Dynamically reshapeable architecture 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] Cross - 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 herein by reference in its entirety.

[0002] Statement regarding federally - sponsored research or development This invention was made with government support under grants 1745303, 1734011, 2012023 awarded by the National Science Foundation, W911NF2010021 and W911NF2010082 awarded by the U.S. Army Research Office, 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 Art

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

Summary of the Invention

[0004] Brief Summary According to aspects of the present disclosure, a method of quantum computer computing is provided. A plurality of neutral atoms are provided. Each of the plurality of neutral atoms is disposed in a corresponding optical trap. Each of the plurality of neutral atoms is prepared in the m F =0 clock state. Pairs of neutral atoms of the plurality of neutral atoms are entangled by directing a laser pulse thereto. The laser pulse is configured to transition the pair of neutral atoms through a Rydberg state. The optical trap corresponding to at least one of the neutral atoms of the pair is adiabatically moved, and a Raman pulse is applied to at least one of the neutral atoms 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, a Raman pulse is applied at the 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 the blockade radius of a target neutral atom of multiple neutral atoms. In various embodiments, at least one neutral atom is entangled with the target neutral atom. In various embodiments, a gate is applied to at least one neutral atom and the target neutral atom.

[0007] In various embodiments, multiple neutral atoms form a two-dimensional array. In various embodiments, at least one neutral atom and a target neutral atom are not adjacent in the two-dimensional array prior to the motion.

[0008] In various embodiments, an optical trap corresponding to at least one neutral atom is generated by directing a beam of light to at least one acoustic-optical deflector (AOD), where adiabatically moving the optical trap corresponding to at least one neutral atom includes changing the driving frequency of at least one AOD. In various embodiments, at least a first subset of optical traps corresponding to multiple neutral atoms is generated by directing a beam of light to a spatial light modulator (SLM).

[0009] According to aspects of this disclosure, a method for quantum computer calculation is provided. A plurality of neutral atoms are provided. Each of the plurality of neutral atoms is placed in a corresponding optical trap. Each of the plurality of neutral atoms is m FIt is prepared in a =0 clock state. Pairs of multiple neutral atoms are entangled by directing a laser pulse toward them, and the laser pulse is configured to cause the pair of neutral atoms to transition 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 breaking the entanglement of the pair. A first region is irradiated, and the first region contains the first atom of the pair, thereby applying 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. An optical trap corresponding to the second atom of the pair is moved adiabatically into the first region. The first region is irradiated, thereby applying rotation to the second atom of the pair.

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

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

[0012] In various embodiments, multiple neutral atoms form a two-dimensional array.

[0013] In various embodiments, an optical trap corresponding to at least one neutral atom is generated by directing a beam of light to at least one acoustic-optical deflector (AOD), where adiabatically moving the optical trap corresponding to at least one neutral atom includes changing the driving frequency of at least one AOD. In various embodiments, at least a first subset of optical traps corresponding to multiple neutral atoms is generated by directing a beam of light to a spatial light modulator (SLM).

[0014] A method for quantum computer calculation is provided according to aspects of this disclosure. A plurality of neutral atoms are provided. Each of the plurality of neutral atoms is placed in a corresponding optical trap. The plurality of neutral atoms comprises a first subset and a second subset. Each neutral atom of the first subset is placed within the blockade radius of the first corresponding neutral atom of the second subset, thereby forming a first plurality of pairs. Each of the plurality of neutral atoms is m F Prepared in a 0-clock state. The first gate is applied to each of the first multiple pairs. The optical trap corresponding to the first subset is adiabatically moved such that each neutral atom of the first subset is within the blockade radius of the second corresponding neutral atom of the second subset, thereby forming a second multiple pair. A Raman pulse is applied to the first subset during this motion. The second gate is applied to each of the second multiple pairs.

[0015] In various embodiments, the first and / or second gates are CZ gates.

[0016] In various embodiments, the 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, the Raman pulse is applied at the midpoint of the motion.

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

[0020] In various embodiments, multiple neutral atoms form a two-dimensional array.

[0021] In various embodiments, an optical trap corresponding to at least one neutral atom is generated by directing a beam of light to at least one acoustic-optical deflector (AOD), where adiabatically moving the optical trap corresponding to at least one neutral atom includes changing the driving frequency of at least one AOD. In various embodiments, at least a first subset of optical traps corresponding to multiple neutral atoms is generated by directing a beam of light to a spatial light modulator (SLM).

[0022] According to aspects of this disclosure, a method for quantum computer calculation is provided. A plurality of neutral atoms are provided. Each of the plurality of neutral atoms is placed in a corresponding optical trap. Each of the plurality of neutral atoms is m F Prepared in a 0-clock state. Multiple neutral atoms are adiabatically moved between a first configuration and a second configuration different from the first configuration. The first array configuration contains at least one pair of neutral atoms within each other's blockade radii. A gate is applied to at least one pair of neutral atoms when they are in the first configuration. Multiple neutral atoms are advanced according to the first Hamiltonian when they are in the second configuration.

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

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

[0025] In various embodiments, multiple neutral atoms form a two-dimensional array.

[0026] In various embodiments, an optical trap corresponding to at least one neutral atom is generated by directing a beam of light to at least one acoustic-optical deflector (AOD), where adiabatically moving the optical trap corresponding to at least one neutral atom includes changing the driving frequency of at least one AOD. In various embodiments, at least a first subset of optical traps corresponding to multiple neutral atoms is generated by directing a beam of light to a spatial light modulator (SLM).

[0027] According to various embodiments, a quantum computer is provided which includes a plurality of optical traps, a plurality of sources of neutral atoms, each of which is disposable in a corresponding one of the plurality of optical traps, and at least one laser, the quantum computer being configured to perform any of the aforementioned methods. [Brief explanation of the drawing]

[0028] A brief explanation of some of the figures in the drawing. [Figure 1A] Figure 1A is a schematic diagram of a quantum information architecture according to an embodiment of the present disclosure. [Figure 1B] Figure 1B is a pair of images of a neutral atom before and after motion, according to an aspect of this disclosure. [Figure 1C] Figure 1C is a graph of parity vibrations of stationary and transported atoms according to an aspect of the present disclosure. [Figure 1D] Figure 1D is a graph of the measured Bell state fidelity as a function of separation speed according to an aspect of the present disclosure. [Figure 2A] Figure 2A is a series of images of neutral atoms showing the generation of a 12-atom 1D cluster state graph according to an aspect of this disclosure. [Figure 2B] Figure 2B is a quantum circuit representation of 1D cluster state preparation and measurement according to an aspect of this disclosure. [Figure 2C] Figure 2C is a graph of the raw, measured stabilizer of the obtained 1D cluster state according to an aspect of this disclosure. [Figure 2D] Figure 2D is a graph state representation of a 7-qubit Steane code according to an aspect of this disclosure. [Figure 2E] Figure 2E shows a circuit for preparing the Steane code logic state according to an aspect of this disclosure. [Figure 2F] Figure 2F shows a pair of graphs of a measured stabilizer and a logical operator according to an aspect of this disclosure. [Figure 3A] Figure 3A shows a graph of the surface code realized according to an embodiment of the present disclosure. [Figure 3B] Figure 3B is a graph of the measured X-plaquette and Z-star stabilizers of the surface code obtained according to an embodiment of the present disclosure. [Figure 3C] Figure 3C is a schematic diagram of the execution of trick code according to an aspect of this disclosure. [Figure 3D] Figure 3D shows the measured X-plaquette and Z-star stabilizer along with the logical operators for two logical qubits, one with and one without error detection, according to aspects of this disclosure. [Figure 4A] Figure 4A shows a hybrid quantum circuit combining coherent atomic transport with analog Hamiltonian progression and digital quantum gates, according to an aspect of this disclosure. [Figure 4B] Figure 4B includes two atomic images illustrating the measurement of entanglement entropy in a many-body Rydberg system via two-copy interferometry according to an aspect of the present disclosure. [Figure 4C] Figure 4C is a graph of the measured half-chain Renyi entanglement entropy after many-body mechanics according to an aspect of this disclosure. [Figure 4D] Figure 4D is a graph of mutual information for various system sizes according to the embodiments of this disclosure. [Figure 4E] Figure 4E is a graph of single-site Renyi entropy according to an aspect of this disclosure. [Figure 5A] Figure 5A is a diagram of a CZ gate according to an embodiment of the present disclosure. [Figure 5B] Figure 5B is a level diagram showing important 87Rb atomic levels according to an aspect of this disclosure. [Figure 5C] Figure 5C is a schematic diagram of an exemplary pulse sequence for running a quantum circuit according to an aspect of the present disclosure. [Figure 6A] Figures 6A to 6D are graphs of atomic loss and atomic retention according to aspects of this disclosure. [Figure 6B] Figures 6A to 6D are graphs of atomic loss and atomic retention according to aspects of this disclosure. [Figure 6C] Figures 6A to 6D are graphs of atomic loss and atomic retention according to aspects of this disclosure. [Figure 6D] Figures 6A to 6D are graphs of atomic loss and atomic retention according to aspects of this disclosure. [Figure 7A] Figures 7A–7C are graphs of pulse fidelity, coherence, and ensemble differences according to aspects of this disclosure. [Figure 7B] Figures 7A–7C are graphs of pulse fidelity, coherence, and ensemble differences according to aspects of this disclosure. [Figure 7C] Figures 7A–7C are graphs of pulse fidelity, coherence, and ensemble differences according to aspects of this disclosure. [Figure 8A] Figure 8A is a schematic diagram of an exemplary pulse sequence according to an aspect of the present disclosure. [Figure 8B] Figure 8B is a graph of an ultrafine coherence sequence according to an aspect of this disclosure. [Figure 8C] Figure 8C is a graph of vibration frequencies according to an aspect of this disclosure. [Figure 9A] Figures 9A to 9D are schematic diagrams illustrating the generation of 1D cluster states, Steane codes, surface codes, and trick codes according to embodiments of this disclosure. [Figure 9B] Figures 9A to 9D are schematic diagrams illustrating the generation of 1D cluster states, Steane codes, surface codes, and trick codes according to embodiments of this disclosure. [Figure 9C]Figures 9A to 9D are schematic diagrams illustrating the generation of 1D cluster states, Steane codes, surface codes, and trick codes according to embodiments of this disclosure. [Figure 9D] Figures 9A to 9D are schematic diagrams illustrating the generation of 1D cluster states, Steane codes, surface codes, and trick codes according to embodiments of this disclosure. [Figure 10A] Figures 10A to 10B are graphs of error estimation according to the embodiments of this disclosure. [Figure 10B] Figures 10A to 10B are graphs of error estimation according to the embodiments of this disclosure. [Figure 10C] Figure 10C is a table of single-qubit (SQ) and two-qubit (TQ) gate errors according to aspects of this disclosure. [Figure 11A] Figures 11A to 11C are graphs of error probability and expected value according to the embodiments of this disclosure. [Figure 11B] Figures 11A to 11C are graphs of error probability and expected value according to the embodiments of this disclosure. [Figure 11C] Figures 11A to 11C are graphs of error probability and expected value according to the embodiments of this disclosure. [Figure 12A] Figures 12A to 12B are graphs that test interferometry measurement using a benchmark problem according to an aspect of this disclosure. [Figure 12B] Figures 12A to 12B are graphs that test interferometry measurement using a benchmark problem according to an aspect of this disclosure. [Figure 13A] Figures 13A–13C are graphs of raw many-body data and numerical modeling of errors according to aspects of this disclosure. [Figure 13B] Figures 13A–13C are graphs of raw many-body data and numerical modeling of errors according to aspects of this disclosure. [Figure 13C] Figures 13A–13C are graphs of raw many-body data and numerical modeling of errors according to aspects of this disclosure. [Figure 14A] Figures 14A–14C are graphs of local observables and entanglement entropies for quantum many-body scars according to aspects of this disclosure. [Figure 14B] Figures 14A–14C are graphs of local observables and entanglement entropies for quantum many-body scars according to aspects of the present disclosure. [Figure 14C] Figures 14A–14C are graphs of local observables and entanglement entropies for quantum many-body scars according to aspects of the present disclosure. [Figure 14D] Figure 14D is a diagram of a constrained Hilbert space according to an aspect of this disclosure. [Figure 15] Figure 15 is a schematic diagram of an apparatus for quantum computing according to an aspect of the present disclosure. [Modes for carrying out the invention]

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

[0030] In various examples, this architecture is used to realize the programmable generation of entangled graph states, such as cluster states and 7-qubit Steane code states. Furthermore, the entangled ancilla array is rounded to realize surface code states with 13 data and 6 ancilary qubits, as well as trick code states on a torus with 16 data and 8 ancilary qubits. This architecture is also used to realize hybrid analog-digital extensions and to measure entanglement entropy in quantum simulations, experimentally observing non-monotonic entanglement dynamics associated with quantum many-body scars. By achieving long-held goals, these results pave the way towards scalable quantum processing and enable novel applications ranging from simulation to metrology.

[0031] A qubit is the fundamental building unit for quantum computers. By analogy to classical bits (each bit being 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 become entangled to construct multi-qubit quantum gates.

[0032] Bits and qubits are encoded, respectively, in the state of a real-world 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 quid (quantum digit) refers to a unit of quantum information that can be realized in a suitable d-level quantum system. A set of quids that can be measured for an N-state can perform an N-level quid.

[0034] A qubit is encoded in a quantum system having 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 isolated in a vacuum. These isolated atoms, ions, and molecules have many distinct quantum states corresponding to different directions of electron spin, nuclear spin, electron orbits, and molecular rotation / vibration.

[0035] In principle, a qubit can be encoded into any pair of quantum states of atoms / ions / molecules. In practice, the key parameters of a qubit are described by its quantum coherence property. Coherence measures the lifetime of the qubit before its information is lost. This has a close analogy to classical bits: if a classical bit is prepared in a 0 state, it may be randomly jumped to 1 after some time due to ambient noise. Quantum mechanically, the same error can occur: |0> may be randomly jumped to |1> after some characteristic time scale. However, qubits can suffer further errors: for example, superposition states.

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

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

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

[0039] An alternative approach is based on stimulated Raman transitions. In this case, the laser field is applied to atoms / ions / molecules. The laser field resonates approximately (but not precisely) with the 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 resolution of the qubits. Atoms / ions / molecules can absorb photons from one frequency component and coherently emit photons from different frequency components, and in doing so, they change their state. This approach benefits from the ability in quantum computers to concentrate the laser field on individual particles or subsets of particles. The laser field can also be applied at high intensity, enabling fairly fast gate operations.

[0040] A neutral atom quantum computer encodes qubits in individual neutral atoms. The 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 a densely focused laser beam that traps individual atoms at its focal point. Alternatively, individual atoms can be trapped in an optical lattice formed by standing waves of laser light, creating a periodic node / antinode structure.

[0041] A typical approach to encoding qubits in neutral atoms is the hyperfine qubit approach, where 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, which is a highly excited Rydberg state. If one atom is excited to a Rydberg state, adjacent atoms are prevented from being excited to a Rydberg state. This conditional behavior forms a criterion for multi-qubit gates such as controlled-NOT gates. The Rydberg states are used temporarily to mediate the multi-qubit gate, and then the atoms are reversed from the Rydberg state back to their ground state levels to preserve their coherence.

[0042] A trapped ion quantum computer uses atomic species that are ionized, meaning they have a net charge. In most cases, many ions are trapped in one large trap potential formed by electrodes in a vacuum chamber. Ions are pulled to the smallest trap potential, but Coulomb repulsion between ions causes them to form a crystalline structure centered at the middle of the trap potential. Most commonly, ions align in a linear chain. Other methods for trapping ions are also possible, such as using optical tweezers or trapping ions individually with localized electric fields having more complex on-chip electrode structures.

[0043] Qubits are encoded in trapped ions in several ways. One common approach is to use the ground-state hyperfine level, as described for neutral atoms. Similar to neutral atoms, in trapped ions with hyperfine qubit encoding, a single qubit gate can be achieved using microwave irradiation or stimulated Raman transitions.

[0044] Unlike in neutral atoms, trapped ion hyperfine qubits heavily rely on stimulated Raman transitions to perform multi-qubit gates. Stimulated Raman transitions can be used both to control the ion's hyperfine state and to change the ion's kinetic state (i.e., to add momentum). This can be understood as absorbing a photon moving in one direction and emitting a photon in the opposite direction, so that the difference in the photon's momentum is absorbed by the ion. Often, many ions are trapped in a single collective trap potential, and since each repels the others, changing the kinetic state 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 quantum computers, individual particles (atoms / ions / molecules) can first be trapped in an array and aligned to a specific configuration. Next, one or more particles are prepared to a desired quantum state. The quantum circuit can then be executed 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 typically be achieved using an observation system that includes an electron-multiplier 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 by detecting fluorescence emitted by the particles in their final state, for example.

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

[0047] This disclosure shows that a neutral atom array can be dynamically reshaped while preserving quantum coherence and entanglement between qubits by storing quantum information in a hyperfine state and moving atoms back and forth within optical tweezers. This approach provides a scalable method for realizing quantum information systems with many qubits and arbitrary programmability, where any qubit can perform entanglement gates with any other qubit in the array. Various quantum information circuits that enhance the programmability and nonlocal coupling achievable by these approaches using high-fidelity 2-qubit RydeBerrigates are described herein. An example of a high-fidelity RydeBerrigate is described herein by reference 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.

[0048] The approaches described herein are naturally adapted to construct graph states, which are an important category of quantum information states defined by stabilizer states or graphs, some of which have nonlocal coupling properties. In particular, this disclosure demonstrates the preparation of high-fidelity 1D cluster states, 7-qubit Steane code quantum error correction codes, and surface code quantum error correction codes.

[0049] To further demonstrate the true non-local capabilities of these approaches, this disclosure implements a periodic boundary condition with 24 qubits by showing qubit entanglement on opposite ends of an array and realizes a trick code on a torus. The trick code is a normative topological error correction code whose physical realization is impractical in other systems due to the required non-local coupling, highlighting the unique capabilities of this approach.

[0050] The approach provided herein also provides a variety of novel tools for analog quantum simulations using Rydberg atoms. As an example, this disclosure demonstrates a quantum many-body quench on two identical many-body copies, and then interferes with the two systems by a gate system protocol, yielding an important quantity in the entanglement entropy-Rydberg atom system that has not been previously measured experimentally.

[0051] It is understood that the approach described herein has various advantages, such as the ability to maintain the coherence of the qubit during movement and the ability to avoid the disruption of entanglement during movement.

[0052] As will be described in more detail below, the methods provided herein enable a variety of computer calculation scenarios. In some scenarios, multiple neutral atoms are moved in parallel between multiple regions in space. For example, an irradiation source may be directed to a first region, and atoms are moved inside or outside that region during the application of pulses by the irradiation source. Similarly, a camera may be directed to an imaging region, and atoms are moved inside and outside that imaging region for imaging. Likewise, atoms may 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 the algorithm or layers of the quantum circuit.

[0053] It is understood that various stabilizer codes inevitably involve the reading of ancilla qubits, and this disclosure enables the physical relocation of ancilla qubits, separate from data qubits, into the imaging region. In this manner, the reading of ancilla qubits can be provided without the destruction of 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 the analog progression of the array as a whole. As used herein, an array of atoms or configuration of multiple atoms refers to the positioning of these atoms relative to one another. It is understood that a particular configuration provides coupling between qubits that enable a particular gate or analog progression that follows a particular Hamiltonian. One advantage of the methods provided herein is that atoms can be moved to the proximal of atoms that were not adjacent in the array. Non-adjacent atoms are those that are not in the unit cell in a regular lattice or are not nearest neighbors in a disordered array. For example, in a rectangular lattice, each atom has eight atoms in its unit cell and therefore eight adjacent atoms (regardless of edges).

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

[0056] In addition to adiabatic motion, dynamic decoupling is applied during motion in several embodiments. As further described below, the π-pulse during motion cancels out dephasing induced by the trap differential optical shift. The trap differential optical shift changes as the atom moves in the trap, sampling different portions of light intensity and thus having different differential optical shifts.

[0057] Generally, the more pulses applied, the greater the decoupling from fluctuations. For example, fluctuations can arise from laser intensity fluctuations at different transition positions of atoms or at different magnetic fields in space.

[0058] In an embodiment where acceleration and deceleration are symmetric, both alter the differential optical shift in the same way. Therefore, in such an embodiment, it is advantageous to apply a π pulse at the midpoint of the motion. In this way, the changes in differential optical 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 to a configuration suitable for analog Hamiltonian evolution according to a given Hamiltonian. The atoms can further be moved backward and forward between such configurations and configurations suitable for the application of digital quantum gates.

[0060] In the following example, such an approach is described for measuring entanglement entropy in a many-body system. However, it is understood that this approach can be used for a variety of further problems. For example, by moving atoms between multiple configurations and performing multiple rounds of analog evolution, it becomes possible to form the largest independent set problem on a graph with nonlocal coupling. Error mitigation can be performed on analog quantum simulators by using digital gates and multiple copies. More generally, by applying gates in this manner, more precise control of analog evolution (such as spin liquids) becomes possible. This control can further be used to perform shadow tomography in complex systems as a method for investigating many-body physics.

[0061] Referring to Figure 1, a quantum information architecture made possible by the coherent transport of neutral atoms is illustrated. Qubits are transported to perform gate entanglement by distant qubits, enabling programmable and nonlocal coupling. The round trip of atoms is performed using optical tweezers, and selective operation is possible due to high parallelism between multiple regions in two dimensions. The inset shows the atomic level used: |0>, |1> qubit states are, 87 Rb's m F |r> refers to the 0 clock state, and |r> is the Rydberg state used to generate entanglement between qubits (Figure 5B). Figure 1B shows an atomic image illustrating the coherent transport of entangled qubits. Using single-qubit and two-qubit gate sequences, atomic pairs are respectively |Φ + >Prepared in a Bell state, then separated by only 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 static measurements, qubit coherence is preserved by using an XY8 dynamic decoupling sequence for 300 μs. Figure 1D is a graph of the measured Bell state fidelity as a function of the separation rate over 110 μm, showing that fidelity is unaffected for motions slower than 200 μs (average separation rate of 0.55 μm / μs). Inset: Standardization due to atomic loss during motion results in constant fidelity, indicating that atomic loss is the dominant error mechanism.

[0062] Quantum information systems are driven by their power derived from controllable interactions that give rise to quantum entanglement. However, the natural local characteristics of these interactions limit the coupling of quantum circuits and simulations. Nonlocal coupling can be engineered through a globally shared quantum data bus, but these approaches are limited in either control or size.

[0063] According to various aspects of the present disclosure, this long-standing problem 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 during quantum operations, and excitations to Rydberg states are used to generate entanglement. Highly parallel operations are enabled by selective qubit operations in separate regions where the qubits are dynamically shuttled. Together, these components enable a powerful quantum information architecture that is used to realize applications such as the generation of entangled states, the creation of topological surfaces and toric code states, and hybrid analog-digital quantum simulations.

[0064] Entanglement transport in atomic arrays In various aspects, the two-dimensional atomic array system described below is used to perform coherent transport and multiple layers of single-qubit and two-qubit gates. Quantum information is 87 stored in a magnetically insensitive clock state within the ground-state hyperfine manifold of Rb atoms. Robust single-qubit Raman rotations (scattering errors per π pulse, approximately 7×10 -5 ) are realized by composite pulses that are robust to pulse errors (FIGS. 7A - B). High-fidelity controlled-Z (CZ) entanglement gates (FIG. 1A) in the hyperfine basis {|0>, |1>} are

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[0065] Figures 1A–D demonstrate the ability to transport qubits over large intervals while preserving entanglement and coherence. Pairs are initialized with an atomic-atomic spacing of 3 μm (Figure 1B), and then into a Bell state.

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[0066] Programmable circuit and graph states To illustrate the ability to generate nonlocal couplings between qubit arrays in parallel, we prepare entangled graph states as follows: a large category of useful quantum information states, ranging from GHZ states and cluster states to quantum error correction codes. The graph states are,

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[0067] Referring to Figure 2, 1D and 2D graph states using dynamic entanglement transport are illustrated. In Figure 2A, the generation of a 12-atom 1D cluster state graph is illustrated by initializing all qubits (vertices) in |+> and applying a controlled-Z gate on the connections (edges) between the qubits. The atomic image shows the configuration for the first and second gate layers. Figure 2B shows the quantum circuit representation of 1D cluster state preparation and measurement. Dynamic decoupling is applied in all quantum circuits (see Methods). Figure 2C shows S i = Z i-1 X i Z i+1 (Regarding edge qubits X1Z2 and Z 11 X 12Figure 2D shows the raw measured stabilizer of the obtained 1D cluster state given by ). Figure 2D shows the graph state display of the 7-qubit Steane code (shading indicates the stabilizer plaquette). Figure 2E shows the Steane code logic executed in four parallel gate layers. L The circuit for adjusting the state is shown. Figure 2F shows |+> L The measured stabilizers and logical operators after preparation are shown. Error detection is performed by post-selecting measurements where all stabilizers are +1. For both 1D cluster states and Steane codes, stabilizers and logical operators are measured using 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 and parallel layer of CZ gates is executed for adjacent atomic pairs, half of the atoms are moved to form new pairs, and then another parallel layer of CZ gates is executed (Figures 2A, B). To examine the resulting 12-qubit cluster state, the stabilizer set {S} is examined by readouts in two measurement settings given by local π / 2 rotations on either an odd or even sublattice before the projection measurement. i}={Z i-1 X i Z i+1 The} is measured. Local rotation is achieved by moving one sub-grid of the qubit to another region, and then performing rotation on the qubit that remained stationary with a uniform beam illuminating the experimental area (Figure 1A, Methods). <s1>This is measured by analyzing the obtained bit-string output and plotting the obtained raw stabilizer measurements (Figure 2C). Across all 12 stabilizers, the average i >=0.87(1) was observed (Figure 2C) (conditioning and measurement SPAM errors are <s1>= 0.91(1) is explained), and in the cluster state, two biseparable entanglements are proven (all <s1>(>0.5). The measured fidelity corresponds to an error of a few percent per operation for the measurement system quantum computer calculation.

[0069] An important category of graph states is quantum error correction (QEC) codes, where the graph state stabilizer is shown as a stabilizer of the QEC code and can be measured to correct errors on encoded logical qubits. In fact, all stabilizer QEC states are equivalent to several graph states up to single-qubit Clifford rotations, and thus the ability to generate arbitrary graph states makes it possible to easily prepare a wide range of QEC states. As an example, the 7-qubit Steane code, a topological color code shown by the graph in Figure 2D, is a logical state |+> L It is prepared by |+>. To prepare this state, all qubits are initialized with |+> and CZ is applied on the connections between qubits (in four parallel layers, see Figure 9B). Then, one of the two sub-grids is rotated to measure the stabilizer (Figure 2E). After the sub-grid rotation, the six graph state stabilizers are X i or Z i It is transformed into six Steane code stabilizers given by the four-body product of . Figure 2F shows the raw measured expected values ​​of these six stabilizers. The seventh graph state stabilizer is logic Z L The logical qubit operator X is anticommutative with it. L It is transformed into and has an eigenvalue of +1 for the graph state |G>. Therefore, in Figure 2F, <X L >=0.71(2) and <Z L >=-0.02(3), and the logical qubit state |+> L The preparation is shown. Furthermore, error detection is performed by post-selecting the measurement results, where all measured stabilizers produce +1 (with a 66(1)% probability that no errors are detected). Using this procedure, the corrected values ​​are obtained.

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[0070] Topological states with an ancilla array Transportable ancillary qubit arrays are also used to mediate quantum operations between distant qubits. Due to their ability to rapidly move arrays of atoms across the entire system, the use of ancillary qubits naturally complements the kinetic capabilities provided herein. Specifically, ancilas are used for state preparation by mediating entanglement between non-interacting physical qubits, followed by projection measurements of the ancillary array (performed concurrently with data qubit measurements), in the form of measurement-system quantum computer calculations. In particular, topological surface codes 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 operation by simply redefining the stabilizer. Since the redefinition is applied in-software, without physical intervention, projection measurements on the ancilas are interchangeable with all operations on the data qubits and can be performed at any time, so all qubits are measured simultaneously.

[0071] Referring to Figure 3, topological surface code and trick code states using a movable ancila qubit array are illustrated. Figure 3A shows the graph state that realizes the surface code. The circuit shows the formation of the graph state using movable ancila qubits; each movement corresponds to the execution of a CZ gate using adjacent data qubits (illustrated in the box). Logic|+> L The state is generated during projection measurement of an ancilla qubit in the X-reference. The schematic diagram on the right shows the logical operators of the stabilizer and code. Figure 3B shows the measured X-plaquette and Z-star stabilizer of the obtained surface code, along with the logical operators with and without error detection (performed in post-selection). Figure 3C illustrates the execution of the trick code. (Top) Two logical qubit species states of the trick code during projection measurement of an ancilla qubit in the X-reference

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[0072] Figure 3A shows the surface code |+> L This shows the preparation of a 19-qubit graph state that generates a logical state. The surface code is defined by an X-plaquette and a Z-star stabilizer, and the logical operator X L (Z L The state is defined as a string of products of X(Z) over the height (width) of the graph. To prepare this state, the ancila is moved to perform a CZ gate by each of its four neighbors, then measured, and the data qubits are projected onto the surface code state. Here the graph state stabilizers are the X-plaquette, Z-star (with a value of ±1 for ±1 of the measurement result of the central ancila), and logic X L It is converted into an operator. Notably, this procedure generates a topologically ordered state in a circuit of a certain depth, and the ancilla values ​​measured here can be used to redefine the stabilizer, which can be handled by software for the actual QEC operation.

[0073] Figure 3B shows the measured expected values ​​and logical operator expected values ​​with and without error detection for the 12 obtained stabilizers. <X L A raw value of >=0.64(3) was observed, and the stabilizer measured for error detection was used.

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[0074] Surface codes can be prepared by other methods, but the transport capabilities provided herein enable periodic boundary conditions and realize trick code states on a torus. For this purpose, the 24-qubit graph state shown in Figure 3C is generated by executing parallel gates in five layers and moving ancilla to their respective regions for reading on a different criterion. The prepared state has seven independent X-plaquettes and seven independent Z-stars (due to the periodic boundary conditions). Furthermore, due to the topological properties of this graph, two independent logical qubits are logical operators that encircle the entire torus along two topologically distinct directions.

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[0075] State adjustment is verified in Figure 3D by measuring the trick code stabilizer. For two encoded logical qubits,

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[0076] Hybrid Analog-Digital Circuit Referring to Figure 4, the dynamic reshaping possibilities for hybrid analog-digital quantum simulations are illustrated. Figure 4A shows a hybrid quantum circuit combining coherent atomic transport with analog Hamiltonian evolution and digital quantum gates. Figure 4B illustrates the measurement of entanglement entropy in a many-body Rydberg system via two-copy interference. Figure 4C shows the measured half-chain Renyi entanglement entropy after many-body dynamics, followed by a quench for two 8-atom systems. |gggg...> Quench (

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

[0078] Figure 4B illustrates the experimental procedure. After initializing both copies with all qubits in |1>, each progressing copy is given the Rydberg-Miltonian H over time t. Ryd The system progresses independently below, giving rise to an entangled many-body state on the {|1>,|r>} criterion (method). Raman and Rydberg π pulses then map |1>→|0> and |r>→|1>, transitioning the entangled many-body state to a long-lasting, non-interacting {|0>,|1>} criterion. Finally, the entanglement entropy is measured using a Bell measurement circuit by rearranging the system and interfering the respective qubits in the first copy, which have one of their identical pairs in the second copy. By measuring the pairs on the Bell criterion, an antisymmetric singlet state is obtained.

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[0079] This method is used to investigate the growth of entangled entropy arising from many-body mechanics (see Methods for testing on further benchmark problems of the technique). Specifically, the progression of two 8-atom copies under the Rydberg-Miltonian is tested and subjected to nearest neighbor blockade constraints. All atoms are in the ground state.

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[0080] While such thermal neutronization dynamics are generally predicted for strongly interacting many-body systems, it has been previously shown that, notably, for certain initial states, this system can avoid thermal neutronization. Reinforced by special non-thermal eigenstates called quantum many-body scars, these states were theoretically predicted to characterize dynamics associated with slow, non-monotonic entanglement growth. Figure 4 shows the initial states initialized by applying local shifts within one sublattice and by performing a global Rydberg π pulse.

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[0081] These observations agree very well with accurate numerical simulations in isolated systems (lines plotted in Figures 4C, E and 14). Furthermore, while single-site purity approaches that of a well-mixed state, overall purity (16-body observable composed of 3-level systems) remains >100× of that of a well-mixed state (see Figure 13), demonstrating the high accuracy and fidelity of this circuit-based technique overall. These results indicate that combining atomic motion, many-body Hamiltonian evolution, and digital quantum circuits yields a powerful novel tool for simulating and investigating the quantum physics of complex systems.

[0082] Discussion and Outlook The experiments described herein illustrate highly parallel coherent qubit transport and entanglement, enabling powerful quantum information architectures. This technique can be expanded along several lines. Local Rydberg excitations on subsets of qubit pairs eliminate unintended residual interactions from atoms, enabling parallel, independent operations on arrays with significantly higher qubit densities. 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, for example, parallel Raman excitations via AOM arrays. Mid-circuit readouts can be performed by moving an ancilla to a different region and imaging it using, for example, an avalanche photodiode array within hundreds of microseconds.

[0083] These methods offer clear potential for realizing scalable quantum error correction. For example, the procedure shown in Figure 3C could be used for syndrome extraction in actual QEC sequences, where ancilla entangles with their data qubit neighbors and is then moved to another region for mid-circuit readout. The entire QEC round can be executed within milliseconds, considerably faster than the measured T2 > 1s, and the projected fidelity improvement theoretically exceeds the surface code threshold (method). Such mid-circuit readout is essential for realizing scalable fault-tolerant quantum computer computation. Furthermore, the ability to reshape and combine arrays enables efficient and parallel execution of traverse entanglement gates between many logic qubits. These techniques also enable the execution of higher-order or non-local error correction codes with more desirable properties. Together, these components could enable novel approaches to universal fault-tolerant quantum computer computation using thousands of physical qubits.

[0084] The dynamically reconfigurable architectures provided herein also open up many novel opportunities for digital and analog quantum simulations. For example, the hybrid approach can be extended to investigating the entire entanglement spectrum, simulating wormhole generation, performing many-body purification, and reconstructing novel non-equilibrium states. Entanglement transport can also give capabilities to metrological applications, such as generating dispersed states for investigating gravity gradients. Ultimately, these approaches can facilitate quantum network connectivity between isolated arrays, paving the way for large-scale quantum information systems and partition quantum metrology.

[0085] method Dynamic Reformation in 2D Tweezers Arrays These experiments use the same apparatus described below. Inside the vacuum cell, 87 Rb atoms are packed from a magneto-optical trap into a backbone array of programmable optical tweezers created by a spatial light modulator (SLM). The atoms are then repositioned in parallel within defect-free target positions in this SLM backbone by further optical tweezers created from a crossed 2D acoustic-optical deflector (AOD). After the repositioning procedure, selected atoms are moved back from the stationary SLM trap into a mobile AOD trap, and these mobile atoms are then moved to their starting positions in the quantum circuit. Throughout this entire process, the atoms are cooled by deflection gradient cooling. Camera images of the atoms at their initial starting positions are taken before running the quantum circuit. Final camera images are taken after the circuit to detect the qubit states |0> (presence of atoms) and |1> (atomic loss after resonant extrusion). Before running the circuit, all data are post-selected to discover 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 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 waveform consists of multiple frequency gradations corresponding to the x and y positions of the column and row, which are independently varied as a function of time to be dynamically directed around atoms trapped in the AOD; sufficient x and y waveforms are calculated by adding predetermined amplitudes and phases for each component together to the time-domain profile of all frequency components. To run the quantum circuit, the positions of the AOD atoms at each gate position are programmed, and then the AOD frequencies are smoothly interpolated (along with a third-order profile) as a function of time between the gate positions. The tertiary profile establishes a constant jerk for the atoms, which allows for motion approximately 5-10 times faster (without heating and loss) than when moving at a constant velocity (linear profile). In the motion protocol, tension, compression, and deformation of the AOD trap array are applied: that is, the rows and columns of the AOD do not cross each other to avoid atomic loss and heating associated with the crossing of two frequency components with each other.

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

[0088] The SLM tweezers beam (830 nm) and AOD tweezers beam (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 lens, the SLM tweezers beam has a constriction of approximately 900 nm (approximately 1000 nm for the AOD beam). When loaded with atoms, the trap depth is approximately 2π × 16 MHz and the radial trap frequency is approximately 2π × 80 kHz. When running a quantum circuit, the trap depth is approximately 2π × 4 MHz and the radial trap frequency is approximately 2π × 40 kHz.

[0089] Raman laser system Fast, high-fidelity single-qubit operations are a crucial component of the quantum circuits demonstrated in this work. For this purpose, m F A high-power 795nm Raman laser system is used to drive the overall single qubit rotation during the =0 clock state. This Raman laser system is based on a dispersive optics instrument. The 795nm light (Toptica TA pro, 1.8W) is phase-modulated by an electro-optical modulator (Qubig), which is driven by a 3.4GHz microwave (Stanford Research Systems SRS SG384) that is amplified by doubling it up to 6.8GHz. The laser phase modulation is converted to amplified modulation to drive the Raman transition using a chirp Bragg grating (Optigrate). 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 the inventors direct frequency and phase control for ultrafine qubit driving.

[0090] The Raman laser irradiates the atomic plane from the side in an oval beam polarized to annularly, with constrictions of 40 μm and 560 μm on the thin axis and high axis, respectively, and the overall average optical power is 150 mW on the atoms. The large vertical spread ensures non-uniformity of <1% across atoms, and shot-to-shot fluctuations in laser intensity are also <1%. For Figures 1-3, the Raman laser operates at 180 GHz blue detuning intermediate state detuning, with a two-photon Raman frequency of 1 MHz and 7 × 10⁻⁶ -5 of π This results in an estimated scattering error per pulse (i.e., one scattering event per 15,000π pulses). For Figure 4, to shorten the duration of the coherent mapping pulse sequence, the Raman laser power is increased, and a smaller blue detuning intermediate state detuning of 63 GHz is used, with a corresponding two-photon Raman frequency of 3.2 MHz, and an estimated scattering error per π pulse of 2 × 10⁻¹⁶. -4 That is the case.

[0091] Robust single qubit rotation For almost all single qubit rotations in the execution of this task (except for the XY8 / XY16 self-correction sequence), robust single qubit rotations are performed in the form of compound pulse sequences. These compound pulse sequences can be highly insensitive to pulse errors such as amplitude or detuning miscalibration. The dominant source of coherent single qubit errors is amplitude drift of ≤1% and non-uniformity across the array; therefore, the “BB1” (broadband 1) pulse sequence is primarily used, which is a sequence of four pulses that perform arbitrary rotations on the Bloch sphere while being insensitive to amplitude errors up to the 6th 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 consistent with the estimated scattering limit are observed (not plotted here), and the scattering limit is approximately 3 × 10⁻¹⁶ per BB1 pulse due to the increased length of the compound pulse sequence. -4 This suggests that it roughly represents the error (of). Testing on randomized benchmark problems may be applied in future tests that further examine single-qubit rotation fidelity.

[0092] Qubit coherence and dynamic decoupling In the 830nm trap, the ultrafine 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 trade-off between minimizing pulse errors while preserving qubit coherence. In various embodiments, there is an exchange between two types of dynamic decoupling sequences: the XY8 / XY16 sequence is a phase-shifted sequence that self-corrects for amplitude and detuning errors. π Composed of pulses, the CPMG-type dynamic decoupling sequence consists of robust BB1 pulses. The CPMG-BB1 sequence is more robust to amplitude errors but suffers from more scattering errors. The sequence decouples a variable number of these different sequences. π By selecting between pulses, the sequence can be empirically optimized for any given experiment, optimizing either single-qubit coherence (including motion) or the final signal. Typically, the decoupling sequence consists of a total of 12–18 pulses. π It is composed of pulses.

[0094] Kinetic effects on atomic heating and loss The following discusses the effects of motion on atomic loss and heating at the harmonic oscillator potential given by a tweezers trap. The motion of the trap potential is equivalent to a non-inertial frame of reference where the harmonic oscillator potential is stationary, but the atom experiences a hypothetical 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 vibrational quantum number increase ΔN is:

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[0095] Several relevant insights can be gleaned from this equation. Firstly, this expression demonstrates the ability to move a large distance D with an equally small increase in time T. Furthermore, in order to maintain a constant ΔN, the motion time is:

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[0096] Here, we compare Equation 2 with experimental observations. In Figure 1D, atomic loss is observed with a movement of 55 μm in 200 μs under a constant negative jerk. This velocity limit is consistent with the above estimation: ω0 = 2π × 40 kHz and x zpf Using 38 nm, ΔN ≈ 6 is predicted for this motion, which corresponds to the onset of specific heating at this motion velocity. More quantitatively, assuming a Poisson distribution with mean N and variable N, some deterministic N max It integrates higher-level groups, and in the process, atoms escape from the trap. From this analysis, atom retention is,

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[0097] Figures 6A and 6B show the motion time T and trap frequency.

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[0098] Further heating and loss between circuits can also be caused by repeated short drops to execute the 2-qubit gate, where the tweezers are short and off to avoid anti-trapping of the Rydberg state and optical shift of the ground-Rydberg transition. However, the drop-re-capture measurements in Figure 6C suggest that the experimentally used 500 ns drop has a negligible effect up to several hundred drops per atom (corresponding to several hundred CZ gates). Atomic loss and heating as a function of the number of drops are well described by the diffusion model, which then reduces the atomic temperature by 2 × (thermal rate)

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[0099] Execute a 2-qubit CZ gate Two-qubit gates and calibrations can be performed using the techniques provided herein. Specifically, a two-qubit CZ gate is performed by two global Rydberg pulses, each pulse having 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 a Rydberg blockade constraint return in reverse from a Rydberg state to a hyperfine qubit manifold, with the phase depending on the state of the other qubit. The numerical values ​​for these pulse parameters are: Δ = -0.377371Ω ξ = -0.621089 × (2π) τ = 0.683201 / [Ω / (2π)] That is the case.

[0100] The experiments in Figures 1-3 were operated with a two-photon Rydberirabi frequency of Ω / 2π = 3.6 MHz, giving theoretical τ = 190 ns and theoretical Δ / (2π) = -1.36 MHz. The negative detuning sign is detuned by approximately 24 MHz under an 8.5 G field. j = Selected to help minimize excitation to the Rydberg state (and for the reduced Clebsch-Gordan coefficient, the desired m j (This experience coupling to a Rydberg laser 3 × lower than the -1 / 2 state). In this operation, a strong blockade is provided between adjacent qubits, and the Rydberg-Rydberg interaction V0 / 2π is in the range of 200 MHz to 1 GHz. In Figure 4, for a 2-qubit gate, Ω / 2π = 4.45 MHz.

[0101] Spurious phase management between CZ gates A two-qubit gate induces both an intrinsic single-qubit phase and a spurious phase primarily induced by a differential optical shift from the 420 nm laser. Under certain configurations, the differential optical shift induced at 420 nm on the hyperfine qubit can be extremely large (>8 MHz), resulting in phase accumulation on the ≈ 6π hyperfine qubit. Therefore, even small percentage-level variations in the 420 nm intensity can lead to significant qubit detuning.

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

[0103] Echoes between CZ gates. To address these various issues, a Raman π pulse is executed between each CZ gate, resonating out of spurious gate-induced phase on the ultrafine qubit (Figure 5). This approach has several advantages. Here, the phase induced at 420 is canceled out by the pair of CZ gates without explicitly applying a further 420 nm pulse to resonate each individual CZ gate, thereby reducing the scattering error of the CZ gates in this operation by approximately twofold. This echo technique, which reduces the scattering error suffered between each gate, roughly compensates for the increased scattering velocity suffered by spreading the optical power over more space in 2D, thereby giving the 2-qubit CZ gate fidelity an equivalent gate fidelity of ≥97.4(2)%. Furthermore, the echoes between CZ gates also cancel out the intrinsic single-qubit phase of the CZ gates, 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 two-qubit gates (Figure 5). In the example where the number of CZ gates is odd, an echo is performed for the last CZ gate.

[0104] Sign of intermediate state detuning. To further suppress the spurious 420-induced phase effect, a 420nm laser is used, 6P 3 / 2 The transition is manipulated to detun the red band (by only 2 GHz). For red detuning, the optical shifts on the |0> and |1> states have 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 trap vibration In a typical Rydberg excitation timescale using optical tweezers, axial trap oscillation frequencies of several kHz are insignificant. However, in a circuit running for about 1.2 ms with a Rydberg pulse throughout, axial trap oscillation can have a significant effect. In particular, axial oscillation causes atoms to produce vibrations inside / outside the Rydberg beam: axial diffusion occurs at an estimated axial temperature of about 25 μK and an axial oscillation frequency of 6 kHz.

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[0106] Bell condition adjustment and fidelity In Figure 1, |Φ + >The bell state is prepared: After initializing the pair of qubits in |00>, the X / (π / 2) pulse-CZ gate-X / (π / 4) pulse is applied. This |Φ is the sum of the groups in |00> and |11>. + >The raw fidelity obtained for the Bell state is averaged by the fitted amplitude of the parity oscillation (example in Figure 1C), which measures off-diagonal coherence. In Figure 1D, in the event of a significant loss from motion, this fidelity estimate is tilted for the measurement of an artificially large population in |11> (since state |1> is detected as a loss); therefore, once the population difference between |11> and |00> is greater than 0.1 (any cutoff at which the effect of loss begins to become significant), |Φ + The population is estimated as a 2x population of |00>. In Figure 1D, for motions slower than 300 μs, a mean raw Bell state fidelity of 94.8(2)% is achieved after the motion. If there is no motion or attempt to preserve coherence for 500 μs (i.e., measured immediately after the preparation of the Bell state), then a raw Bell state fidelity of 95.2(1)% is measured (not plotted here).

[0107] Analysis of the source of errors The following details some of the measured and estimated sources of error for the entire sequence (particularly trickcode preparation, the deepest exemplary circuit). The total single-qubit fidelity after running the entire sequence is approximately 96.5% for the trickcode circuit, which was measured by embedding the entire experiment in the Ramsay sequence: i.e., a Raman π / 2 pulse is run, all motion and decoupling are performed, and then a final π / 2 pulse is run in which the variability phase measures all contrasts. The single-qubit fidelity is quantitatively described in Figure 10C as consisting of known single-qubit errors.

[0108] The estimated contributions to the 2-qubit gate error are summarized in Figure 10C. These estimates arise from numerical simulations in QuTiP with experimental parameters. The effects of intermediate-state scattering and Rydberg decay are included in the Lindbladmaster equation solver via the collapse operator. Other error contributions include finite-temperature random Doppler shift and position fluctuations, as well as per-laser pulse fluctuations, all of which are simulated using classical Monte Carlo sampling of experimental parameters. The experimental parameters used in the simulation are as follows: 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 second Rydberg state = 24 MHz, radial and axial trap frequencies (ω r ,ω z The frequency is 2π × (40,6) kHz, and the temperature T = 20 μK. This modeling can also be used to project future performance; by estimating a 10x increase in the available 10¹³ nm intensity and estimating that the atoms are cooled to a temperature of 2 μK, a CZ gate fidelity of 99.7% above the surface code threshold is projected. Alkaline earth atoms may also provide another pathway to high-fidelity operations for quantum error correction.

[0109] To understand how various single-qubit and two-qubit errors contribute to graph state fidelity, speculative simulations of the quantum circuits used for graph state preparation are performed (Figures 10A, B). The Clifford characteristics of the circuits are utilized, allowing for efficient numerical evaluation and random sampling of many possible error realizations. The simulations are performed under a realistic error model where ambient depolarization noise and atomic loss rates are measured experimentally (see Figure 10C). The resulting stabilizer and logic qubit expectations agree well with those measured experimentally.

[0110] Rydberg beam shape formation and homogeneity The Rydberg beam is shaped into a top-hat of variability size through wavefront control using a phase profile on a spatial light modulator (SLM). This capability allows the beam profile height to be adapted to the experimental area size of any given experiment, thereby maximizing the 10¹³ nm light intensity and CZ gate fidelity. Rydberg beam homogeneity is optimized until per-peak heterogeneity is less than <1%. To this end, all aberrations are corrected up to the vacuum chamber window, which results in a few percent of atomic heterogeneity causing imperfections in the final window. To further optimize homogeneity, aberration correction is adjusted to the top-hat up to the phase profile on the SLM surface (Fourier plane) via Zernike polynomial correction. This procedure reduces per-peak heterogeneity to <1% over the 40–50 μm range at the atomic plane.

[0111] Graph layout generation and optimization The following outlines how graph layouts are optimized for cluster states, Steane codes, surface codes, and trick codes. The optimization in this example is inductive, and other optimal circuits can be designed using atomic spatial arrangements and AOD orbitals. Figure 9 shows exemplary graphs and the process for creating them. These are the results of optimizations for several parameters: (1) Minimize the number of parallel 2-qubit gate layers. (2) Minimize the total distance traveled by the moving atoms. (3) All moving atoms in one sublattice (all graphs realized here are divided into two parts) facilitate the final local rotation of one sublattice. (4) Minimize the vertical spread of the array and the number of separate rows (to maximize the intensity of 10¹³ and minimize sensitivity to beam heterogeneity between rows). (5) When ordering gates, apply 2-qubit gates as early as possible in the circuit. If a gate layer induces a bit flip (X error), the error can propagate between subsequent gates (resulting in a Z error on other qubits), so the gate should be in the earliest possible layer.

[0112] Local (sublattice) hyperfine rotation Local rotation is performed using a horizontally propagating 420 nm beam to a hyperfine reference, which can be used to impose differential light of several MHz on the hyperfine qubits, thereby achieving fast Z rotations. To achieve the local Y(π / 2) rotation used throughout this work, one sublattice of the atom is moved outside the 420 nm beam, and then the following pulses are applied: [global Y(π / 4)]-[local Z(π)]-[global Y(π / 4)]. This achieves a Y(π / 2) rotation on one sublattice and a Z(π) rotation on the other sublattice (which is then negligible if it is interchangeable with the immediate measurement in Z reference). To apply Y(π / 2) to the other sublattice of the atom, an additional global Z(π) is added between the two Y(π / 4) pulses (performed by jumping the Raman laser phases). Further locally focused beams may be provided to perform local Raman control of the hyperfine qubit state. However, moving atoms (even just 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 for approximately half of the qubits.

[0113] Local Rydberg initialization Localized Rydberg control is used to test the mechanics of multi-body scars.

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[0114] A 50 MHz biased optical shift is significantly larger than the Rydberg-Rabi frequency Ω / 2π = 4.45 MHz, resulting in a Rydberg population over <1% of undesirable regions. Figure 14B shows the time point t=0 using this approach.

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[0115] Rydberg Hamiltonian In Figure 4, the dynamics under the many-body Rydberg-Miltonian in Equation 3 are considered.

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[0116] In Equation 3,

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[0117] Coherent Mapping Protocol The coherent mapping protocol is provided to transition a general many-body state based on {|1>, |r>} criteria to a long-lasting, non-interacting {|0>, |1>} criterion. To achieve this mapping, immediately after Rydberg dynamics, a Raman π pulse is applied to map |1>→|0>, followed by a subsequent Rydberg π pulse to map |r>→|1>.

[0118] Even for perfect Raman and Rydberg π pulses (on isolated atoms), there are three important sources of fidelity associated with this mapping process: (1) Any group in a blockade-violating state (i.e., two adjacent atoms are both in |r>) is a strongly shifted non-resonant group with respect to the final Rydberg π pulse. Thus, this group of atoms remains in the Rydberg state and is lost. (2) For example, long-range interactions from neighboring nearest neighbors detune the final Rydberg π pulse from resonance, thereby reducing pulse fidelity. Since long-range interactions are not the same for all many-body microstates, this effect cannot be mitigated by a simple shift in detunement. (3) The diffusing in this state is Raman π Throughout the pulse duration, these primarily arise from Doppler shifts between the ground states |0>, |1>, and the Rydberg state |r>. While these random on-site detunings also exist during many-body dynamics, by turning off the Rydberg drive Ω, the system is allowed to accumulate phase freely, making it particularly susceptible to diffusing errors.

[0119] The above error mechanism is mitigated as follows: To minimize the error from (1), many-body dynamics is used.

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[0120] The overall Raman beam is Raman π During pulses, it induces an optical shift-induced phase shift of approximately π on |0> for |r> and |1>. Similarly, the overall 420 nm laser also exhibits Rydberg π A light-shift-induced phase shift of approximately π is induced between |0> and |1> during the pulse. Since the measurement performed here is interferometric (i.e., the singlet state being measured is invariant under the overall rotation), it is not affected by these overall phase shifts, which can be measured and described where relevant.

[0121] Measurement of 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 pair of the pair, then apply a CZ gate, and then apply an overall X / (π / 2) rotation. In other measurements, the local X / (π / 2) is realized by performing an overall X / (π / 4) rotation, then a local Z(π) rotation, and then an overall X / (π / 4). However, for this singlet measurement circuit, the first X / (π / 4) overlaps when the singlet state is invariant under the overall rotation, so for the local X / (π / 2), only the local Z(π) and then the second overall X / (π / 4) are applied. This efficiently realizes X / (π / 2) on one qubit up to Z(π) on the other qubit (not shown in the circuit diagram of FIG. 4). Under this simplification, |Ψ - > → |00> for mapping can be roughly understood as the inverse of the Bell state preparation circuit, which is exactly how the parameters of the Bell measurement are calibrated.

[0123] Calibration of interferometry and testing in benchmark problems. To verify the interferometry measurement (and to investigate proper calibration), this is tested in a benchmark problem separately from the many-body dynamics and coherent mapping protocols. This test in the benchmark problem is performed by preparing independent qubits in the same, variable single-qubit superposition (by the overall Raman pulse of the fluctuating time) and ensuring that the interferometry rarely produces |00> for all initial product states of the fluctuations (FIG. 12A). This is an important test step in the benchmark problem because a small mis-calibration of the Bell measurement can result in lower fidelity (i.e., higher entropy) for different initial product states, thereby producing additional spurious signals in the entanglement entropy measurement. This measurement is particularly sensitive to the single-qubit phase just before the final X / (π / 2) pulse (induced by the CZ gate and canceled by the overall Z(θ) pulse).

[0124] Additional many-body data and details To test a method for measuring the entanglement entropy in a many-body system using a benchmark problem, in Fig. 12B, two proximal atoms are initialized in |1> and the entanglement dynamics are tested after resonantly exciting them to the Lieb state during a variable time t. Under the conditions of the Lieb blockade, this excitation gives rise to two-particle Rabi oscillations between |11> and the entangled state [Number] (upper panel of Fig. 12B). The state purity of this two-particle system is measured by performing Bell measurements on pairs of atoms from two identical copies. Locally, the measured purity of the one-particle subsystem reduces to a value of ≈0.5 when the system enters the maximally entangled |W> state, at which point the reduced density matrix of each individual atom is maximally mixed. In contrast, the purity of the overall two-particle state remains high. The observation that the overall state purity is higher than the local subsystem purity is a distinct signature of quantum entanglement.

[0125] For the data shown in Figs. 4C and 4E, the data are subtracted by a wide range of classical entropy. This fixed time-independent offset is given by the entropy per particle, i.e., (the overall entropy at quench time t = 0) × (subsystem size) / (overall system size). In Fig. 13A, the raw entanglement entropy measurements are shown together with the numerical values, indicating the size of the wide range of classical entropy contributions. In the plot, to account for the fact that the Raman π pulse interrupts the last 10 ns of the Lieb evolution, the theoretical curve is delayed by 10 ns, which is done to keep the coherent mapping gap as short as possible and to minimize Doppler dephasing. Further in Fig. 13B, the measured overall purity is plotted and compared with a numerical simulation incorporating experimental errors (Fig. 13C).

[0126] Figure 14 shows further many-body data for an 8-chain system with the same parameters as those used in the main text. The measured single-site entropy for each site is shown in Figure 14A.

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[0127] Referring to Figure 5, the CZ gate echo, atomic-level structure, and typical pulse sequence are illustrated. As shown in Figure 5A, the two-qubit gate, in addition to applying a controlled Z operation between the two qubits, also induces a single-qubit phase Z(ζ) in both qubits, consisting of the intrinsic phase of the CZ gate and additional spurious phases from the 420 nm Rydberg laser, to which pulses are applied to trap off. 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 which is adjacent to another qubit in a dark state |0> with respect to the Rydberg laser). As shown in the figure, any additional undesirable Z(ζ) is offset by applying a π pulse between the pair of CZ gates. This echo procedure eliminates 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. Further Y(π) propagates through the CZ gate in a known manner, multiplying a specific stabilizer by a -1 sign, which simply redefines the sign of the stabilizer and logic qubit. Figure 5B shows the key elements used. 87 This is a level diagram showing the Rb atomic level. 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 operation. Figure 5C is a schematic diagram of an exemplary pulse sequence for running the quantum circuit.

[0128] Referring to Figure 6, the motion characterization and multiple drop recaptures are illustrated. In Figure 6A, atomic retention is given as a function of the average separation rate 2D / T, with a 0.7% background loss subtracted (as plotted in Figure 1D to separate the Bell pair). The inset in Figure 1D shows (atomic retention) (without subtracting background loss). 2 It is standardized by the following. The dark curve is calculated using experimental parameters and Equation 2, N max Set =26, 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 to normal single-qubit rotation as a function of pulse area error. Any BB1(θ,φ) rotation on a Bloch sphere at an angle θ around axis φ is given by four pulses:

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[0130] Referring to FIG. 8, the effect of axial trap vibrations on the echo fidelity of 420 nm Rudolph pulses is illustrated. FIG. 8A illustrates the noise correlation measurement of the 420 nm Rudolph laser pulse intensity. In the blue detuned configuration used only in this figure, the 420 nm laser induces an 8 MHz differential optical shift on the ultrafine qubit and consequently a 32π phase accumulation between 2 μs pulses (the CZ gate is 400 ns in total). A small fluctuation in the 420 nm laser intensity results in a large fluctuation in the phase accumulation of the ultrafine qubit, thus causing significant dephasing. The echo sequence shown here examines the correlation of the accumulated phase between two 420 nm pulses separated by the variability time τ, and thus gives information on how far apart the 420 nm pulses can be separated in time while more appropriately responding to fluctuations outside the 420 nm intensity. FIG. 8B is a graph of the ultrafine coherence (a surrogate for echo fidelity) versus the gap time τ between two 420 nm pulses. The echo fidelity first decreases due to the decoherence of the 420 nm intensity, but then increases again, indicating that the correlation of the 420 nm intensity is non-monotonic. Damping the vibrations is

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[0131] Referring to Figure 9, an exemplary schematic diagram of motion is provided. The schematic diagram shows the gate-by-gate generation of (Figure 9A) 1D cluster state, (Figure 9B) Steane code, (Figure 9C) surface code, and (Figure 9D) trick code in parallel comparison. All these various graph states are generated in the same way, and encoding the desired circuit is a matter of arranging atoms at different starting positions and applying appropriate AOD waveforms. To realize the desired circuit, the atomic layout and orbitals are inductively optimized in the manner described in the Method text. Figure 9C also shows the definition of a surface code stabilizer.

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

[0133] Referring to Figure 11, the characteristics of the encoded logical state are illustrated. Figure 11A shows this work for raw measurement and for performing error correction and error detection in post-processing (all logical states). L This provides a summary of logical error probabilities for various error correction graphs created in (located at). Error correction for Steane codes is performed by a Steane code decoder, and for surface and trick codes, it is performed using a minimum-weighted perfect fit algorithm. For equally spaced trick codes, if the correction is unclear, the logical qubit is not rejected, and therefore, the interval d=2 logical qubit does not change under the correction procedure. The observed fidelity is comparable to similar displays in recent experiments using other platforms. Figure 11B shows logical error probabilities for surface codes with correction and detection performed in post-processing similar to Figure 11A. L Indicates the lifespan of the state. After state adjustment, |+> L The state is held for a period of time with variability before projection measurement, and two π A pulse is applied for dynamic decoupling (the lifetime is, for example, 128 as shown in Figure 7B). π (This can be further significantly extended by applying pulses). Here, some experimental parameters are those in Figure 11A, and therefore slightly different compared to the higher error rate at time 0. Figure 11C shows the logical qubit state |0> L This shows a logical π / 2 rotation on the Steane code for preparing the sphere. The Steane code, surface code, and trick code all have a transverse single-qubit Clifford operation on a logical qubit (including in-software rotation of the grid), and the transverse rotation is performed in parallel with the overall Raman laser, and since the physical single-qubit fidelity is high, this is a high-fidelity operation in the system. Here, a logical π / 2 rotation for the Steane code is shown as an example, but various reference states along the fundamental axis of the logical Bloch sphere can be realized for all of these codes.

[0134] Referring to Figure 12, the test of the interferometric measurement on a benchmark problem is illustrated. To test the gate-based interferometric technique on the benchmark problem, a variable single-particle pure state is prepared (by applying a resonant Raman pulse of variability length), and the system is then reconstructed, with the interferometric circuit applied to one of the pairs. The interferometric circuit transforms the symmetric triplet state into an antisymmetric singlet state |Ψ| while converting the symmetric triplet state into other computer-calculated states. - > is converted to the computer-calculated reference state |00>. The output state of one of the resulting pairs is plotted in the left panel. The |00> state is rarely observed (1.95(2)% measurement), and the measurement fidelity is independent of the initial state. This low probability P of observing |00> 00 2P 00 -1 = 0.961 (3) corresponds to a high extracted single-particle purity (Figure 12A, right panel). This measurement is a useful benchmark because interferometric miscalibration can lead to significant state dependence of observed purity, which impairs the validity of the many-body entanglement entropy measurement. Test of the benchmark problem for entanglement entropy measurement using Bell state arrays. (Figure 12B, top) |11> under a Rydberg pulse with variability duration and

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[0135] Referring to Figure 13, the raw many-body data and numerical modeling of errors are provided. Figure 13A shows the raw measured Renyi entropy, without subtracting the broad classical entropy, as a function of subsystem size for quenching from |rgrgrgrg> and |gggggggg>. The Renyi entropy of the 4-atom subsystem is the same underlying data as the half-chain entanglement entropy plotted in Figure 4D. In the previous example, the data was subtracted for a fixed offset given by the classical per-particle entropy, corresponding to the time=0 offset for each subsystem size. The broad classical entropy offset is slightly larger for the |rgrgrgrg> quench due to the non-uniform fidelity of both preparing |r> and mapping |r>→|r>. Figure 13B shows the raw overall purity after the |gggggggg> quench. Overall purity is a substitute for the fidelity sensitivity of the entire process. This 16-body observable, composed of a 3-level system, is a well-mixed state of 8 qubits (1 / 2 8 The predicted purity for ) is left as >100× (see inset). For scale comparison, the single particle purity is also plotted to the power of 8 to show what the overall purity would be if the measurement result for one of each pair were not corrected. Figure 13C shows a three-level system with various simulated error sources.

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[0136] Referring to FIG. 14, local observables and entanglement entropy for quantum many-body scars are illustrated. FIG. 14A shows the experimentally measured single-site entropy for each site in an 8-atom chain when quenching from a scarred

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[0137] Formation of particle arrays 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 within the atom. The relevant energy shift within the atom from the induced dipole, averaged over the optical oscillation period, 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 are therefore trapped at a local intensity maximum (for detuned red, i.e., longer wavelength trap light). The AC Stark shift is proportional to the intensity of the light. Thus, the shape of the intensity field is the shape of the relevant atomic trap. Optical tweezers utilize this principle by focusing a laser to a constriction on the micron scale, where individual atoms are trapped at the focal point. Two-dimensional (2D) arrays of optical tweezers are generated, for example, by irradiating a spatial light modulator (SLM) that provides a computer-generated hologram to the wavefront of the laser field. The 2D array of optical tweezers overlaps with the cloud of laser-cooled atoms within the magneto-optical trap (MOT). The precisely focused optical tweezers operate in a "collision blockade" region where a single atom is loaded from the MOT, and pairs of atoms are ejected for optically assisted collisions, ensuring that at most one atom is loaded onto the tweezers, although the loading is expected, so there is approximately a 50-60% probability that a single atom will be loaded into the trap.

[0138] To prepare a deterministic atomic array, a real-time feedback procedure identifies randomly loaded atoms and rearranges them into a pre-programmed geometric structure. Atomic rearrangement requires moving atoms in tweezers, which can be smoothly carried out to minimize heating by deflecting the laser beam by an adjustable angle controlled by the frequency of the acoustic waveform applied to the AOD crystal, for example, using an acoustic-optical deflector (AOD). Dynamic tuning of the acoustic frequency translates to smooth movement of the optical tweezers. Multi-frequency acoustic waves generate an array of laser deflections, which, after being focused through a microscope objective lens, form an array of optical tweezers with adjustable position and amplitude, both controlled by the acoustic waveform. Atoms are rearranged using a further set of dynamically moving tweezers placed over the top of the SLM tweezers array.

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

[0140] Parallel classification can be achieved by using two acoustic-optical deflectors (AODs) to generate multiple tweezers capable of picking up particles from an existing particle trap structure, moving them simultaneously, and releasing them elsewhere. This may involve moving particles around within a single trap structure (e.g., a tweezers array) or transporting and classifying particles from one trap system to another (e.g., between one tweezers array and another type of optical / magnetic trap). This classification is flexible and allows for the programmed placement of each particle. Each movable trap is formed by an AOD, and its position is dynamically controlled by the frequency components of the radio frequency (RF) driven field for the AOD. Since the RF drive of the AOD can be controlled in real time and may include any combination of frequency components, it is possible to generate any grid of traps (e.g., lines of arbitrarily placed traps) by changing the number, magnitude, and distribution of frequency components in the RF driven field of the AOD, move rows or columns of the grid, and add or remove rows and columns of the grid.

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

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

[0143] Referring to Figure 15, a schematic diagram of the apparatus 1500 for quantum computer computation according to an aspect of the present disclosure is provided. As shown in Figure 15, using a beam generated by a light source 1502 (e.g., a coherent light source; in some exemplary embodiments - a monochromatic light source), the SLM 1504 forms an array of trap beams (i.e., a tweezers array), which is imaged onto the trap surface 1508 in a vacuum chamber 1510 by a train of optical components including elements 1506a, 1506c, 1506d and a high numerical aperture (NA) objective lens 1506e, in the exemplary embodiment shown in Figure 15. Other suitable trains of optical components may be used as readily understood by those skilled 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 having non-parallel directions (e.g., orthogonal directions) of sound wave transmission generate dynamically moving classification beams. Using a series of optical components such as those shown in Figure 15 (elements 1517, 1506b, 1506c, 1506d, and 1506e), the classification beam overlaps with the trap beam. It is understood that the same result can be achieved using other series of optical components. For example, sources 1502 and 1512 could be a single source, and the trap beam and classification beam could be generated by a beam splitter.

[0144] The dynamic motion of the steering beam is achieved using two non-parallel AODs 1514, 1516 arranged in series. In the exemplary embodiment shown in Figure 15, one AOD defines the direction of the “row” (“horizontal” - 'X' AOD) and the other AOD defines the direction of the “column” (“vertical” - 'Y' AOD). Each AOD is driven by an arbitrary RF waveform originating from an arbitrary waveform generator 1520, which is generated in real time by a computer 1522 that processes a feedback routine after analyzing an image of the positions where atoms are loaded. When each AOD is driven by 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, a single AOD trap can be advanced to overlap with any SLM trap.

[0145] In Figure 15, laser 1502 irradiates SLM 1504 with a beam of light. SLM 1504 can be controlled by computer 1522 to generate a beam pattern ("trap beam" or "tweezers array"). The beam pattern is focused by lens 1506a, passes through mirror 1506b, and is parallelized by lens 1506c on mirror 1506d. The reflected light passes through objective lens 1506e to focus the optical tweezers array in the vacuum chamber 1510 on the trap surface 1508. The laser light from the optical tweezers array continues to pass through objective lens 1524a, through dichroism mirror 1524b, and is detected by charge-coupled device (CCD) camera 1524c.

[0146] The vacuum chamber 1510 may be illuminated by a further light source (not shown). Fluorescence from atoms trapped on the trap surface also passes through the objective lens 1524a but is reflected by the dichroic mirror 1524b to an electron-amplified CCD (EMCCD) camera 1524d. In this example, the laser 1512 directs a beam of light to AODs 1514, 1516. The AODs 1514, 1516 are driven by an arbitrary wave generator (AWG) 1520, which is then controlled by a computer 1522. The intersecting AODs 1514, 1516 emit one or more beams as described above, which are directed to the focusing lens 1517. The beams then enter the same set of optical components 1506b...1506e as described above with respect to the optical tweezers array and focus onto the trap surface 1508.

[0147] It is understood that an optical tweezers array suitable for the use described herein can be fabricated using an alternative set of optical components.

[0148] The descriptions of various aspects of this disclosure are provided for illustrative purposes only and are not intended to be exhaustive or limiting to the aspects disclosed. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the aspects described. The terms used herein have been chosen to best describe the principles of the aspects, their practical applications or technological advancements to the art available on the market, or to enable those else skilled in the art to understand the aspects disclosed herein. ​

Claims

1. A step of providing multiple neutral atoms, where each of the multiple neutral atoms is placed in a corresponding optical trap; Each of the multiple neutral atoms is m F The process involves adjusting the clock state at 0; A process of entangling pairs of neutral atoms by directing a laser pulse to pairs of neutral atoms of multiple neutral atoms, wherein the laser pulse is configured to cause the pairs of neutral atoms to transition through a Rydberg state; and A process of causing an optical trap corresponding to at least one neutral atom of the pair to undergo adiabatic motion, applying a Raman pulse to at least one neutral atom during the motion, thereby moving the neutral atoms of the pair relative to each other without breaking the entanglement of the pair. A method for performing quantum computer calculations, including [the following].

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

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

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

5. The process of moving an optical trap corresponding to at least one neutral atom within the blockade radius of a target neutral atom of multiple neutral atoms. The method according to claim 1, further comprising:

6. A process of entangling at least one neutral atom with a target neutral atom. The method according to claim 5, further comprising:

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

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

9. The method according to claim 8, wherein at least one neutral atom and the target neutral atom are not adjacent in the two-dimensional array prior to the motion.

10. The method according to claim 1, wherein an optical trap corresponding to at least one neutral atom is generated by directing a beam of light to at least one acoustic-optical deflector (AOD), and the step of causing the optical trap corresponding to at least one neutral atom to undergo adiabatic motion includes changing the driving frequency of at least one AOD.

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

12. A step of providing multiple neutral atoms, where each of the multiple neutral atoms is placed in a corresponding optical trap; Each of the multiple neutral atoms is m F The process involves adjusting the clock state at 0; The process involves directing a laser pulse towards a pair of neutral atoms, thereby entangling the pairs of neutral atoms, where the laser pulse is configured to cause the pairs of neutral atoms to transition through a Rydberg state; A step of causing an optical trap corresponding to at least one neutral atom of the pair to undergo adiabatic motion, thereby moving the neutral atoms of the pair relative to each other without breaking the entanglement of the pair; A step of irradiating a first region, wherein the first region contains the first atom of the pair, thereby applying rotation to the first atom of the pair; A step of causing the optical trap corresponding to the first atom of the pair to undergo adiabatic motion outward from the first region; The process of causing the optical trap corresponding to the second atom of the pair to undergo adiabatic motion toward the first region; and A step of irradiating a first region, thereby applying rotation to the second atom of the pair. A method for performing quantum computer calculations, including [the following].

13. The process of applying a Raman pulse to at least one neutral atom during the motion. The method according to claim 12, further comprising:

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

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

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

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

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

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

20. A step of providing a plurality of neutral atoms, each of which is placed in a corresponding optical trap, the plurality of neutral atoms comprising a first subset and a second subset, where each neutral atom of the first subset is placed within the blockade radius of the first corresponding neutral atom of the second subset, thereby forming a first plurality of pairs; Each of the multiple neutral atoms is m F The process involves adjusting the clock state at 0; The process of applying a first 2-qubit gate to each of the first multiple pairs; The steps of causing an optical trap corresponding to a first subset to undergo adiabatic motion such that each neutral atom of the first subset is within the blockade radius of the second corresponding neutral atom of the second subset, thereby forming a second set of pairs, and applying a Raman pulse to the first subset during the motion; and The process of applying a second 2-qubit gate to each of the second set of pairs. A method for performing quantum computer calculations, including [the following].

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

22. A step of causing an optical trap corresponding to the first subset to undergo adiabatic motion into an imaging region that does not include the second subset; and A process of illuminating the imaging area and measuring the state of the first subset. The method according to claim 20, further comprising:

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 a Raman pulse is applied at the midpoint of the motion.

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

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

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

28. The method according to claim 20, wherein an optical trap corresponding to at least one neutral atom is generated by directing a beam of light to at least one acoustic-optical deflector (AOD), and the step of causing the optical trap corresponding to at least one neutral atom to undergo adiabatic motion includes changing the driving frequency of at least one AOD.

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

30. A step of providing multiple neutral atoms, where each of the multiple neutral atoms is placed in a corresponding optical trap; Each of the multiple neutral atoms is m F The process involves adjusting the clock state at 0; The process of causing multiple neutral atoms to undergo adiabatic motion between a first arrangement and a second arrangement different from the first arrangement, wherein the first array arrangement includes at least one pair of neutral atoms within each other's blockade radii; The step of applying a gate to at least one pair of neutral atoms when they are in the first configuration; and A process of advancing multiple neutral atoms according to the first Hamiltonian when in the second configuration. A method for performing quantum computer calculations, including [the following].

31. The process of applying a Raman pulse to at least one neutral atom during the motion. The method according to claim 30, further comprising:

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

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

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

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

36. The method according to claim 30, wherein an optical trap corresponding to at least one neutral atom is generated by directing a beam of light to at least one acoustic-optical deflector (AOD), and the step of causing the optical trap corresponding to at least one neutral atom to undergo adiabatic motion includes changing the driving frequency of at least one AOD.

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

38. Multiple optical traps; Multiple neutral atoms, where each of the multiple neutral atoms is placed in a corresponding one of the multiple optical traps; and At least one laser, where at least one laser is Each of the multiple neutral atoms is m F Prepared in a =0 clock state, and Multiple neutral atoms are created by transitioning pairs of neutral atoms through the Rydberg state. It entangles pairs of neutral atoms. It is configured in such a way. A quantum computer that includes; A quantum computer configured to cause an optical trap corresponding to at least one neutral atom of a pair to undergo adiabatic motion, and to apply a Raman pulse to at least one neutral atom during this motion, thereby causing the neutral atoms of the pair to move relative to each other without breaking the entanglement of the pair.

39. Multiple optical traps; A plurality of neutral atoms comprising a first subset and a second subset, where each of the plurality of neutral atoms is located in a corresponding one of a plurality of optical traps, and each neutral atom of the first subset is located within the blockade radius of the first corresponding neutral atom of the second subset, thereby forming a first plurality of pairs; and At least one laser, where at least one laser penetrates each of several neutral atoms m F It is configured to be prepared in a =0 clock state. A quantum computer that includes, Apply a gate to each of the first pairs; Each neutral atom in the first subset corresponds to the second neutral atom in the second subset. The optical tracks corresponding to the first subset are within the atomic blockade radius. The particles are subjected to adiabatic motion, thereby forming a second set of pairs; During the motion, a Raman pulse is applied to the first subset; and Apply a gate to each of the second set of pairs. It is configured in such a way. Quantum computer.

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