, thereby moving the neutral atoms of the pair relative to each other without destroying the entanglement of the pair A method of performing quantum computer calculations, including.

2. The method according to claim 1, wherein the Raman pulse is applied at the midpoint of the movement.

3. The method according to claim 1, wherein the adiabatic movement has a constant jerk.

4. The method according to claim 1, wherein the adiabatic movement has an average speed of less than 0.55 μm / μs.

5. A step of moving an optical trap corresponding to at least one neutral atom within the blockade radius of a target neutral atom among the plurality of neutral atoms The method according to claim 1, further including.

6. A step of entangling at least one neutral atom and a target neutral atom The method according to claim 5, further including.

7. A step of applying a gate to at least one neutral atom and a target neutral atom The method according to claim 5, further including.

8. The method according to claim 7, wherein the plurality of neutral atoms form a two-dimensional array.

9. The method according to claim 8, wherein at least one neutral atom and a target neutral atom are not adjacent within the two-dimensional array before the movement.

10. The method according to claim 1, wherein an optical trap corresponding to at least one neutral atom is generated by directing a light beam at at least one acousto-optic deflector (AOD), and the step of adiabatically moving in the optical trap corresponding to at least one neutral atom includes changing the drive frequency of at least one AOD.

11. The method according to claim 1, wherein at least a first subset of optical traps corresponding to the plurality of neutral atoms is generated by directing a light beam at a spatial light modulator (SLM).

12. A step of providing a plurality of neutral atoms, wherein each of the plurality of neutral atoms is disposed in a corresponding optical trap; Preparing each of a plurality of neutral atoms in the m F =0 clock state; A step of entangling pairs of neutral atoms by directing laser pulses at pairs of neutral atoms, where the laser pulses are configured to transition the pairs of neutral atoms through a Rydberg state; A step of causing adiabatic motion in an optical trap corresponding to at least one neutral atom of the pair, thereby moving the neutral atoms of the pair relative to each other without destroying the entanglement of the pair; A step of irradiating a first region, where the first region contains a first atom of the pair therein, thereby applying rotation to the first atom of the pair; A step of causing adiabatic motion of the optical trap corresponding to the first atom of the pair to the outside of the first region; A step of causing adiabatic motion of the optical trap corresponding to the second atom of the pair to the first region; and A step of irradiating the first region, thereby applying rotation to the second atom of the pair A method for performing quantum computer calculations, including the above steps.

13. A step of applying a Raman pulse to at least one neutral atom during the motion The method according to claim 12, further including the above step.

14. The method according to claim 13, where the Raman pulse is applied at the midpoint of the motion.

15. The method according to claim 12, where the adiabatic motion has a constant jerk.

16. The method according to claim 12, where the adiabatic motion has an average velocity of less than 0.55 μm / μs.

17. The method according to claim 12, where a plurality of neutral atoms form a two-dimensional array.

18. The method according to claim 12, where the optical trap corresponding to at least one neutral atom is generated by directing a light beam at at least one acousto-optic deflector (AOD), and the step of causing adiabatic motion in the optical trap corresponding to at least one neutral atom includes changing the driving frequency of at least one AOD.

19. The method according to claim 12, where at least a first subset of the optical traps corresponding to a plurality of neutral atoms is generated by directing a light beam at a spatial light modulator (SLM).

20. A step of providing a plurality of neutral atoms, wherein each of the plurality of neutral atoms is disposed in a corresponding optical trap, the plurality of neutral atoms includes a first subset and a second subset, and each neutral atom of the first subset is disposed within the blockade radius of a first corresponding neutral atom of the second subset, thereby forming a first plurality of pairs; The step of preparing each of a plurality of neutral atoms in the m F =0 clock state; A step of applying a first two-qubit gate to each of the first plurality of pairs; A step of adiabatically moving the optical trap corresponding to the first subset so that each neutral atom of the first subset is within the blockade radius of a second corresponding neutral atom of the second subset, thereby forming a second plurality of pairs, and applying a Raman pulse to the first subset during the movement; and A step of applying a second two-qubit gate to each of the second plurality of pairs A method for performing quantum computer calculations, including

21. The method according to claim 20, wherein the first and / or second two-qubit gate is a CZ gate.

22. A step of adiabatically moving the optical trap corresponding to the first subset to an imaging region that does not include the second subset; and A step of irradiating the imaging region to measure the state of the first subset The method according to claim 20, further including

23. The method according to claim 20, wherein the optical traps corresponding to the first subset are moved simultaneously.

24. The method according to claim 20, wherein the Raman pulse is applied at the midpoint of the movement.

25. The method according to claim 20, wherein the adiabatic movement has a constant jerk.

26. The method according to claim 20, wherein the adiabatic movement has an average speed of less than 0.55 μm / μs.

27. The method according to claim 20, wherein the plurality of neutral atoms form a two-dimensional array.

28. The method according to claim 20, wherein the optical trap corresponding to at least one neutral atom is generated by directing a light beam to at least one acousto-optic deflector (AOD), and the step of adiabatically moving the optical trap corresponding to at least one neutral atom includes changing the driving frequency of at least one AOD.

29. The method according to claim 20, wherein at least a first subset of the optical traps corresponding to the plurality of neutral atoms is generated by directing a light beam to a spatial light modulator (SLM).

30. A step of providing a plurality of neutral atoms, where each of the plurality of neutral atoms is disposed in a corresponding optical trap; The step of preparing each of a plurality of neutral atoms in an m F =0 clock state; A step of adiabatically moving a plurality of neutral atoms between a first arrangement and a second arrangement different from the first arrangement, where the first array arrangement includes at least one pair of neutral atoms within each other's blockade radius; A step of applying a gate to at least one pair of neutral atoms when in the first arrangement; and A step of evolving a plurality of neutral atoms according to a first Hamiltonian when in the second arrangement A method of performing quantum computer calculations, including.

31. A step of applying a Raman pulse to at least one neutral atom during the movement The method according to claim 30, further including.

32. The method according to claim 31, wherein the Raman pulse is applied at the midpoint of the movement.

33. The method according to claim 30, wherein the adiabatic movement has a constant jerk.

34. The method according to claim 30, wherein the adiabatic movement has an average speed of less than 0.55 μm / μs.

35. The method according to claim 30, wherein the plurality of neutral atoms form a two-dimensional array.

36. An optical trap corresponding to at least one neutral atom is generated by directing a beam of light to at least one acousto-optic deflector (AOD), and the step of adiabatically moving the optical trap corresponding to at least one neutral atom includes changing the driving frequency of at least one AOD. The method according to claim 30.

37. The method according to claim 30, wherein at least a first subset of the optical traps corresponding to the plurality of neutral atoms is generated by directing a beam of light to a spatial light modulator (SLM).

38. A plurality of optical traps; A plurality of neutral atoms, where each of the plurality of neutral atoms is disposed in a corresponding one of the plurality of optical traps; and At least one laser, where the at least one laser is Prepare each of a plurality of neutral atoms in the m F =0 clock state, and configured to entangle pairs of neutral atoms of a plurality of neutral atoms by transitioning pairs of neutral atoms through a Rydberg state as such; A quantum computer, including; A quantum computer configured to adiabatically move an optical trap corresponding to at least one neutral atom of the pair, apply a Raman pulse to at least one neutral atom during the movement, and thereby move the neutral atoms of the pair relative to each other without destroying the entanglement of the pair.

39. A plurality of optical traps; A plurality of neutral atoms including a first subset and a second subset, where each of the plurality of neutral atoms is disposed in a corresponding one of the plurality of optical traps, and each neutral atom of the first subset is disposed within the blockade radius of a first corresponding neutral atom of the second subset, thereby forming a first plurality of pairs; and At least one laser, where the at least one laser is configured to prepare each of a plurality of neutral atoms in an m F = 0 clock state, A quantum computer comprising: Applying a gate to each of the first plurality of pairs; Adiabatically moving the optical trap corresponding to the first subset so that each neutral atom of the first subset is within the blockade radius of a second corresponding neutral atom of the second subset, thereby forming a second plurality of pairs; Applying a Raman pulse to the first subset during the movement; and Applying a gate to each of the second plurality of pairs is configured to Quantum computer. ​

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