Fault-tolerant quantum computing

Abstract

Error detection in a quantum computer is provided. The quantum computer includes a plurality of qubits encoding a plurality of data qudits and an ancilla qudit. The qubits encoding the plurality of data qudits are arranged in a grouping, where the qubits encoding each of the plurality of data qudits are within an interaction distance of an interaction state of the qubit encoding the ancilla qudit. A leakage error into an interaction state of a first data qudit of the plurality of data qudits is detected by detecting the state of the ancilla qudit. Error correction in a quantum computer is also provided. The quantum states of the plurality of qudits are selected such that an angular momentum selection rule prevents mixing between the selected quantum states during leakage error into a non-interacting state of one of the plurality of qudits. The leakage error is corrected by optical pumping of the non-interacting state, which preserves the coherence of the selected quantum state in the absence of leakage error.

Description

[Technical field]

[0001] REFERENCE TO RELATED APPLICATIONS This application claims the benefit of U.S. Provisional Application No. 63 / 194,012, filed May 27, 2021, which is incorporated by reference in its entirety. [Background technology]

[0002] background Aspects of the present disclosure relate to fault-tolerant quantum computer computation, and more specifically, to systems and methods for error correction in quantum computers, such as those implemented using Rydberg atoms. Summary of the Invention

[0003] Quick Overview According to aspects of the present disclosure, systems, methods and computer program products are provided for fault-tolerant quantum computing.

[0004] In various aspects, a method of error detection in a quantum computer is provided. The quantum computer includes a plurality of qubits encoding a plurality of data qudits and an ancilla qudit. The qubits encoding the plurality of data qudits are arranged in a grouping, where a qubit encoding each of the plurality of data qudits is within an interaction distance of an interaction state of a qubit encoding the ancilla qudit. A leakage error of a first data qudit of the plurality of data qudits into the interaction state is detected by detecting a state of the ancilla qudit.

[0005] In some embodiments, each of the plurality of data qudits and the ancilla qudits is encoded in an atomic state of a neutral atom, hi some embodiments, the plurality of data qudits is encoded in an atomic state of a first species of neutral atom and the ancilla qudit is encoded in an atomic state of a second species of neutral atom.

[0006] In some aspects, each of the plurality of data qudits and ancilla qubits corresponds to a qubit.

[0007] In some embodiments, the interacting state is a Rydberg state.

[0008] In some embodiments, the grouping is a grouping of seven qudits.

[0009] In some embodiments, the grouping is a grouping of three qudits.

[0010] In various embodiments, a method of error correction in a quantum computer is provided. The quantum computer includes a plurality of qubits encoding a plurality of qudits. The quantum states of the plurality of qudits are selected such that an angular momentum selection rule precludes mixing between the quantum states selected during a leakage error into a non-interacting state of one of the plurality of qudits. The leakage error is corrected by optical pumping of the non-interacting state, which optical pumping preserves the coherence of the selected quantum state in the absence of the leakage error.

[0011] In some embodiments, each of the plurality of qudits is encoded in an atomic state of a neutral atom.

[0012] In some embodiments, selecting a quantum state of the plurality of qudits includes: selecting a first qudit state having a first magnetic quantum number and a second qudit state having a second magnetic quantum number, the first and second magnetic quantum numbers having opposite signs. In some embodiments, correcting leakage errors further includes: coherently transitioning atoms in the first qudit state to a first shelving state before the optical pumping; coherently transitioning atoms in the second qudit state to a second shelving state before the optical pumping; coherently transitioning a population of atoms in the first shelving state to the first qudit state after the optical pumping; coherently transitioning a population of atoms in the second shelving state to the second qudit state after the optical pumping, where the optical pumping does not transition atoms outside of the first shelving state, and the optical pumping transitions atoms from any ground state other than the first shelving state to the second shelving state.

[0013] In some aspects, each of the plurality of qudits corresponds to a qubit.

[0014] In various embodiments, a method for performing a controlled gate in a quantum computer is provided. The quantum computer includes a plurality of qubits encoding at least one target qudit and at least one control qudit. Conditionally, the qubits encoding the at least one target qudit are coherently transitioned from a plurality of states to corresponding shelving states, each selected from a first plurality of shelving states, according to a control state of the at least one control qudit, and the at least one control qudit excludes such transition when the control state is in an interacting state. The plurality of states is a subset of possible qudit states, and each possible qudit state can be populated by a decay process from at most one of the first plurality of shelving states. Conditionally, according to a control state of at least one control qudit, a qubit encoding at least one target qudit is coherently transitioned from a first plurality of states to a corresponding shelving state selected from a second plurality of shelving states if an error occurs during a transition from the first plurality of states to the corresponding shelving state, where the at least one control qudit precludes the transition when the control state is in an interaction state. Any of the qubits in the plurality of states are modified. Conditionally, according to a control state of at least one control qudit, a qubit encoding at least one target qudit is coherently transitioned from a shelving state of the first plurality of shelving states to a corresponding state of a shelving state from the plurality of states. Any qubit encoding a target qudit that is not in a qudit state is non-coherently transitioned to a corresponding qudit state.

[0015] In some embodiments, each of the plurality of qudits is encoded in an atomic state.

[0016] In some embodiments, each of the at least one target qudit and the at least one control qudit corresponds to a qubit.

[0017] In some embodiments, modifying any of the plurality of qubits comprises applying a unitary operation, hi some embodiments, the unitary operation is an X-gate.

[0018] In various aspects, a system is provided that includes a confinement system and a detector. The confinement system is configured to align a plurality of particles into an array, the plurality of particles being configured to encode a plurality of data qudits and an ancilla qudit, and the confinement system is further configured to align the plurality of particles encoding the plurality of data qudits into a grouping, where the particles encoding each of the plurality of data qudits are within an interaction distance of an interaction state of the particle encoding the ancilla qudit. The confinement system includes a laser source and a source of atomic cloud aligned to generate a plurality of confinement regions, the atomic cloud may be positioned to at least partially overlap the plurality of confinement regions. The detector is configured to detect the state of the ancilla qudit, thereby detecting a leakage error of a first data qudit of the plurality of data qudits into the interaction state.

[0019] In some embodiments, the array is two-dimensional.

[0020] In various embodiments, a system is provided that includes a confinement system and a plurality of laser sources. The confinement system is configured to align a plurality of particles into an array, the plurality of particles being configured to encode a plurality of data qudits and an ancilla qudits. The confinement system includes a first laser source aligned to generate a plurality of confinement regions and a source of an atomic cloud, the atomic cloud being arranged to at least partially overlap the plurality of confinement regions. The second laser source is configured to drive each of the plurality of particles into one of a plurality of quantum states, the plurality of quantum states being selected such that an angular momentum selection rule prevents mixing between the plurality of quantum states during leakage errors of the plurality of particles into a non-interacting state. The third laser source is configured to optically pump the non-interacting state, the optical pumping preserves the coherence of the plurality of quantum states in the absence of leakage errors.

[0021] In some embodiments, the array is two-dimensional. [Brief description of the drawings]

[0022] Brief description of some figures of the drawing [Figure 1A] FIG. 1A is a schematic diagram of an ancillary and data atom using seven-qubit encoding, according to an embodiment of the present disclosure. [Figure 1B] FIG. 1B is a schematic diagram of a circuit for implementing a procedure for measuring a stabilizer operator according to an embodiment of the present disclosure. [Figure 1C] FIG. 1C is a level diagram illustrating an example encoding of a qubit in a superfine clock state of 87Rb, according to an embodiment of the present disclosure. [Figure 1D] FIG. 1D is a schematic diagram of an ancillary and data atom using three-qubit encoding, according to an embodiment of the present disclosure. [Figure 1E] FIG. 1E is a schematic diagram of a circuit for measuring a stabilizer operator according to an embodiment of the present disclosure. [Figure 2A]FIG. 2A is a level diagram illustrating a Rydberg blockade mechanism according to an embodiment of the present disclosure. [Figure 2B] FIG. 2B illustrates a protocol for performing a multi-qubit entangled Rydberg gate according to an embodiment of the present disclosure. [Diagram 3] FIG. 3 illustrates the reordering of physical gates in performing a logical CCZ operation, according to an aspect of the present disclosure. [Figure 4] FIG. 4 is a schematic diagram of a circuit for implementing Steane's Latin rectangular encoding method according to an embodiment of the present disclosure. [Diagram 5] FIG. 5 illustrates an optical pumping protocol for converting non-Rydberg leakage errors to Pauli-Z errors according to an embodiment of the present disclosure. [Figure 6] FIG. 6 is a schematic diagram of a circuit for measuring a stabilizer according to an embodiment of the present disclosure. [Figure 7] FIG. 7 illustrates a pulse sequence for target atoms in a bias-preserving CNOT gate according to an embodiment of the present disclosure. [Figure 8] FIG. 8 is a schematic diagram of a circuit using an ancilla qubit and multiple Rydberg states to eliminate X-type errors resulting from symmetric qubit decay, according to an embodiment of the present disclosure. [Figure 9] FIG. 9 is a schematic diagram of a circuit that provides a pieceable, fault-tolerant implementation of a Toffoli gate in a repetitive code, according to an embodiment of the present disclosure. [Figure 10] FIG. 10 is a schematic diagram of a circuit implementing a logical Hadamard gate using a logical Toffoli gate combined with a fault-tolerant measurement on the X bias, in accordance with an embodiment of the present disclosure. [Figure 11] FIG. 11 is a relevant level diagram for performing error correction using neutral alkaline earth Rydberg atoms, according to an embodiment of the present disclosure. [Figure 12] FIG. 12 is a graph of branching ratios for BBR transitions outside of a stretched Rydberg state, according to an embodiment of the present disclosure. [Figure 13] FIG. 13 is a schematic diagram of a circuit for detecting atom loss according to an embodiment of the present disclosure. [Figure 14] FIG. 14 is a schematic diagram of a circuit using two ancilla qubits and multiple Rydberg states to implement a bias-preserving Toffoli gate according to an embodiment of the present disclosure. [Figure 15A] FIG. 15A is a schematic diagram of an ancillary and data atom using seven-qubit encoding, according to an embodiment of the present disclosure. [Figure 15B] FIG. 15B is a schematic diagram of an ancillary and data atom using three-qubit encoding, according to an embodiment of the present disclosure. [Figure 16] FIG. 16 is a schematic diagram of an ancillary and data atom using three-qubit encoding on a square lattice geometry, according to an embodiment of the present disclosure. [Figure 17] FIG. 17 is a schematic diagram of an apparatus for fault-tolerant quantum computing according to an embodiment of the present disclosure. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0023] Detailed Description Neutral atomic arrays have emerged as a promising platform for quantum information processing. However, a limitation to the application of these systems is the ability to perform error-correcting quantum operations. One important remaining obstacle for large-scale quantum processing in such systems relates to the finite lifetime of atomic Rydberg states during entanglement operations. To entangle qubits in these systems, atoms are typically excited into Rydberg states, which can decay or give rise to various correlated errors. Because Rydberg state decay errors can give rise to many possible leakage channels and correlated errors outside the computational subspace, they cannot be directly addressed by traditional methods of fault-tolerant quantum computing.

[0024] This disclosure provides a detailed analysis of the effects of these error sources in neutral atom quantum computing and proposes hardware-efficient fault-tolerant quantum computing schemes that mitigate them. By using a specific structure of the error model, the multi-level nature of atoms, and bipolar selection rules, this disclosure provides novel and demonstrably efficient methods for dealing with the most significant errors associated with the decay of atomic qubits to states outside the computational subspace. These advances enable significant reductions in resource costs for fault-tolerant quantum computing compared to alternative, general-purpose schemes, even when these novel types of errors are offset. The experimental feasibility of these protocols is demonstrated in 87 Rb, 85 Rb or 87 We illustrate this through a concrete example with a qubit encoded on Sr atoms. The protocols provided herein can soon be implemented using state-of-the-art neutral atom platforms with qubits encoded on both alkali and alkaline earth atoms.

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

[0026] Neutral atomic systems have emerged as a promising platform for quantum information processing. The exceptional coherence time of their ground state enables long-lasting quantum memories, while fast, high-fidelity quantum operations can be achieved by individually treating atoms with laser pulses and coupling them to highly excited Rydberg states. Furthermore, many individual neutral atoms can be deterministically aligned with arbitrary geometric structures in two- and three-dimensional systems. Experiments have demonstrated quantum operations in large arrays of atoms for applications ranging from quantum computing to quantum simulation and quantum metrology. Some advances in the dynamic reconfiguration of atoms have even led to the realization of logical qubits that are encoded in color, surface or trick codes, a key step in implementing quantum error correction (QEC) on neutral atomic platforms.

[0027] Although current experiments already describe a remarkable level of quantum control, experimental imperfections such as Rydberg state decay ultimately limit the depth of accessible quantum manipulation. Therefore, it is essential to consider quantum error correction (QEC) protocols to scale up computational sizes. In particular, such protocols should be fault-tolerant and protect against significant sources of errors occurring in either the computation, error detection, and encoding and decoding states. Fault-tolerant protocols for general quantum platforms do not address the specific errors present in the Rydberg atomic setting. Indeed, Rydberg-atom QEC faces a seemingly formidable difficulty: the Rydberg state can decay into multiple other states, which can result not only in leakage errors from the computational space, but also in high-weight correlated errors from subsequent undesirable blockade effects.

[0028] To address these issues, this disclosure describes the effects of these inherent errors and how to exploit the unique capabilities of the structure of Rydberg systems and error models to design hardware-efficient fault-tolerant quantum computing (FTQC) schemes that handle these errors despite the aforementioned difficulties. This tailored FTQC approach can also be significantly more resource efficient than common alternatives, which often require many qubits and quantum operations with smaller threshold errors than are achievable in near-term experiments to perform non-Clifford logical operations either directly or using state excerpts. The high overhead associated with such protocols is why experimental descriptions of QEC have so far been limited to only one or two logical qubits.

[0029] This disclosure first provides a detailed description of the errors arising from the finite lifetime of Rydberg states or from imperfections in Rydberg laser pulses from the perspective of QEC. A method for performing hardware-efficient fault-tolerant quantum computing (FTQC) is provided while addressing the inherent sources of errors in the neutral Rydberg atom platform (FIG. 1).

[0030] Second, this disclosure shows that nine atoms - seven data qubits and two ancilla qubits - are sufficient to encode each logical qubit in a fault-tolerant, seven-qubit Steane code based performance of a universal set of fault-tolerant quantum operations is provided.

[0031] It is shown that for atomic species with sufficiently large nuclear spins and high fidelity ground state operations, leading-order fault-tolerant quantum computations can be achieved using a simple three-atom iterative code. Since the three-qubit iterative code does not correct any Pauli-X errors, it cannot be used for FTQC in typical settings. However, since the error model for the Rydberg atomic settings does not include any Pauli-X errors at the leading order (as shown below), the iterative code is applicable in these platforms. Therefore, the term "leading-order fault-tolerant" is used when describing the Ryd-3 protocol to clearly emphasize this point. Both the seven-atom and the three-atom codes can be implemented on scalable geometries where the atoms are arranged in a triangular lattice configuration (see Fig. 1A, 1D), allowing their proof and application in near-term implementations.

[0032] Various aspects of hardware-efficient FTQC are based on several important observations. First, a realistic error model is provided to show that by using the bipolar selection rule, the Rydberg blockade effect and optical pumping techniques, the complex leakage errors associated with Rydberg atomic damping can be reduced to simple Pauli-Z type errors (Figure 1C). Second, it is shown that all of the logic gates of the universal set for logic state preparation, stabilizer measurements and the seven-qubit code can be implemented as sequences of physical gates that are commutative with Pauli-Z errors up to single-qubit unitaries at the beginning and end of the operation. Thus, any Rydberg gate error cannot propagate up to other qubits within a single stabilizer measurement or logic operation and can be efficiently detected and corrected using significantly fewer entanglement gates than existing general-purpose schemes (Figure 1B, Tables 1 and 2). Third, to adopt an even more compact three-atom code for this error model, it is provided that all error correction and logic operations can be performed in a bias-preserving manner-i.e., Pauli-X and Y errors cannot appear at any stage of the computation. For atomic species with sufficiently high nuclear spin, this can be achieved by designing novel laser pulse sequences to entangle gates between Rydberg atoms, which can be used to perform bias-preserving controlled-NOT (CNOT) and Toffoli gates (see Fig. 1E and Fig. 7). Fourth, by testing the qubit-connectivity required to perform all error correction and logic operations, it is shown that both the seven-atom and three-atom codes can be performed on scalable geometries where the atoms are arranged in a triangular lattice configuration (Fig. 1A,D), allowing their verification and testing in upcoming experiments.

[0033] The present disclosure provides an important advance over conventional methods by introducing a sufficiently efficient approach to handle leakage of qubits outside the computational subspace. For traditional QEC proposals, such leakage is one of the most difficult and expensive types of errors to detect and handle, making it undesirable to encode qubits in large multi-level systems such as neutral atoms. The methods provided herein for handling these leakage errors utilize techniques based on optical pumping, so that the multi-level structure of each atom can be utilized as part of the redundancy required for QEC. Although the focus of this specification is on neutral atom-based quantum information processes, these techniques are adaptable to many other hardware platforms, for example, they may also greatly facilitate the correction of leakage-type errors in superconducting qubits or trapped ions. For the Rydberg atom system described herein, a method is provided that converts all highest order errors to Pauli-Z type errors (see FIG. 1B), enabling the development of particularly efficient FTQC protocols.

[0034] Referring to FIG. 1A, an architecture for FTQC using Rydberg atoms according to an embodiment of the present disclosure is illustrated.

[0035] Figure 1A shows the atomic geometric layout for FTQC using a seven-qubit encoding. Data (D, 101) and ancilla (A, 102) atoms are arranged at the vertices of a triangular lattice, with seven data atoms comprising a logical qubit (dotted hexagon 103). The grey dotted line 104 indicates the required Rydberg interaction range.

[0036] FIG. 1B illustrates a circuit that implements the procedure for measuring the stabilizer operator, X1X2X3X4, for the seven-qubit code supported on the four data atoms highlighted in FIG. 1A. Optical pumping (OP, 105) is performed after the fully controlled phase gate (106) to correct leakage to other basis states. An ancilla qubit A2 (107) measures the stabilizer eigenvalues, and ancilla qubit A1 (108) is used to detect and correct Rydberg leakage errors (109). In this way, all gate errors are converted to Pauli-Z type errors and do not propagate to other qubits.

[0037] FIG. 1C shows 87 A level diagram showing an example encoding of a qubit in a hyperfine clock state of Rb. The dominant intrinsic errors for this encoding arise from blackbody radiation (BBR, 110), radiation damping (RD, 111) and intermediate state scattering (112). Their effects can be determined via the dipole selection rule (113), and the associated leakage errors can be corrected by exploiting the Rydberg blockade effect or optical pumping.

[0038] FIG. 1D illustrates a geometric layout for quantum computing with the highest degree of fault tolerance using a three-atom encoding. Data 114 and ancillary 115 atoms are arranged at the vertices of a triangular lattice, with three data atoms comprising a logic qubit (116). In this case, different blockade radii, R B,1 and R B,2 Two Rydberg states with (117 and 118 respectively) are required.

[0039] Figure 1E shows a circuit for measuring the stabilizer operator X1X2 of the repeating code supported on the two data atoms highlighted in Figure 1D. By combining a novel entanglement pulse sequence with Rydberg leakage correction and optical pumping, a bias-preserving CNOT gate is implemented (see Figure 7), allowing the implementation of QEC without introducing X or Y errors at any point in the computation.

[0040] static magnetic field

number

[0041] Although this procedure provides an efficient scheme for entangling two or a few atoms, for large-scale quantum computer calculations, the finite lifetime of the Rydberg state presents a significant source of error even when the rest of the experimental setup is perfect. This lifetime is determined by several contributions. First, interaction with a blackbody photon can induce a transition from the nS state to a nearby Rydberg n'P state of higher or lower energy; such an error is then referred to as a blackbody radiation induced (BBR) error. Second, spontaneous emission of the optical frequency lattice can result in radiative decay (RD) to the low-lying P state, which quickly relaxes to the ground state set.

[0042] Furthermore, when using a multiphoton Rydberg excitation scheme for Rydberg pulses, another inherent source of error during the Rydberg gate is photon scattering from intermediate states. These error channels are illustrated in FIG. 1C.

[0043] For the purposes of QEC, these errors can be formally described as follows: BBR errors result in quantum jumps from the qubit |1> state to the Rydberg P state (corresponding to leakage errors) and Pauli-Z errors within the qubit ensemble, while RD and intermediate state scattering can also result in quantum jumps from |1> to the Rydberg nS state or other hyperfine ground states. Selection rules and branching ratios determine the relative error probabilities. In addition to these intrinsic errors, errors in the experimental setup, such as Rydberg pulse imperfections or finite atomic temperatures, are also considered. These experimental errors fall into a subset of the RD error model, and therefore can be addressed using the techniques provided herein. Throughout this work, it is assumed that rotations within the hyperfine ensemble have a much higher fidelity than Rydberg pulses, as is typically the case. Such errors can also be constrained to higher orders by using existing experimental methods, such as composite pulse sequences, or can be addressed by incorporating conventional QEC techniques, such as chaining.

[0044] Referring to FIG. 2A, the Rydberg blockade mechanism according to the present disclosure is illustrated. Δ is the Rydberg laser detuning and the Rydberg interaction strength

number

[0045] FIG. 2B shows the construction of a multi-qubit entangled Rydberg gate R(C1,C2,...,C a ; T1,T2,...T b ) shows a protocol for performing a resonant π pulse.

number

[0046] The labels on the arrows indicate the ordering of the pulses. This Rydberg gate can be realized by conjugating all the reference qubits and one target qubit by Pauli-X manipulation, or

number

[0047] Pauli-Z Error Reduction To guard against the above mentioned errors, we use three crucial observations (see Fig. 1C). First, note that the quantum jump from |1> to the Rydberg state associated with the BBR can be detected via the Rydberg blockade effect by using nearby ancilla qubits, and then converted to a Pauli-Z type error by incoherently re-jumping the Rydberg state back to the ground state ensemble or by ejecting the Rydberg atom and replacing it with a new atom prepared in the |1> state. Second, the quantum jump from |1> to a ground state sublevel outside the qubit subspace can be corrected via optical pumping techniques. This is particularly useful in that, unlike alternative proposals for correcting leakage errors, we do not require any qubit measurements for feed-forward correction.

[0048] Third, for atomic species with sufficiently large nuclear spin, the dipole selection rule prevents the elongated Rydberg states from decaying to certain ground-state sublevels. By exploiting this multilevel structure of neutral atoms together with high-fidelity manipulation of the hyperfine states, it is ensured that RD and intermediate-state scattering errors do not result in |1>→|0> transitions, thereby eliminating X- and Y-type errors from the error model. This reduction in error types can significantly ease the resource requirements for FTQC. [Table 1]

[0049] Table 1 provides a comparison of resource costs for fault-tolerant measurements of all stabilizers for correcting Pauli errors. The numbers in parentheses indicate the maximum number of operations required in the unlikely scenario that an error is detected. Details on how to obtain the gate counts for the Ryd-7 and Ryd-3 protocols can be seen below. [Table 2]

[0050] Table 2 provides a comparison of resource costs for the most expensive fault-tolerant logic operations. CCZ denotes a three-qubit controlled-controlled phase gate, and H denotes a single-qubit Hadamard gate. The numbers in brackets indicate the maximum number of operations required in the unlikely scenario that an error is detected. For the Rydberg protocol, the gate counts shown assume a blockade radius of 3d, where d is the nearest-neighbor lattice space. Details of how to obtain the gate counts and blockade radius requirements for the Ryd-7 and Ryd-3 protocols can be found below.

[0051] Fault Tolerant Protocols Here we describe two FTQC protocols to handle these inherent errors in the neutral Rydberg atom platform. The first is based on a 7-qubit Steane code, and the second uses a 3-qubit iterative code; the latter is more compact and efficient but has further experimental requirements such as more complex encoding of control and logic operations over multiple Rydberg states.

[0052] Note that to realize the 7-qubit code (Ryd-7), logic state preparation, stabilizer measurements and a universal set of logic gates (Hadamard and Toffoli) can be performed up to single-qubit unitaries at the beginning and end of operations using only controlled-phase (CZ) or controlled-controlled-phase (CCZ) gates. For example, while stabilizer measurements are typically presented as sequences of CNOT gates between data and ancillary atoms, these CNOT gates can be constructed by conjugating CZ and Hadamard gates on the target qubit. Mapping each Rydberg gate error to a Pauli-Z error ensures that it is interchangeable with all subsequent entanglement gates in logic operations or stabilizer measurements and therefore does not spread to other qubits (Figure 1B). The resulting single-qubit X or Z error can be corrected by the 7-qubit code in subsequent rounds of QEC. This eliminates the need for flag qubits that would otherwise be necessary to prevent error diffusion. To further reduce the resource cost for experimental execution, further use is made of the structure of the Rydberg error model, the stabilizer measurement circuit and the logical operation of the 7-qubit code. For example, one important observation is that leakage errors to other Rydberg states do not need to be corrected after the entire Rydberg gate but can be postponed until the end of the stabilizer measurement (e.g., Fig. 1B). This allows for the minimization of the number of intermediate measurements required for each FTQC component, which is typically a limiting factor in state-of-the-art neutral atom experiments.

[0053] The simplified error model introduced by the transformation of all Rydberg gate errors into Pauli-Z errors motivates the use of a 3-qubit iteration code instead of a 7-qubit code (Ryd-3) to design a fault-tolerant protocol of the highest order. In this case, the stabilizer measurement circuit also consists of a CNOT gate on the data atoms controlled by the ancillary. However, the respective CNOT implementation must be modified: when a CZ gate is conjugated with a Hadamard gate as in Fig. 1B, the Pauli-Z type errors occurring during the CZ gate are transformed into Pauli-X errors after Hadamard. Such errors can no longer be corrected by the iteration code.

[0054] Further errors, such as radiative decay of the control qubit prior to manipulation of the target qubit, can result in diffuse and correlated errors.

[0055] These errors can be addressed via a protocol for directly implementing the CNOT gate in a bias-preserving manner, and these implementations do not generate any Pauli-X and -Y errors for the highest orders (Figures 7 and 8). The protocol provided herein is particularly useful for the analysis of the rich multilevel structures of atoms with large nuclear spins (I ≥ 5 / 2, e.g. 85 Rb, 133 Cs, 87Sr...) as well as further Rydberg states for shelving. Furthermore, the fact that pulses between specific (hyperfine) levels can be performed with very high fidelity is reinforced, so that the highest order errors only include Rydberg state decay or Rydberg pulse imperfections. This assumption is particularly important if bias-preserving CNOT gates cannot be performed in any qudit system with a finite number of levels without having such structure in the error model. To avoid this, it is shown that a pulse sequence directly performs a hyperfine Pauli-X gate on the target qubit only if there is a nearby Rydberg atom (there is no need for a subsequent Hadamard gate), and any errors during this sequence can be mapped to Pauli-Z errors. Furthermore, correlated errors due to control atom decay can be prevented by using multiple control atoms, so that if one atom decays, the remaining atom(s) still ensure proper gate operation on the target atom. This bias-preserving CNOT protocol can be straightforwardly generalized to perform bias-preserving Toffoli operations, enabling highest order fault-tolerant execution of each operation of the three-atom repeat code. Throughout this disclosure, the term "highest order fault-tolerant" is used in reference to the Ryd-3 protocol, where the framework provided herein does not inherently address all single qubit errors, but existing experimental techniques such as composite pulse sequences can be used in conjunction with the protocols provided herein to suppress such errors to higher orders (see below).

[0056] The protocol provided herein is preferable to the general-purpose FTQC proposal. In particular, the number of physical qubits and gates required for both approaches provided herein is dramatically reduced (Tables 1 and 2). For example, as seen in Table 2, which performs the highest cost operation from the logic gate setup, the Ryd-7 protocol requires only two ancillar qubits compared to 72 ancillars in Yoder, et al. See TJ Yoder, R. Takagi, and IL Chuang, Universal fault-tolerant gates on concatenated stabilizer codes, Phys. Rev. X vol. 6, p. 031039 (2016). Similarly, Ryd-7 uses up to 60 2-qubit gates (in case of error detection) to perform this logic operation instead of 1416 gates as in Chao, et al. See R. Chao and BW Reichardt, Fault-tolerant quantum computation with few qubits, Quantum Information vol. 4, p. 42 (2018). Such significant reductions are possible for the protocols provided herein because both the special structure of the error model and the unique capabilities of the Rydberg setting are leveraged.

[0057] Certain single qubit errors addressed in Chao and Yoder are not corrected in the protocols provided herein (e.g., Pauli-X errors caused by rotations in the hyperfine set). However, Chao and Yoder did not consider additional types of errors, such as leakage errors, which are corrected by the protocols provided herein. Indeed, incorporating leakage correction would further increase the resource cost for the earlier proposals. Thus, Tables 1 and 2 should be interpreted as a comparison of the cost of ensuring fault tolerance for the highest order source of error in a given setting. In the case of Chao and Yoder, these errors include all single qubit Pauli errors, not leakage errors, and in a Rydberg system, leakage errors must be addressed at the highest order, but certain single qubit errors can be ignored. Such a significant reduction in cost is possible for the protocols provided herein, since both the special structure of the error model and the unique capabilities of the Rydberg setting are enhanced.

[0058] Experimental Run For a scalable implementation of the FTQC protocol presented here, it is important to consider the geometry of the atoms. Also, the Rydberg entanglement gates have a blockade radius R B Since each protocol can only be executed between atoms in R B (In units of d, this is the minimum atom-atom separation). It is shown that both the Ryd-7 and Ryd-3 protocols can be naturally implemented when the atoms are placed at the vertices of a triangular lattice as shown in Fig. 1A,D. For both protocols, the required Rydberg gates are determined by the blockade radius (R for Ryd-7). B or for Ryd-3, the larger radius R B.1 ) is larger than 3d. This requirement could be further reduced in both cases if it were possible to move atoms during certain operations while preserving the coherence of the hyperfine ground state, a capability that is now available.

[0059] Each component of the FTQC scheme provided herein may be implemented in upcoming experiments. High fidelity control and entanglement is available in neutral alkali atomic systems. Near-deterministic loading of atoms into the lattice structures shown in Figures 1A, 1D is available in two and three dimensions.

[0060] To perform QEC according to the protocols provided herein, a key requirement is the ability to measure individual qubits and / or detect the Rydberg ensemble and perform feed-forward corrections. One approach to perform fast measurements of individual qubit states in neutral atomic arrays is to use an array with two atomic species, where the data atoms are encoded in one atomic species and the ancillary atoms are encoded in the other species so that they can be easily measured. Alternatively, this fast qubit state detection can be performed in a single-species array using resonant photon scattering on a cyclic transition, if the atoms can be moved far enough away from the rest of the array to mitigate the effects of crosstalk while preserving the quantum coherence of the remaining atoms. Fast detection schemes can be illustrated in experiments with large atomic ensembles using Rydberg electromagnetically induced transparency (EIT) technology and can be integrated with tweezer array platforms currently used for quantum computer calculations. In these procedures, the Rydberg blockade effect is transformed to remove features in the absorption spectrum, and the collectively enhanced Rabi frequency allows for ultrafast detection in microseconds.

[0061] Finally, although this disclosure focuses primarily on neutral alkali atoms, alkaline earth atoms may also be used for Rydberg-based quantum computing. Clock transitions in these atoms allow for high-fidelity qubit encoding, and the large nuclear spin in fermionic species is particularly advantageous for the protocols provided herein.

[0062] Error channels in the Rydberg atom. The dominant error mechanisms for quantum operations involving Rydberg atoms (Figure 1C) are analyzed below. Since the dominant errors in single qubit operations can be suppressed to higher orders via composite pulse sequences, the focus can be primarily on errors arising during Rydberg-mediated entanglement operations. The decay channels of Rydberg states include blackbody radiation induced (BBR) transitions to lower lying states and spontaneous radiative decay (RD) transitions. Depending on the specific choice of atomic species, another source of error for Rydberg gates can be scattering from intermediate states when two- or multiple-photon excitation schemes are used; this can be achieved by using a 10-nm lattice excitation scheme, which is a 10-nm lattice excitation scheme. 87 Rb or 85 This is true for the excitation of Rb to the Rydberg nS state. These effects are hypothesized to be the major source of errors that arise during the entanglement operation and to contribute to the highest order error model in the total error probability.

[0063] Error modeling for BBR transitions. If a BBR transition occurs on one of the atoms during an entanglement gate, this signals that this atom started in the |1> state, since |0> is not coupled to |r>. Such a procedure corresponds to a quantum jump. The resulting states are mainly nearby Rydberg states |r'>, which are compatible with the bipolar selection rule. Due to the relatively long lifetime of the Rydberg states, it can be assumed that the atom will not decay again within the period of several Rydberg gate operations, since these are higher order processes. In this case, since state |r'> is not de-excited in ensuring the operation, one serious consequence of the BBR quantum jump is that the remaining Rydberg operations on the atoms within the interaction range are affected by blockade, potentially resulting in multiple correlated Pauli-Z type errors. Even in the less intuitive case where no quantum jump occurs during the gate operation, the state of the atom is still modified due to the evolution under a non-Hermitian Hamiltonian: it is more likely that the atom will start outside the |0> state.

[0064] For the purposes of QEC, it is useful to represent the decay channel in the Claus operator form, where the time evolution of the density operator is

number

number

number

number

[0065] During entanglement operations, these BBR errors can produce correlated errors. For example, in the Rydberg gate shown in FIG. 2B, if the control qubits are all in the |0> state, the target qubit can only suffer from a BBR error. a Z b For a gate, the possible correlated errors are those that occur on one of the qubits along with Z-type errors on some or all of the remaining qubits included in that gate, as expressed by the Claus map M r' Or it may include one of M0.

[0066] The rate of BBR transition from a given Rydberg state nL to another specific state n'L' can be calculated from the Planck distribution of photons at a given temperature T and Einstein coefficients for the corresponding transition. 87For the Rb atom, there are four dominant final states associated with these BBR errors; these are illustrated in Figure 1C. The total rate of BBR transitions, summed over all possible final states, is given in Equation 3.

number

[0067] In Equation 3, k B is the Boltzmann constant, c is the speed of light, and n eff The energy:

number

[0068] Error modeling for RD transitions. The spontaneous radiative event corresponding to the RD transition can be modeled as a quantum jump involving the emission of an optical wavelength photon. However, unlike BBR, the resulting state is a low-lying P state, which decays rapidly back to the ground state set. 87 For the stretched Rydberg state of Rb, the RD transition is almost entirely a two- or four-photon decay process to one of the five states in the ground-state ensemble shown in Fig. 1C. For the purposes of QEC, we consider separately the cases of decay to the qubit |1> state and to one of the other ground-state sublevels. Since a spontaneous emission event can occur at any time during the Rydberg laser pulse, the first type of decay can result in a final state that is a superposition of |1> and |r>. When averaging over all possible decay times during the entire pulse, it is found that these errors can be modeled using a combination of Z-type errors and leakage to the |r> state, where the Claus operators

number

[0069] At the same time, the decay to one of the other ground state sublevels shown in Fig. 1C leads to leakage outside the computational subspace (without affecting the Rydberg operations on the neighboring atoms) similar to the traditional QEC setup. That is, for each hyperfine state |f>≠|1>, the Claus operator

number

number

[0070] As in the case of BBR, the absence of quantum jumps results in a population of atoms that are shifted toward the |0> state, which can be modeled using Pauli-Z errors. RD errors can also result in correlated errors if they occur during the first-order entanglement gate illustrated in FIG. 2B. For example, if the control qubit undergoes an RD transition, the target qubit Rydberg pulse can become resonant. In this case, possible correlated errors can include one of the aforementioned Claus maps occurring on one of the qubits, along with Pauli-Z and / or |r><1| errors on some or all of the remaining qubits involved in such a gate.

[0071] As noted above, the rate of the BBR transition depends on the temperature T and n eff However, the total RD rate is temperature independent. Due to the reduced overlap between atomic orbitals, it is

number

[0072] Errors from intermediate state scattering. When using multiphoton excitation to couple the |1> state to the Rydberg state, scattering from intermediate states can introduce another important intrinsic source of error. + Using the polarization and dividing the intermediate state into P with the lowest possible n 3 / 2 By choosing , the intermediate state scattering channels form a subset of the RD channels - as shown in Fig. 1C, they can only cause decay 112 to the qubit |1> state or to the two other hyperfine ground states. 3 / 2Although this is still true up to the highest order when using intermediate states with higher n such as |1> and |0>, (highly unlikely) four-photon processes could potentially result in mixing between the qubit states |1> and |0>. Therefore, whenever intermediate state scattering is not explicitly mentioned in the following sections, it is assumed that it is integrated with the RD error. This error rate can be suppressed by increasing the intermediate laser detuning in the multiphoton transition, while also increasing the laser power.

[0073] Experimental imperfections Although the processes of BBR, RD and intermediate state scattering constitute the dominant errors for the Rydberg-mediated collective gate, it is also important to consider other types of errors, such as technical imperfections in the experimental setup. The most significant errors of this kind are atomic losses and fluctuations in the laser phase, intensity and frequency. These errors can be treated together with the other errors mentioned above, since all Rydberg laser fluctuations can be modeled using Pauli-Z errors and leakage to |r> states. Finite atomic temperatures, which produce velocity broadening and Doppler widths on the Rydberg transition, similarly produce Pauli-Z errors and leakage to |r> states. Temperature-induced positional broadening causes similar errors, and due to the robustness of blockade-based gates, these errors can even become negligible with sufficiently large interaction strengths. On the other hand, atomic losses form a more complicated version of leakage errors (referred to as erasures in the quantum information literature). However, as shown below, such errors can be efficiently handled in the framework of the present invention as well. In certain cases, the special nature of these errors can be further enhanced to improve QEC efficiency.

[0074] Experimental imperfections can also affect the hyperfine qubits used to store quantum information and perform single-qubit gates. However, these mainly result in Pauli-Z errors and leakage into other hyperfine states, which we group together with the error types mentioned above. Moreover, they tend to be significantly smaller sources of error than for two-qubit gates. By choosing magnetically insensitive transitions for the qubit states, the highest order errors arising from magnetic field fluctuations are eliminated. However, Z-type detuning errors can also result from differential light shifts from optical traps. Thus, although finite atomic temperatures, fluctuation tweezer dynamics and atomic heating can cause detuning, these can be mitigated by applying standard dynamical decoupling sequences to achieve qubit coherence times T2 of about 1 second. Other hyperfine m F Leakage into the qubit state can also occur due to Raman scattering from the tweezer light, but these effects can be largely suppressed for periods larger than 10 s by sufficiently detuning the tweezer light. The qubit state is F The bit-flip X and Y error rates from tweezing stimulated scattering are even smaller because they are separated by θ = 2 (nuclear spin-flip transitions). Finally, because the qubit transitions are at microwave frequencies and microwave phase stability can be exceptional on Raman lasers used for single-qubit operations, temperature-induced Doppler effects, which can in principle produce Z-type errors, can be neglected.

[0075] At the same time, as noted above, certain experimental imperfections associated with hyperfine rotation are not directly corrected by the protocols provided herein, but may be minimized or suppressed through other mechanisms, such as composite pulse sequences. For example, the major source of single-qubit gate errors in recent experiments includes laser amplitude drift or pulse miscalibration, which can result in X-, Y-, and Z-type errors. However, these coherent errors can be significantly suppressed by using composite pulse sequences. In particular, the BB1 ​​pulse sequence suppresses pulse amplitude errors to the sixth order. On the other hand, the error rates associated with phase noise in single-qubit gates are typically quite small: for example, 171 Yb + Phase noise in hyperfine qubits has been shown to limit the coherence to an order of magnitude of 5,000 s. Other sources of frequency fluctuations include a 4 ms delay for Rb qubits.

number

[0076] Error Channel Summary It is shown herein that the multi-level nature of neutral atoms gives rise to various complexities in the error model, such as the possibility for many decay channels and Rydberg leakage errors that affect many subsequent operations, resulting in correlated errors of high weight. Despite these complexities, one important feature of the error model provided herein makes it substantially simpler than the all-Pauli error setting tested in the more general setting - no Pauli-X or Y type errors are introduced during the Rydberg gate. Indeed, in the following section, it is shown how all the additional leakage and correlated errors in the error model can be converted to Z type errors, which is used to design an FTQC protocol with substantially reduced resource costs.

[0077] FTQC with 7-qubit Steane code Once the error model for Rydberg operations is established, fault-tolerant schemes are provided to detect and correct these errors and perform a universal set of logical operations. The key concept of this construction is the ability to introduce an ancilla qubit to convert all errors described in the previous sections to Pauli-Z type errors by using blockade effects, bipolar selection rules and optical pumping (see FIG. 1C). Protocols where only BBR errors are significant (in the limit of higher Rydberg principal quantum number n) are described first, as the error model and QEC mechanism are easier to understand in this case. The universal gate set developed herein includes logical Hadamard gates and logical controlled-controlled phase (CCZ) or Toffoli gates. A more general case involving both BBR and RD errors is described. Then, the resource cost of these protocols is compared to other fault-tolerant computational schemes, and ideas for scalable computation are discussed. The final scheme presented in the following section is referred to as Ryd-7. Throughout this section, 87 A qubit encoded in Rb is used as an example to illustrate the protocol.

[0078] Various equivalent definitions of FTQC are given in the literature for traditional error models, but to accommodate the possibility of Rydberg leakage errors - i.e. any Rydberg ensemble remaining after gating - the following more rigorous one must be applied:

[0079] Interval-d QEC codes have order (p tot ) t It is fault-tolerant up to t, with single-qubit Pauli errors of at most t, where

number

[0080] This final requirement is important because any remaining Rydberg population may block future Rydberg gates.

[0081] Below, the code spacing d=3 and

number

[0082] It is clear to show that any interval-3 code that satisfies the above properties is fault-tolerant.

[0083] In the following, the term data qubit is used to refer to a physical qubit used to encode a logical qubit, and ancilla qubit is used to refer to a physical qubit used to perform stabilizer measurements or detect errors.

[0084] FTQC with BBR errors 1. Qubit Encoding The quantum code in this example is based on the 7-qubit Steane code, which uses a logical state encoding derived from the classical binary Hamming code:

number

[0085] The stabilizer operator for this code is

number

[0086] In equation 8 and the rest of the manuscript where appropriate, the tensor product symbols and qubit indices are omitted, and the jth operator in each product is assumed to act on qubit j. Measurements of the stabilizers g1, ... g6 allow for the unique identification and correction of single-qubit X and Z errors. For example, the absence of any error implies that all stabilizers g j = +1, and the Z error on the first qubit corresponds to g3 = -1 and g for all j ≠ 3. j =+1. The error can then be corrected via an appropriate single-qubit gate.

[0087] 2. Error detection and correction To fault-tolerantly detect and correct errors associated with BBR events, one must be able to handle both Rydberg leakage and Pauli-Z errors. For the former case, even though leakage errors in traditional QEC settings can be particularly difficult to detect and correct, the specific form of leakage caused by BBR errors makes them much easier to identify - one can use ancilla and blockade effects to detect leaky Rydberg ensembles. Specifically, nearby ancilla qubits can be in the state

number

[0088] Once detected, such errors can be converted to atom loss errors or Z-type errors. The conversion of errors to atom loss errors reinforces the fact that the Rydberg atom naturally ejects itself due to the anti-trapping potential of the tweezers and can be directly ejected in about 100 ns by pulsing a weak ionizing electric field (about 10 V / cm), which removes the ion and the electron. The exact location of the ejected atom can be determined by the following atom loss protocol outlined below; the error can then be corrected by replacing the ejected atom with a new atom prepared in the |1> state (thereby converting it to a Z-type error) and applying another round of QEC. To reduce the need to apply the atom loss protocol, a preventative step can be added after every entanglement gate, which incoherently pumps any remaining population of some of the most likely Rydberg states back into the qubit |1> state. This procedure is described further below, along with more details on the conversion of Rydberg population errors.

[0089] For fault-tolerant error detection and correction, it is important to note that the ancillary used to look for the Rydberg ensemble may also suffer from BBR errors. This can be resolved by repeating the detection protocol upon discovery of a BBR error and also using a multi-step measurement procedure on the ancillary qubits. Such a protocol is assumed below when using an ancillary to detect the Rydberg ensemble.

[0090] To detect and correct Pauli errors in a fault-tolerant manner, the stabilizers are measured in a manner that is robust against possible errors during the detection procedure. Since the Steane code is a CSS code, the stabilizers for this 7-qubit code are either products of Pauli-X operators or products of Pauli-Z operators. A non-fault-tolerant method for measuring the products of four Pauli-X operators (stabilizers g1, g2 or g3) uses four controlled phase gates conjugated by Hadamard (Figure 1B). Since Rydberg gate errors can occur during this protocol, a second ancillary qubit is used to detect BBR errors after each entanglement operation and to convert them to Z-type errors if detected.

[0091] Any Z errors that occur during the Rydberg gate (or from the transformation of BBR errors) are exchanged with the remaining CZ operations. Thus, p tot The only errors that can occur during one round of stabilizer measurements up to the highest order in consist of Pauli errors acting on the ancillary qubits and on one of the data qubits (FIG. 1B). By resetting the ancillary qubit when a −1 measurement result is obtained and repeating the measurement protocol, the effect of errors on the ancillary qubit can be eliminated. A similar method can be used for the Z stabilizer.

[0092] In this method, after each round of stabilizer measurements, we introduce at most one physical qubit X or Z error while totAccurate stabilizer eigenvalues ​​can be obtained up to the highest order in

[0093] The above description presents the simplest form of a fault-tolerant stabilizer measurement protocol in which Rydberg state detection is performed after every physical gate, but in practice this is not necessary. In fact, all such detection operations must be performed after the stabilizer X α X β X γ X δ (where the Rydberg gates are applied to the data atoms in the order α, β, γ, δ), the only possible correlated errors that can occur are the X β X γ X δ , X γ X δ or X δ For the stabilizer of Equation 8, all of these errors are

number

number

[0094] 3. Logical Operations Logical Hadamard, Pauli and S gates. One particular advantage of the Steane code is the transversality of logical Hadamard, Pauli and S = diag(1,i) gates. Specifically, the logical Hadamard simply consists of a Hadamard on each physical qubit:

number

[0095] These operations can be performed without populating Rydberg states and therefore without introducing Rydberg gate errors. Similar decompositions exist for the S-gate and the Pauli gates X, Y and Z.

[0096] Logical controlled phase gates. Controlled phase gates in Steane codes are also transversal:

number

[0097] Therefore, logical controlled phase operations can be performed by performing only seven physically controlled phase operations and looking for BBR errors between each physical controlled phase gate (to convert them to Z-type errors). This eliminates the possibility of correlated multi-qubit errors within a single logical qubit.

[0098] Logical Toffoli gate. To implement the Toffoli gate fault-tolerantly and complete the universal gate set, we implement a logical CCZ gate, where the target qubit is conjugated with a Hadamard gate. Although this gate is not transversal in the Steane code, it can be further decomposed into a product of physical CCZ gates in a round-robin fashion:

number

number

number

[0099] Referring to Figure 3, the rearrangement of physical gates in performing logical CCZ operations is illustrated. For each logical qubit, only the first three data qubits are shown since no other data qubits are included in the logical gate.

number

[0100] Although the physical implementation of the CCZ gate is not transversal, the physical gates can be rearranged if all of them are swapped for one another. In doing so, some but not all of the intermediate Rydberg ensemble detection steps can be eliminated, and the total number of measurement operations can be reduced, as has been done for fault-tolerant stabilizer measurements. Specifically, the 3-qubit physical Rydberg gate of the protocol can be implemented in 9 groups of 3,

number

number

number

number

number

[0101] Hadamard and CCZ gates together form a universal gate set for quantum computer computation, and thus a scheme is described herein for constructing arbitrary quantum operations over code space in a fault-tolerant manner against BBR errors.

[0102] 4. Logic state preparation Referring to Figure 4, for the Steane code, the logic |0> L The protocol for preparing the conditions is illustrated.

[0103] Finally, in a fault-tolerant manner, L The state can be prepared. The most obvious preparation of this state uses the Steane's Latin rectangular encoding method, the circuit of which is shown in FIG. 4. In the Rydberg setting, the controlled-NOT gate is replaced by a Rydberg controlled phase gate in which the target qubit is conjugated with a Hadamard gate. totSince the Z-errors associated with Rydberg gates up to the highest order in are commutative with controlled phase manipulations, there is at most one Pauli-Z error among the three data qubits initially in the |+> state and at most one Pauli-X error among the four data qubits initially in the |0> state. This can be a two-qubit error, but it is correctable because the Steane code identifies and corrects X and Z errors separately. In this procedure, it is assumed that the Rydberg population resulting from the BBR errors is detected after the respective physical entanglement gates and these errors are converted to Z-errors if necessary. In this method, by applying one round of stabilizer measurements and error correction, it is possible to obtain a Pauli-Z error of (p tot Logic with Pauli error on at most one physical qubit (up to the highest degree in L The status is obtained.

[0104] FTQC with BBR and RD errors To deal with RD errors and intermediate state scattering, two new classes of leakage errors must be considered: (1) leakage to the original Rydberg state |r> and (2) leakage to other hyperfine ground states, also referred to herein as non-Rydberg leakage. The first class of errors is similar to quantum jumps in the BBR error model and can be detected and corrected in the same way using an ancilla qubit. In the following sections, this error is grouped together with the BBR error and collectively referred to as Rydberg leakage errors.

[0105] Referring to FIG. 87 An optical pumping protocol for converting non-Rydberg leakage errors to Pauli-Z errors in Rb atoms is illustrated. First, a π pulse

number

number

[0106] As shown above, leakage to other states in the hyperfine ensemble can be prevented by using a novel optical pumping protocol shown in Fig. 5, e.g. 87 Using optical pumping on Rb, it can be converted to a Pauli-Z type error. One crucial property of this optical pumping procedure is that it does not affect the qubit coherence in the absence of errors.

[0107] Furthermore, we note that while leakage in traditional QEC setups can be particularly difficult to handle, requiring additional entanglement gates or ancilla qubits, the specific multi-level structure of neutral atoms allows for the efficient correction of these errors. Notably, this optical pumping can be performed without the need for qubit measurements and feedforward corrections, allowing for efficient implementation in experiments.

[0108] Correction of non-Rydberg leakage errors can be integrated into the fault-tolerant protocol of the previous section by performing this procedure between the Rydberg entanglement gates. Thus, the protocol from the previous section is fault-tolerant against general intrinsic Rydberg decay errors. Furthermore, considering this sufficient error model, which includes both BBR and RD events, we can consider the |1> state and the stretched ground state when dealing with Rydberg leakage errors.

number

number

[0109] Algorithm 1: On the Rydberg 7-qubit code

number

number

number

number

number

number

number

[0110] Algorithm 2: Fault-tolerant logic CZ for the Rydberg 7-qubit code. 1. Apply single-qubit Z-gates to all physical control and target qubits. 2. For each j=1,2,...,7: a. 2-qubit Rydberg gate R(C j ;T j ) is applied. b. Use an ancillary qubit A1 to detect the Rydberg ensemble. c. If a Rydberg leak is detected, it is classified as a non-Rydberg leak.

number

[0111] Algorithm 3: Fault-tolerant logic CCZ for the Rydberg 7-qubit code ABC . 1. X-gates for all physical qubits

number

number

number

number

number

number

[0112] Although the above discussion focuses on intrinsic RD errors, non-intrinsic errors, described above as “experimental imperfections”, can also be integrated into the FTQC protocol. Specifically, errors arising from Rydberg laser imperfections, such as intensity and phase fluctuations, only produce Pauli-Z errors and single-qubit Rydberg leakage errors, so they are already addressed within the current framework. Similarly, atom losses can be detected by using an ancilla qubit and implementing a small leakage detection circuit. In this case, if a reservoir of atoms is available, atom loss errors can be converted to single-qubit Pauli-X or Z errors, for example, by replacing the lost atom with a new atom initialized to the |0> state.

[0113] Comparison against alternative fault-tolerant quantum computing protocols To illustrate the significance of the Ryd-7 FTQC protocol provided herein and to highlight the importance of considering specific error models when designing QEC approaches, the model provided herein is compared below to an alternative general-purpose FTQC scheme. Specifically, the cost of measuring the stabilizer and performing fault-tolerant logic operations can be compared using as metrics the number of 2- and 3-qubit entanglement operations required for physical qubits and the minimum number of ancillary qubits required. Details of how these numbers can be obtained for the Ryd-7 protocol are provided below.

[0114] Table 1 compares the minimum number of 2-qubit gates and ancilla qubits required for fault-tolerant stabilizer measurements (and associated error correction) in various QEC proposals. Results for general-purpose FTQC protocols for 7- and 15-qubit CSS / Hamming codes are based on a flagged syndrome extraction procedure. For each protocol, the resource cost for not having any errors is shown separately from the worst-case cost in the presence of errors (numbers in parentheses), since the former case is typically much more likely. Although the number of ancilla qubits required is the same for all cases, the protocols provided herein are found to require the minimum number of entanglement operations in any case, even if leakage, an additional type of error not considered in alternative approaches, must be detected.

[0115] Similarly, Table 2 shows this comparison for fault-tolerant logical CCZ gates, where the improvement is significant. A general-purpose implementation of this non-Clifford gate for three logical qubits in a 7-qubit Steane code is given by Yoder; this implementation requires only a moderate number of physical 2- and 3-qubit gates, but it requires a significant overhead of 72 additional ancilla qubits, making experimental illustration very difficult. On the other hand, Chao's proposal for a fault-tolerant Toffoli gate using the [[15,7,3]] code significantly reduces the ancilla qubit count, but the number of physical entanglement operations is considerable. The protocol provided herein uses only two ancilla qubits compared to the 72 required in Yoder, while using significantly fewer entanglement operations (e.g., about 60 2-qubit gates) than Chao (1416 2-qubit gates), even in the unlikely scenario that an error must be corrected. Although the protocol presented here uses more three-qubit entanglement gates than Yoder, such gates are straightforward to implement, much like two-qubit CZ gates in the Rydberg atom setting.

[0116] These results clearly demonstrate the advantages of considering hardware-specific error models when designing FTQC schemes and leveraging the unique capabilities of the Rydberg setting. In particular, the required number of entanglement gates or ancilla qubits can be further dramatically reduced, even if additional errors not considered in traditional settings have to be corrected.

[0117] Scalable Execution Further details regarding scalable implementations of the protocols provided herein, such as possible physical qubit geometric layouts, resource trade-offs and residual error rates, are provided below.

[0118] Geometric Considerations. One particular advantage of the Rydberg atomic platform is its flexibility, allowing for arbitrary geometric arrangements of atoms. Motivated by experimental descriptions of the nearly deterministic loading and rearrangement of neutral atoms onto a regular lattice structure, we provide herein a scalable FTQC architecture in which logical qubits form a coarser lattice on top of a lattice of physical atoms. For the Ryd-7 scheme, one natural layout in a two-dimensional atomic array could include placing physical atoms at the vertices of a triangular lattice (FIG. 1A). In this geometry, hexagonally shaped logical qubits (dotted hexagons, 103) form a coarser triangular lattice, and ancilla qubits (A, 102) are placed on the edges of this coarser lattice to mediate error correction and logic gates. Fault-tolerant universal quantum computer computations can be performed if nearest-neighbor logical qubits can be entangled; physical entanglement gates can be achieved by using a blockade radius R B Since this can only be done between atoms within the B In testing the physical gates required to perform the logical operations on the 7-qubit code, the requirement in this case is R B >3d (gray dotted line, 104). B This requirement for can be further reduced if atoms can be moved between certain logical operations while preserving coherence between the hyperfine ground states.

[0119] Resource trade-offs. For any experiment, resource trade-offs can be made to minimize the total logic error probability. For example, if the time for one round of measurement is significantly larger than a typical gate time (as is the case for certain atomic setups), it may be desirable to reduce the number of measurement shots required at the expense of performing additional operations. This can be achieved by turning any possible Rydberg leakage errors into non-Rydberg leakage ones.

number

[0120] Improvements. The FTQC protocol presented in this section relies on a selection rule that imposes restrictions on the possible RD error channels. Specifically, the attenuation channels arising from RD up to the highest order in error probability |0><1| have been ignored. 87 Rb: about 10 -3 Although this is already a reasonable assumption, considering that |r> is numerically determined to be ∇|r|, several approaches can be taken to further suppress the probability of such an error. First, due to the lower branching ratio from |r> to this stretched state, it is possible that the population in the |0> state before and after each entanglement gate is more likely to branch out from the stretched ground state

number

[0121] Highest degree fault tolerance with iterative codes. Considering that all Rydberg errors can be transformed into Z-type, one can naturally ask whether we still need a 7-qubit Steane code sufficient to detect and correct these errors; in particular, one could try to simply use a 3-qubit repetition code in the X basis to detect and correct Z-type errors. In such a code, the logical states are

number

number

[0122] However, the straightforward application of such iterative codes for FTQC is difficult even with this biased noise model, as it must be possible to implement all physical gate and logic gate procedures in the encoding, decoding and stabilizer measurements that do not introduce Pauli-X or Y type errors at any stage - i.e., each gate must be implemented in a bias-preserving manner.

[0123] This requirement can be easily satisfied for certain physical gates such as Rydberg controlled phases or collective gates (after all leakage errors are mapped to Pauli-Z type), but is much harder to satisfy for other gates. Specifically, measuring the stabilizer of Equation 13 requires implementing a controlled-NOT (CNOT) gate as shown in Figure 1E.

[0124] Referring to Figure 6, a circuit is provided for measuring the stabilizer X1X2 for a repeat code. A CNOT gate must be implemented between the ancilla qubit and the data qubits 1 and 2. The standard implementation of the CNOT gate using a Rydberg controlled phase gate conjugated with a single-qubit Hadamard gate on the target qubit is not bias-conserving (601), since a Z error on the target qubit during the controlled phase gate becomes an X error once the final Hadamard gate is applied.

[0125] On the Bloch sphere

number

[0126] Since an identity gate cannot be smoothly coupled to CNOT while remaining within the set of bias-preserving operations, bias-preserving CNOT gates are not possible between two qubits encoded in systems where the underlying Hilbert space is finite dimensional. In the setup provided here, the no-go theorem is avoided using the special fact that certain pulses in this finite dimensional atomic system - pulses between hyperfine states - can be performed with such high fidelity that the highest order errors arise only from Rydberg pulse imperfections and Rydberg state decay. This allows us to develop novel laser pulse sequences for entanglement of Rydberg atoms that directly perform CNOT or Toffoli gates while preserving noise biases.

[0127] The protocols provided herein can be applied to any atomic species with sufficiently high nuclear spin (I≧5 / 2). For the sake of concreteness, 85 The exemplary case of Rb is used to illustrate the protocol.

[0128] Bias-preserving CNOT in the Rydberg atom setting Referring to FIG. 85 A pulse sequence for a target atom in a bias-preserving CNOT gate between Rb atoms is illustrated. The Rydberg pulse is resonant if and only if there is no nearby Rydberg population; otherwise the Rydberg level is shifted due to the blockade effect (dotted level). This pulse sequence eliminates the target atom X error in the standard implementation of CNOT shown in Figure 6.

[0129] Step 1: Rydberg state stretched from qubit state

number

number

number

number

number

[0130] Step 2: From the qubit state to the Rydberg state

number

[0131] Step 3: Apply a resonant π-pulse between the |0> and |1> ground states (arrow 703, dark dashed line).

[0132] Step 4: Repeat step 701, but use -π pulses instead of π pulses for all transitions (arrows 704, light dashed lines).

[0133] Step 5: Incoherently drive any remaining Rydberg population to the elongated ground state (arrow 705, solid line). J +m I >0 (respectively <0), F=m F =I+3 / 2 (F=-m F =I+3 / 2), which is fast and F=m F =I+1 / 2 (F=-m F =I+1 / 2).

[0134] Step 6: Using optical pumping techniques, F >0 (respectively, m F <0>) to the qubit state |1> (|0>) (arrow 706).

[0135] The standard implementation of the CNOT gate in a Rydberg system is not bias-conserving, as shown in Figure 6. In particular, considering the error model for the Rydberg gate, an X error on the target qubit can be induced in two ways.

[0136] First, the target qubit may experience a Rydberg error (e.g., radiative damping) directly during the controlled phase gate, resulting in a Pauli-Z error that is converted to an X error after the Hadamard gate (arrow 601 in FIG. 6).

[0137] Alternatively, the control atom could decay from the Rydberg state to the ground state at some point during the controlled phase gate so that the target qubit Rydberg pulse that should have been blocked is now resonant during the controlled phase gate. This would result in a two-qubit correlated error between the control and target atoms, where the target atom experiences an X-type error.

[0138] Here we begin by introducing a novel entanglement gate pulse sequence for Rydberg atoms to address the target atom X-error. In this discussion, it is first assumed that the Rydberg pulses on the target atom are either all resonant or all blocked; i.e., the possibility of neighboring Rydberg atoms decaying during the target atom sequence is ignored. This effect is then subsequently included, and correlated errors are eliminated by introducing an ancilla qubit and utilizing two Rydberg states with different blockade radii.

[0139] To eliminate the target atom X error, it is desirable to design an entanglement gating protocol that uses the Rydberg state to conditionally directly exchange the |0> and |1> populations without any change of criteria from the Hadamard gate. This can be achieved for atomic species with sufficiently high nuclear spin (I ≥ 5 / 2).

[0140] 85 Qubits encoded in Rb clock states

number

[0141] The first step of the procedure (arrow 701) is to divide the population of qubit states |1> (respectively |0> to Rydberg states

number

number

number

number

[0142] Then, in a third step, the population of qubit states is subjected to a π pulse

number

[0143] After step 4, if no Rydberg errors occur, the atomic state is either the original qubit state (identity map) if there are no neighboring Rydberg populations, or the opposite qubit state otherwise.

number

[0144] Then, after the pumping steps (5 and 6), the resulting state can be verified to be the same as in the error-free case up to Z-type local errors (e.g., |0><0|, |1><1|). As before, error channels (e.g., phase, frequency and intensity variations) due to intermediate state scattering and other Rydberg pulse imperfections can be captured by the error models provided herein, including BBR and RD errors.

[0145] 8, the use of an ancilla qubit and multiple Rydberg states to eliminate X-type errors resulting from symmetric qubit decay according to an embodiment of the present disclosure is illustrated. The atoms are arranged on a line such that atom T is in the middle and the spacing between adjacent atoms is

number

[0146] Steps (a) and (c): Apply C as a control and A as a target to the CNOT gate. This applies a π pulse to atom C.

number

number

[0147] Step (b): Apply a three-atom gate between C, A and T. This is done by applying a π pulse to both atom C and atom A.

number

number

[0148] Table 3 shows the Rydberg transitions used to implement a bias-preserving CNOT gate between two atoms C and T as shown in FIG. 8. Within each step, one Rydberg transition (

number

number

[0149] Having eliminated the X-errors arising from target qubit Rydberg errors, we now proceed to address the second type of potential X-error arising from control qubit decay. The key here is to utilize multiple Rydberg atoms (e.g., a control atom and an ancilla atom) to blockade the target atom when the control is in the |1> state; in this way, if one of the atoms decays, the remaining Rydberg atom(s) can still ensure (up to the highest order in the total error probability) that the Rydberg pulse on the target atom does not go into resonance. For the simplest case, a bias-preserving CNOT gate can be implemented with one ancilla qubit. Assuming that the control (C), target (T) and ancilla (A) atoms are evenly spaced along the line and the target atom is between the control and ancilla atoms; the ancilla atom is initialized in state |0>. The blockade radius R B,1 and R B,2 Two pairs of Rydberg states |r 1,± > and |r 2,± > can be used, R B,1 >2d and d <R B,2 <2d, where d is the spacing between adjacent atoms (between C and T or T and A); thus, atoms C and A have a blockade radius R B,1 Within, R B,2 and the adjacent atoms have a blockade radius R B,2 A fully bias-preserving CNOT gate between the reference and target atoms then consists of a three-step procedure as illustrated in Figure 8, followed by correction of Rydberg leakage errors (as described below) and optical pumping to eliminate non-Rydberg leakage errors (see Figure 5). In each step of Figure 8, the Rydberg transitions that are addressed are listed in Table 3.

[0150] This protocol is robust to symmetric atom decay errors, such that the Rydberg pulse for atom T is resonant only if neither C nor A are excited to a Rydberg state, up to the highest order in the total error probability. This can only occur if the C state is in the |0> state: First, it can be seen that if C starts in the |0> state, then the state of T is not flipped, since A must also remain |0>. On the other hand, if C starts in the |1> state and no decay event occurs during step (a), then after this step |C,A>=|1,1>. Since the Rydberg pulse for T is prevented in step (b), its state is flipped. Finally, if C starts in the |1> state but decays during the first step, then after this step |C,A>=|1,1> or |1,0>. Since the Rydberg pulse for T is still prevented in step (b), its state is flipped. Finally, the Rydberg decay error in step (c) results in a projection of the form |0><0| or |1><1|, which can be expressed in terms of Z error.

[0151] In this way, any possible source of X-error arising from the CNOT gate has been eliminated up to the highest order in the total error probability. The protocol can also be generalized to implement a bias-preserving Toffoli gate as described herein. Alternatives that produce suppression at higher orders are discussed below.

[0152] In the bias-preserving implementation of CNOT provided herein, two sets of Rydberg states |r 1,± > and |r 2,± The ability to couple atoms to |r| allows atom C to interact with atom A during steps (a) and (c) of FIG. 8 but not during step (b). Alternatively, this tunability of interactions allows for a single set of tractable Rydberg states |r| where atoms can be rearranged while preserving coherence between the hyperfine ground states. 1,± >. In this case, atoms may be transferred between steps (a) and (b) to further separate C, T and A from one another, and the spacing between C and A may be reduced by R B,1and the spacing between any of them and atom T is R B,1 The atoms may then return to their original configuration after step (b) to allow for interaction between C and A during step (c).

[0153] Highest degree fault tolerance with iterative codes. The bias-preserving operations described above allow direct execution of each component of the three-atom repeat code to perform quantum computations with the highest degree of fault tolerance on the Rydberg setting. In particular, logic states can be prepared or measured fault-tolerantly in the X-base by preparing or measuring across each atom. Stabilizer measurements can be achieved using the circuit of FIG. 1E, where each controlled-NOT gate is made in the bias-preserving manner described above; for robustness against errors made in this circuit, the stabilizer measurements must be repeated if either g1 or g2 is measured to be −1.

[0154] Logical operations of the universal set can be achieved by implementing logical Toffoli and Hadamard gates, as in the 7-qubit case, using the bias-preserving pulse sequences shown above. Although not strictly necessary, implementations of logical controlled phase and CCZ gates are also provided herein. These gates may be useful to simplify the implementation of certain quantum algorithms, as they do not require novel bias-preserving pulse sequences and can be implemented using standard methods for implementing the Rydberg-mediated entanglement gate illustrated in FIG. 2B.

[0155] Referring to FIG. 9, a synthesizable, fault-tolerant implementation of the Toffoli gate on a repetition code is provided.

[0156] Logical Toffoli Gates. One important feature of the encoding is that the logical |0> L (each |1> L) state consists of an equal superposition of even (odd) numbers of physical qubits and states in the |1> state:

number

[0157] From this observation, we can see that the Toffoli gate CCX has logical control qubits A and B and a logical target qubit C. ABC It can be seen that can be implemented as a product of nine physical Toffoli gates:

number

[0158] Each physical Toffoli gate can be implemented in a bias-preserving manner as described previously, resulting in at most one physical Z error in each logical qubit, and it is assumed that Rydberg and non-Rydberg leakage errors are converted to possible Z errors after each physical gate. However, in this case, a Z error on a control qubit A or B is interchangeable with the remaining Toffoli gate, but a Z error on one of the physical qubits of C can spread to multiple Z errors in A or B after the following Toffoli gate if not corrected. To handle this, the physical gates are arranged as shown in FIG. 9 to perform error correction after all three physical Toffoli operations by measuring the stabilizer; this is followed by a synthetic possible fault-tolerant implementation of the non-transversal gate. In this way, after the entire logical gate, there is at most one physical qubit Z error per logical qubit involved.

[0159] Referring to FIG. 10, a logical Hadamard implementation of a repetitive code using logical Toffoli gates is provided.

[0160] Logical Hadamard gates. Unlike Steane codes, iterative codes are not CSS codes and their logical Hadamard gates are not transversal. However, logical Hadamard gates can be implemented using logical Toffoli gates combined with fault-tolerant measurements in the X-base, as shown in Figure 10. Logical Hadamard gates combined with logical Toffoli or CCZ gates form a universal set of logical operations.

[0161] Logically controlled phase gates. Logically controlled phase operations in a 3-qubit code are performed on each pair of physical qubits (j A , k B ), where j A and k B belongs to the respective encodings of logical qubits A and B:

number

[0162] To correct errors occurring between gates, any Rydberg population should be removed and an optical pumping scheme should be applied to convert non-Rydberg leakage errors into possible Z-errors after each physical control phase operation. Since Rydberg gates can only produce Z-errors that are exchanged with all physical CZ gates that are executed, the stabilizer only needs to be measured after the entire logic operation (so it does not spread to higher weight errors).

[0163] A logical CCZ gate. Similarly, a logical Control-Control-Z operation between logical qubits A, B, and C

number

number

[0164] As in the case of logical CZ, Rydberg and non-Rydberg leakage errors should be converted into possible Z-errors after each physical gate. Even though logical CCZ is not transversal, this implementation is fault-tolerant to the highest order since any given physical gate can produce at most one physical qubit Z-error per logical qubit; note that Z-errors do not propagate to become multi-qubit errors since they are commutative with the remaining gates applied. Although CCZ gates are not strictly necessary for the universal gate set assuming the highest order fault-tolerant implementation of the logical Toffoli gates, it requires fewer resources to implement than the logical Toffoli when it uses the standard and simpler Rydberg gate R(C1,C2;T) instead of the more complex bias-preserving CNOT pulse sequence (see Table 2). This operation can therefore be useful to reduce the resource cost of certain quantum algorithms.

[0165] Scalable Execution Some key issues for scalable implementation of the Ryd-3 protocol, such as geometric layout, resource requirements and potential improvements, are described below.

[0166] Geometry layout. Based on the logic gate implementation, stabilizer measurements and the underlying bias-preserving CNOT given in the previous sections, a convenient geometry is found to be to place the data and ancilla atoms on the vertices of a triangular lattice shown in Figure 1D, where three data atoms contain the logic qubits. In this configuration, the logic qubits form a coarser triangular lattice, similar to that in Ryd-7. For different blockade radii R B.1 >R B.2Two Rydberg states with θ = 0 are required to implement a bias-preserving CNOT gate. Based on previously described fault-tolerant stabilizer measurements and the interaction range required to perform logical operations, it is found that a larger blockade radius must be larger than 3d (117 in Figure 1D), where d is the nearest neighbor spacing on a square lattice; this is required for some of the physical gates in the logical CCZ and Toffoli gates. On the other hand, a smaller blockade radius R B.2 should be strictly d ~ 2d for efficient execution of bias-preserving CNOT and fault-tolerant stabilizer measurements (118 in Figure 1D). Requirement R B.1 Details on how to obtain >3d can be seen below.

[0167] Alternatively, the data and ancilla atoms may be arranged on the vertices of a square lattice in an alternative manner. In this case, the blockade radius requirement is R B.1 >3.61d and d <R B.2 <2d. Experimental progress that allows the rearrangement of atoms while preserving the coherence of the hyperfine ground state, for both triangular and square lattice geometries, has been demonstrated in R B.1 We further simplify the requirement for the blockade radius R B.2 This can be used to eliminate the need for a second set of Rydberg states having

[0168] Resource comparison. Here we compare the resource cost of the Ryd-3 protocol with the Ryd-7 approach and with alternative general-purpose schemes. Compared to the 7-qubit approach, we find that the number of entanglement gates required for the extraction of all stabilizers for error correction is significantly reduced due to the smaller number of data atoms and stabilizers per logical qubit, without a substantial increase in the number of ancillaries required (Table 1). On the other hand, the cost of implementing a logical CCZ gate is essentially the same as in Ryd-7, but the number of gates required for the logical Hadamard is larger (Table 2), since the Hadamard gates are not transversal using the iterative code. Note that each CNOT gate in the stabilizer measurement is transformed into two two-atom entanglement gates and one three-atom entanglement gate in the bias-preserving implementation; this is reflected in Tables 1 and 2 (further details on obtaining the Ryd-3 resource costs can be seen below).

[0169] However, the number of gates required is still very modest compared to logical operations in other universal FTQC gate sets. Consequently, the substantial resource cost reduction and improved efficiency for stabilizer measurements in the use of fewer atoms makes the three-atom approach very promising for near-term implementations.

[0170] Although bias-preserving CNOT suppresses X-type errors up to the highest order, the amount of bias preservation is ultimately limited by the decay rate of the elongated Rydberg D state to the qubit state. To further suppress these errors, one can shelve to an elongated Rydberg state with higher angular momentum, which has a lower decay rate to the qubit state. Alternatively, atomic species with higher nuclear spin can also be used, where the qubit state has a larger |Δm F Similarly, high |Δm F The magnetic field can also be increased in the experimental setup to suppress the rate of transitions with |.

[0171] To achieve suppression beyond the highest order, more Rydberg shelving states and more ancillars may then be used in the target atom pulse sequence of FIG. 7 to suppress the effects of symmetric atomic damping.

[0172] The Ryd-3 hardware-tuned FTQC approach inherently addresses errors due to Rydberg pulse imperfections in addition to those caused by finite Rydberg state lifetimes, since these errors are included in a subset of radiation decay errors. As with Ryd-7, the Ryd-3 approach can also be enhanced to further protect against atom loss errors, at the expense of additional physical operations, by incorporating an atom loss detection scheme described below during the Rydberg operations.

[0173] Experimental Run In the following sections, further issues are described regarding how the FTQC protocol provided herein can be implemented in upcoming experiments. Neutral alkali atomic systems can achieve near-deterministic trapping, loading and rearrangement of tens to hundreds of atoms into two-dimensional lattice structures, such as the triangular lattice required for the protocols provided herein. Furthermore, high fidelity manipulation within the ground state ensemble and two- and three-atom Rydberg blockade mediated entanglement gates are possible. Blockade interactions between Rydberg atoms separated by three times the lattice spacing are also possible, which is the interaction range required for both of the protocols provided herein.

[0174] Measurement and feedforward correction To perform QEC, a key component is the ability to measure the state of the ancilla qubit and / or detect the Rydberg population as well as perform feed-forward corrections. Several approaches can be considered. First, rapid measurement of the ancilla qubit state can be achieved by using two different atomic species for the data and ancilla atoms. In this approach, the ancilla atoms can still interact with the data atoms when both are coupled to the Rydberg state, but they can be measured independently without perturbing the data atom state.

[0175] Alternatively, another method for rapidly measuring individual qubit states is to drive cycling transitions (e.g. 87 In Rb

number

[0176] To estimate the maximum rate of coherent transport before atom loss and heating become significant, one can consider a harmonic oscillator potential in which the atoms are trapped (i.e., optical tweezers). The average energy increase for the atoms is

number

number

number

number

[0177] Alternatively, measurement of the ancilla qubit state can be achieved by using two different atomic species for the data and ancilla atoms (e.g., two different isotopes of the same atom or two different atomic species). In this approach, the ancilla atoms can still interact with the data atoms when both are coupled to the Rydberg state, and they can be measured independently without perturbing the data atom state.

[0178] Finally, fast detection schemes using Rydberg electromagnetically induced transparency (EIT) with atomic ensembles can be used. These can be utilized to identify Rydberg ensembles after entanglement gating. These schemes can be integrated into the tweezers array platform by creating larger extended traps at selected locations that contain optically dense atomic ensembles.

[0179] In this approach, the Rydberg blockade effect leads to sharp features in the absorption spectrum of the weak EIT probe beam depending on whether nearby Rydberg atoms are present. Due to the collectively enhanced Rabi frequency, the detection time can be reduced to about 6 μs, equivalent to the duration of the entanglement gate. Therefore, this ultrafast nondestructive Rydberg atom detector provides a promising implementation for the measurements and feedforward corrections required for the protocol provided herein.

[0180] Implementation with alkaline earth(-like) atoms Referring to FIG. 87 A relevant level diagram is provided for implementing the FTQC protocol provided herein with a neutral alkaline earth Rydberg atom, such as Sr. The qubit is elongated. 1 Encoded in the S0 ground state. 5S nS, 3 The transition to the S1 Rydberg state is achieved by first changing one of the qubit states 3 The clock state can be driven by coherently mapping to the P0 clock state and then exciting the clock state to the Rydberg state (R). Optical pumping to correct non-Rydberg leakage is performed in two states by driving the P1 transition and then the P2 transition. State readout and strong cooling for state initialization can be performed in two states by driving the P1 transition and then the P2 transition.

number

[0181] This disclosure is primarily focused on developing FTQC protocols for neutral alkali atoms coupled to Rydberg states. Alkaline earth(-like) atoms, such as Sr and Yb, can also be used for Rydberg-based quantum computer calculations. The following description shows how the methods provided herein can also be applied to such settings. The focus is on specifics. 87 Although for the example of Sr, the discussion provided here is general for fermionic species of alkaline earth(-like) atoms.

[0182] For alkaline earth(-like) atoms, 1 Since the S0 ground state has no electron orbital or spin angular momentum, the only source of degeneracy is the nonzero nuclear spin (which can be very large, e.g. 87 For Sr I=9 / 2).

[0183] For the protocols provided herein, the most convenient qubit encoding is the stretched basis state:

number

number

number

[0184] During these entanglement operations, atoms in the Rydberg state can experience various errors such as BBR transitions, RD or intermediate state scattering. For alkaline-earth(-like) atoms, the resulting Clauss operators can undergo Pauli-Z errors and quantum jumps to Rydberg states, as enabled by dipole selection, 1 S0 ground state or metastable 3 It can be described by the P0 state.

[0185] Following the approach provided herein for alkali atoms, all such errors must be converted to Pauli-Z errors and the FTQC protocol provided herein applied. By using ancilla atoms and blockade effects, quantum jumps to Rydberg states can be corrected in the same manner as for alkali atoms. However, metastable 3 Due to the presence of the P level, the correction of non-Rydberg leakage errors becomes more complicated and the optical pumping has to be done in two states (see Fig. 11): (1) 3 P 0,2 Triplet excitation from the 3 σ to S1 state + - Use polarized light to see all 3 P state 3 Pump them back into the P1 set; these states are 1 It decays back to the S0 ground state. (2) Narrow line cooling transition

number

[0186] After these two steps, all non-Rydberg leakage errors are mapped to errors |1><1|, which is explicit for Pauli-Z errors. Pauli-X errors can in principle be expressed as

number

[0187] This disclosure provides a comprehensive analysis of the dominant error channels that arise in quantum computer calculations using neutral Rydberg atoms. Although the multi-level nature of the atom and the complex decay channels for the Rydberg states result in many additional types of errors that are not considered in traditional QEC settings, the specific structure of the error model allows the design of a hardware-efficient FTQC protocol based on 7-qubits and a hardware-tuned 3-qubit code with significantly reduced overhead compared to general-purpose schemes. These results provide the ability to convert the complicated error model to Pauli-Z errors by introducing ancillary atoms and utilizing the Rydberg blockade effect, the bipolar selection rule, and a novel scheme for optical pumping. Novel laser pulse sequences for implementing bias-preserving CNOT and Toffoli gates are provided to use the 3-atom repeat code. Scalable geometric structure layouts are provided for both protocols.

[0188] Compared to alternative general-purpose FTQC protocols, the hardware-efficient approach for the Rydberg system provided herein allows for an order of magnitude improvement in resource overhead in terms of the number of physical gates or ancillaries required. While this disclosure focuses on a particular implementation, the teachings provided herein are translatable to other quantum computing platforms, such as trapped ions and superconducting qubits.

[0189] It is understood that the present disclosure can be combined with topological codes such as surface codes or color codes. In an exemplary embodiment, the techniques provided herein are applied to handle Rydberg and non-Rydberg leakage errors and then to apply such topological codes. After eliminating all Rydberg specific leakage errors using the FTQC protocols provided herein, these codes can be coupled with alternative QEC approaches to handle any higher order Pauli-X or Y type errors or to further suppress logic error rates to even higher orders.

[0190] Numerical computation of branching ratios and transition rates This section contains the stretched Rydberg state 87 About Rb70S 1 / 2 , m J = 1 / 2, m I The results of numerical computations of the branching ratios for the BBR and RD transitions from =3 / 2 are shown.

[0191] 1. Blackbody radiation induced transition Referring to FIG. 12, m J = 1 / 2, m I Stretched 70S with =3 / 2 1 / 2 From the state J = 3 / 2 to different P states 87 The branching ratios of the BBR transitions between the Rydberg states of Rb are plotted (open circles) or m J =1 / 2 (black diamond).

[0192] To quantify the relative probability of transitions to different nearby Rydberg P states, the rate W(nL → n'L') of BBR transitions from a given Rydberg state nL to another Rydberg state n'L' is computed using the Planck distribution of photons at a given temperature T and the Einstein coefficients for the corresponding transitions:

number

number

[0193] In the above formula:

number

[0194] The present disclosure relates to 87 The stretched Rydberg state 70S of Rb 1 / 2 , m J = 1 / 2, m I An analytical formula was used to numerically compute the radiative dipole matrix elements for single-photon BBR transitions from =3 / 2. The corresponding transition rates were then computed using Equation 19 to obtain the total BBR rate Γ BBR These were normalized by (see Eq. 3) to obtain the branching ratio.

[0195] m J =3 / 2 and m J The branching ratios of the P states with .DELTA.=1 / 2 are plotted in Fig. 12 as open circles and closed diamonds, respectively. Indeed, the atoms are found to decay mainly to the 69P and 70P states, as illustrated in Fig. 1C.

[0196] 2. Radiation damping As shown in FIG. 1C, 87 Stretched 70S of Rb 1 / 2 , m J = 1 / 2, m I The radiative decay transitions from the =3 / 2 Rydberg state are almost entirely two- or four-photon decay processes to one of five states in the ground state ensemble; this fact was important to convert all Rydberg errors to Z-type for fault-tolerant quantum computer calculations. To justify this, the branching ratios for multiphoton spontaneous radiative processes were numerically computed by evaluating the ratio of the individual transition rates for each decay channel, which is given by the Einstein A coefficients in Equation 20. Due to the cube-dependence of these coefficients on the transition frequencies, the major contributions come from dipole-enabled transitions to proximal states of the ground state ensemble. The dipole matrix elements for such transitions are

number

number

[0197] By computing the radiating dipole matrix elements, 87 70S for Rb 1 / 2 , m J = 1 / 2, m I The branching ratios for the RD process outside the elongated Rydberg state of θ = 3 / 2 have been evaluated. [Table 4]

[0198] Table 4 shows the 70S 1 / 2 , mJ = 1 / 2, m I On the radiative decay process from the elongated Rydberg state of .DELTA.=3 / 2. 87 We provide branching ratios for each of the Rb ground state transitions, which account for transitions up to four-photon radiative processes. The contributions from higher order transitions are 2.5 × 10 -4 is less than.

[0199] The results of this computation are shown in Table 4. Indeed, the branching ratios for the remaining three states are each significantly lower than those for the five dominant transitions, about 10 -3 It is found that, when the total error probability is already very small, these three processes (especially the minimum m F =-2) is highly unlikely.

[0200] An example of the solution of the master equation for radiation damping. Above, the Claus operator corresponding to the spontaneous radiation event from the Rydberg state |r> to the qubit |1> is

number

number

[0201] The master equation for this driven three-level system is (

number

number

number

number

number

number

number

number

[0202] Where:

number

[0203] Then, up to the highest order in γ / Ω, the Claus operators

number

number

[0204] Conversion of Rydberg leakage into Pauli error. Once a Rydberg leakage error is detected, it can be converted into an atom loss error by ejecting the Rydberg atom, which is done naturally by the anti-trapping potential from the tweezers and can be enhanced by applying a pulse of a weak ionizing electric field. The exact location of the ejected atom can be determined by following the atom loss protocol outlined below and illustrated in FIG. 13. In this case, since the error has already occurred, the atom loss protocol does not need to be applied in a robust manner. The ejected atom can then be replaced by a new atom prepared in the |1> state.

[0205] This process is called operator identity.

number

[0206] To reduce the need to apply atomic loss correction circuits, a precautionary step can be added after every entanglement gate that incoherently re-pumps any remaining population in some most probable Rydberg state into the |1> qubit state. This re-pumping can be done in three steps: 1. |1> and the stretched ground state

number

number

[0207] Although the above description focuses on the most probable final state |r'> for a BBR error, other BBR errors can be corrected by extending step 2 to cover these states.

[0208] By proactively applying this procedure, a large fraction of Rydberg leakage errors can be converted to Z-type errors without the need for the atomic loss correction circuitry of FIG.

[0209] Atom Loss Error Referring to FIG. 13, a circuit for detecting atom loss is shown.

[0210] As mentioned above, if trapping is incomplete or if the trapping laser needs to be turned off during Rydberg excitation (e.g., in various embodiments, 87 Neutral atom settings can also suffer from atom loss errors (as is typically done for Rb). Fortunately, such errors can be detected and corrected even within the FTQC framework provided herein, at the expense of one ancilla qubit and some extra gates for each operation. In particular, atom loss events can be detected by applying the circuit of FIG. 13 for each data qubit after using an optical pumping technique to correct leakage outside the computational subspace. An ancilla measurement then produces +1 in the presence of an atom loss and −1 if no such error occurs. Once detected, an atom loss error can be converted to a single-qubit Pauli-Z or -X type error if a reservoir of atoms is available, for example by replacing the lost atom with a new atom initialized to the |0> state.

[0211] The steps required to establish robustness against errors occurring during this circuit are described below. As in the case of fault-tolerant Rydberg leakage detection described below, a multi-step ancillary measurement protocol is again employed to protect against ancillary errors in FIG. 13, which requires two positive ancillary measurements to confirm an atomic loss error. On the other hand, any phase flip error on a data qubit cannot propagate more than a single physical qubit error per logical qubit in the universal gate set implementation for Ryd-7 or Ryd-3. Leakage errors (Rydberg or non-Rydberg) can be handled by repeating the respective re-pumping procedure after applying the atomic loss detection circuit. Thus, by integrating this circuit into the implementation of the fault-tolerant stabilizer measurements and logical operations described above, atomic loss errors in the FTQC protocol provided herein can also be handled.

[0212] Note that this circuit can be used for atomic loss after correcting for leakage to atomic states outside the computational subspace by using blockade effects and optical pumping techniques. This approach does not distinguish between atomic loss and leakage to other hyperfine states, and so can also be used to suppress any residual hyperfine leakage errors.

[0213] Fault-Tolerant Detection of Rydberg Leakage Errors. As mentioned above, for fault-tolerant error detection and correction, it is important to deal with any errors that may occur on the ancillary qubits that are used to look for the Rydberg population. This is done using a multi-step measurement procedure to detect leakage on ancillary qubits: 1. Implementing a Hadamard gate on the Ancillary 2. Checking whether the ancilla is in the |1> state (e.g. by coupling |1> to a cycling transition and detecting fluorescence); 3. Running the X-Gate on the Ancillary 4. |1> Recheck the group This can be done by using:

[0214] If neither the second nor the last step yields |1>, then the ancilla atom must have experienced a leakage error. In such a case, any possible ancilla atom Rydberg error is converted to a possible Z-type error. Similarly, since the Rydberg pulse can potentially produce a phase-flip error on the ancilla qubit, if a Rydberg leakage error is detected by the ancilla, the detection protocol must be repeated one more time to ensure that the result did not result from such an error.

[0215] Once a Rydberg leakage error is detected, it can be converted to a phase flip error by sending the Rydberg state to |1>.

[0216] Error syndromes with deferred measurements. It is described above how Rydberg leakage detection can be postponed in the Ryd-7 stabilizer measurements and controlled phase gate protocol to facilitate experimental implementation. This relied on the ability to use the stabilizer measurements to distinguish possible correlated errors that may result from the postponed detection of the Rydberg leakage error. Details are now given on how to use the error syndromes to identify the corresponding correlated error in each case. As mentioned above, for the Steane code the stabilizer:

number

[0217] Table 5 shows the error syndromes used to distinguish correlated errors resulting from delayed detection of Rydberg leaks during measurements of the X4X5X6X7 stabilizer in the Ryd-7 FTQC protocol. Since all possible correlated errors are products of Pauli-X errors, Table 5 shows that

number

[0218] For the stabilizer measurements, measurements of g1 on qubits 4, 5, 6, 7 are considered (without loss of generality) using a circuit of the form shown in FIG. 1B. If a Rydberg leakage error occurs on the ancilla atom at any point, the data atoms do not suffer from any correlated errors. On the other hand, if a data atom suffers from a Rydberg leakage error during the circuit, the possible correlated errors that may occur are X5X6X7, X6X7 or X7. These errors are due to the fact that the 7-qubit code

number

[0219] For the case of logical CCZ gates, the 27 physical Rydberg gates are divided into three groups:

number

number

number

[0220] Implementing a bias-preserving Toffoli gate Figure 8 illustrates how an ancilla atom can be used to eliminate X-type errors arising from contrast atom decay in the implementation of a bias-preserving CNOT gate. Similarly, a bias-preserving Toffoli gate can be implemented by utilizing two ancilla atoms present on either side of the target atom. This protocol is illustrated in Figure 14.

[0221] Referring to FIG. 14, a circuit is illustrated for performing a bias-preserving Toffoli gate between control atoms C1, C2 and a target atom T using two ancilla qubits and multiple Rydberg states. The ancilla atoms (A1 and A2) are chosen to be on either side of the target atom. The dotted box shows the most natural bias-preserving three-qubit gate for a Rydberg system, where a π-pulse

number

number

[0222] As in the bias-conserving CNOT case, the choice of Rydberg state varies throughout the procedure. During the third gate in Fig. 14, we place the atom in |r 2,± > and using an ancilla atom on the opposite side of the target atom ensures that the ancilla atoms do not interact with each other via Rydberg blockade during this gate; this is important in case one of the ancilla atoms experiences a radiative decay transition during this gate. On the other hand, all of the other entanglement gates in Figure 14 are in the Rydberg state |r 1,± >. Two control atoms may interact with each other during these four gates if the spacing between them is less than one blockade radius, which is different from the case for the third gate. This may be tolerated since a Rydberg error may occur during at most one of these four gates, so that at least one ancilla atom will have a correct interaction with the target atom during the third gate.

[0223] Computing resource costs for the Rydberg FTQC protocol Details on how to obtain resource costs for the Ryd-7 and Ryd-3 protocols shown in Tables 1 and 2 are provided below.

[0224] For the Ryd-7 protocol, each stabilizer measurement requires four 2-qubit Rydberg gates in the absence of errors (see Algorithm 1); thus, 24 2-qubit gates are required to measure all stabilizers. If errors occur, the worst case scenario for stabilizer measurements is when the first 5 stabilizers all have +1 eigenvalues ​​and the truly last stabilizer is measured to be -1. In this case, g4, g5 and g6 need to be remeasured, requiring 12 more 2-qubit gates. The logical CCZ gates for Ryd-7 are implemented in the absence of errors using 27 physical 3-qubit gates, as described in Algorithm 3. The worst case error in this case is the final group of Fig. 3.

number

[0225] In the Ryd-3 protocol, each of the two stabilizer measurements requires two bias-preserving CNOT gates (Figure 1E), and each bias-preserving CNOT gate is decomposed into two two-atom gates and one three-atom entanglement gate. Thus, in the absence of errors, a stabilizer measurement requires eight two-qubit gates and four three-qubit gates. If an error occurs, the worst-case scenario is when the second stabilizer is measured to be -1; in this case, both stabilizers would need to be measured again, doubling the gate cost. The Ryd-3 CCZ gates can be performed in a round-robin fashion in the same way as the Ryd-7 CCZ, which is bias-preserving and uses 27 physical three-qubit gates.

[0226] Finally, the Ryd-3 Hadamard gate consists of a fault-tolerant, bias-preserving Toffoli gate followed by a single-qubit measurement and rotation (Figure 10). The synthesizable fault-tolerant Toffoli gate in the Ryd-3 code consists of nine physical bias-preserving Toffoli gates and two rounds of error correction. As mentioned above, each round of error correction includes eight two-atom Rydberg gates and four three-atom Rydberg gates. With the data atoms in each logical qubit indexed as in Figure 15B, the logical Toffoli gate CCX ABC is implemented between the three qubits A, B, and C highlighted in bold, the number of Rydberg gates required to implement each physical Toffoli gate is B.1 It depends on the blockade radius R B.1 If is greater than 3.61d, then each physical Toffoli gate can be implemented using two ancilla atoms (one on each side of the target atom) and five three-atom Rydberg gates; this can be done for any physical reference atom C in FIG. i and any ancilla A j The spacing between is always equal to the blockade radius R B.1Since the entanglement gates can be implemented directly, each physical Toffoli gate contains five three-atom Rydberg gates, so the total number of gates (including the QEC step) is 16 two-atom gates and 53 three-atom gates in the absence of errors. B.1 If reduction to >3d is desired, there are two physical Toffoli gates (choice j A =l C = 1, k B =2 and j A =l C = 3, k B = 2), where one of the physical reference atoms and one of the ancilla atoms (corresponding to 2 in Figure 15B) B The spacing between A and A3) is too large to directly implement the Rydberg entanglement gate required for a physical Toffoli gate. Instead, instead of the first (respectively second) three-atom Rydberg gate involving A3, a Rydberg gate with the same two control atoms and one of the ancilla atoms A1 or A2, whichever is not included in the remainder of the FIG. 14 circuit, is implemented, followed by a bias-preserving CNOT gate between the ancilla and A3 (preceded respectively). Both A1 and A2 are 2 B , 1 A , 2 A , 3 A Since A3 and A4 are within the blockade radius of A3, these gates can be implemented directly. In this way, four extra two-atom gates are needed for the logic Toffoli (j A =l C = 1, k B For physical Toffoli with =2, two and j A =l C = 3, k B= 2), which increases the total number of gates in the absence of errors to 20 2-atom gates and 53 3-atom gates, as shown in Table 2. With errors, the worst-case scenario is when the final stabilizer measurement in the second round of QEC yields -1, in which case the stabilizer needs to be measured again; this adds another 8 2-atom gates and 4 3-atom gates to the total resource cost.

[0227] 15A,B, exemplary labeling of atoms for each of the Ryd-7 and Ryd-3 FTQC protocols used to derive the gate number and blockade radius requirements are provided. Data atoms are indicated with numbers and ancilla atoms are indicated by an "A".

[0228] Referring to Figure 15A, in the Ryd-7 protocol, each logical qubit consists of seven data atoms (dotted hexagons). For each data atom, a number is used to indicate which physical qubit of the 7-qubit logical state the atom encodes. This labeling allows the blockade radius R B is defined by the interaction range required to perform a logical CCZ gate between three adjacent logical qubits, such as A, B and C. Using the specific CCZ protocol given in Algorithm 3, the blockade radius requirement is then R B >3.61d, where d is the spacing between nearest neighbors on the lattice; this is the physical atomic A and 1 C , which is determined by the spacing between R (thinner light grey dotted line 1501). However, by using a different set of physical CCZ gates to implement the logical CCZ, this requirement can be met by B >3d (thicker dark grey dotted line 1502).

[0229] Referring to Figure 15B, in the Ryd-3 protocol, each logical qubit consists of three data atoms (dashed triangles). For each data atom, a number is used to indicate which physical qubit of the 3-qubit logical state the atom encodes. This labeling allows for a larger blockade radius R B.1 is determined by the interaction range required to perform a logical Toffoli gate between three adjacent logical qubits, such as A, B, and C. In this case, R B.1 There are two possibilities for R B.1 >3.61d (lighter light grey dotted line 1503) or R B.1 >3d (thicker dark grey dotted line 1504). If a larger blockade radius of 3.61d could be realized, the resource cost for logical Toffoli and Hadamard gates could be reduced by four 2-qubit entanglement gates compared to the numbers shown in Table 2.

[0230] Computation of the Rydberg blockade radius requirement for the Rydberg FTQC protocol To obtain the blockade radius requirement for the Rydberg FTQC protocol, one must identify each physical qubit with atoms on a lattice and then determine the maximum spacing between two atoms that must interact with each other during a Rydberg gate. When the underlying atoms are arranged in a triangular lattice, Figures 15A,B show a convenient identity for each of the Ryd-7 and Ryd-3 codes. In these figures, numbers are used to label the index of the data atoms within each logical qubit. (The index of a physical qubit within each logical qubit is the position counted from the left of such qubit in the definition of the logical state; see Eqs. 6 and 7 for the 7-qubit code or Eq. 12 for the 3-qubit code.)

[0231] In the Ryd-7 protocol, the blockade radius is defined by the interaction range required to perform a logical CCZ gate between three adjacent logical qubits, such as A, B and C.

number

number

number

[0232] The above discussion defines the blockade radius requirement for Ryd-7 as R B >3.61d (the thinner light grey dotted line 1501 in FIG. 15A). In fact, by modifying the implementation of the logical CCZ gate, we can simplify this requirement to R B Further simplification to >3d is possible (darker dark grey dotted line 1502 in FIG. 15A).

[0233] In the Ryd-3 protocol, the blockade radius R B.1 is determined by the interaction range required to implement logical Toffoli gates between adjacent logical qubits (e.g., A, B, and C in FIG. 15B). There are two possibilities in this case. To directly implement all physical bias-preserving Toffoli gates using the circuit of FIG. 14, B The distance between A and A3 is R B.1 must be less than;

number

[0234] Blockade radius simplification for Ryd-7 In the Ryd-7 protocol, the blockade radius requirement is R B =3.61d to R B To simplify to ∑ = 3d, we must modify the implementation of the logical CCZ operation. Recall that Algorithm 3 implements a logical CCZ gate using 27 physical CCZ gates between the first three physical qubits of every logical qubit. This round-robin decomposition makes use of Equation 11, which is derived here:

number

[0235] To begin the derivation, first we consider that the logical states of a 7-qubit code have well-defined parity: |1> the number of physical qubits in the state is always |0> L is even for |1> L It then follows that the logical CCZ gates can be executed in a round-robin fashion that sufficiently involves all the physical qubits.

number

[0236] This means that the round-robin execution is A , k B , l C ), and the number of such triplets is increased so that all logical qubits are |1> L is odd if at least one logical qubit is in the |0> logical state. L To simplify this to Equation 32, g4 = Z4Z5Z6Z7 is the stabilizer for the 7-qubit code, so j A and k B For each choice of

number

number

[0237] The blockade radius requirement is R B =3.61d to R B To simplify to =3d, in this derivation for one of the logical qubits, say qubit C, we change the product in Eq.

number

number

[0238] Therefore, the 27 physical CCZ gates in Algorithm 3 can be replaced by the 27 CCZ gates used in the right-hand side of Equation 36.

[0239] Considering the geometric layout of the individual atoms in each logical qubit shown in FIG. 15A, the interaction range required to perform the logical CCZ operation using these 27 gates is smaller than the interaction range required to perform the 27 gates of Algorithm 3. Furthermore, all of the physical qubits

number

number

number

number

[0240] Square lattice geometry for Ryd-3 Referring to FIG. 16, a square lattice geometry for the Ryd-3 FTQC protocol is illustrated. Data (numbered) and ancilla (A) atoms are arranged at the vertices of a square lattice in an alternating fashion, with three data atoms comprising a logical qubit (dotted box). The number on each data atom indicates the index of such atom within the respective logical qubit; this is relevant for performing stabilizer measurements and logical operations. Two Rydberg states with different blockade radii are required to perform bias-preserving CNOT and Toffoli gates. The larger the blockade radius R B.1 teeth

number

number

number

number

[0241] With these blockade radii, the protocol described above can be directly applied to perform all logical operations. The higher density of ancilla atoms in this arrangement makes it possible to perform all physical Toffoli gates in logical Toffoli operations directly using the circuit of FIG. 14 without the need for additional ancilla atoms or CNOT gates (as is the case for two physical Toffoli operations under a triangular lattice geometry). In this way, for a square lattice geometry, the number of two-qubit entanglement operations required for logical Hadamard or Toffoli operations can be reduced by four compared to the number shown in Table 2.

[0242] Optical pumping procedure for bias-preserving CNOT To implement the bias-preserving CNOT pulse sequence shown in Fig. 7, the optical pumping procedure in the final step is F Pump only the >0 state to the |1> state, and m F It is important to pump only <0 states to |0> states. This requirement is essential to ensure that CNOT does not produce any X or Y type errors. For the magnetic field region typically used in alkali Rydberg experiments, F This state selectivity may not be obvious to implement, since the level separation between the states can be much smaller than the linewidth of the laser used for optical pumping. To address this difficulty, the Rydberg state can be used as a shelving state (because of its long lifetime) to obtain m F <0 (respectively m F 7, the undesired pumping of the |1> (|0>) state to the |1> (|0>) state can be avoided. FOptical pumping of the >0 state to the |1> state is 85 This can be done for Rb as follows: 1. |1> state and stretched ground state

number

number

number

[0243] |F=2,m F =-2> A state can be assembled only if a Rydberg error occurs up to the highest order in the total error probability in one of the earlier steps of the bias-preserving CNOT, so the Rydberg state

number

[0244] Next, m F A similar procedure can be applied to pump the <0> state to |0>. In this latter case, for all m F Since the >0 population has already been transitioned to the |1> state, there is no need to shelv the Rydberg state population.

[0245] An exemplary device for fault-tolerant quantum computing includes a two-dimensional array of optical tweezers configured to provide confinement for atoms. Rearrangement of atoms to form a desired defect-free array with any geometry can be provided using a two-dimensional AOD as described below. A laser is provided to excite the atoms from their electronic ground state to a Rydberg state (a highly excited electronic state), where the atoms interact with each other via strong van der Waals interactions. Readout of the atomic states is provided via fluorescence imaging. This allows detection of atoms in the ground state, while atoms in the Rydberg state are detected as lost (due to the anti-trapping effect of the optical tweezers).

[0246] Formation of arrays of particles using optical tweezers Optical trapping of neutral atoms is a powerful technique for isolating atoms in a vacuum. Atoms are polarized, and the oscillating electric field of a light beam induces an oscillating electric dipole moment in the atoms. The relevant energy shift in the atoms from the induced dipole, averaged over the period of the light oscillation, is called the AC Stark shift. Based on the AC Stark shift induced by light that is detuned (i.e., offset in wavelength) from the atomic resonance transition, atoms are attracted to light below the resonance frequency and thus are trapped at a local intensity maximum (for detuned red, i.e., longer wavelength trapping light). The AC Stark shift is proportional to the intensity of the light. The shape of the intensity field is therefore the shape of the relevant atom trap. Optical tweezers exploit this principle by focusing a laser into a micron-scale constriction, where individual atoms are trapped at the focus. Two-dimensional (2D) arrays of optical tweezers can be generated, for example, by illuminating a spatial light modulator (SLM) that imparts a computer-generated hologram to the wavefront of the laser field. The 2D array of optical tweezers overlaps with a cloud of laser-cooled atoms in a magneto-optical trap (MOT). The tightly focused optical tweezers operate in the "collision blockade" regime where a single atom is loaded from the MOT and pairs of atoms are ejected for optically assisted collisions to ensure that at most one atom is loaded into the tweezers, but since the loading is probabilistic, there is about a 50-60% chance that the trap will have a single atom loaded.

[0247] To prepare deterministic atomic arrays, a real-time feedback procedure identifies randomly loaded atoms and rearranges them into preprogrammed geometric structures. Atomic rearrangement requires moving atoms in tweezers, which can be facilitated to minimize heating, for example by using an acousto-optical deflector (AOD) to deflect the laser beam by an adjustable angle controlled by the frequency of the acoustic waveform applied to the AOD crystal. Dynamic tuning of the acoustic frequency translates into smooth movement of the optical tweezers. Multi-frequency acoustic waves generate an array of laser deflections, which, after focusing through a microscope objective, form an array of optical tweezers with adjustable positions and amplitudes, both controlled by the acoustic waveform. The atoms are rearranged by using an additional set of dynamically moving tweezers overlaid on top of the SLM tweezers array.

[0248] Exemplary Hardware Optical tweezers arrays constitute a powerful and flexible way to build large systems composed of individual particles. Each optical tweezers traps a single particle, including but not limited to individual neutral atoms and molecules for applications in quantum technology. Loading individual particles into such tweezers arrays is a stochastic process, where each tweezers in the system is loaded with a single particle with a finite probability p<1, e.g., p~0.5, for many neutral atom tweezers runs. To compensate for this random loading, real-time feedback can be obtained by measuring which tweezers are loaded and then sorting the loaded particles into programmable geometric structures. This can be done by moving one particle at a time or in parallel.

[0249] Parallel sorting can be achieved by using two acousto-optical deflectors (AODs) to generate multiple tweezers that can pick up particles from an existing particle trapping structure, move them simultaneously, and release them elsewhere. This can include moving particles around within a single trapping structure (e.g., a tweezer array) or transporting and sorting particles from one trapping system to another (e.g., between one tweezer array and another type of optical / magnetic trap). This sorting is flexible and allows for programmed placement of each particle. Each movable trap is formed by an AOD, and its position is dynamically controlled by the frequency components of a radio frequency (RF) driving field for the AOD. Because the RF driving of the AOD can be controlled in real time and can include any combination of frequency components, it is possible to create any grid of traps (such as a line of arbitrarily positioned traps), move the rows or columns of the grid, and add or remove rows and columns of the grid by varying the number, magnitude, and distribution of frequency components in the RF driving field of the AOD.

[0250] In an exemplary embodiment, the optical tweezers array is generated using liquid crystal on a silicon spatial light modulator (SLM), which can programmatically generate flexible arrangements of tweezers.These tweezers are fixed in space for a given experimental sequence, and individual atoms are loaded stochastically, so that each tweezers is loaded with a probability of p 0.5.The fluorescent image of the loaded atoms is taken to identify in real time which tweezers are loaded and which are empty.

[0251] After detecting which tweezers are loaded, the movable tweezers, overlapping the optical tweezers array, can dynamically reposition the atoms from their starting positions to fill the target location of the trap with near uniform packing. The movable tweezers are generated using a pair of crossed AODs. These AODs can be used to either move one atom at a time to fill the target location or generate a single movable trap that moves many atoms in parallel.

[0252] Referring to FIG. 17, a schematic diagram of an apparatus 1700 for fault-tolerant quantum computing according to an embodiment of the present disclosure is provided. As shown in FIG. 17, using a beam generated by a light source 1702 (e.g., a coherent light source; in some exemplary embodiments—a monochromatic light source), an SLM 1704 forms an array of trapping beams (i.e., a tweezer array), which, in the exemplary embodiment shown in FIG. 17, is imaged onto a trapping surface 1708 within a vacuum chamber 1710 by an optical train including elements 1706a, 1706c, 1706d and a high numerical aperture (NA) objective lens 1706e. Other suitable optical trains may be used as would be readily understood by one of ordinary skill in the art. Using a beam generated by a light source 1712 (e.g., a coherent light source; in some exemplary embodiments—a monochromatic light source), a pair of AODs 1714 and 1716 with non-parallel (e.g., orthogonal) directions of acoustic wave propagation generate dynamically movable sorting beams. The sorting beam is overlapped with the trapping beam using an optical series such as that shown in Figure 17 (elements 1717, 1706b, 1706c, 1706d and 1706e). It will be appreciated that other optical series may be used to achieve the same result. For example, sources 1702 and 1712 may be a single source, with the trapping and sorting beams being generated by a beam splitter.

[0253] Dynamic movement of the steering beam is achieved using two non-parallel AODs 1714, 1716 arranged in series. In the exemplary embodiment shown in FIG. 17, one AOD defines the "row" ("horizontal" - 'X' AOD) direction and the other defines the "column" ("vertical" - 'Y' AOD) direction. Each AOD is driven by an arbitrary RF waveform from an arbitrary waveform generator 1720, which is generated in real time by a computer 1722 that processes a feedback routine after analyzing the image of the position where the atoms are loaded. When each AOD is driven with a single frequency component, a single steering beam ("AOD trap") is generated in the same plane 1708 as the SLM trap array. The frequency of the X AOD drive determines the horizontal position of the AOD trap and the frequency of the Y AOD drive determines the vertical position; in this way, the single AOD trap can be stepped to overlap any SLM trap.

[0254] In Figure 17, laser 1702 shines a beam of light onto SLM 1704. SLM 1704 can be controlled by computer 1722 to generate a pattern of beams (the "trapping beam" or "tweezer array"). The beam pattern is focused by lens 1706a, passes through mirror 1706b, and is collimated by lens 1706c on mirror 1706d. The reflected light passes through objective lens 1706e to focus the optical tweezer array in vacuum chamber 1710 on trapping plane 1708. The optical tweezer array laser light continues through objective lens 1724a, passes through dichroic mirror 1724b, and is detected by charge-coupled device (CCD) camera 1724c.

[0255] The vacuum chamber 1710 may be illuminated by an additional light source (not shown). Fluorescent light from atoms trapped on the trapping surface also passes through the objective lens 1724a but is reflected by the dichroic mirror 1724b to an electron multiplied charge-coupled device (EMCCD) camera 1724d.

[0256] In this example, a laser 1712 directs a beam of light to AODs 1714, 1716. The AODs 1714, 1716 are driven by an arbitrary wave generator (AWG) 1720, which in turn is controlled by a computer 1722. The crossed AODs 1714, 1716 emit one or more beams as described above, which are directed to a focusing lens 1717. The beams then enter the same optical train 1706b...1706e as described above for the optical tweezer array and are focused onto the trapping plane 1708.

[0257] It will be appreciated that alternative optical series components may be used to create an optical tweezers array suitable for use as described herein.

[0258] Excitation of atoms to Rydberg states in an optical tweezers array In separate optical tweezers on the micrometer-length scale, atoms in their ground electronic state have negligible van der Waals interactions. Fortunately, neutral atoms offer a remarkable way to switch on the strong interactions by coherently exciting the atoms into their Rydberg states.

[0259] The properties of atomic states scale dramatically with the principal quantum number. A Rydberg state is a highly excited electronic state of an atom, in which one of the atom's electrons has a high principal quantum number, n, in the range of 30-100. In the classical picture of an atom, this situation corresponds to one (negatively charged) electron orbiting far away from the (positively charged) ion core on atomic length scales, thus forming an oscillating electric dipole. Two atoms excited to the same Rydberg state can exhibit very strong dipole-dipole interactions over distances of tens of microns. Interaction energy

number

number

[0260] Consider an ideal two-level atom with a ground state |g> and a Rydberg state |r>. These two states are laser-coupled with a coupling strength set by the angular Rabi frequency Ω, also called the Rabi flop, which is the inverse of the duration of a Rabi cycle, which is the periodic absorption and stimulated emission of quanta of energy by the two-level atom in the presence of an oscillatory driving field. The Rabi frequency is proportional to the strength of the coupling between the light and the atomic transition and the amplitude of the electric field of the light. For two such atoms, also called Rydberg atoms herein, the van der Waals interaction energy V vdw is negligible compared to the laser coupling strength, i.e.

number

number

number

number

number

number

number

[0261] Several implementations of optical excitation from atomic ground states to target Rydberg states are available. Direct laser excitation using single-photon transitions is the simplest. Wavelengths for such transitions in Rydberg atoms are typically in the ultraviolet. For example, 87The single-photon wavelength for Rb is 297 nm. Ultraviolet lasers pose significant experimental difficulties, for example due to material decomposition and the unavailability of optical fibers and low-loss optics. Alternatively, two-photon laser excitation can be used to couple the atomic ground state to the target Rydberg state via an intermediate electronic excited state by irradiation of the atom from opposite faces with two counterpropagating laser beams.

[0262] Consistent with the above description, the term "blockade" is used herein to refer to the phenomenon in which a laser-stimulated transition from a first state (e.g., ground state) to an excited state of an atom in an interacting atom pair cannot be achieved (is blocked) due to a mismatch between the laser frequency and the shifted energy level of the excited state, where the energy level shift is induced electrically or magnetically. For example, blockade can be achieved by dipole-dipole interactions between two adjacent atoms, where one atom is excited to a Rydberg state.

[0263] Detuning from a resonance with an excited state The coherent evolution of two atoms under laser excitation from the ground state |g> to the Rydberg state |r> is given by the Hamiltonian

number

number

number

[0264] Furthermore, in a two-photon laser excitation scheme, two excitation lasers, typically with one frequency in the blue range of the optical spectrum, e.g., 420 nm, and another frequency in the red or infrared, e.g., 1013 nm, are excited into an intermediate state (

number

[0265] It is understood that in various embodiments, the pulse sequences described herein can be generated by computer control of a laser source. Similarly, the detection of the conditions described herein can be performed by various methods known in the art and provided to a computer controller. Thus, it is understood that in various embodiments, computer instructions can be provided to perform the control and detection steps described herein.

[0266] It may also be appreciated that a variety of methods may be used to read out the state of the array of atoms. For example, a quantum gas microscope may be used to determine whether each atom in the array is in an excited state or a ground state, as described in Browaeys, et al., Many-Body Physics with Individually-Controlled Rydberg Atoms, DOI: 10.1038 / s41567-019-0733-z (available at https: / / arxiv.org / abs / 2002.07413), which is incorporated by reference in its entirety.

[0267] The present disclosure may be embodied as a system, method, and / or computer program product. The computer program product may include a computer-readable storage medium(s) having computer-readable program instructions thereon for causing a processor to implement aspects of the present disclosure.

[0268] A computer readable storage medium may be a tangible device that can hold and store instructions for use by an instruction execution device. A computer readable storage medium may be, for example, but not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the above. A non-exhaustive list of more specific examples of computer readable storage medium includes the following: portable computer diskettes, hard disks, random access memories (RAMs), read-only memories (ROMs), erasable programmable read-only memories (EPROMs or flash memories), static random access memories (SRAMs), portable compact disk read-only memories (CD-ROMs), digital versatile disks (DVDs), memory sticks, floppy disks, punch cards or mechanically encoded devices such as raised structures in grooves with instructions recorded on the grooves, and any suitable combination of the above. As used herein, a computer-readable storage medium is not to be understood as a transitory signal per se, such as electric waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide or other transmission medium (e.g., light pulses through a fiber optic cable), or electrical signals transmitted through wires.

[0269] The computer readable program instructions described herein may be downloaded from a computer readable storage medium to the respective computing / processing device or to an external computer or storage device via a network, such as the Internet, a local area network, a wide area network, and / or a wireless network. The network may include copper transmission cables, optical fiber transmission, wireless transmission, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface in each computing / processing device receives the computer readable program instructions from the network and transfers the computer readable program instructions for storage in a computer readable storage medium within the respective computing / processing device.

[0270] The computer readable program instructions for carrying out the operations of the present disclosure may be either source code or object code written in any combination of assembler instructions, instruction set architecture (ISA) instructions, machine instructions, machine dependent instructions, microcode, firmware instructions, state setting data, or one or more programming languages, such as object-oriented programming languages ​​such as Smalltalk, C++, and traditional procedural programming languages ​​such as the "C" programming language or similar programming languages. The computer readable program instructions may run entirely on the user's computer, partially on the user's computer as a stand-alone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be to an external computer (e.g., through the Internet using an Internet service provider). In some embodiments, electronic circuitry including, for example, a programmable logic circuit, a field programmable gate array (FPGA), or a programmable logic array (PLA), can execute computer readable program instructions using state information of the computer readable program instructions to personalize the electronic circuitry to carry out aspects of the present disclosure.

[0271] Aspects of the present disclosure are described herein with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the present disclosure. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer readable program instructions.

[0272] These computer readable program instructions may be provided to a processor of a general purpose computer, special purpose computer or other programmable data processing apparatus to manufacture a machine, and the instructions executing via the processor of the computer or other programmable data processing apparatus create means for performing the functions / acts identified in the flowchart and / or block diagram block(s). These computer readable program instructions may also be stored in a computer readable storage medium that may direct a computer, programmable data processing apparatus and / or other device to function in a particular manner, and a computer readable storage medium having instructions stored therein includes an article of manufacture that includes instructions for performing aspects of the functions / acts identified in the flowchart and / or block diagram block(s).

[0273] The computer readable program instructions may also be loaded into a computer, other programmable data processing apparatus or other device to cause a series of operational steps to be executed on the computer, other programmable apparatus or other device to create a computer-implemented process, the instructions executing on the computer, other programmable apparatus or other device performing the functions / acts identified in the flowchart and / or block diagram block(s).

[0274] The flowcharts and block diagrams in the figures illustrate the architecture, functions and possible implementation operations of the systems, methods and computer program products according to various aspects of the disclosure. In this regard, each block in the flowcharts or block diagrams may represent a module, segment or portion of instructions that includes one or more executable instructions for performing a particular logical function(s). In some alternative implementations, the functions noted in the blocks may occur out of the order noted in the figures. For example, two blocks shown in succession may in fact be executed substantially simultaneously, or the blocks may sometimes be executed in reverse order depending on the functionality involved. It is also noted that each block of the block diagrams and / or flowchart representations, and combinations of blocks in the block diagrams and / or flowchart representations, may be executed by a special purpose hardware-based system that performs a particular function or act or a combination of special purpose hardware and computer instructions.

[0275] The description of various aspects of the present disclosure is given for illustrative purposes, but is not intended to be exhaustive or limited to the disclosed aspects. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described aspects. The terms used herein are selected to best explain the principles of the aspects, practical applications or technical improvements to the technology found in the market, or to enable other skilled in the art to understand the aspects disclosed herein.

Claims

1. 1. A method of error correction in a quantum computer, the quantum computer including a plurality of qubits encoding a plurality of qudits, the method comprising: selecting quantum states of the plurality of qudits such that an angular momentum selection rule prevents mixing between the selected quantum states during leakage of one of the plurality of qudits to a non-interacting state; and correcting the leakage error by optical pumping of the non-interacting state, where the optical pumping preserves the coherence of the selected quantum state in the absence of leakage error; wherein each of the plurality of qudits is encoded in an atomic state of a neutral atom, and selecting quantum states of the plurality of qudits includes selecting a first qudit state having a first magnetic quantum number and a second qudit state having a second magnetic quantum number, the first and second magnetic quantum numbers having opposite signs.

2. The step of correcting the omission error comprises: coherently transitioning atoms in the first qudit state to a first shelving state prior to optical pumping; coherently transitioning atoms in the second qudit state to a second shelving state prior to optical pumping; coherently transitioning a population of atoms in the first shelving state to a first qudit state after optical pumping; Coherently transferring the ensemble of atoms in the second shelving state to the second qudit state after optical pumping. Further comprising: optical pumping does not transition atoms outside the first shelving state; Optical pumping transitions the atoms from any ground state other than the first shelving state to the second shelving state; The method of claim 1.

3. The method of claim 1 or 2, wherein each of the plurality of qudits corresponds to a qubit.

4. a confinement system configured to align a plurality of particles into an array, the plurality of particles configured to encode a plurality of data qudits and an ancilla qudit; The confinement system includes a first laser source and a source of an atomic cloud aligned to generate a plurality of confinement regions, the atomic cloud being disposed to at least partially overlap the plurality of confinement regions; a second laser source configured to drive each of the plurality of particles into one of a plurality of quantum states, the plurality of quantum states being selected such that an angular momentum selection rule precludes mixing between the plurality of quantum states during leakage of one of the plurality of particles into a non-interacting state; a third laser source configured to optically pump the non-interacting state, where the optical pumping preserves coherence of the multiple quantum states in the absence of leakage errors; wherein each of the plurality of qudits is encoded in an atomic state of a neutral atom, and selecting a quantum state of the plurality of qudits includes selecting a first qudit state having a first magnetic quantum number and a second qudit state having a second magnetic quantum number, the first and second magnetic quantum numbers having opposite signs.

5. The system of claim 4 , wherein the array is two-dimensional.

Images