Reconfigurable architecture for parallel quantum operations on neutral atomic arrays

Abstract

A quantum processor is provided. A first plurality of neutral atoms are provided in an active zone, each in a respective optical trap of a first array. A first logical qubit is encoded on the first plurality of neutral atoms by a first laser and a second laser. The first plurality of neutral atoms are illuminated by at least the first laser or the second laser while in the active zone, thereby applying a gate to the first logical qubit. The first plurality of neutral atoms are adiabatically transferred from the active zone to a readout zone, each into a respective optical trap of a second array. The first plurality of neutral atoms are illuminated by a third laser while in the readout zone. An image of the first plurality of neutral atoms is captured while in the readout zone, thereby determining the state of the first logical qubit.

Description

[Technical Field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS This application claims the benefit of U.S. Provisional Application No. 63 / 482,702, filed February 1, 2023, and U.S. Provisional Application No. 63 / 604,545, filed November 30, 2023, each of 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, and W911NF2010021 and W911NF2010082 awarded by the US Army Research Office, and grants N00014-15-1-2846 and N00014-15-1-2761 awarded by the US Office of Naval Research, and DE-SC0021013 awarded by the US Department of Energy. The government has certain rights in this invention.

[0003] FIELD OF THE DISCLOSURE Embodiments of the present disclosure relate to quantum computing, and more particularly to a dynamically reconfigurable architecture for parallel quantum operations on neutral atomic arrays. [Prior art documents] [Non-patent literature]

[0004] [Non-Patent Document 1] Levine et al., Parallel Implementation of High-Fidelity Multiqubit Gates with Neutral Atoms, Phys. Rev. Lett., Issue 123, Volume 17, https: / / link.aps.org / doi / 10.1103 / PhysRevLett.123.170503 Summary of the Invention [Means for solving the problem]

[0005] According to an embodiment of the present disclosure, there is provided a quantum processor including a first array of optical traps disposed in an active zone, a second array of optical traps disposed in a readout zone, a first laser configured to illuminate the active zone and drive transitions to Rydberg states, a second laser configured to illuminate the active zone and drive transitions between hyperfine states, a third laser configured to illuminate the readout zone, a fourth laser configured to adiabatically transfer neutral atoms between the optical traps in the active zone and the optical traps in the readout zone, and a camera configured to capture images of the readout zone. The quantum processor is configured to provide a first plurality of neutral atoms within the active zone, each within a respective optical trap of a first array; encode a first logical qubit onto the first plurality of neutral atoms with a first laser and a second laser; illuminate the first plurality of neutral atoms while within the active zone with at least the first laser or the second laser, thereby applying a gate to the first logical qubit; adiabatically transfer the first plurality of neutral atoms from the active zone to a readout zone, each into a respective optical trap of a second array; illuminate the first plurality of neutral atoms while within the readout zone with a third laser; and capture an image of the first plurality of neutral atoms while within the readout zone, thereby determining the state of the first logical qubit.

[0006] In some embodiments, the quantum processor further includes a third array of optical traps disposed in the storage zone, and the fourth laser is further configured to adiabatically transfer neutral atoms between the optical traps in the active zone and the optical traps in the storage zone. The quantum processor is further configured to adiabatically transfer a first plurality of neutral atoms from the active zone to the storage zone, each into a respective optical trap in the third array, with the fourth laser after said encoding, and to adiabatically transfer a first plurality of neutral atoms from the storage zone to the active zone, each into a respective optical trap in the first array, with the fourth laser before applying the gate.

[0007] In some embodiments, the quantum processor is further configured to provide a second plurality of neutral atoms within the active zone, encode a second logical qubit onto the second plurality of neutral atoms with the first laser and the second laser, and arrange the first and second plurality of neutral atoms within the active zone such that each neutral atom of the first plurality of neutral atoms is within a blockade radius of exactly one corresponding neutral atom of the second plurality of neutral atoms before applying the gate. Illuminating the first plurality of neutral atoms while within the active zone further illuminates the second plurality of neutral atoms, thereby applying the gate to the first logical qubit and the second logical qubit.

[0008] According to an embodiment of the present disclosure, a method for performing quantum computing is provided. A first array of optical traps is disposed in an active zone of a quantum processor. A second array of optical traps is disposed in a readout zone of the quantum processor. A first plurality of neutral atoms is provided in the active zone, each in a respective optical trap of the first array. A first logical qubit is encoded on the first plurality of neutral atoms by a first laser and a second laser, the first laser being configured to illuminate the active zone and drive transitions to Rydberg states, and the second laser being configured to illuminate the active zone and drive transitions between hyperfine states. The first plurality of neutral atoms are illuminated while in the active zone by at least the first laser or the second laser, thereby applying a gate to the first logical qubit. The first plurality of neutral atoms are adiabatically transferred from the active zone to a readout zone, each into a respective optical trap of the second array. The first plurality of neutral atoms are illuminated while in the readout zone by a third laser. An image of the first plurality of neutral atoms is captured while in the readout zone, thereby determining the state of the first logical qubit.

[0009] In some embodiments, the first plurality of neutral atoms are adiabatically moved by a fourth laser from the active zone to a storage zone of the quantum processor after said encoding, and into respective optical traps of a third array, and the first plurality of neutral atoms are adiabatically moved by a fourth laser from the storage zone to the active zone, and into respective optical traps of the first array, before applying the gate.

[0010] In some embodiments, a second plurality of neutral atoms is provided within the active zone. A second logical qubit is encoded on the second plurality of neutral atoms by a first laser and a second laser. Prior to applying the gate, the first plurality of neutral atoms and the second plurality of neutral atoms are arranged within the active zone such that each neutral atom of the first plurality of neutral atoms is within the blockade radius of exactly one corresponding neutral atom of the second plurality of neutral atoms. Illuminating the first plurality of neutral atoms while they are within the active zone also illuminates the second plurality of neutral atoms, thereby applying the gate to the first logical qubit and the second logical qubit.

[0011] In various embodiments, the gate is a transversal CNOT gate. In various embodiments, applying the gate includes applying a single pulse of the first laser.

[0012] In various embodiments, adiabatically transferring the first plurality of neutral atoms from the active zone to the storage zone, adiabatically transferring the first plurality of neutral atoms from the storage zone to the active zone, and adiabatically transferring the first plurality of neutral atoms from the active zone to the readout zone each includes applying a Raman pulse during the transfer. In various embodiments, the Raman pulse is applied at a midpoint of the transfer. In various embodiments, adiabatically transferring the first plurality of neutral atoms from the active zone to the storage zone, adiabatically transferring the first plurality of neutral atoms from the storage zone to the active zone, and adiabatically transferring the first plurality of neutral atoms from the active zone to the readout zone each has a constant jerk.

[0013] In various embodiments, the first array, second array, and / or third array of optical traps are two-dimensional arrays.

[0014] In various embodiments, a beam of light is directed from a fourth laser to at least one acousto-optic deflector (AOD), and adiabatically moving the neutral atoms includes varying a drive frequency of the at least one AOD.

[0015] In various embodiments, the first array, second array, and / or third array of optical traps are generated by directing a beam of light onto a spatial light modulator (SLM).

[0016] In various embodiments, the first plurality of neutral atoms are transferred simultaneously.

[0017] In various embodiments, the third array has a higher density than the first array.

[0018] In various embodiments, encoding the first logical qubit includes applying a CSS code, hi various embodiments, the CSS code is selected from a surface code, a color code, a Steane code, and a hypergraph product LDPC code. [Brief explanation of the drawings]

[0019] [Figure 1] 1 is a schematic diagram of a quantum information architecture according to an embodiment of the present disclosure. [Figure 2] FIG. 1 is a level diagram illustrating important 87Rb atomic levels according to embodiments of the present disclosure. [Figure 3] FIG. 1 is a schematic diagram of a toric code implementation according to an embodiment of the present disclosure. [Figure 4] FIG. 1 is a schematic diagram of a quantum processing unit (QPU) according to an embodiment of the present disclosure. [Figure 5] FIG. 1 is a schematic diagram of a logical qubit illustrating efficient control according to an embodiment of the present disclosure. [Figure 6]FIG. 1 is a schematic diagram of a logical qubit illustrating the application of a transversal CNOT gate according to an embodiment of the present disclosure. [Figure 7] 1 is a schematic diagram of a portion of a processor core according to an embodiment of the present disclosure. [Figure 8] FIG. 1 is a schematic diagram of a portion of a processor core suitable for use in implementing repetition codes according to embodiments of the present disclosure. [Figure 9] FIG. 1 is a schematic diagram of a portion of a processor core suitable for use in implementing a surface code according to an embodiment of the present disclosure. [Figure 10] FIG. 1 is a schematic diagram of a method for active feedforward QEC according to an embodiment of the present disclosure. [Figure 11] FIG. 1 is a schematic diagram of an apparatus for quantum computing according to an embodiment of the present disclosure. [Figure 12A] FIG. 1 illustrates a programmable logic processor based on a reconfigurable atomic array according to an embodiment of the present disclosure. [Figure 12B] FIG. 1 illustrates a programmable logic processor based on a reconfigurable atomic array according to an embodiment of the present disclosure. [Figure 12C] FIG. 1 illustrates a programmable logic processor based on a reconfigurable atomic array according to an embodiment of the present disclosure. [Figure 13A] FIG. 1 illustrates a transversal entanglement gate between two surface codes according to an embodiment of the present disclosure. [Figure 13B] FIG. 1 illustrates a transversal entanglement gate between two surface codes according to an embodiment of the present disclosure. [Figure 13C] FIG. 1 illustrates a transversal entanglement gate between two surface codes according to an embodiment of the present disclosure. [Figure 13D] FIG. 1 illustrates a transversal entanglement gate between two surface codes according to an embodiment of the present disclosure. [Figure 13E] FIG. 1 illustrates a transversal entanglement gate between two surface codes according to an embodiment of the present disclosure. [Figure 14A]1 illustrates a fault-tolerant logic algorithm according to an embodiment of the present disclosure. [Figure 14B] 1 illustrates a fault-tolerant logic algorithm according to an embodiment of the present disclosure. [Figure 14C] 1 illustrates a fault-tolerant logic algorithm according to an embodiment of the present disclosure. [Figure 14D] 1 illustrates a fault-tolerant logic algorithm according to an embodiment of the present disclosure. [Figure 14E] 1 illustrates a fault-tolerant logic algorithm according to an embodiment of the present disclosure. [Figure 15A] FIG. 10 illustrates scaling and mid-circuit feedforward in a zoned logical processor according to an embodiment of the present disclosure. [Figure 15B] FIG. 10 illustrates scaling and mid-circuit feedforward in a zoned logical processor according to an embodiment of the present disclosure. [Figure 15C] FIG. 10 illustrates scaling and mid-circuit feedforward in a zoned logical processor according to an embodiment of the present disclosure. [Figure 15D] FIG. 10 illustrates scaling and mid-circuit feedforward in a zoned logical processor according to an embodiment of the present disclosure. [Figure 15E] FIG. 10 illustrates scaling and mid-circuit feedforward in a zoned logical processor according to an embodiment of the present disclosure. [Figure 16A] FIG. 1 illustrates a complex logical circuit using 3D codes according to an embodiment of the present disclosure. [Figure 16B] FIG. 1 illustrates a complex logical circuit using 3D codes according to an embodiment of the present disclosure. [Figure 16C] FIG. 1 illustrates a complex logical circuit using 3D codes according to an embodiment of the present disclosure. [Figure 16D] FIG. 1 illustrates a complex logical circuit using 3D codes according to an embodiment of the present disclosure. [Figure 16E]FIG. 1 illustrates a complex logical circuit using 3D codes according to an embodiment of the present disclosure. [Figure 16F] FIG. 1 illustrates a complex logical circuit using 3D codes according to an embodiment of the present disclosure. [Figure 17A] FIG. 1 illustrates a logical two-copy measurement according to an embodiment of the present disclosure. [Figure 17B] FIG. 1 illustrates a logical two-copy measurement according to an embodiment of the present disclosure. [Figure 17C] FIG. 1 illustrates a logical two-copy measurement according to an embodiment of the present disclosure. [Figure 17D] FIG. 1 illustrates a logical two-copy measurement according to an embodiment of the present disclosure. [Figure 18A] FIG. 1 illustrates a neutral atom quantum computer architecture according to an embodiment of the present disclosure. [Figure 18B] FIG. 1 illustrates a neutral atom quantum computer architecture according to an embodiment of the present disclosure. [Figure 18C] FIG. 1 illustrates a neutral atom quantum computer architecture according to an embodiment of the present disclosure. [Figure 18D] FIG. 1 illustrates a neutral atom quantum computer architecture according to an embodiment of the present disclosure. [Figure 19A] FIG. 1 illustrates one-qubit Raman addressing according to an embodiment of the present disclosure. [Figure 19B] FIG. 1 illustrates one-qubit Raman addressing according to an embodiment of the present disclosure. [Figure 19C] FIG. 1 illustrates one-qubit Raman addressing according to an embodiment of the present disclosure. [Figure 19D] FIG. 1 illustrates one-qubit Raman addressing according to an embodiment of the present disclosure. [Figure 20A] FIG. 1 illustrates a circuit mid-readout and feedforward according to an embodiment of the present disclosure. [Figure 20B] FIG. 1 illustrates a circuit mid-readout and feedforward according to an embodiment of the present disclosure. [Figure 20C]FIG. 1 illustrates a circuit mid-readout and feedforward according to an embodiment of the present disclosure. [Figure 20D] FIG. 1 illustrates a circuit mid-readout and feedforward according to an embodiment of the present disclosure. [Figure 20E] FIG. 1 illustrates a circuit mid-readout and feedforward according to an embodiment of the present disclosure. [Figure 20F] FIG. 1 illustrates a circuit mid-readout and feedforward according to an embodiment of the present disclosure. [Figure 20G] FIG. 1 illustrates a circuit mid-readout and feedforward according to an embodiment of the present disclosure. [Figure 20H] FIG. 1 illustrates a circuit mid-readout and feedforward according to an embodiment of the present disclosure. [Figure 21A] FIG. 10 illustrates further surface signature data according to an embodiment of the present disclosure. [Figure 21B] FIG. 10 illustrates further surface signature data according to an embodiment of the present disclosure. [Figure 21C] FIG. 10 illustrates further surface signature data according to an embodiment of the present disclosure. [Figure 21D] FIG. 10 illustrates further surface signature data according to an embodiment of the present disclosure. [Figure 21E] FIG. 10 illustrates further surface signature data according to an embodiment of the present disclosure. [Figure 21F] FIG. 10 illustrates further surface signature data according to an embodiment of the present disclosure. [Figure 22A] FIG. 1 illustrates surface code preparation and data decoding according to an embodiment of the present disclosure. [Figure 22B] FIG. 1 illustrates surface code preparation and data decoding according to an embodiment of the present disclosure. [Figure 22C] FIG. 1 illustrates surface code preparation and data decoding according to an embodiment of the present disclosure. [Figure 23A] FIG. 1 illustrates [[8,3,2]] and hypercube coding according to an embodiment of the present disclosure. [Figure 23B]FIG. 1 illustrates [[8,3,2]] and hypercube coding according to an embodiment of the present disclosure. [Figure 23C] FIG. 1 illustrates [[8,3,2]] and hypercube coding according to an embodiment of the present disclosure. [Figure 24A] FIG. 10 illustrates sampling data for a further [[8,3,2]] circuit in accordance with an embodiment of the present disclosure. [Figure 24B] FIG. 10 illustrates sampling data for a further [[8,3,2]] circuit in accordance with an embodiment of the present disclosure. [Figure 24C] FIG. 10 illustrates sampling data for a further [[8,3,2]] circuit in accordance with an embodiment of the present disclosure. [Figure 25A] FIG. 1 illustrates a theoretical exploration of a hypercube IQP circuit according to an embodiment of the present disclosure. [Figure 25B] FIG. 1 illustrates a theoretical exploration of a hypercube IQP circuit according to an embodiment of the present disclosure. [Figure 25C] FIG. 1 illustrates a theoretical exploration of a hypercube IQP circuit according to an embodiment of the present disclosure. [Figure 26A] FIG. 10 illustrates further Bell basis measurement results according to an embodiment of the present disclosure. [Figure 26B] FIG. 10 illustrates further Bell basis measurement results according to an embodiment of the present disclosure. [Figure 26C] FIG. 10 illustrates further Bell basis measurement results according to an embodiment of the present disclosure. [Figure 26D] FIG. 10 illustrates further Bell basis measurement results according to an embodiment of the present disclosure. [Figure 26E] FIG. 10 illustrates further Bell basis measurement results according to an embodiment of the present disclosure. [Figure 26F] FIG. 10 illustrates further Bell basis measurement results according to an embodiment of the present disclosure. DETAILED DESCRIPTION OF THE INVENTION

[0020] Large-scale quantum computers have the potential to solve problems that are intractable with classical processors. While there has been exciting progress in the development of small- and medium-scale quantum processors and their applications for studying the physical phenomena of complex quantum systems in regimes that are difficult to simulate classically, it is unclear whether and how truly large-scale quantum processors, whose value outweighs the cost of construction, can be built and applied to solving general-purpose, computationally difficult problems. For example, current estimates of the resources required to realize one of the best-known high-value applications, the Shor factorization algorithm, are estimated at approximately 5,000 logical qubits and 10 -12 Alternatively, using conventional error correction methods, 20 million superconducting qubits with a realistic gate error rate (0.1%) could be used, which is nearly six orders of magnitude larger than the scale of currently available well-controlled systems.

[0021] This will require the development of novel, unconventional approaches to practical-scale quantum computing, likely involving a synergistic combination of new hardware architectures that dramatically reduce the cost of error correction and computation, new resource-efficient approaches to developing algorithms co-designed with the hardware, and a large-scale engineering effort to build practical systems. Furthermore, all such large-scale developments will also require significant efforts for verification and validation.

[0022] To address these and other shortcomings of alternative approaches, the present disclosure provides systems and methods for controlling millions of physical qubits using efficient parallel, fault-tolerant quantum operations. These systems are applicable to the resource-efficient implementation of practical algorithms aimed at practical-scale quantum computing.

[0023] A quantum bit (qubit) is the fundamental building block of a quantum computer. Analogous to classical bits (each bit is either 0 or 1) used to store information in conventional computers, a qubit can occupy two distinct states, labeled |0> and |1>, or any quantum superposition of the two states. In various applications, multiple qubits are entangled to build multi-qubit quantum gates.

[0024] Bits and qubits are each encoded into the state of a real physical system: for example, a classical bit (0 or 1) might be encoded into whether a capacitor is charged or discharged, or whether a switch is "on" or "off."

[0025] The term qudit (quantum digit) denotes a unit of quantum information that can be realized in an appropriate d-level quantum system. A collection of qubits that can be measured into N states can implement an N-level qudit.

[0026] A qubit is encoded in a quantum system that has two (or more) different quantum states. There are many physical realizations that may be adopted. 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 different quantum states that correspond to different orientations of electron spin, nuclear spin, electron orbitals, and molecular rotation / vibration.

[0027] In principle, a qubit may be encoded into any pair of atomic / ionic / molecular quantum states. In practice, an important parameter of a qubit is described by its quantum coherence property. Coherence measures the lifetime of the qubit before the qubit information is lost. Coherence is similar to that of a classical bit: if a classical bit is prepared in the 0 state, after a while, the 0 state can be randomly flipped to 1 by environmental noise. In quantum mechanics, the same error can occur: |0> can randomly flip to |1> after some characteristic timescale. However, a qubit can also suffer from additional errors, for example, the superposition state (|0> + |1>) / √2 can randomly flip to (|0> - |1>) / √2. In a real quantum computer, qubits must be encoded into quantum states with long coherence properties.

[0028] A quantum computer can generally accommodate many qubits, each encoded in its own atom / molecule / ion, etc. Rather than simply accommodating qubits, a quantum computer should be able to (1) initialize the qubits, (2) manipulate the qubits' states in a controlled manner, and (3) read out the qubits' final states. Regarding qubit operations, they are typically divided into two types: one type of qubit operation is a so-called one-qubit gate, which refers to an operation applied to a qubit individually. This may, for example, flip the qubit's state from |0> to |1>, or bring |0> into a superposition state (|0> + |1>) / √2. The second required type of qubit operation is a multi-qubit gate, which acts on two or more qubits collectively, including entangled qubits. Multi-qubit gates are realized by some form of interaction between the qubits. Different quantum computing platforms (with different physical encodings of qubits) rely on different physical mechanisms for both one-qubit and multi-qubit gates, depending on the physical system that stores the qubits.

[0029] In various quantum computer implementations, qubits are encoded into energy levels close to the two ground states of an atom, ion, or molecule. An example of this is the hyperfine qubit. Such qubits are encoded into two electronic ground states that differ by the relative orientation of the nuclear spin with respect to the outer electron spin. Such state pairs can be selected to be particularly robust / insensitive to environmental perturbations, leading to long coherence times. These states are energy-split by the hyperfine interaction energy of the atom / ion / molecule, which is the interaction energy between the nuclear spin and the electron spin. The robustness of a qubit can be understood as the particularly stable energy split between the two states. For this reason, such states are called clock states, because a stable energy split can form an excellent frequency standard and thus the basis for atomic clocks. Typical hyperfine splitting between the states of these qubits is in the frequency range of 1 to 13 GHz.

[0030] To perform one-qubit gates on such hyperfine qubits, coherent microwave radiation at the exact frequency of the energy splitting between the states can be applied. However, this approach has two drawbacks. First, microwaves cannot be applied to just one qubit without affecting nearby qubits. This is because qubits are typically encoded on particles that are only a few microns apart, and microwaves cannot be focused on such small scales due to their long wavelength. Second, the intensity of microwaves is quite limited, and therefore the maximum speed of one-qubit gates is correspondingly limited.

[0031] An alternative approach is based on stimulated Raman transitions. In this case, a laser field is applied to an atom / ion / molecule. The laser field is nearly (but not perfectly) resonant with an optical transition from one of the ground states to an optically excited state. The laser contains multiple frequency components separated in frequency by an amount exactly equal to the hyperfine splitting of the qubit. The atom / ion / molecule absorbs a photon from one frequency component and coherently emits it at a different frequency component, changing its state in doing so. This approach benefits from the ability to focus the laser field on individual particles or subsets of particles within a quantum computer. Also, the laser field can be applied at high intensity, enabling faster gate operation.

[0032] Neutral atom quantum computers encode qubits into individual neutral atoms. Neutral atoms are trapped in a vacuum chamber and levitated by a trapping laser. The trapping laser is most commonly an individual optical tweezers, which is an individual, highly focused laser beam that traps an individual atom at its focal point. Alternatively, individual atoms may be trapped in an optical lattice formed from a standing wave of laser light that generates a periodic structure of nodes and antinodes.

[0033] A typical approach for encoding qubits in neutral atoms is the hyperfine qubit approach, where two ground states separated by several GHz form a qubit. Multi-qubit gates in neutral atom quantum computers are realized using a third atomic state, a highly excited Rydberg state. When an atom is excited into a Rydberg state, neighboring atoms are prevented from being excited into the Rydberg state. This conditional behavior forms the basis for multi-qubit gates, such as controlled-NOT gates. The Rydberg state is temporarily used to mediate the multi-qubit gate, and then the atoms are returned from the Rydberg state to the ground state to preserve their coherence.

[0034] Trap ion quantum computers use atomic species that are ionized, meaning they have a net electric charge. In most cases, many ions are trapped in one large trapping potential created by electrodes in a vacuum chamber. The ions are attracted to the minimum of the trapping potential, but Coulomb repulsion between the ions causes them to form a crystalline structure centered in the center of the trapping potential. Ions most commonly line up in linear chains. Other methods of trapping ions are possible, such as using optical tweezers or more complex on-chip electrode structures to trap ions individually in local electric fields.

[0035] Qubits can be encoded into trapped ions in several ways. One common approach is to use the ground-state hyperfine level, as described for neutral atoms. As with neutral atoms, in trapped ions with hyperfine qubit encoding, one-qubit gates may use microwave radiation or stimulated Raman transitions.

[0036] Unlike neutral atoms, trapped ion hyperfine qubits rely heavily on stimulated Raman transitions to implement multi-qubit gates. Stimulated Raman transitions may be used not only to control the ion's hyperfine state, but also to change the ion's state of motion (i.e., add momentum). This can be understood as absorbing a photon traveling in one direction and emitting a photon in a different direction, such that the difference in the photons' momentum is absorbed by the ion. Because many ions are trapped in a single collective trapping potential and often repel each other, changing the state of motion of one ion affects the other ions in the system, and this mechanism forms the basis of multi-qubit gates.

[0037] According to various embodiments of a quantum computer, individual particles (atoms / ions / molecules) can first be captured in an array and arranged in a specific configuration. Next, one or more particles are prepared in a desired quantum state. A quantum circuit can then be implemented 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 states of the particles can be read out to observe the results of the quantum circuit. Readout can typically be accomplished using an observation system that includes an electron-multiplying CCD (EMCCD) camera image to detect the loaded position of the particles and a second camera image to read out the final state of the particles, for example, by detecting fluorescence emitted by the particles at their final state.

[0038] Quantum information platforms rely on interactions between qubits to either perform quantum gates or run analog many-body simulations. However, qubits often interact locally, which limits the connectivity of circuits or analog simulations and constrains the computations possible. While some platforms can communicate in a nonlocal manner by using shared buses (e.g., trapped ions), these shared-bus approaches are limited to small systems and therefore still require a way to dynamically move qubits to truly scale up the platform.

[0039] Neutral atomic arrays store quantum information in hyperfine states and can be dynamically reconfigured by shuttling atoms with optical tweezers while preserving quantum coherence and entanglement between qubits. This approach provides a scalable method for implementing quantum information systems with large numbers of qubits and arbitrary programmability—any qubit can perform an entanglement gate with any other qubit in the array. Various quantum information circuits are described herein that utilize high-fidelity two-qubit Rydberg gates to exploit the programmability and nonlocal connectivity achievable with these approaches. Examples of high-fidelity Rydberg gates are described in Levine et al., Parallel Implementation of High-Fidelity Multiqubit Gates with Neutral Atoms, Phys. Rev. Lett., No. 123, Vol. 17, https: / / link.aps.org / doi / 10.1103 / PhysRevLett.123.170503, incorporated herein by reference.

[0040] As described in more detail below, the methods provided herein enable various computational scenarios. In some scenarios, multiple neutral atoms are moved in parallel between multiple regions in space. For example, an illumination source may be directed at a first region, and atoms are moved into and out of that region during application of pulses by the illumination source. Similarly, a camera may be directed at an imaging region, and atoms are moved into and out of that imaging region for imaging. Similarly, atoms may be moved into and out of the blockade radius of other atoms, thereby allowing gates to be applied to different groups of atoms at different stages of an algorithm or different layers of a quantum circuit.

[0041] Of course, various stabilizer codes involve readout of an ancilla qubit, and the present disclosure allows for physical relocation of the ancilla qubit to an imaging region separate from the data qubit, in this way readout of the ancilla qubit may occur without destroying the data qubit.

[0042] More broadly, an array of atoms may be moved between multiple arrangements to facilitate both digital gates between different selections of atoms and analog evolution of the entire array. As used herein, an array of atoms or an arrangement of multiple atoms refers to the relative positioning of the atoms with respect to one another. It will be understood that a particular arrangement provides connectivity between qubits that enables a particular gate or analog evolution according to a particular Hamiltonian. One advantage of the methods provided herein is that atoms may be moved to the vicinity of atoms that were not nearby in the array. Non-neighbor atoms are atoms that are not within a unit cell in a regular lattice or are not nearest neighbors in an irregular array. For example, in a rectangular lattice, each atom has eight atoms within its unit cell and therefore eight neighbors (ignoring the edges).

[0043] As further defined below, to maintain entanglement, the atoms are moved adiabatically. As used herein, the term adiabatic transfer refers to transfer that avoids transitions of the target atom within its trap. For example, transfer is considered adiabatic if the first time derivative of the target atom's acceleration is not greater than a predetermined value. Typically, adiabatic transfer is defined as a transfer where jerk < (atom size) x (trap frequency) 3 In physics, jerk or jolt is the term given to the rate at which an object's acceleration changes with respect to time.

[0044] In addition to adiabatic transfer, in some embodiments, dynamic decoupling is applied during transfer. As explained further below, a π pulse during transfer cancels the dephasing induced by the differential light shift of the trap. A π pulse is a pulse that rotates the state of the qubit by π radians on the Bloch sphere. The differential light shift of the trap changes (depending on the acceleration) as the atom moves through the trap and thus samples different portions of the light intensity and therefore experiences different differential light shifts.

[0045] Generally, the more pulses applied, the more decoupling there is from fluctuations, which may result, for example, from fluctuations in laser intensity at different displacement positions of the atoms, or from different magnetic fields in space.

[0046] In embodiments where acceleration and deceleration are symmetric, they both change the differential light shift in the same way. Therefore, in such embodiments, it is advantageous to apply a π pulse at the midpoint of the motion. In this way, the changes in differential light shift induced by acceleration and deceleration cancel each other out.

[0047] Referring to Figure 1, a quantum information architecture enabled by the coherent transport of neutral atoms is shown. Qubits are transported to perform entanglement gates with distant qubits, enabling programmable, non-local connectivity. Atom shuttling is performed using optical tweezers, and high parallelism in two dimensions and across multiple zones allows selective manipulation. The inset shows the atomic levels used, where |0> and |1> qubit states are 87 Rb m F = 0, and |r> is the Rydberg state used to generate entanglement between qubits, which are further described with respect to FIG. 2.

[0048] Figure 2 shows the important 87 This is a level diagram showing the atomic levels of Rb. The Rydberg excitation scheme from |1> to |r> consists of two-photon transitions driven by 420 nm and 1013 nm lasers. A DC magnetic field of B=8.5 G is applied throughout this study.

[0049] As mentioned above, quantum information systems derive their power from controllable interactions that generate quantum entanglement. However, the naturally local nature of the interactions limits the connectivity of quantum circuits and simulations. Nonlocal connectivity can be engineered through global shared quantum data buses, but these approaches are limited in either control or size.

[0050] According to various embodiments of the present disclosure, this long-standing challenge is addressed by dynamically reconfigurable arrays of entangled neutral atoms shuttled by optical tweezers in two spatial dimensions. Hyperfine states are used to store and transfer quantum information between quantum operations, and excitation to Rydberg states is used to generate entanglement. Selective qubit operations in different zones, between which qubits are dynamically shuttled, enable highly parallel operations. Taken together, these elements enable a powerful quantum information architecture that can be used to realize applications including the generation of entangled states, the generation of topological surface states and torus-coded states, and hybrid analog-digital quantum simulation.

[0051] Within this architecture, programming a specific quantum circuit involves control of only a few optical degrees of freedom. Any tweezer position in space is controlled by a computer-generated hologram, hundreds of atoms are dynamically reconfigured in parallel by two waveforms in a 2D acousto-optic deflector (AOD), and qubit operations are achieved by pulsing a light beam. This flexible optical control enables advanced quantum circuits with only a few classical controls. This architecture enables an inherently scalable approach, where larger codes do not require an increase in the number of classical controls.

[0052] A variety of quantum circuits can be realized with this approach, including quantum error correction (QEC) codes such as surface codes and Steane codes, and the fidelity of our work already rivals state-of-the-art experiments on other platforms. Furthermore, parallelized nonlocal connectivity is used to generate torus code states on a torus.

[0053] Figure 3 shows the implementation of a torus code state encoding two protected qubits obtained using a movable ancillary qubit array. Above, the two-logical qubit product state of the torus code upon projective measurement of the ancillary qubit in the X basis.

number

[0054] Referring to Figure 4, a quantum processing unit (QPU) according to the present disclosure is shown. The design centers around efficient classical control of many logical qubits in parallel using a light beam. One-qubit logic gates can be realized transversally, for example, by illuminating all physical qubits in the same logical qubit block with a light beam. Two-qubit logic gates can also be realized transversally by interlacing two logical arrays of qubits and applying a global light pulse to entangle each of the pair.

[0055] Neutral atomic systems have the potential for practical-scale computing; for example, millions of identical neutral atomic qubits may be trapped in a millimeter-scale spatial region. A key challenge is the classical control required to assemble these qubits into large-scale quantum processors. Full programmability of a single physical qubit generally requires extremely complex classical control techniques to operate on millions of qubits. In contrast, the architecture provided herein enables full programmability of a single logical qubit while requiring only a few classical controls per logical qubit. This makes it possible to reach practical scale by encoding logical qubits into blocks that can be efficiently controlled in parallel. Using advanced optical microscope systems (such as those utilized in modern industrial-scale lithography) with high numerical apertures, wide fields of view exceeding several millimeters, and appropriately scaled trapping laser powers, direct capture and manipulation of more than a million qubits is possible. Further scaling is possible by creating tens to hundreds of such processing units, each under its own microscope objective, and connecting these units together using photonic links and / or optical lattice transport. This allows for sufficient space, resolution, and power density to implement high-fidelity control of 10M qubits or more.

[0056] QPU 400 is segmented into several important zones: storage zone 411, entanglement zone 412, readout zone 413, atom loading zone 404, and remote entanglement zone 405. Storage zone 411, entanglement zone 412, and readout zone 413 form processor core 401, which in some embodiments, fits into a footprint of 0.5-5 mm. 4 ~10 6 Fresh atoms are continuously reloaded from a remote atom loading zone 404, and a remote entanglement zone 405 (using optical interconnect and / or lattice transport) delivers the remote Bell pair entanglement resource.

[0057] In the storage zone 411, idle logical qubits are stored for long periods of time, utilizing long qubit coherence times and high-fidelity one-qubit gates; therefore, error correction cycles are only required before logical two-qubit gates. Assuming one-tenth of threshold performance for coherence times of 10-100 seconds, roughly 1% one-qubit dephasing errors can be tolerated before d rounds of error correction. This corresponds to a tolerable storage time of about 0.1-1 seconds before correction is required. Due to the all-to-all connectivity provided by the currently described architecture, idle logical qubits can simply be archived in the storage zone, safe from additional errors. Logical qubits are thus stored in dense blocks, shuttled out when needed by the algorithm, and error-corrected only before two-qubit gates, significantly reducing error correction overhead. In various exemplary devices, atoms are stored in the dense storage zone at approximately 1 / (2 μm) 2 and is stored in the active zone at a density of approximately 1 / (10 μm) 2 are stored at a density of

[0058] The active logical qubit is operated in the active zone 412. By utilizing qubit transport, all combinations of two-qubit gates can be performed within a defined region of space. This greatly reduces the complexity of classical control. For example, all two-qubit gates can be performed using a single global light beam, which is dramatically simpler than calibrating each individual qubit. This extraordinary degree of parallelism of logical qubit control is a major advantage of this architecture over alternative approaches, such as those involving individual control of atomic qubits.

[0059] The readout zone 413 allows selective readout of a subset of qubits mid-circuit without perturbing the other qubits. This readout occurs in parallel with a global beam and camera, again requiring only one set of classical controls.

[0060] Outside the core processor 401, atoms are constantly reloaded from a load zone 404 and transported to the core processor to execute circuits of arbitrary length. Remote bell pairs with other processing units are created using optical links and / or optical lattice transport 405 and shuttled to the core processor 401 to generate remote entanglement. This allows for the interconnection of 10-100 single processing units into a single, error-corrected, practical-scale quantum computer.

[0061] The architecture given above allows for mid-circuit readout. In particular, this architecture may be paired with high-speed imaging and classical control loops in the readout zone. Furthermore, various methods may be used to suppress crosstalk errors and detect / correct losses. By continuously reloading atoms and further suppressing crosstalk, circuit depths of arbitrary length may be achieved.

[0062] To connect multiple units, many high-fidelity, long-distance Bell pairs may be generated in parallel using lattice transport and / or photonic links.

[0063] It will be appreciated that this architecture is suitable for storing logic states through repeated in-circuit measurement and correction. Furthermore, surface-encoded logic qubits may be implemented, for example, by moving an auxiliary from a storage zone reservoir, entangled with a data qubit for syndrome extraction, and moving it to the readout zone. This allows for fast in-circuit readout and feedback while preserving the coherence of the data qubit. In various embodiments, the data qubit is protected by spacing the imaging zones about 50 microns apart, thereby suppressing crosstalk from the readout beam and light scattered by the auxiliary atoms.

[0064] In various embodiments, a fast classical control loop uses auxiliary measurements to determine errors in data qubits and to detect and correct loss of qubits. The missing qubits may then be replaced by reservoir atoms. To achieve surface code distances several times larger than the largest codes created in alternative systems, local detuning patterns may be utilized for space-efficient use of the entanglement zone.

[0065] The currently described architecture can be used to implement algorithms using logical qubits. The zoned approach combined with efficient parallel optical control of many logical qubits enables the construction of large-scale processors. In an exemplary use case, approximately 10 logical qubits are encoded in the active zone and transferred to the storage zone. After encoding all logical qubits, the algorithm is executed with the appropriate logical 1-qubit and logical 2-qubit gates. The flexible, localized 1-qubit control required for the logical 1-qubit gates is implemented with Raman light from a 2D AOD illuminating a grid of single code blocks. The logical 2-qubit gates are realized transversally in the entanglement zone. In-circuit readout is used for the non-Clifford gate teleportation sequence followed by fast feedback for the logical 1-qubit rotation.

[0066] While specific operating parameters are given below as examples, it will be understood that improved fidelity in two-qubit gate errors may be achieved through various further optimizations. For example, increasing the power and detuning of the Rydberg laser reduces laser scattering errors and suppresses other errors due to increased gate speed. Cooling atoms to the motional ground state (thereby suppressing Doppler dephasing errors) and utilizing 10 times higher laser power theoretically results in gate fidelity exceeding 99.8%. Further improvements can be made by continuing to increase the laser power, but alternative means, such as single-photon excitation to the Rydberg P state or alkaline-earth-based systems, are also available. Processor speed can be increased to a logical qubit cycle time of approximately 10 microseconds by increasing collection efficiency, utilizing cavity-based or ensemble-based readout schemes, or by increasing the transfer speed with deeper optical tweezers.

[0067] To reach circuits of any depth, the atoms may be continuously reloaded. Thus, some embodiments use loading into a remote magneto-optical trap (MOT) and transporting the atoms in an optical lattice conveyor belt.

[0068] In various embodiments, crosstalk during readout is suppressed by keeping the ancillary atoms away from the data qubits.

[0069] Further scaling of quantum processors can be achieved by connecting two or more microscope objectives either by atomic transport or optical communication links. In various embodiments, the first approach utilizes the novel capabilities of atomic rearrangements combined with the use of optical lattice conveyor belts to coherently transport qubits and distribute entanglement between multiple active optical control regions. In various embodiments, the second approach utilizes the novel capabilities of atomic rearrangements combined with the use of optical lattice conveyor belts to coherently transport qubits and distribute entanglement between multiple active optical control regions. 4 It utilizes photon-mediated entanglement between different atomic array nodes with more than qubits. High entanglement rates can be achieved with parallel nanophotonic or bulk optical cavities, and the large size of the atomic array can provide further parallelism. This approach also enables modular construction of quantum processor units that can be flexibly rewired and linked together.

[0070] Referring to FIG. 5, a schematic diagram of a logical qubit is provided illustrating efficient control of a single logical qubit through parallelized optical control of the physical qubit blocks that make up the logical qubit. Logical qubits 501, 502, 503, and 504 each consist of 13 atomic qubits (shown as circles). It will be understood that the number and arrangement of qubits is purely exemplary. Various qubit blocks are known in the art and suitable for use as described herein. For example, 2D surface codes and 2D color codes are particularly suitable due to their high threshold and extremely simple 2D structure. However, various other codes, including transversal CNOTs, such as 3D color codes and 3D torus codes, are available. In various embodiments, a single laser beam is configured to illuminate a given logical qubit when positioned in an active zone (e.g., active zone 412) of a processor. This is illustrated by beam 511 illuminating logical qubit 501. In some embodiments, a single laser beam is configured to illuminate multiple logical qubits when positioned in an active zone of a processor (e.g., active zone 412), as illustrated by beam 513 illuminating logical qubits 503, 504, and 505.

[0071] This structure allows the application of transversal logic gates: to apply a transversal logic gate to one logical qubit, a corresponding physical qubit gate is performed for each physical qubit in the block that makes up the logical qubit.

[0072] For example, to perform a transversal one-qubit gate, the same one-qubit rotation is applied to each physical qubit in the block by illuminating the entire spatial block (e.g., 501) with one beam (e.g., 511) that covers all physical qubits. In various embodiments, this is achieved by creating a grid of Raman beams using a crossed AOD device to illuminate a grid of one surface code. An example of this one-qubit is shown by beams 511 and 512, where surface code blocks 501 and 502 (connected grids of 13 atoms) are illuminated with a beam exiting a microscope objective and entering the atomic plane. In this example, two logic blocks 511 and 512 are illuminated in parallel. While this can be advantageous in various use cases, one code block at a time may also be illuminated.

[0073] This structure enables the application of transversal logical multi-qubit gates between two or more logical qubits. A transversal logical multi-qubit gate is a logical gate in which each atomic (i.e., physical) qubit of one logical qubit is coupled to only one atomic qubit of another logical qubit, and therefore errors do not propagate to other atomic qubits by propagation. Referring to FIG. 6, a schematic diagram of a logical qubit illustrating the application of a transversal controlled-NOT (CNOT) gate is provided. Similar to FIG. 5, each of the plurality of logical qubits consists of 13 atomic qubits (shown as circles). To perform a transversal CNOT for two logical qubits (e.g., 601, 602), a physical-qubit CNOT is performed for each pair of two logical blocks. The architecture provided herein enables moving groups of atoms in parallel to efficiently perform a transversal logical CNOT between any two logical qubit blocks.

[0074] In particular, a logical qubit block is picked up with a cross AOD and moved to interlace with another logical qubit in the same 2D plane, which is stored in a different set of optical tweezers (e.g., a backbone SLM grid). When interlaced, each atomic qubit of one logical qubit is within the blockade radius of exactly one corresponding atomic qubit of the other logical qubit. A single pulse of a global Rydberg laser is applied (e.g., beam 611). This realizes transversal CNOT between the two logical qubits in one parallel step. Transversal CNOT may be performed simultaneously for multiple logical qubits in parallel, as shown in FIG. 6.

[0075] A transversal CNOT is a permissible fault-tolerant operation between any two Calderbank-Shor-Steane (CSS) codes, a broad class of codes that encompasses surface codes, color codes (e.g., Steane codes), hypergraph product low-density parity-check (LDPC) codes, etc. The key intuition is that a CNOT propagates X for a first qubit to X for a second qubit, and Z for a first qubit to Z for a second qubit. For CSS codes, the logical qubit operator is the product of X and Z, and a logical CNOT is formed by the product of physical qubit CNOTs. Thus, a logical X for a first logical qubit propagates to a logical X for a second logical qubit, and a logical Z for a first logical qubit propagates to a logical Z for a second logical qubit. This follows the rules for CNOT at the logical qubit level. Therefore, it implements a transversal logical CNOT.

[0076] More specifically, a controlled NOT can be performed bitwise on any CSS code.

number

number

number

number

number

number

number

number

number

number

number

number

[0077] In this way, the bitwise CNOT generates the encoded CNOT for every encoded qubit in the block.

[0078] Without transversal CNOTs, entanglement operations between logical qubits often must be performed by braiding or lattice surgery, which are significantly less efficient than transversal CNOTs. For example, while both braiding and lattice surgery require d rounds of stabilizer measurements to be truly fault-tolerant, no rounds of stabilizer measurements are required to make a transversal CNOT fault-tolerant—the fault-tolerance of a transversal CNOT is already guaranteed by the fact that it is transversal. As mentioned above, transversal gates are inherently fault-tolerant because errors cannot propagate from one qubit in a block to another qubit in the same block. Therefore, braiding and lattice surgery are much more resource-intensive in that they require multiple rounds of stabilizer measurements, are slower, and have lower thresholds.

[0079] The threshold of the 2Q gate for performing transversal CNOT is given by the threshold for perfect syndrome extraction, which should be approximately 10% (surface code), while the threshold of the 2Q gate for iterative syndrome extraction is approximately 1% (surface code). The ability to perform this transversal CNOT efficiently by using only a few classical controls in a code-size independent manner is important for simplifying the classical controls required for building large-scale quantum computers.

[0080] Atomic movement and parallel optical control of logical qubit blocks greatly simplifies the control required to implement logical quantum computation. In various embodiments, logical qubits are multiplexed into a grid, with each logical qubit block behaving as if it were one large atom. To perform a one-qubit rotation on a logical qubit block, the logical qubit block is illuminated with a single beam. To perform CNOT between two logical qubit blocks, the logical qubit blocks are moved together and pulsed with a single Rydberg beam. With this highly efficient parallelized control, logical qubit algorithms can be performed on the logical qubits.

[0081] In an exemplary embodiment, a two-qubit CZ gate is implemented by two global Rydberg pulses, each with a detuning Δ and length τ, with a phase jump ξ between the two pulses. The pulse parameters are chosen so that a pair of qubits that are close together and under the Rydberg blockade constraint will return from the Rydberg state to a hyperfine qubit manifold with a phase that depends on the state of the other qubit.

[0082] 7, a schematic diagram of a portion of a processor core, such as processor core 401, is provided with exemplary measurements and an arrangement of qubits. In this example, storage zone portion 701 measures 145×40 μm, active zone portion 702 measures 145×40 μm, and readout zone portion 703 measures 145×20 μm. Storage zone portion 701 is separated from active zone portion 702 by a 20 μm buffer. Active zone portion 702 is separated from readout zone portion 703 by a 20 μm buffer.

[0083] In this example, the active zone portion 702 has 50 positions spaced 16 μm apart in one dimension and 10 μm apart in the other. Each dot represents an atomic qubit, and thus, in this example, when interlaced to perform bitwise operations, the qubits are placed close to each other. The storage zone portion 701 has 250 positions spaced 5 μm apart in one dimension and 4 μm apart in the other. Thus, the storage zone portion 701 has a higher density than the active zone portion 702.

[0084] Alternative arrangements are provided in Figures 8-9. For example, the arrangement of Figure 8 is suitable for use in implementing a repeat code. In another example, the arrangement of Figure 9 is suitable for use in implementing a surface code.

[0085] Referring to Figure 10, a schematic diagram of a method for active feedforward QEC is provided. In this example, a portion of a quantum processor (as shown in Figure 4) is shown including a reservoir 1001 (such as load zone 404), an active zone 1002 (such as active zone 412), and a readout zone 1003 (such as readout zone 411). Ancillary qubits are continuously replenished (1004) from reservoir 1001 to replace ancillary qubits that are moved (1005) from active zone 1002 to readout zone 1003 to be measured. Using the zoned architecture provided herein allows for a complete QEC round within 1 ms.

[0086] Dynamic reconfiguration in 2D tweezers arrays An exemplary experiment utilizes the apparatus described below: In a vacuum cell: 87Rb atoms are loaded from a magneto-optical trap into a backbone array of programmable optical tweezers generated by a spatial light modulator (SLM). The atoms are realigned in parallel to defect-free target positions within this SLM backbone by additional optical tweezers generated from crossed 2D acousto-optic deflectors (AODs). After the realignment procedure, selected atoms are transferred from the static SLM trap back to the movable AOD trap, and then these movable atoms are moved to their starting positions within the quantum circuit. During this entire process, the atoms are cooled with polarization gradient cooling. Before running the quantum circuit, camera images of the atoms are taken in their initial starting positions. After the circuit, final camera images are taken to detect the qubit states |0> (presence of the atom) and |1> (loss of the atom after resonant pushout). All data is postselected by finding the complete rearrangement of the AOD and SLM atoms before running the circuit. In some embodiments, each atom remains in a single static or single movable trap throughout the duration of the quantum circuit.

[0087] The crossed AOD system consists of two independently controlled AODs (AA Opto Electronic DTSX-400) for x- and y-control of the beam position. Both AODs are driven by independent arbitrary waveforms generated by a dual-channel arbitrary waveform generator (AWG) (Spectrum Instrumentation M4i.6631-x8) and then amplified by independent MW amplifiers (Minicircuits ZHL-5W-1). The time-domain arbitrary waveforms consist of multiple frequency tones corresponding to the x- and y-positions of columns and rows, which are independently varied as a function of time to dynamically steer the atoms trapped in the AODs. The complete x- and y-waveforms are calculated by summing the time-domain profiles of all frequency components, with a given amplitude and phase for each component. To execute the quantum circuit, the positions of the AOD atoms at each gate location are programmed, and the AOD frequency is smoothly interpolated (with a cubic profile) as a function of time between gate locations. The cubic profile imposes a constant jerk on the atoms, which allows them to move approximately 5-10 times faster (without heating and loss) than if they were moving at a constant velocity (a first-order profile). The movement protocol applies extension, compression, and translation of the AOD trap array, i.e., the rows and columns of the AOD never cross each other to avoid the atomic loss and heating associated with two frequency components crossing each other.

[0088] The AOD tweezer strength is homogenized throughout the atom's trajectory to minimize dephasing induced by the time-varying magnitude of the differential optical shift. To this end, a reference camera is used in the image plane to measure and homogenize the strength of each AOD tweezer at each gate location by varying the amplitude of each frequency component, and the amplitude of each individual frequency component is interpolated during movement between two locations.

[0089] The SLM tweezer beam (830 nm) and the AOD tweezer beam (828 nm) are generated by two separate free-running titanium sapphire lasers (M Squared, 18-W pump). When projected through a 0.5 NA objective, the SLM tweezers have an approximate waist of approximately 900 nm (1000 nm for the AOD). When loading atoms, the trap depth is approximately 2π × 16 MHz, and the radial trap frequency is approximately 2π × 80 kHz. When running quantum circuits, the trap depth is approximately 2π × 4 MHz, and the radial trap frequency is approximately 2π × 40 kHz.

[0090] Raman laser system Fast, high-fidelity single-qubit operations are a crucial element of the quantum circuits demonstrated in this work. To this end, a high-power 795 nm Raman laser system is used to F This Raman laser system is based on dispersive optics. 795 nm light (Toptica TA pro, 1.8 W) is phase-modulated by an electro-optic modulator (Qubig) driven by a 3.4 GHz microwave (Stanford Research Systems SRS SG384) that is doubled and amplified to 6.8 GHz. The laser's phase modulation is converted to amplitude modulation using a chirped Bragg grating (Optigrate) to drive the Raman transition. The IQ control of the SG384 is used to control the frequency and phase of the microwave, which is imprinted onto the laser's amplitude modulation and thus allows for direct frequency and phase control of the hyperfine qubit drive.

[0091] The Raman laser transversely illuminates the atomic plane with a circularly polarized elliptical beam with waists of 40 μm and 560 μm on the thin and tall axes, respectively, with a total average optical power of 150 mW at the atom. The large vertical spread ensures less than 1% inhomogeneity across the atom, and shot-to-shot fluctuations in laser intensity are also less than 1%. The Raman laser is operated with a blue-detuned intermediate state detuning of 180 GHz, with a two-photon Rabi frequency of 1 MHz and a 7×10 -5 This results in an estimated scattering error per π pulse of 1000 π (i.e., 1 scattering event per 15000 π pulses).

[0092] Qubit coherence and dynamical decoupling In an 830 nm trap, the coherence of a hyperfine qubit is T2 * The experimental setup is characterized by T = 4 ms (not plotted here), T = 1.5 s (XY16 with 128 total π pulses), and T = 4 s (including atom loss). The experiments described herein are performed in a DC magnetic field of 8.5 Gauss. Coherence is further enhanced with detuned optical tweezers (with the trap depth held constant, the differential spectroscopic shift in the tweezers decreases as 1 / Δ, where 1 / T decreases as 1 / Δ). 3 This can be further improved by using a finite element (decreasing as ) and shielding against magnetic field fluctuations. For practical QEC operation, the loss of atoms can be detected in a hardware-efficient manner, and atoms can then be replaced from a reservoir, which could in principle be continuously reloaded by an MOT to reach circuits of arbitrary depth.

[0093] The transport sequence is accompanied by a dynamic decoupling sequence. The number of pulses used is a trade-off between preserving qubit coherence and minimizing pulse errors. In various embodiments, two types of dynamic decoupling sequences are alternated: an XY8 / XY16 sequence composed of individual π pulses with alternating phases that self-correct for amplitude and detuning errors, and a CPMG-type dynamic decoupling sequence composed of robust BB1 pulses. The CPMG-BB1 sequence is more robust to amplitude errors but incurs additional scattering errors. By selecting from these different sequences and a variable number of decoupling π pulses, the sequence can be empirically optimized for any given experiment, optimizing either single-qubit coherence (including transfer) or the final signal. Typically, the decoupling sequence consists of a total of 12 to 18 π pulses.

[0094] The effect of migration on atomic heating and loss In the following we consider the effect of movement on the loss and heating of atoms in the harmonic oscillator potential imposed by a tweezer trap. The motion of the trapping potential is equivalent to a non-inertial reference frame in which the harmonic oscillator potential is at rest but the atoms are subjected to an apparent 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 given by

number

number

number

number

number

[0095] Several important insights can be gained from this equation. First, it describes the ability to travel long distances D in relatively small increments of time T. Furthermore, to maintain a constant ΔN, the travel time T ∝ ω0 -3 / 4 Also, to perform multiple moves k for deep circuits, ΔN ∝ k / T 4 can be estimated, suggesting that the number of transfers could be increased from 5 to 80, for example, by slowing each transfer from 200 μs to 400 μs. The transfer speed could be further increased by different a(t) profiles, but quantum speed limits, necessitated by finite resources such as trap depth, eventually prevent indefinitely fast movement of qubits in the array.

[0096] Equation 3 is then compared with experimental observations. Atom loss is observed with a movement of 55 μm in 200 μs under constant negative jerk. This velocity limit is consistent with the estimates above, ω = 2π × 40 kHz and x zpf Using = 38 nm, we predict ΔN ≈ 6 for this migration, corresponding to the onset of appreciable heating at this migration rate. More quantitatively, a Poisson distribution with mean N and variance N is assumed, and some critical N is reached before atoms leave the trap. max From this analysis, the retention of atoms is

number

[0097] Additional heating and loss in the circuit can also be caused by repeated short drops to implement the two-qubit gate, where the tweezers are briefly turned off to avoid anti-trapping of the Rydberg state and photo-shifting of the ground-Rydberg transition. However, drop-recapture measurements suggest that the 500 ns drops used experimentally have a negligible effect up to several hundred drops per atom (corresponding to several hundred CZ gates). The loss and heating of atoms as a function of the number of drops is well described by a diffusion model, which reduces the temperature of the atoms by a factor of two (the thermal velocity

number

[0098] Implementation of a two-qubit CZ gate Two-qubit gates and calibrations may be implemented using the techniques provided herein. In particular, a two-qubit CZ gate is implemented by two global Rydberg pulses, each with a detuning Δ and length τ, with a phase jump ξ between the two pulses. The pulse parameters are selected so that a pair of qubits that are close together and under the Rydberg blockade constraint will return from the Rydberg state to a hyperfine qubit manifold with a phase that depends on the state of the other qubit. The numerical values ​​of these pulse parameters are as follows: Δ=-0.377371Ω ξ=-0.621089×(2π) τ=0.683201 / [Ω / (2π)]

[0099] An exemplary experiment is performed at a two-photon Rydberg-Rabi frequency of Ω / 2π = 3.6 MHz, giving a theoretical τ = 190 ns and a theoretical Δ / (2π) = -1.36 MHz. The negative detuning sign corresponds to a m detuned by approximately 24 MHz under a field of 8.5 G. j is chosen to help minimize excitation to the Rydberg state at m = +1 / 2 (and the desired m due to the reduced Clebsch-Gordan coefficient). j (This results in three times less coupling to the Rydberg laser than the =-1 / 2 state.) In this work, the Rydberg-Rydberg interaction V0 / 2π is in the range of 200 MHz to 1 GHz, providing strong blockade between closely spaced qubits.

[0100] Managing spurious phase in CZ gates. The two-qubit gate induces both an intrinsic one-qubit phase and a spurious phase induced primarily by the differential optical shift from the 420 nm laser. Under certain configurations, the 420 nm-induced differential optical shift of the hyperfine qubit can be extremely large (greater than 8 MHz), resulting in a phase accumulation on the hyperfine qubit of approximately 6π. Thus, even small percentage-level fluctuations in the 420 nm intensity can lead to significant qubit dephasing.

[0101] This 420-induced phase problem can be addressed by performing an echo sequence: after the CZ gate, the 1013 nm Rydberg laser is turned off, a Raman π pulse is applied, and then the 420 nm laser is pulsed again to cancel the 420-induced phase during the CZ gate. This method echoes out the 420-induced phase, but at the cost of a two-fold increase in 420-induced scattering error, which is the main source of error in two-qubit CZ gates.

[0102] Echoes Between CZ Gates. To address these various issues, a Raman π pulse is implemented between each CZ gate to echo cancel the spurious phase induced by the hyperfine qubit gate. This approach has several advantages. The 420-nm induced phase is now canceled by the pair of CZ gates without explicitly applying an additional 420 nm pulse to echo each individual CZ gate, thereby reducing the scattering error of our CZ gate by approximately a factor of two. This echo technique, which reduces the scattering error introduced during each gate, nearly compensates for the increased scattering rate introduced by spreading the optical power over a wider 2D space, thereby providing gate fidelity comparable to the two-qubit CZ gate fidelity of over 97.4(2)%. Furthermore, the echoes between CZ gates also cancel the intrinsic one-qubit phase of the CZ gates, eliminating errors in the calibration of this parameter and canceling spurious one-qubit phases induced by all other gates, such as the approximately 0.01 rad phase induced by pulsing off the trap for 500 ns for a two-qubit gate. If the number of CZ gates is odd, an echo for the last CZ gate is performed.

[0103] To further suppress the spurious 420-induced phase effects, the 420 nm laser is 3 / 2 The laser is operated red-detuned (2 GHz) from the transition. For red-detuning, the optical shifts in the |0> and |1> states are of the same sign, minimizing the differential optical shift, whereas for blue-detuning below 6.8 GHz, the optical shifts in the |0> and |1> states have opposite signs, amplifying the differential optical shift.

[0104] Sensitivity to axial trapped vibrations On the time scale of typical Rydberg excitation with optical tweezers, axial trap oscillation frequencies of a few kHz are insignificant. However, for circuits run for as long as 1.2 ms with Rydberg pulses throughout, axial trap oscillations can have significant effects. In particular, axial oscillations cause atoms to oscillate in and out of the Rydberg beam, resulting in an axial spread of 1.2 ms at an estimated axial temperature of about 25 μK and an axial frequency of 6 kHz.

number

[0105] Rydberg beam shaping and uniformity The Rydberg beam is shaped into a top hat of variable size by wavefront control using the phase profile of a spatial light modulator (SLM). This capability allows the height of the beam profile to be tailored to the size of the experimental zone for any given experiment, thereby maximizing the intensity and CZ gate fidelity of the 1013 nm light. The uniformity of the Rydberg beam is optimized until the peak-to-peak inhomogenity is less than 1%. To this end, all aberrations are corrected down to the vacuum chamber window, resulting in a few percent of atomic inhomogenity attributable to defects in the final window. To further optimize uniformity, aberration corrections are adjusted in the top hat by Zernike polynomial corrections to the phase profile in the SLM plane (Fourier plane). This procedure reduces the peak-to-peak inhomogenity to less than 1% over a 40-50 μm range in the atomic plane.

[0106] Coherent Mapping Protocol A coherent mapping protocol is presented to transfer general many-body states in the {|1>, |r>} basis to the long-lived, non-interacting {|0>, |1>} basis. To achieve this mapping, a Raman π pulse is applied immediately after the Rydberg dynamics to map |1> → |0>, and then a subsequent Rydberg π pulse is applied to map |r> → |1>.

[0107] Even for perfect Raman and Rydberg π-pulses (for isolated atoms), there are three important sources of infidelity associated with this mapping process. (1) All populations in the blockade-breaking state (i.e., two adjacent atoms both at |r>) are strongly shifted off-resonance for the final Rydberg π-pulse, and therefore this atomic population is lost, remaining in the Rydberg state. (2) For example, long-range interactions from next-nearest neighbors detune the final Rydberg π pulse from resonance, thus reducing the fidelity of the pulse. Because the long-range interactions are not the same for all many-body microscopic states, this effect cannot be mitigated by a simple shift in detuning. (3) Dephasing of the states arises primarily from Doppler shifts between the ground states |0>, |1> and the Rydberg state |r> over the duration of the Raman π pulse. Although these random on-site detunings also exist during many-body dynamics, turning off the Rydberg drive Ω allows the system to freely accumulate phase, making us particularly sensitive to dephasing errors.

[0108] The above error mechanism is mitigated as follows: To minimize the error from (1), the multibody dynamics is

number

number

[0109] A global Raman beam induces a phase shift during a Raman π pulse, induced by an optical shift of approximately π for |0> and |1> relative to |r>. Similarly, a global 420 nm laser induces a phase shift during a Rydberg π pulse, induced by an optical shift of approximately π between |0> and |1>. Although the measurements performed here are interferometric (i.e., the singlet state being measured is invariant under global rotation) and therefore unaffected by these global phase shifts, these phase shifts can be measured and accounted for accordingly.

[0110] Formation of particle arrays using optical tweezers Optical trapping of neutral atoms is a powerful technique for isolating atoms in a vacuum. Atoms are polarizable, and the oscillating electric field of a light beam induces an oscillating electric dipole moment in the atoms. The associated energy shift in the atom from the induced dipole, averaged over the oscillation period of the light, is called the AC Stark shift. Based on the AC Stark shift induced by light detuned (i.e., wavelength offset) from the atom's resonance transition, atoms are attracted to light below the resonance frequency and thus trapped in a local intensity maximum (for red-detuned, i.e., longer wavelength trapping light). The AC Stark shift is proportional to the light intensity. Therefore, the shape of the intensity field is the shape of the associated atom trap. Optical tweezers exploit this principle by focusing a laser beam to a micron-scale waist, where individual atoms are trapped at the focus. A two-dimensional (2D) array of optical tweezers is generated, for example, by illuminating a spatial light modulator (SLM) that imprints a computer-generated hologram onto the wavefront of a laser field. The 2D array of optical tweezers is then superimposed on a cloud of laser-cooled atoms in a magneto-optical trap (MOT). The tightly focused optical tweezers operate in a "collision blockade" regime, where single atoms are loaded from the MOT while pairs of atoms are released by light-assisted collisions, ensuring that the tweezers are at best loaded with a single atom; however, loading is probabilistic, such that the trap is loaded with a single atom approximately 50–60% of the time.

[0111] To prepare deterministic atomic arrays, a real-time feedback procedure identifies randomly loaded atoms and rearranges them into preprogrammed shapes. Rearranging the atoms requires, for example, using an acousto-optic deflector (AOD) to move the atoms within tweezers that can be smoothly steered to minimize heating by deflecting a laser beam by an adjustable angle controlled by the frequency of an acoustic waveform applied to the AOD crystal. Dynamic adjustment of the acoustic frequency translates into smooth movement of the optical tweezers. Multi-frequency acoustic waves create an array of laser deflections that, after focusing through a microscope objective, form an array of optical tweezers with adjustable position and amplitude, both controlled by the acoustic waveform. The atoms are rearranged by using an additional set of dynamically moving tweezers superimposed on top of the SLM tweezers array.

[0112] Exemplary Hardware Optical tweezers arrays provide a powerful and flexible way to build large-scale systems composed of individual particles. Each optical tweezers captures a single particle, including, but not limited to, individual neutral atoms and molecules, for applications in quantum technology. Loading such tweezers arrays with individual particles is a stochastic process, with each tweezers in the system filling with a single particle with a finite probability p < 1, e.g., p ≈ 0.5 in the case of many neutral atom tweezers implementations. To compensate for this random loading, real-time feedback can be obtained by measuring which tweezers are loaded and then sorting the loaded particles into a programmable shape. This can be done by moving one particle at a time or in parallel.

[0113] Parallel sorting can be achieved by using two acousto-optic deflectors (AODs) to create multiple tweezers that can pick up particles from an existing particle trapping structure, simultaneously move them, and release them at another location. This can involve moving particles within a single trapping structure (e.g., a tweezer array) or transporting and sorting particles from one trapping system to another (e.g., between one tweezer array and another type of optical / magnetic trap). This sorting is flexible, allowing for programmed positioning 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) driving field for the AOD. Because the RF driving of the AOD can be controlled in real time and can include any combination of frequency components, it is possible to create an arbitrary grid of traps (such as a line of arbitrarily positioned traps), move rows or columns of the grid, and add or remove rows and columns of the grid by varying the number, magnitude, and distribution of the frequency components of the AOD's RF driving field.

[0114] In an exemplary embodiment, an optical tweezers array is fabricated using a liquid crystal-on-silicon spatial light modulator (SLM), which can programmatically create flexible arrangements of tweezers. These tweezers are fixed in space for a given experimental sequence and stochastically loaded with individual atoms, such that each tweezers is loaded with probability p≈0.5. Fluorescence images of the loaded atoms are taken to identify in real time which tweezers are loaded and which are empty.

[0115] After detecting which tweezers are loaded, a movable tweezers overlapping the optical tweezers array can dynamically reposition the atoms from their starting positions to fill the target array of traps with near-unity filling. The movable tweezers are made of a pair of crossed AODs. These AODs can be used to create a single movable trap that moves one atom at a time to fill the target array, or to move many atoms in parallel.

[0116] Referring to FIG. 11 , a schematic diagram of an apparatus 1100 for quantum computing according to an embodiment of the present disclosure is provided. As shown in FIG. 11 , using a beam generated by a light source 1102 (e.g., a coherent light source—in some exemplary embodiments, a monochromatic light source), an SLM 1104 forms an array of trapping beams (i.e., a tweezer array) that is imaged onto a trapping plane 1108 within a vacuum chamber 1110 by an optical train that, in the exemplary embodiment shown in FIG. 11 , includes elements 1106 a, 1106 c, 1106 d and a high numerical aperture (NA) objective lens 1106 e. As will be readily recognized by those skilled in the art, other suitable optical trains may be used. Using a beam generated by a light source 1112 (e.g., a coherent light source—in some exemplary embodiments, a monochromatic light source), a pair of AODs 1114 and 1116 having non-parallel (e.g., orthogonal) directions of acoustic wave propagation generate dynamically movable sorting beams. The sorting beam is superimposed with the trapping beam by using an optical train such as the one shown in Figure 11 (elements 1117, 1106b, 1106c, 1106d, and 1106e). Of course, other optical trains can be used to achieve the same result. For example, light sources 1102 and 1112 can be a single light source, and the trapping and sorting beams are generated by beam splitters.

[0117] Dynamic movement of the steering beam is accomplished by using two non-parallel AODs 1114, 1116 arranged in series. In the exemplary embodiment shown in FIG. 11 , one AOD defines the “row” direction (“horizontal”—the “X” AOD) and the other defines the “column” direction (“vertical”—the “Y” AOD). Each AOD is driven with an arbitrary RF waveform from an arbitrary waveform generator 1120, which is generated in real time by a computer 1122 that processes a feedback routine after analyzing an image of where the atoms are being loaded. When each AOD is driven with a single frequency component, a single steering beam (“AOD trap”) is generated in the same plane 1108 as the SLM trap array. The frequency of the X AOD drive determines the horizontal position of the AOD trap, and the frequency of the Y AOD drive determines the vertical position; in this way, the single AOD trap can be steered to overlap any SLM trap.

[0118] In Figure 11, laser 1102 projects a beam of light onto SLM 1104. SLM 1104 can be controlled by computer 1122 to generate a beam pattern (the "trapping beam" or "tweezer array"). The beam pattern is collected by lens 1106a, passes through mirror 1106b, and is collimated by lens 1106c onto mirror 1106d. The reflected light passes through objective lens 1106e to focus the optical tweezers array onto trapping plane 1108 within vacuum chamber 1110. The optical tweezers array laser light continues through objective lens 1124a, passes through dichroic mirror 1124b, and is detected by charge-coupled device (CCD) camera 1124c.

[0119] The vacuum chamber 1110 may be illuminated by an additional light source (not shown). Fluorescence from atoms trapped at the trapping plane also passes through the objective lens 1124a but is reflected by the dichroic mirror 1124b and onto an electron-multiplying CCD (EMCCD) camera 1124d. In this example, a laser 1112 directs a beam of light to the AODs 1114, 1116. The AODs 1114, 1116 are driven by an arbitrary waveform generator (AWG) 1120, which is further controlled by a computer 1122. The crossed AODs 1114, 1116 emit one or more beams as described above, which are directed to a focusing lens 1117. The beams then enter the same optical train 1106b...1106e as described above for the optical tweezers array and are focused onto the trapping plane 1108.

[0120] It will be appreciated that alternative optical trains may be used to generate optical tweezers arrays suitable for use as described herein.

[0121] Exemplary Logic Processor Based on Atomic Arrays Figure 12 shows a programmable logic processor based on a reconfigurable atomic array. Figure 12A is a schematic diagram of the logic processor segmented into three zones: storage, entanglement, and readout (see Figure 18 for a detailed layout). Logical one-qubit and two-qubit operations are realized transversally by efficient parallel operations. Transversal CNOT is realized by interlacing two logical qubit grids and executing a single global entanglement pulse that excites the atoms into a Rydberg state. The physical qubits are trapped in optical tweezers. 87The Rb atom's hyperfine ground state is encoded in the Rb atom's hyperfine ground state. Figure 12B shows a fully programmable one-qubit rotation implemented using Raman excitation through a 2D AOD. Parallel grid illumination delivers the same instruction to multiple atomic qubits. Figure 12C shows the readout and feedforward in-circuit. The imaging histogram shows high-fidelity state discrimination (imaging time 500 μs, readout fidelity ≈ 99.8%), and the Ramsey fringe indicates that the qubit's coherence is not affected by measurements of other qubits in the readout zone (error probability p ≈ 10 -3 ). The FPGA performs real-time image processing, state decoding, and feedforward (as discussed further with respect to FIG. 15).

[0122] The logic processor architecture shown in FIG. 12A is segmented into three zones 1201, 1202, and 1203 (as further shown in FIG. 4). Storage zone 1201 is used for high-density qubit storage, characterized by long coherence times and no entanglement gate errors. Entanglement zone 1202 is used for parallel logic qubit encoding, stabilizer measurements, and logic gate operations. Finally, readout zone 1203 allows for in-circuit readout of desired logic or physical qubits without disturbing the coherence of computational qubits still in operation. This architecture allows for individual qubits trapped in optical tweezers to be dynamically reconfigured mid-computation while preserving qubit coherence. 87 It is implemented using an array of Rb atoms.

[0123] Physical qubits are encoded into clock states (T2 > 1 s) within a ground-state hyperfine manifold and stored in an optical tweezers array created by a spatial light modulator (SLM). In an exemplary embodiment, a 280-atom qubit system is provided that combines high-fidelity two-qubit gates enabled by fast excitation of interacting atomic Rydberg states via robust Rydberg blockade with arbitrary connectivity enabled by atomic transport through a 2D acousto-optic deflector (AOD). The AOD uses frequency multiplexing to incorporate just two voltage waveforms (one for each axis) to create a large dynamically programmable grid of light. Fully programmable local one-qubit rotation is achieved by qubit-specific parallel Raman excitation via an additional 2D AOD (Figure 12B). In-circuit readout is enabled by translating the selected qubit approximately 100 μm to the readout zone 1203 and illuminating it with a focused imaging beam, resulting in high-fidelity imaging and negligible decoherence of the stored qubit (Figure 12C). In-circuit images are collected with a CMOS camera and sent to the FPGA for real-time decoding and feedforward.

[0124] The logic processor allows control of individual logical qubits as the basic unit, rather than individual physical qubits. During most error-corrected operations, the physical qubits of a logic block are supposed to perform the same operation, and this instruction can be distributed in parallel over only a few control lines. This approach naturally multiplexes with optical techniques. For example, to realize a logical 1-qubit gate, a Raman 2D AOD (Figure 12B) can be used to create a grid of light beams that simultaneously illuminate the physical qubits of a logic block with the same instruction. Such gates are transversal, meaning that operations act independently on the physical qubits of a code block. Furthermore, this transversal property implies that the gate is inherently fault-tolerant, meaning that errors cannot propagate within the code block, thereby preventing physical errors from propagating to logical faults. A similar approach can realize logical entanglement gates. In particular, a grid generated by a moving 2D AOD can be used to pick up two logical qubits, interlace them in the entanglement zone, and then pulse a single global Rydberg pump laser to realize a physical entanglement gate for each pair of blocks (Fig. 12A, Fig. 13A). This process realizes a high-fidelity, fault-tolerant, transversal CNOT in a single parallel step.

[0125] Improving entanglement gates with code distance Figure 13 shows a transversal entanglement gate between two surface codes. Figure 13A is a diagram of a transversal CNOT gate between two d = 7 surface codes based on parallel atomic transport. Figure 13B illustrates the concept of correlated decoding. Physical errors propagate between physical qubit pairs during the transversal CNOT gate, creating correlations that can be exploited for improved decoding. These correlations, resulting from the propagation of deterministic errors (as opposed to correlated error events), are accounted for by adding edges and hyperedges connecting the decoding graphs of the two logical qubits. Figure 13C shows the populations of an entangled d = 7 surface code measured in the XX and ZZ bases. Figure 13D shows the Bell pair errors measured as a function of code distance for both conventional decoding (top) and correlated decoding (bottom). The Bell errors are estimated at the population average of ZZ and XX parity. To shorten the signature distance, selected atoms in the grid are removed from the grid to ensure consistent experimental conditions (for d=3, four logical bell pairs are generated in parallel), as shown in Figure 13E. Error bars represent the standard error of the mean. For additional surface signature data, see Figures 21 and 22.

[0126] An important property of QEC codes is that for error rates below a certain threshold, performance should improve with system size, which is related to the so-called code distance. This property can be experimentally verified by reducing the idling error of the code. Neutral atomic qubits can be stored unused for long periods with few errors, and the central challenge is to improve entanglement operations with code distance. To address this need, the present disclosure enables transversal CNOT gates using logic qubits encoded in two surface codes (Figure 13). The surface codes have stabilizers that are used to detect and correct errors without corrupting the logic state. The stabilizers are X-rays running horizontally (vertically) along the lattice. L (Z L ) form a 2D lattice of 4-body plaquettes of X and Z operators that commute with the logical operators (Figure 13E). By measuring the stabilizers, we can detect the presence of errors in the physical qubits, decode them (infer what kind of errors occurred), and decode them simply by software.

number

[0127] To test the performance of these logical entanglement gates, logical qubits are initialized by preparing two blocks 1301, 1302 of physical qubits in the |+> and |0> states, respectively, and performing one round of stabilizer measurements in parallel operation. Although this state preparation is non-fault-tolerant (nFT) above d=3, the error suppression of transversal CNOTs can still be explored. In particular, two logical qubits are initialized by preparing two blocks 1301, 1302 of physical qubits in the |+> and |0> states, respectively, and performing one round of stabilizer measurements in parallel operation. L > and |0 LA parallel move 1303 is performed and a global laser is pulsed (1304) to perform a transversal CNOT. Then, a logic Bell state stabilizer

number

number

[0128] For decoding and correcting logical states, strong correlations are observed between the stabilizers of the two blocks (Fig. 21) due to the physical error propagation between codes in the transversal CNOT (Fig. 13B). These correlations are used to improve performance by joint decoding of the logical qubits, realized by a joint decoding graph containing edges and hyperedges (Fig. 13B) connecting the stabilizers of the two logical qubits. Using this correlation decoding procedure, a population of approximately 0.95 is obtained for X L X L base and Z L Z L basis (Fig. 13C), showing entanglement between d = 7 logical qubits.

[0129] Examining performance as a function of code size (Figure 13D) reveals that logical Bell pairs improve as code distance increases, indicating improved entanglement operations. In contrast, when using conventional decoding, i.e., independent minimum-weight perfect matching within both codes, fidelity decreases with code distance. This is partly due to nFT state preparation, the effect of which is partially mitigated by correlation decoding.

[0130] Fault-tolerant logic algorithms FIG. 14 illustrates a fault-tolerant logic algorithm according to an embodiment of the present disclosure. FIG. 14A illustrates a circuit for preparation of a logical GHZ state. Ten color codes are encoded into a non-fault-tolerant (nFT), and then a parallel transversal CNOT between the computation logic qubit and the auxiliary logic qubit performs FT initialization. The auxiliary logic qubit is moved to storage, and a four-logical qubit GHZ state is created between the computation qubit. A logical Clifford operation is applied before readout to probe the GHZ state. FIG. 14B illustrates SPAM infidelity of a logical qubit without (nFT) and with (FT) transversal CNOT-based flagged preparation compared to physical qubit state preparation and measurement (SPAM). FIG. 14C illustrates the fidelity of logical GHZ without (nFT), with (FT), and with (EDFT) post-selection on flags and a stabilizer on the computation logic qubit, corresponding to error detection. Figure 14D shows the GHZ fidelity as a function of the sliding-scale error detection threshold (converted to the probability of accepted repeats) and the number of successful flags for the circuit. Figure 14E shows the density matrix of the GHZ state for a four-logical qubit (with up to three flag errors) measured by full state tomography including all 256 logical Pauli strings.

[0131] Figure 15 illustrates a zoned logic processor with scaling and mid-circuit feedforward according to an embodiment of the present disclosure. Figure 15A illustrates atoms in the storage zone 1501 and entanglement zone 1502 and the method for creating and entangling 40 color codes using 280 physical qubits. Figures 15B and 15C show the 40 color codes prepared with an nFT circuit. Twenty transversal CNOTs are used to fault-tolerantly prepare 20 of the 40 codes, and their fidelity is plotted. The logical decoherence is smaller than the physical idling decoherence experienced during the encoding step. Figure 15D illustrates mid-circuit measurement and feedforward for logical entanglement teleportation. The middle 1503 of the three logical qubits is measured in the X basis, and by applying mid-circuit conditional locally pulsed logical S rotations 1504 and 1505 to the other two logical qubits 1506 and 1507, the state |0 L 0 L > + |1 L 1 L > is prepared. Figure 15E shows the measured parity of the logical qubit with and without feedforward, showing that feedforward recovers the intended state with 77(2)% Bell fidelity (83(4)% ZZ parity, not plotted). No mid-circuit readout turns off the mid-circuit readout, and the middle logical qubit is in state |+ L >. Postselection (error detection in the last measurement) with a full stabilizer of only two computational logic qubits results in a feedforward Bell fidelity of 92(2)% (not plotted). In Figure 15D, three of the additional blocks are flag qubits, and four others are primed but unused in this circuit.

[0132] The exemplary logic algorithms described herein are constructed from transversal gates that are inherently fault-tolerant. Fault-tolerant state preparation is used below to provide programmable logic algorithms.

[0133] In one example, a two-dimensional d=3 color code is used, which is akin to the surface code but has the complete Clifford group, i.e., Hadamard (H),

number

[0134] State initialization is benchmarked (Fig. 14B). On average, over five computational logic qubits, using fault-tolerant initialization (post-selection with no error detected by the auxiliary logic flag) results in |0 L Initialization fidelity is

number

number

[0135] Because not all nontrivial syndromes are equally likely to cause algorithm failure, partial postselection can be performed in which events of the syndromes most likely to cause algorithm failure, as given by the weights of correlated matching across the algorithm, are discarded. Figure 14D shows the fidelity of GHZ measured as a function of this sliding threshold, which translates into the percentage of accepted experimental repeats, continuously adjusting the trade-off between the algorithm's probability of success and its fidelity; for example, discarding just 50% of the data improves GHZ fidelity to approximately 90%. As discussed below, for certain applications, refining the sample can be advantageous in improving the algorithm's performance.

[0136] To measure all 256 logical Pauli strings in a fault-tolerant manner, a full GHZ state tomography is performed (Fig. 14E).

[0137] The use of a zoned architecture allows the circuit to scale to larger numbers straightforwardly without increasing the number of controls by encoding logical qubits, computing them, moving them to storage, and then accessing the storage appropriately. This process is illustrated in Figures 15A and 15B, where 10 color codes are created, computed in a parallel transversal CNOT, moved to storage, and then additional qubits are accessed from storage. Repeating this process four times creates 40 color codes with 280 physical qubits (Figure 15C), at the expense of a slow idling error of approximately 1% logical decoherence per additional encoding step. These storage idling errors are primarily due to the global Raman π pulse applied for dynamical decoupling of atoms in the entanglement zone, which can be significantly reduced by zone-specific Raman control.

[0138] Since mid-circuit readout is an important component of logic algorithms, this disclosure presents a fault-tolerant entanglement teleportation circuit. From the fault-tolerant prepared color code, we can obtain the GHZ state |0 of a three-logic qubit. L 0 L 0 L > + |1 L 1 L 1 L > (Fig. 15D, Fig. 15E) is created. The X-basis measurement in the middle of the circuit of the central quantum qubit is |+ L > measured as |0 L 0 L > + |1 L 1 L > and |- L > measured as |0 L 0 L > - |1 L 1 L Similar to the magic state teleportation circuit, we create |0 by applying a logical S gate to the first and third logical qubits, which are conditioned in real time on the state of the middle logical qubit.L 0 L > + |1 L 1 L The measurement in Figure 15E shows that without the feedforward step <X L X L and <Y L Y L > does indeed vanish, but we show that applying feedforward corrections recovers 77(2)% Bell-state fidelity, limited by imperfections in the original underlying GHZ state. We compare this experiment to a case where the middle logic qubit does not read out mid-circuit, but instead |+ L By repeating this experiment with post-selection that >, a similar Bell fidelity of 75(2)% was obtained, indicating high-fidelity performance of the readout and feedforward operations.

[0139] Complex logic circuits using 3D codes Figure 16 illustrates a complex logic circuit using a 3D code according to an embodiment of the present disclosure. Figure 16A illustrates an [[8,3,2]] block code that can be transversally implemented with {CCZ, CZ, Z, CNOT} gates within each block and transversal CNOTs between blocks. LBy preparing >10 logical qubits, implementing layers of {CCZ, CZ, Z} alternating with interblock CNOT gates, and measuring in the X basis, a classically hard sampling circuit is realized with the logical qubits. Figure 16B shows the measured sampling results for a circuit with 12 logical qubits, 8 logical CZ gates, 12 logical CNOT gates, and 8 logical CCZ gates. By increasing the number of error detections, the measured distribution converges to the ideal distribution. Figure 16C shows a circuit containing 48 logical qubits with 228 logical CZ / CNOT gates and 48 logical CCZ gates. Figure 16D shows the running time of a classical simulation for calculating the probability of individual bit strings, with the bottom plot estimated based on the computational complexity of matrix multiplication. Figure 16E shows the normalized XEB measured as a function of sliding-scale error detection for 3, 6, 12, 24, and 48 logical qubits. For all sizes, finite XEB scores are achieved that improve with increasing error detections. The figure shows the connectivity of 48 logical qubits, with logical triplets entangled on a 4D hypercube. Figure 16F shows the scaling of the raw (1601) and completely misdetected (1602) XEBs of Figure 16E. The physical upper bound fidelity (1603) is calculated using the best measured physical gate fidelity (see below and Figure 24 for scaling discussion). An [[8,3,2]] cube is entangled on a 4D hypercube to achieve the physical connectivity of a 7D hypercube.

[0140] When using 2D codes such as surface codes, non-Clifford operations cannot be easily implemented, and Clifford circuits can be easily simulated, requiring relatively expensive techniques for non-trivial computation. In contrast, 3D codes allow non-Clifford operations to be realized transversally, but at the expense of the transversal H. However, these constraints do not imply that classically hard or useful quantum circuits cannot be realized transversally or efficiently. To address these constraints, the present disclosure provides efficient implementations of classically hard algorithms co-designed with specific error-correcting codes. In particular, a fast scrambling circuit used for native non-Clifford operations (CCZ) is provided using small 3D codes.

[0141] Exemplary embodiments use small 3-dimensional [[8,3,2]] codes (Fig. 16A) with various attractive features. These codes encode three logical qubits per block, feature d=2 (d=4) in the Z (X) basis, which suggests error detection (correction) capability for Z(X) errors, and can realize transversal CNOT between blocks. Physical {T,S} rotations (T is

number

[0142] These transversal operations enable the realization of logic algorithms that are difficult to simulate classically. More specifically, these circuits can be mapped to Instantaneous Quantum Polynomial (IQP) circuits. Sampling from the output distribution of such circuits is known to be classically difficult in some cases, suggesting that quantum devices could be exponentially faster than classical computers for this task.

[0143] Figure 16B shows an example implementation of a 12 logical qubit sampling circuit, where all logical blocks are |+ L >, and implement a scramble circuit with 28 logical entanglement gates, and all logical qubits are measured in the X basis. Figure 16B shows the 2 12 The figure shows the probability of observing each of the 4096 possible logical bitstream outcomes, demonstrating that as more error detection (post-selection) is applied in post-processing, the distribution more closely reproduces the ideal theoretical distribution. To characterize the overlap of the distributions, the cross-entropy benchmark (XEB) is used, where XEB is a weighted sum of the measured probability distribution and the ideal calculated distribution, normalized so that XEB = 1 corresponds to a perfect reproduction of the ideal distribution and XEB = 0 corresponds to a uniform distribution that occurs when the circuit is overwhelmed by noise.

[0144] Consistent with Figure 16B, applying error detection increases the XEB of the 12 logical qubit circuit from 0.156(2) to 0.616(7) (Figure 16E). XEB is a benchmark of superior fidelity for IQP circuits.

[0145] Scaling to larger systems and circuit depths is provided. To ensure high logical circuit complexity, nonlocal connectivity is used to entangle logical triplets on hypercube graphs up to 4D, resulting in fast scrambling. Entangled systems of 3, 6, 12, 24, and 48 logical qubits are explored, yielding finite XEB scores that improve with increasing error detections in all cases ( Figures 16E and 16F ). Finite XEBs indicate successful sampling, and the improvement with error detection demonstrates the benefits of using logical qubits. While this improvement comes at the expense of measurement time due to error detections, improving sample quality cannot be substituted by simply generating more samples. Therefore, improving the XEB score offers significant practical benefits. For 48 logical qubits and hundreds of nonlocal logical entanglement gates, we obtain XEBs of approximately 0.1, roughly an order of magnitude higher than previous physical qubit implementations of similar complexity, demonstrating the advantages of our logical encoding.

[0146] Assuming the best measured physical fidelity, the estimated upper limit of an optimized physical qubit implementation of this system is also significantly below the measured logical XEB (line 1603 in Figure 16F). In small physical instances, values ​​much below this upper limit are measured. In addition to the error detection benefits, the logical circuit appears to be significantly more tolerant to coherent errors and exhibits essentially digital behavior, with only imperfect fidelity (see, e.g., Figure 24A), consistent with theoretical predictions. With respect to the logical algorithm, performance is optimized by optimizing the expectation of the stabilizer (rather than the complex sampled output), providing further benefits to the logical implementation.

[0147] An exemplary 48-qubit circuit, corresponding to the connectivity of physical qubits in a 7D hypercube, contains up to 228 logical 2-qubit gates and 48 logical CCZ gates. Simulation of such logical circuits is challenging due to the high connectivity and large number of non-Clifford gates (which make tensor networks inefficient). To benchmark the circuits, they are structured to take advantage of an efficient simulation method that takes approximately 2 seconds to calculate the probability of each bit string (Figure 16D). Modeling noise in logical circuits is even more complex because the circuits are composed of 128 physical qubits and 384 T gates, thereby making experimentation with logical algorithms essential to understand and optimize performance.

[0148] Quantum simulation using logical qubits Figure 17 shows logical two-copy measurements according to embodiments of the present disclosure. Figure 17A shows the same scrambled circuit performed on two copies of 12 logical qubits and then measured in a Bell basis to extract information about the state. The Z-basis measurement is corrected with an [[8,3,2]] decoder (when perfect error detection is not applied). Figure 17B shows measured entanglement entropy as a function of subsystem size, demonstrating the expected Page curve behavior for highly scrambled states, improving with increasing error detection. Figure 17C shows measured and simulated Bell magic (associated with non-Clifford operations) as a function of the number of applied CCZ gates, performed on two copies of a scrambled six-logical qubit system. Figure 17D shows measurements and zero-noise extrapolation of a Pauli string using logical qubits. This plot illustrates the 4-bit entanglement entropy, which has only five discrete values ​​for a digital circuit. 12The graph shows the absolute expectation values ​​of Pauli strings; Pauli strings with the same theoretical value are grouped together. Analysis with sliding-scale error detection achieves an improvement towards the theoretical expectation (squared), while also improving towards purity of 1. Extrapolating to perfect purity allows the expectation to be extrapolated, approximating the ideal value better (the darker areas are the uncertainty in the statistical fit).

[0149] Logic qubits are also used as tools for quantum simulation. In particular, Bell-basis measurements (Figure 17A) performed on two copies of a quantum state provide a powerful tool that can efficiently extract many properties of an unknown state. This two-copy technique plots the measured entanglement entropy in a scrambled system in Figure 17B. The characteristic Page curve associated with a maximally entangled, highly scrambled, yet globally pure, state is observed. These measurements also reveal a final-state purity of 0.74(3), compared with the measured XEB of 0.616(7) in Figure 16F, consistent with XEB being a good proxy for final-state fidelity. Because entropy near zero is exponentially faster to measure, error detection here significantly improves the signal-to-noise ratio, despite the overhead of postselection (Figure 26).

[0150] 2 copy measurements are 4 NIt can also be used to simultaneously extract information about all Pauli strings. Using this property and an analytical technique known as Bell difference sampling, the amount of additive Bell magic in the circuit is experimentally evaluated and directly verified as a function of the number of applied logical CCZs (Figure 17C). This measure of magic associated with non-Clifford operations quantifies the number of T gates (assuming a decomposition into T) required to realize a quantum state by observing the probability that the sampled Pauli strings commute with one another. Furthermore, combining encoded qubits with two-copy measurements enables further error mitigation techniques. As an example, Figure 17D shows the 4-copy error detection with sliding-scale error detection. 12 Figure 1 shows the measured absolute expectation values ​​of all logical Pauli strings. In the two-copy measurement, the overall system purity for each error detection threshold is measured, allowing for the extrapolation of the expectation values ​​to the unit-purity (zero noise) case. This procedure evaluates the averaged Pauli expectation values ​​with a relative accuracy of approximately 10% of the ideal theoretical value over several orders of magnitude.

[0151] Exemplary System Overview Figure 18 shows a neutral atom quantum computer architecture according to an embodiment of the present disclosure. Figure 18A shows the experimental layout featuring optical tools including a static SLM trap 1801 and a 2D moving AOD trap 1802, a global Raman one-qubit laser beam 1803 and a local Raman one-qubit laser beam 1804, a 420 nm Rydberg beam 1805 and a 1013 nm Rydberg beam 1806, and imaging systems for both global and local imaging. Figure 18B shows the experimental layout featuring optical tools including a static SLM trap 1801 and a 2D moving AOD trap 1802, a global Raman one-qubit laser beam 1803 and a local Raman one-qubit laser beam 1804, a 420 nm Rydberg beam 1805 and a 1013 nm Rydberg beam 1806, and imaging systems for both global and local imaging. 87 The level structure of the Rb atom is shown, along with the relevant atomic transitions used in this example.

[0152] Figure 18C shows the control infrastructure used to program the quantum circuit, featuring multiple arbitrary waveform generators (AWGs). In particular, the transfer and Raman 2D AODs are controlled by two waveforms 1807, 1808 and 1809, 1810 (one for the x-axis and one for the y-axis), respectively. An additional AWG is used in first-in, first-out (FIFO) mode for rearrangement before the circuit begins, and then control of the transfer AOD is switched to the transfer AWG. All AWGs (except the rearrangement AWG) are synchronized to a jitter of less than 10 ns. During Rydberg gates, traps are briefly pulsed off by TTL. An FPGA processes images from the camera in real time and sends control signals to the Raman 2D AOD for localized, single-qubit control, in this example.

[0153] Figure 18D shows an example array layout featuring entanglement, storage, and readout zones. The zones can be directly reprogrammed and repositioned for different applications and specific tweezer site locations. The tweezer beam and local Raman control are projected out-of-plane. The entire field of view of the objective lens is 400 μm in diameter, resulting in virtually no tweezer deformation near the edges of the processor. In a two-qubit Rydberg gate, atoms are spaced approximately 2 μm or less apart within the gate site, and the gate sites are spaced so that atoms in different gate sites are no closer than 10 μm within the gate. At n = 53 and a two-photon Rabi frequency of 4.6 MHz, the blockade radius is approximately 4.3 μm, so that nearby atoms are well inside the blockade and distant atoms are well outside the blockade.

[0154] To carry out this experiment, several key features enable quantum circuits that are programmable with both physical and logical qubits: 87A cloud containing Rb atoms is loaded into a magneto-optical trap in a glass vacuum cell 1811, and the atoms are then stochastically loaded into a programmable static array of 852 nm traps 1801 generated with a spatial light modulator (SLM), and then rearranged in a set of moving 850 nm traps 1802 generated by a pair of crossed acousto-optic deflectors (AOD, DTSX-400, AA Opto-Electronic) to achieve a defect-free array. The atoms are imaged by a 0.65-NA objective (Special Optics) 1812 onto a CMOS camera (Hamamatsu ORCA-Quest C15550-20UP) 1813, chosen for its fast electronic readout time.

[0155] The state of the qubit is T2>1s, 87 m of the Rb ground-state manifold FThe signal is encoded in a hyperfine clock state at n = 0, and fast, high-fidelity single-qubit control is performed by two-photon Raman excitation (Figure 18B). A global Raman channel illuminating the entire array is used for global rotation (Rabi frequency ∼1 MHz, resulting in a rotation of ∼1 μs using a combined pulse technique) and dynamic decoupling throughout the circuit (typically one global π pulse per transfer). Fully programmable local single-qubit rotation is achieved with the same Raman light, but redirected through local channels focused on the target atom by an additional set of 2D AODs 1814. Entanglement gates (270 ns duration) between clock qubits are performed by fast two-photon excitation using 420 nm and 1013 nm Rydberg beams to the n = 53 Rydberg state, utilizing time-optimal two-qubit gate pulses. During computation, atoms are rearranged in the AOD traps to enable arbitrary connectivity. Inter-circuit readout is achieved by side illumination with a locally focused 780 nm imaging beam 1815, and scattered photons are collected on the CMOS 1813 and processed in real time by a Field Programmable Gate Array 1816, FPGA (Xilinx ZCU102) with feedforward control signal output.

[0156] The quantum circuit is programmed with a control infrastructure consisting of five arbitrary waveform generators (AWGs) (Spectrum Instrumentation) shown in Figure 18C, synchronized to a jitter of less than 10 ms. A two-channel reordering AWG is used to reorder the atoms into a defect-free arrangement before the circuit, one channel of the Rydberg AWG is used for the entanglement gate pulses, four channels of the Raman AWG are used for IQ (in-phase and quadrature) control of the 6.8 GHz source (a reference for the global phase of all qubits) and pulse shaping for the global and local Raman drives, two channels of the Raman AOD AWG are used to display tones that create a programmable optical grid for local single-qubit control, and two channels of the moving AOD AWG are used to control the positions of all atoms in the circuit. The AOD is at the heart of an efficient control method: two voltage waveforms (one for the X axis and one for the Y axis) control many physical or logical qubits in parallel; each row and column of the grid simply corresponds to a single frequency tone, and these tones are superimposed on the waveform delivered to the AOD (amplified by a Minicircuits ZHL-5W-1+). The phase relationship between the tones is chosen to minimize interference.

[0157] Programming the Circuit Selection of Zone Parameters. For simplicity, the entanglement zone is fixed for each example described herein. This conveniently allows, for example, switching between surface and [[8,3,2]] code experiments without additional calibration. The entanglement zone profile, realized by the 420 nm and 1013 nm Rydberg "top-hat" beams generated by the SLM phase profile, is chosen to be homogeneous over a 35 μm-high region. Because the Rydberg beams propagate longitudinally, the entanglement zone is longer than it is tall. The top-hat is optimized to be homogeneous over a horizontal extent of approximately 250 μm. Higher regions are possible, at the cost of reduced laser intensity and more difficult homogenization. The 250 μm width of the zone used here is set by the bandwidth of the AOD's deflection efficiency. The readout zone is placed opposite the storage zone to further minimize decoherence of atoms in the entanglement zone.

[0158] In a two-qubit Rydberg (n = 53) gate, atoms are spaced approximately 2 μm apart within a "gate site," resulting in an interaction strength of approximately 450 MHz or greater between pairs, much greater than the 4.6 MHz Rabi frequency. Due to the use of Rydberg blockade, the gate is largely independent of the exact distance between atoms. Therefore, precise interatomic positioning is not required. The gate sites are spaced so that atoms in different gate sites are no closer than 10 μm during the gate, resulting in negligible long-range interactions. Throughout this example, four gate sites are provided vertically (five for the surface code experiment) and 20 gate sites are provided horizontally, simultaneously gated on as many as 160 qubits (see Figure 18D). Under various conditions, with appropriate calibration, two-qubit gate fidelity within the range of F = 99.3%–99.5% is measured. When Rydberg gates are performed within the entanglement zone, no errors are observed in the storage zone atoms. Even though the tail of the top-hat Rydberg excitation beam is only suppressed to about 0.1 times its intensity, the two-photon drive is significantly off-resonant due to the ~20 MHz 1013 optical shift detuning that exists for atoms in the entanglement zone. A physical CZ gate is natively realized, and a physical H gate is added when implementing CNOT. Minimal two-qubit crosstalk is observed between gate sites when probed with long benchmark sequences. While some small crosstalk may exist, likely due to decay to the Rydberg P state, this should be significantly suppressed in practical operation here thanks to the ~200 μs duration between gates, during which time the Rydberg atom should either fly away or decay back to the ground state.

[0159] Shuttling and Transfer. In an exemplary embodiment, the SLM tweezers are static, though they can have any position. The AOD tweezers are movable, but with some constraints. In particular, the AOD array creates a rectangular grid (although not all sites need to be filled). During atom transfer operations, they are used only to stretch, compress, and translate the AOD trap array; atoms move within rows and columns, never crossing rows and columns. By shuttling atoms within the AOD tweezers and then transferring them appropriately between the AOD tweezers and the SLM tweezers, arbitrary qubit movement and reordering is achieved. Gates are performed for atom pairs in both the AOD-AOD trap and the AOD-SLM trap, and no difference in gate performance is observed as measured by randomized benchmarks.

[0160] Free-space shuttling of atoms within AOD tweezers (without transfer) incurs essentially no fidelity cost (other than time overhead). In various embodiments, photodiodes are used to calibrate and homogenize the 2D deflection efficiency of the 2D AOD to percent-level uniformity across the entire area used. The atomic trajectories and echo sequences are designed to cancel out residual path-dependent nonuniformities. For example, atoms can be moved 100 μm away to achieve a remote entanglement gate, and a Raman π pulse is performed before returning the atoms, so that the differential optical shift accumulated during the return movement cancels the initial movement. The movement is realized with a third-order profile, with a characteristic free-space transit time between gates of approximately 200 μm, and acoustic lensing effects from the AOD are estimated to be negligible. The 1013 laser is pulsed off between movements to eliminate loss effects from the large optical shift. Note that the 10 13 induced differential optical shift for the hyperfine qubit is only on the kHz scale, but its effect is cancelled out by the echo.

[0161] Transferring atoms between tweezers poses additional challenges. The infidelity of each transfer, encompassing both transport relaxation and loss, is measured to be approximately 0.1% or less. To achieve this performance, during the SLM-to-AOD transfer, the intensity of the AOD tone is boosted (with a quadratic intensity profile, when possible) corresponding to the appropriate site, over a time period of 100–200 μs, to a trapping depth approximately twice the SLM trapping depth, and then the AOD trap is moved 1–2 μm further away over a time period of 50–100 μs.

[0162] During subsequent movements, the AOD trap depth remains at this double value. To transfer atoms from the AOD to the SLM, the reverse process is performed. During these transfer processes, the differential optical shift for the transferred atoms is dynamically changing, which can result in a large unechoed phase shift. Therefore, the circuit is designed to echo the transfer pair with a properly selected π pulse. When the transfer pair cannot be echoed, a one-cycle XY4 or XY8 dynamic decoupling is performed during the transfer. Low-loss transfer is highly sensitive to the alignment of the AOD and SLM grids. Small optical distortions between the AOD tweezer grid and the SLM tweezer grid are corrected by fine-tuning the individual SLM grid tweezers, which can be arbitrarily positioned to overlap the AOD trap as seen by the image-plane reference camera. It is important to adjust the SLM but not the AOD, because even slight adjustments of the individual AOD tones outside the frequency comb can cause beats and atom loss.

[0163] Dynamic decoupling and local gating. In an exemplary circuit design, the echo sequence is designed to cancel as many deleterious points as possible. Dynamic decoupling has an odd number of π pulses between CZ gates (whenever possible) because this cancels both systematic and spurious contributions to the one-qubit phase with the echoes. Appropriate X(π) and Z(π) rotations are used to create a global

number

[0164] Localized one-qubit gates using the Raman AOD are realized at arbitrary locations in space above both the AOD atoms and the SLM atoms. Target logical qubit blocks are addressed by illuminating the grid of logical blocks. Arbitrary patterns of rotation for the qubit grid (e.g., in preparation for color coding) are realized row-by-row serialization, with the x-coordinates of the targets in each row illuminated simultaneously. The duration of each row is 5-8 μs (several tens of μs for arbitrary patterns of rotation), which can be significantly faster, as discussed in the next section. For simplicity, rotations are calibrated at 80-160 specific sites across the array, and rotations are performed at any spot using the nearest calibrated value.

[0165] Using calibrated local one-qubit gates and two-qubit gates in the entanglement zone, the entire circuit is defined by the appropriate trapping SLM phase profile and waveforms of several AWG channels and TTL pulse generators, which then program complex and varied circuits on hundreds of physical qubits.

[0166] Programmable one-qubit gate Figure 19 illustrates one-qubit Raman addressing according to an embodiment of the present disclosure. Figure 19A illustrates two possible implementations of a local one-qubit gate, i.e., at the two-photon Rabi frequency Ω. Raman shows the X(θ) rotation (1901) in resonance with and the Z(θ) rotation (1902) out of resonance with

number

number

number

number

[0167] Figure 19C shows the calibration procedure used to equalize the Rabi frequencies across a 220 μm x 35 μm array. Position calibration is shown for 80 sites, with close proximity

number

number

[0168] To enable individual one-qubit gates, the same Raman laser system as in the global rotation scheme is used to illuminate only selected atoms using a pair of crossed AODs. The focused beam waist in the plane of the atoms is 1.9 μm, which is large enough to be robust against atomic position fluctuations and small enough to prevent crosstalk to neighboring atoms spaced approximately 6 μm or more apart. For Raman excitation, polarization must be carefully considered. Unlike the global path, the local beam propagation direction is perpendicular to the atom's quantization axis (set by the external magnetic field). Therefore, the apparent magnetic field responsible for driving the transition

number

number

number

[0169] Versions of π and σ ±Both versions of σ and σ are realized above, but in this example the polarization uniformity is affected by the sharp wavelength edge of the dichroic after the AOD, resulting in reduced polarization sensitivity, resulting in an off-resonance σ ± A dressing procedure is used. Furthermore, for most circuits, local rotations are performed row by row (only one Y tone at a time), which allows arbitrary fine adjustment of the X coordinate and power at each site to equalize and calibrate the rotation (Figure 19B). Calibrations are performed using the procedure in Figure 19C, and these calibrations are stable on lunar timescales.

[0170] To quantify fidelity, randomized benchmarking was performed at 16 sites with 0, 10, 20, 30, 40, and 50 localized (per site) scores.

number

number

[0171] Mid-circuit readout and feedforward Figure 20 illustrates a circuit mid-readout and feedforward according to an embodiment of the present disclosure. Figure 20A shows a single-shot 500 μs local image in the readout zone, with a peak corresponding to approximately 50 photons collected by a CMOS camera. Figure 20B shows the atomic transitions and pulse sequence used for local imaging of the ancillary qubit. The large spatial separation between the entanglement zone and the readout zone, plus the optical shift of the data qubit trap, suppresses data qubit errors. Loss of atoms in the readout zone during local imaging is prevented by using a five-fold higher trap depth, and the ancillary qubit trap and local imaging light are pulsed to directly image on resonance, avoiding the adverse effects of the large trap optical shift.

[0172] Figure 20C is a diagram of the components involved in the readout and feedforward steps of the circuit. Atom detection and logic state decoding are performed using FPGA 2001, which then outputs conditional TTL 2002 to gate local Raman pulse 2003, which is performed on the logic qubits in the entanglement zone. Figure 20D is a diagram of the approximate timing of the feedforward cycle of the circuit. First, the F=2 population is pushed out (in 10 μs), and then the remaining F=1 population is locally imaged for 500 μs. 24 rows of pixels covering the readout zone are read out to the FPGA in 200 μs, after which processing is performed. Finally, a conditional TTL output based on the decoded state gates the local Raman pulse on or off. The entire readout and feedforward cycle takes less than 1 ms and can be sped up by optimizing the local imaging and camera readout.

[0173] Figures 20E-20G show characterization of the data qubit error probability during local imaging. To quantify the impact of the local imaging beam on the data qubit coherence for very long illuminations, Figure 20E shows the data qubit error probability (fraction of the population depumped from F = 2 to F = 1) as a function of the duration of local imaging up to 20 ms. Figure 20F shows the data qubit error probability after 20 ms of local imaging as a function of the detuning of the local imaging beam, demonstrating the suppression of errors both red-detuned and blue-detuned from the data qubit transition. Figure 20G shows that, equivalently, increasing the trap depth of the data qubit enables the suppression of decoherence caused by the local imaging beam. Because qubits in the readout zone are imaged while their qubit traps are pulsed off, any optical shift of the data qubit transition from the trap directly contributes to the relative detuning. Figure 20H shows that for a long 10.5 ms local beam illumination using optimal local imaging parameters, a 0.7(1)% increase in data qubit error is observed during the XY8 dynamic decoupling sequence, suggesting an error probability of approximately 0.034(5)% for data qubits in the 500 μs mid-circuit readout image used in this example.

[0174] To perform in-circuit readout of a selected qubit without affecting the other qubits, a local imaging beam is focused into a readout zone, spatially separated by approximately 100 μm from the entanglement zone. The local imaging beam consists of a 780 nm circularly polarized beam with a near-resonant component from F = 2 to F' = 3 and a small repump component. This beam is transmitted through the side of a glass vacuum cell and co-propagates with the global Raman and 1013 nm Rydberg beams (Figure 19A). A cylindrical lens is used to shape the beam to a focused beam waist of 30 μm in the plane of the atomic array and 80 μm out of the plane. After moving a portion of the atoms into this readout zone, local pushing of the F = 2 ground-state manifold population is performed (by turning off the repump laser frequency), followed by local imaging of the remaining F = 1 population.

[0175] As shown in Figure 20A, an average of approximately 50 photons are collected per imaged atom. To prevent rapid loss of atoms during in-circuit imaging (which, unlike global imaging schemes, lacks multi-axis cooling), deep (approximately 5 mK) traps are used (to help hold the atoms), and these traps are stroboscopically pulsed on and off out of phase with the local imaging light to prevent adverse effects of deep traps, such as non-uniform light shifts and fluctuating dipole-force heating (Figure 20B). A double-Gaussian fit to the two distributions in Figure 14A extracts imaging fidelity of over 99.9%. Since this fitting can lead to an overestimation of the imaging fidelity (e.g., due to loss of atoms during imaging), the total SPAM error (measured by the amplitude of the Ramsey fringes) is compared between local and global imaging for the same state preparation sequence, extracting an error 0.14(5)% higher for local imaging, which, taking these into account, leads to an estimated local imaging fidelity of around 99.8%.

[0176] Several design considerations facilitate local imaging in the readout zone while preserving the coherence of the data qubit within the entanglement zone (Figures 20E–20G). The primary sources of decoherence are photon rescattering from the locally imaged atoms and beam reflections and tails of the local imaging beam impinging on the data qubit. As shown in Figure 12C, for the 500 μs mid-circuit imaging used in this study, unchanged coherence of the data qubit (identical within error bars) is achieved with the local imaging light on as well as without it. To understand these effects more quantitatively, the error probability of the data qubit within the entanglement zone is measured while the local imaging beam is on in the readout zone for up to 20 ms, at a higher intensity than that used for local imaging in this study. Decoherence is suppressed by optically shifting the 780 nm transition of the data qubit to differ by several tens of MHz from the transition of the locally imaged qubit, as examined in Figures 20F–20G. Data qubit decoherence is further suppressed by the large spatial separation between the readout zone and the entanglement zone, where the intensity from the tail of the local imaging beam's Gaussian distribution rapidly drops off. Even with a large separation, stray light reflections (e.g., from the glass cell window and other optical elements) can strike the data qubit region. To mitigate this effect, the reflections are shifted far away from the atomic array by angling the local imaging beam when it strikes the glass cell window. The estimated effect of rescattered photons from the imaged atoms is negligible, especially with the added relative detuning. Taking all these factors into account, the data qubit decoherence rate is suppressed to approximately 0.1% or less per 500 μs of local imaging exposure, as shown in Figure 20H.

[0177] The complete in-circuit readout and feedforward cycle, including local extrusion, local imaging, camera pixel readout, decoding of the logic qubit state on the FPGA, and local Raman pulses gated on or off by conditional triggers, occurs in just under 1 ms (Figure 20D). This approach to in-circuit readout and feedforward can be modified to enable in-circuit readout on the scale approaching 100 μs. This method can be straightforwardly extended to perform many rounds of measurement and feedforward, where groups of auxiliary atoms are successively transported to the readout zone throughout the entire deep quantum circuit.

[0178] Figure 21 shows additional surface code data. Figure 21A is a diagram of a Bell-state circuit and a surface code with d=7. Figure 21B illustrates the transversal CNOT and physical error propagation rules. Figure 21C shows the covariance of 48 measured stabilizers in both bases. Correlations near the diagonal correspond to nearby stabilizers within each block. Due to error propagation in the transversal CNOT, strong correlations are also observed with stabilizers in other blocks. Figure 21D shows an upper bound on the Bell-pair infidelity (as opposed to the estimated Bell-pair error in Figure 13D), showing improvement with increasing code distance. Figure 21E shows the probability of no error detection for each of the 96 measured stabilizers, demonstrating agreement when compared with theoretical values ​​from empirically selected error rates (experimental mean = 77%, theoretical mean = 82%). Note that the X-basis logical qubit 1 and the Z-basis logical qubit 2 have higher stabilizer error probabilities (lower expected values ​​compared to when no transversal CNOT is implemented) due to error propagation in the transversal CNOT. Using the empirical error rates corresponding to the agreement between the measured stabilizer data and theory in Figure 21E, simulations of Bell pair error improvement as a function of code distance (as shown in Figure 21F) closely match the experiments. The empirical error rates used are consistent with the 99.3% two-qubit gate fidelity measured for this large array and the approximately 4% data qubit decoherence error (aggregated across the entire circuit and measured with the Ramsey method). These dephasing error rates are dominated by the complex transfer sequence during the preparation of the two successive surface codes and are significantly smaller for iterative error correction experiments.

[0179] FIG. 22 illustrates surface code preparation and data decoding according to an embodiment of the present disclosure. FIG. 22A shows the surface code stabilizers for two independent d=7 codes after state preparation. The entire transfer circuit corresponding to the transversal CNOT is implemented, and the transversal entanglement gate pulse is turned off. The mean stabilizer probability of success for a total of 96 stabilizers is 83%. The high probability of success of the stabilizers for the two independent codes in both the X and Z bases indicates that topological surface codes were prepared (and FIG. 21 shows that the topological surface codes were preserved in the transversal CNOT). Physical fidelity was slightly lower during this measurement due to calibration drift, and therefore these results slightly underestimate performance compared to the data in FIGS. 13 and 21. Figure 22B shows the logical pairing error while optimizing the decoder by (inversely) scaling the weights of the inter-qubit edges and hyperedges connecting the stabilizers of two logical qubits (higher values ​​correspond to smaller pairing weights). More specifically, the probability p of the error mechanism corresponding to the inter-qubit edge / hyperedge is scaled, and the weights are

number

[0180] 8.3.2 Circuit Implementation Figure 22 shows [[8,3,2]] and hypercube coding according to an embodiment of the present disclosure. Figure 23A shows a state preparation circuit for [[8,3,2]] code, in which two 4-qubit GHZ states are simultaneously prepared and then entangled. This is shown in Figure 23B, where the logic state |- L1 , + L2 , - L3We initialize the [[8,3,2]] code with >. Figure 23B shows a 4D hypercube circuit implemented for 48 logical qubits (128 physical qubits). The circuit is drawn at the block level, with each block consisting of 3 logical qubits and 8 physical qubits. The first intra-block gate layer is a global T † The local gate patterns and the corresponding logic gates they implement within each code block are shown in the inset. Figure 23C illustrates the movement of code blocks throughout the circuit and the use of the processor's zoned architecture.

[0181] First, eight [[8,3,2]] code blocks are prepared in the entanglement zone, and atoms for state preparation after eight additional code blocks are loaded into the storage zone. Then, code blocks in the entanglement zone are picked up to run three transversal CNOT layers, interlaced with adjacent blocks. Two groups of eight code blocks are then swapped, and the same procedure is repeated with the second group of code blocks. Then, the first group of code blocks is returned to the entanglement zone to run the final parallel transversal CNOT, interleaved with the atoms of the first group. Layers of CNOT gates connect the code blocks to build a 4D hypercube of 16 blocks of [[8,3,2]] code.

[0182] The [[8,3,2]] code block is |- L , + L , - L > state, which is initialized to two 4-qubit GHZ states (corresponding to the [[4,2,2]] code), i.e.,

number

[0183] A set of transversal gates for the [[8,3,2]] code is enabled as follows: The interblock transversal CNOT follows immediately from the fact that the [[8,3,2]] code is a CSS code. Two logical qubits L i and L j CZ gate in the block between Li,Lj ) is the logical qubit L k The surfaces S and S corresponding to † For example, in FIG. † Gate pattern, i.e., S1S3 † S5 † Considering applying S7, this is X L1 =X1X2X3X4 to X L1 '=-Y1X2Y3X4, which is X L1 '= L1 Z L2 is equal to x L2 '=X L2 Z L1 This means that the CZ is realized between logical qubits 1 and 2. This procedure is why T, T † This can also be used to understand how the pattern of realizes CCZ between three encoded qubits. The CCZ gate is X L3of

number

[0184]

number

[0185] Physically rearranging the atoms to swap qubits like 4⇔8 and 3⇔7 also results in X (by multiplying it with a global X stabilizer). L1 =X1X2X3X4 to X L1 '=X1X2X7X8 or instead X L1 '=X L1 X L2 Similarly, Z L2 '=Z L2 Z L1 We can see that by tracing the permutation of the qubits, i.e., realizing CNOT. Finally, since these 3D codes are CSS codes, they do not have a transversal H, but can be initialized and measured in either the X or Z basis, effectively allowing for H gates at the beginning or end of the circuit.

[0186] The intra-block logical entanglement gates are applied block by block, and any intra-block gate combination can be realized. For conceptual simplicity, only two specific local Raman patterns are applied within a layer. The first is T †The gate combination CCZ is given by applying L1,L2,L3 CZ L1,L2 CZ L1,L3 CZ L2,L3 Z L1 Z L2 Z L3 The second gate combination is T in the top row and T in the bottom row. † CCZ given by applying L1,L2,L3 CZ L2,L3 CZ L1,L3 Z L3 The circuit alternates between layers of intra-block transversal entanglement gates and layers of extra-block transversal CNOTs, entangling the logic blocks over a hypercube of up to 4D (see Figure 23). The control and target qubits are kept the same throughout the circuit for conceptual simplicity, allowing a local physical H gate for the target qubit to be compiled with the intra-block gate layer, although the control-target direction can be chosen arbitrarily. Intra-block logical entanglement gates are applied in such a way that they trivially commute through and do not cancel out the application of previous entanglement gates. For the Clifford states realized in the rest of this example, the stabilizers are (e.g., physical H gates instead of H)

number

number

[0187] Implementation of physical qubit circuits To compare the logical qubit algorithms provided herein with analogous circuits on physical qubits, a concrete implementation of a sampling / scrambling circuit on physical qubits is provided and experimentally realized using the same set of physical gates, Clifford + T, as used in the logical circuit. Each [[8,3,2]] block is replaced by a 3-physical qubit block, and the "intra-block" CCZ gates are replaced with 6 CNOT and 7 {T, T}. † The CZ decomposes into} gates, and a "transversal" CNOT is implemented directly between three-qubit blocks. CZ can be compiled into CCZ implementations, but this has a trivial impact on analysis and inference. These physical circuits are complex: 48 qubits with 48 CCZ and 228 two-qubit gates (realized with logical qubits) decompose into an effective 516 two-qubit gates (384 when the CZ gates are compiled into CCZ). In practical implementations of these circuits, the accumulation of coherent errors resulted in vanishing XEBs of the physical circuits. These experiments revealed that the logical circuit equivalents significantly outperformed the physical circuits, thereby providing direct evidence that, for this particular sampling circuit, the logical algorithm outperformed the physical algorithm.

[0188] More quantitatively, an upper bound is calculated using a concrete physical implementation and assuming optimistic performance. The best-measured fidelities are assumed: 99.4% SPAM, 99.91% local one-qubit gate fidelity (Figure 19), 99.55% two-qubit gate fidelity, and T = 2s. The total number of entangled gate pulses is counted against the CZ gate, the total number of compiled local one-qubit gates, and the estimated circuit duration, which are used to calculate the estimates presented in Figure 16F. This analysis is confirmed for small-scale circuit implementations. For short three-qubit circuits, XEB is benchmarked at approximately 0.87, below the estimated three-qubit upper bound of approximately 0.92 for physical circuits. Although Figure 16F plots the fidelity estimates for the physical qubits rather than XEB, XEB and fidelity are expected to be closely related.

[0189] Empirically, logical circuits appear to be significantly more tolerant to coherent errors. In particular, logical circuits implement essentially digital operations, and small coherent errors do not significantly shift / distort the bit-string distribution, but only appear to degrade overall fidelity (see, e.g., the congruence in Figure 24A). This contrasts with physical implementations, where coherent errors are found to dramatically alter the shape of the bit-string distribution, e.g., changing its relative amplitude. [[8,3,2]] Circuits are optimized only by optimizing the expected value of the stabilizer, not by directly optimizing the XEB or two-copy results. When implementing complex circuits, stabilizers serve as useful intermediate fidelity benchmarks for both optimizing circuit design and ensuring proper execution, especially in regimes where the output distribution or other observable properties cannot be calculated. These complex circuits appear to perform significantly better with logical qubits than with physical qubits.

[0190] Figure 24 shows the sampled data for a further [[8,3,2]] circuit. Figure 24A shows the overlap of the sampled data for 12 misdetected qubits with the theoretical distribution (the same data as in Figure 16B for the fully misdetected case). Progressive zoom-in shows the overlap of the sampled data for 12 misdetected qubits with the theoretical distribution (the same data as in Figure 16B for the fully misdetected case). -4 The agreement between theory and experiment is shown to a probability level of 0.616(7). This error-detected dataset consists of 23,545 shots (the raw dataset is 138,626 shots). Simultaneous measurements were performed on two groups of 12 logical qubits; only one of the two 12-logical qubit groups is plotted here, with an XEB of 0.69(1), while in Figures 16E, 16F, and 24B the two logical groups are averaged, yielding a measured XEB of 0.616(7).

[0191] Figure 24B shows the same data as Figure 16F, but additionally plots the purity measured by two-copy measurements. The measured XEB is slightly below the measured purity, providing evidence that XEB is a good proxy for fidelity. Under error detection, the logical XEB of these IQP circuits should be a good proxy for fidelity. Interestingly, since the circuits applied at the physical level are not IQP, they may behave differently with respect to the raw, uncorrected data. Without error detection, not all errors result in logical errors, and therefore the circuits may behave differently from IQP and be amenable to different scaling. For systems with 3, 6, and 12 logical qubits, multiple systems are measured in parallel and the results are averaged together. The preparation of the [[8,3,2]] code states creates these states on a cube, but does not have CNOTs between two pairs of qubits in the first step, and therefore does not have the full gate connectivity of the cube. Instead, we can interpret these CNOTs as having been included but compiled away because they commute with the state. This is ignored when plotting the physical qubit connectivity, which is derived from entangling a 3D cube with the connectivity of a 4D hypercube to achieve a 7D hypercube. Figure 24C shows sliding-scale error detection data for a 48-qubit XEB. The point with perfect postselection, with all stabilizers perfect, returns only 8 samples, and therefore this point is omitted from the plot in Figure 24C for clarity.

[0192] Figure 25 shows a theoretical exploration of hypercube IQP circuits. Figure 25A illustrates the anti-concentration property of the circuit. A circuit is said to be anti-concentrated if its output distribution is approximately uniformly spread among all outcomes, without concentration of probability in a subset of bit strings. This property is important for many proofs of classical hardness, and it is therefore desirable for sampling circuits to be anti-concentrated. The plot shows that the output distribution of a random hypercube circuit (randomized intrablock operations and randomized control / target in the extrablock CNOT layer) becomes anti-concentrated as the dimension of the hypercube is increased, and that XEB (which captures the output collision probability) converges to a uniform-IQP value of 2 (here using a Clifford circuit—a circuit composed only of random CZ and Z). This suggests that sampling from an ideal output distribution is classically hard. In general, the hypercube IQP circuit ensemble converges to a uniform IQP ensemble with uniform total variation distance as the depth and hypercube dimension are increased. The specific circuit instance implemented in the experiment also rapidly deconverges as the hypercube dimension increases.

[0193] Figure 25B shows a single layer of the hypercube circuit, which allows for an efficient tensor network contraction scheme that allows for the evaluation of ideal and experimental XEB values. The final out-of-block CNOT layer is immediately followed by a measurement, which can be incorporated into a non-unitary tensor that is contracted between the two halves of the system (the control and target of the final CNOT layer). This contraction scheme reduces memory requirements by half the system size, which allows for the evaluation of the bit string amplitudes of a 48-qubit experiment. This simulation approach can be made significantly more expensive by applying additional out-of-block operations within the two subsystems, which forces intra-subsystem tensor blocking, increasing memory and runtime requirements (Figure 16D).

[0194] Figure 25C explores whether the circuit family can be "spoofed" with less expensive approximation simulations that achieve moderately high XEB scores, here investigated for a 24-qubit system using full state-vector simulation. The spoofing algorithm works by independently sampling the two halves of the system (two groups of 12 qubits), essentially removing the last layer of CNOT. This further reduces simulation complexity, as each half could, in principle, be simulated independently with the efficient approach from Figure 25B. The plot shows that the spoofed XEB of a 24-qubit non-Clifford circuit can be exponentially reduced by extending the circuit with additional gate layers (similar to the approach used to degrade the performance of efficient hypercube contractions) for a given extension of the circuit.

[0195] Figure 26 shows further Bell basis measurements. Figure 26A shows the results of a 4-qubit measurement in a 6-qubit logic circuit. 6 |tr(Pρ)| for all Pauli strings P 2 as a function of the stabilizer post-selection threshold (the number of correct stabilizers across 6 × 2 logical qubits). Region 2601 (2602) shows the histogram of |tr(Pρ)| 2= 0.0625(0). The separation between the histograms improves as more postselection is applied. Figure 26B shows the signal-to-noise (purity divided by the statistical uncertainty of purity) as a function of the sliding-scale error detection (converted to an acceptable percentage) for a two-copy measurement of 12 logical qubits, where subsystem size 1 indicates a single logical qubit in one copy and subsystem 12 indicates all logical qubits. For subsystem size 1, the signal remains unchanged (maximally mixed), but as the number of iterations decreases, the signal-to-noise ratio worsens as data is discarded. In contrast, for global purity, a purity near 1 improves signal-to-noise because the measurement is faster. Figures 26C and 26D show the entanglement entropy when analyzing the circuit as a physical Bell basis measurement, as opposed to a logical Bell basis measurement. For logical entanglement entropy calculations, an average is taken over all possible subsystems of that given subsystem size, which, thanks to the connectivity of high-dimensional hypercubes, behaves very similarly to, for example, continuous subsystems. For physical qubit entanglement entropy calculations, a random selection is made from the possible subsystems. Figure 26C shows 6 logical (16 physical) qubits per copy, and Figure 26D shows 12 logical (32 physical) qubits per copy. Finite sampling imposes a noise floor for very high entanglement entropy values. Figure 26E shows entanglement entropy measurements, similar to Figure 17B, but as a function of logical subsystem size. Figure 26F shows the logical circuit used to benchmark magic. For 1 CCZ, U1 is included and U0 is omitted; for 2 CCZ, U0 is included and U1 is omitted; and for 3 CCZ, both U0 and U1 are included.

[0196] The description of various embodiments of the present disclosure is presented for illustrative purposes and is not intended to be exhaustive or limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terms used herein were selected to best explain the principles of the embodiments, practical applications, or technical improvements to technology found in the market, or to enable those skilled in the art to understand the embodiments disclosed herein. [Explanation of symbols]

[0197] 400 QPUs 401 processor cores 404 Atomic Road Zone 405 Remote Entanglement Zones, Optical Links and / or Optical Lattice Transport 411 Storage Zone 412 Tangled Zone, Active Zone 413 Read Zone 501 logical qubits 502 logical qubits 503 Logical Qubits 504 logical qubits 505 logical qubits 511 Beam 512 Beam 513 Beam 601 Logical Qubits 602 Logical Qubits 701 Storage Zone 702 Active Zone 703 Readout Zone Part 1001 reservoir 1002 Active Zone 1003 read zone 1100 equipment 1102 Light source, laser 1104 SLM 1106a Element, lens 1106b Element, mirror 1106c Element, lens 1106d element, mirror 1106e objective lens 1108 Capture Plane 1110 Vacuum Chamber 1112 Light source, laser 1114 AOD 1116 AOD 1117 Element, condenser lens 1120 Arbitrary Waveform Generator, AWG 1122 Computer 1124a Objective Lens 1124b Dichroic mirror 1124c Charge Coupled Device (CCD) Camera 1124d Electron-Multiplying CCD (EMCCD) Camera 1201 Storage Zone 1202 Tangle Zone 1203 Read Zone Block 1301 Block 1302 1303 Parallel Movement 1501 Storage Zones 1502 Tangle Zone 1503 Logical Qubits 1504 Logical S Rotation 1505 Logical S Rotation 1506 Logical Qubits 1507 Logical Qubits 1601 Raw XEB 1602 Fully error detected XEB 1603 Physical Upper Limit Fidelity 1801 Static SLM Trap 1802 2D Moving AOD Trap 1803 Global Raman single-qubit laser beam 1804 Localized Raman single-qubit laser beam 1805 420nm Rydberg beam 1806 1013nm Rydberg beam 1807 Waveform 1808 waveform 1809 waveform 1810 waveform 1811 Glass Vacuum Cell 1812 Objective Lens 1813 CMOS camera 1814 2D AOD 1815 780nm imaging beam 1816 Field Programmable Gate Array 1901 Resonant X(θ) rotation 1902 Off-resonance Z(θ) rotation 2001 FPGA 2002 Conditional TTL 2003 Localized Raman Pulse 2601 area 2602 area

Claims

1. a first array of optical traps disposed in the active zone; a second array of optical traps disposed in the readout zone; a first laser configured to illuminate the active zone and drive a transition to a Rydberg state; a second laser configured to illuminate the active zone and drive transitions between hyperfine states; a third laser configured to illuminate the readout zone; a fourth laser configured to adiabatically transfer neutral atoms between the optical traps in the active zone and the optical traps in the readout zone; a camera configured to capture an image of the readout zone; Including, providing a first plurality of neutral atoms within the active zone, each within a respective optical trap of the first array; encoding a first logical qubit into the first plurality of neutral atoms with the first laser and the second laser; illuminating the first plurality of neutral atoms while within the active zone with at least the first laser or the second laser, thereby applying a gate to the first logical qubit; adiabatically transferring the first plurality of neutral atoms from the active zone to the readout zone, each into a respective optical trap of the second array; illuminating the first plurality of neutral atoms while within the readout zone with the third laser; A quantum processor configured to capture an image of the first plurality of neutral atoms while in the readout zone, thereby determining a state of the first logical qubit.

2. further comprising a third array of light traps disposed in the storage zone; the fourth laser is further configured to adiabatically transfer neutral atoms between the optical traps in the active zone and the optical traps in the storage zone; the quantum processor: adiabatically transferring the first plurality of neutral atoms from the active zone to the storage zone with the fourth laser after the encoding, each into a respective optical trap of the third array; 2. The quantum processor of claim 1, further configured to adiabatically transfer the first plurality of neutral atoms from the storage zone to the active zone with the fourth laser, each into a respective optical trap in the first array, before applying the gate.

3. providing a second plurality of neutral atoms within the active zone; encoding a second logical qubit into the second plurality of neutral atoms with the first laser and the second laser; further configured, prior to applying the gate, to position the first and second plurality of neutral atoms within the active zone such that each neutral atom of the first plurality of neutral atoms is within a blockade radius of exactly one corresponding neutral atom of the second plurality of neutral atoms; 3. The quantum processor of claim 1, wherein illuminating the first plurality of neutral atoms while within the active zone further illuminates the second plurality of neutral atoms, thereby applying the gate to the first logical qubit and the second logical qubit.

4. 4. The quantum processor of claim 3, wherein the gate is a transversal CNOT gate.

5. The quantum processor of claim 4 , wherein applying the gate comprises applying a single pulse of the first laser.

6. 6. The quantum processor of claim 1, wherein adiabatically moving the first plurality of neutral atoms from the active zone to the storage zone, adiabatically moving from the storage zone to the active zone, and adiabatically moving from the active zone to the readout zone each comprises applying a Raman pulse during the movement.

7. The quantum processor of claim 6 , wherein the Raman pulse is applied at a midpoint of the movement.

8. 8. The quantum processor of claim 6 or 7, wherein adiabatically moving the first plurality of neutral atoms from the active zone to the storage zone, adiabatically moving from the storage zone to the active zone, and adiabatically moving from the active zone to the readout zone each has a constant jerk.

9. 9. A quantum processor according to claim 1, wherein the first array, the second array, and / or the third array of optical traps are two-dimensional arrays.

10. 10. The quantum processor of claim 1, further comprising at least one acousto-optic deflector (AOD), wherein the fourth laser is configured to direct a beam of light toward the at least one AOD, and wherein adiabatically moving neutral atoms comprises varying a drive frequency of the at least one AOD.

11. 11. A quantum processor according to any one of claims 1 to 10, wherein the first array, the second array, and / or the third array of optical traps are generated by directing a beam of light onto a spatial light modulator (SLM).

12. The quantum processor of claim 1 , wherein the first plurality of neutral atoms are moved simultaneously.

13. 13. The quantum processor of claim 2, wherein the third array has a higher density than the first array.

14. 14. The quantum processor of claim 1, wherein encoding the first logical qubit comprises applying a CSS code.

15. 15. The quantum processor of claim 14, wherein the CSS code is selected from a surface code, a color code, a Steane code, and a hypergraph product LDPC code.

16. 16. The quantum processor of claim 1, further comprising an FPGA configured to receive the image of the first plurality of neutral atoms and to calculate the state of the first logical qubit.

17. 1. A method of performing quantum computing, comprising: providing a first array of optical traps disposed in an active zone of a quantum processor; providing a second array of optical traps disposed in a readout zone of the quantum processor; providing a first plurality of neutral atoms within the active zone, each within a respective optical trap of the first array; encoding a first logical qubit into the first plurality of neutral atoms with a first laser and a second laser, the first laser configured to illuminate the active zone and drive transitions to Rydberg states, and the second laser configured to illuminate the active zone and drive transitions between hyperfine states; illuminating the first plurality of neutral atoms while within the active zone with at least the first laser or the second laser, thereby applying a gate to the first logical qubit; adiabatically transferring the first plurality of neutral atoms from the active zone to the readout zone, each into a respective optical trap of the second array; illuminating the first plurality of neutral atoms while within the readout zone with a third laser; capturing an image of the first plurality of neutral atoms while in the readout zone, thereby determining a state of the first logical qubit; A method comprising:

18. adiabatically transferring the first plurality of neutral atoms from the active zone to a storage zone of the quantum processor by a fourth laser after the encoding, each into a respective optical trap of a third array; adiabatically transferring the first plurality of neutral atoms from the storage zone to the active zone with the fourth laser before applying the gate, each into a respective optical trap in the first array; 20. The method of claim 17, further comprising:

19. providing a second plurality of neutral atoms within the active zone; encoding a second logical qubit into the second plurality of neutral atoms with the first laser and the second laser; prior to applying the gate, arranging the first and second plurality of neutral atoms within the active zone such that each neutral atom of the first plurality of neutral atoms is within a blockade radius of exactly one corresponding neutral atom of the second plurality of neutral atoms; further comprising 18. The method of claim 17, wherein illuminating the first plurality of neutral atoms while in the active zone further illuminates the second plurality of neutral atoms, thereby applying the gate to the first logical qubit and the second logical qubit.

20. 20. The method of claim 19, wherein the gate is a transversal CNOT gate.

21. 21. The method of claim 20, wherein applying the gate comprises applying a single pulse of the first laser.

22. 22. The method of claim 19, wherein each of the steps of adiabatically transferring the first plurality of neutral atoms from the active zone to the storage zone, adiabatically transferring from the storage zone to the active zone, and adiabatically transferring from the active zone to the readout zone comprises applying a Raman pulse during the transfer.

23. 23. The method of claim 22, wherein the Raman pulse is applied at a midpoint of the movement.

24. 24. The method of claim 22 or 23, wherein the steps of adiabatically transferring the first plurality of neutral atoms from the active zone to the storage zone, adiabatically transferring from the storage zone to the active zone, and adiabatically transferring from the active zone to the readout zone each have a constant jerk.

25. 25. The method of any one of claims 17 to 24, wherein the first array, the second array, and / or the third array of optical traps are two-dimensional arrays.

26. 26. The method of any one of claims 17 to 25, further comprising the step of directing the beam of light from the fourth laser to at least one acousto-optic deflector (AOD), and wherein the step of adiabatically moving the neutral atoms comprises varying a drive frequency of the at least one AOD.

27. 27. The method of any one of claims 17 to 26, wherein the first array, the second array, and / or the third array of optical traps are generated by directing a beam of light onto a spatial light modulator (SLM).

28. 28. The method of any one of claims 17 to 27, wherein the first plurality of neutral atoms are transferred simultaneously.

29. 29. The method of any one of claims 18 to 28, wherein the third array has a higher density than the first array.

30. 30. The method of claim 17, wherein encoding the first logical qubit comprises applying a CSS code.

31. 31. The method of claim 30, wherein the CSS code is selected from a surface code, a color code, a Steane code, and a hyper-graph product LDPC code.

Images