Advancing Neutral Atom Quantum Computing: Unleashing the Potential of Alkaline-Earth Atoms and Beyond

WOMANIUM Global Quantum Media Project Initiative — Winner of Global Quantum Media Project

Introduction

Alkaline-earth atoms, including Strontium and Ytterbium, have emerged as promising candidates for neutral atom quantum computing due to their unique characteristics. With two valence electrons like helium, these atoms enable efficient manipulation and control, making them ideal for qubit applications. While Strontium is better suited for precision clocks and many-body dynamics, Ytterbium stands out as a particularly favorable element for neutral atom qubits, leading to significant advancements in this field. This article explores the concept of Erasure Conversion and its application to a metastable Ytterbium neutral atom qubit, showcasing the remarkable progress and potential of using alkaline earth metals to revolutionize neutral atom quantum computing.

Exploring Alkaline-Earth Atoms

1. Broad and Narrow Transitions

The presence of two valence electrons leads to a distinct level structure characterized by the possibility of their spins being either anti-aligned (resulting in a singlet state with zero total electron spin) or aligned (giving rise to a triplet state). Transitions between singlet and triplet states are infrequent, or “forbidden,” compared to transitions within the singlet or triplet states. Consequently, two types of optical transitions are generally observed: those allowed between singlet and triplet states, and those occurring within the singlet or triplet states. The narrow and forbidden transitions offer a notable advantage in laser cooling applications. The broad transitions enable rapid cooling from room temperature to lower temperatures, while the narrow transitions facilitate cooling to extremely low temperatures by leveraging the fundamental limit of the transition’s line length, known as the Doppler Temperature.

Ytterbium Energy Level Diagram, Image Credit: [1]

2.Highly Coherent Nuclear Spin Qubits

Another remarkable feature of alkaline-earth atoms is observed in their ground state when the two valence electrons are anti-aligned, resulting in an absence of electronic spin. When an isotope possesses a nuclear spin in this ground state, it remains remarkably isolated and exhibits minimal sensitivity to magnetic or optical fields. Consequently, alkaline-earth atoms serve as exceptionally stable and robust qubits, making them highly desirable for quantum computing applications.

3. Optical Clock Transitions and Metastable Qubits

The true ground state of alkaline-earth atoms is a singlet state, while a lower energy triplet state, although technically not stable, exhibits an exceptionally long lifetime. Ytterbium atoms in this metastable triplet state possess a lifetime of several tens of seconds. Since this state still lacks a net electronic spin, it can be employed as a qubit. The utilization of this metastable triplet state offers significant advantages for specific applications in quantum information processing.

4. Optical Transitions between Rydberg States

In alkaline-earth atoms, upon transitioning to the Rydberg state, an ion becomes trapped within the Rydberg atom, with one valence electron remaining in the outer shell. The optical transitions of this trapped ion within the Rydberg atom provide a rich avenue for conducting intriguing experiments and achieving diverse functionalities.

By harnessing the unique properties of alkaline-earth atoms, particularly their two-valence-electron configuration, researchers can explore a plethora of applications ranging from laser cooling to highly coherent qubits and optical transitions between Rydberg states.

Two Key Aspects

0. The Rationale Behind the Qubit of Interest

It is essential to comprehend the motivation underlying the interest in the aforementioned qubit, which is predicated on a theoretical concept. This concept in question is known as “Efficient Quantum Error Correction with Erasure Conversion.”

1. The Novel Neutral Atom Qubit Demonstrated using Ytterbium’s Metastable State

A novel neutral atom qubit that showcases significant potential in the field of quantum computing. This remarkable achievement is centered around the utilization of Ytterbium’s metastable state. By exploiting the unique characteristics and properties of this metastable state, a novel qubit configuration is successfully engineered. This breakthrough represents a significant advancement in neutral atom quantum computing.

Error Correction in Quantum Computing

Errors pose a major challenge in quantum computing. If we had perfect qubits capable of executing flawless gates, we could efficiently factor large numbers. However, the presence of qubit decoherence and gate errors hinders our progress. Fortunately, quantum error correction provides a viable path forward. By redundantly encoding the information we aim to compute onto multiple qubits, we can mitigate errors. This approach allows us to recover the desired state by utilizing the remaining error-free information in the encoded qubits.

Error correction is a well-established concept in classical computing and communication. Let us consider a noisy communication channel as an example. To transmit the message “0,” we can send the sequence “000,” and for “1,” we transmit “111.” The receiver employs a majority vote mechanism to determine the intended message. For instance, the received sequence “000” would be decoded as “0” (even though the middle state got flipped to “1”). However, if the received sequence is “111,” an incorrect decoding of “0” occurs due to two errors. Nevertheless, we can correct any single error that may have occurred.

Quantum error correction involves encoding logical qubits into multiple physical qubits. For example, we can encode the superposition α|0⟩ + β|1⟩ as α|000⟩ + β|111⟩. However, a challenge arises when performing the majority vote operation on these qubits. Directly measuring the qubits to determine whether there are more zeros or ones would collapse the superposition and destroy the quantum information. Peter Shor introduced a crucial insight: instead of measuring the qubits directly, we should carefully measure the desired property of interest, namely, the relative parity of two qubits. By using additional qubits called ancillas, we can perform checks on the relative states of these qubits. Through careful measurement of the ancilla qubits, we can ascertain whether the first two qubits are the same and whether the second two qubits are the same. Importantly, this measurement does not reveal the values of the individual qubits (0 or 1), allowing us to identify errors without destroying the superposition. However, this specific code can only correct bit flips, where |0⟩ becomes |1⟩ and |1⟩ becomes |0⟩, but not other types of quantum errors like phase flips (e.g., (|0⟩ + |1⟩) becoming (|0⟩ — |1⟩)). Nonetheless, there are generalizations of this code that can handle such errors.

Error Correction, Image Credit: [1]

Error Threshold

An essential concept in error correction is the threshold, which relates to the number of errors that can be corrected. In classical repetition codes, where 0 is encoded as 000, we can correct one error but fail if two or more qubits have errors. By employing more copies and constructing larger codes (e.g., encoding 0 as 0^d with d copies), we can correct up to (d-1)/2 errors. However, this also creates more opportunities for errors to occur. While we can use arbitrarily large codes to correct an increasing number of errors, it does not guarantee successful information transmission.

The crucial factor is the threshold error rate. When the physical error rate is below the threshold, we can achieve reliable transmission with sufficient redundancy. Above the threshold, error correction becomes ineffective. The logical error probability that the receiver obtains the correct state is directly proportional to the ratio of the error rate to the threshold. Errors are exponentially suppressed with distance below the threshold, emphasizing the significance of being sufficiently below it.

Threshold, Image Credit: [1]

Fault-Tolerant Thresholds in Quantum Circuits

The threshold property extends to quantum circuits, where errors predominantly occur during gate operations rather than solely within the stored qubits. Analyzing code performance requires consideration of errors induced during ancilla qubit measurements and error checking processes. This consideration increases the redundancy cost, as larger codes with more qubits not only have more qubits susceptible to errors but also introduce additional opportunities for errors to occur during gate operations. Currently, the surface code represents the best-known threshold for a generic error model, estimated at approximately 1%. While this threshold is considerably lower than 50%, it aligns with the error rates observed in many current qubit platforms.

Fault-Tolerant Threshold, Image Credit: [1]

Enhancing Thresholds through Structured Noise

Efforts to raise thresholds constitute a significant area of research in quantum computing and quantum error correction. While it remains uncertain whether improved error-correcting codes with higher thresholds will emerge, alternative approaches can be explored. One such approach involves designing qubits that exhibit biased noise, where certain types of errors are suppressed intrinsically in the qubit’s design. This restriction in error models may allow for a higher threshold. Biased noise has been widely studied in recent years, particularly in relation to superconducting qubits like the Kercat qubit. It has also inspired investigations into different ways of encoding qubits using neutral atoms and ions. However, incorporating biased noise into quantum circuit-level operations can be challenging, as it requires specialized gates that avoid converting one type of error into another.

Qubit encoding, stabilization and implementation. Image Credit: [2]

Erasure Errors: Error Localization

Considering a unique noise model called erasure errors provides an interesting perspective. Erasure errors involve qubits where the occurrence of an error is immediately known, such as when a qubit “lights up” or emits fluorescence. Erasure errors are easier to correct than other types of errors. In a classical repetition code, if erasure errors are detected, they can be excluded from the majority vote, allowing for successful correction. Generalizing this concept to a distance-d code, we can correct up to d-1 errors. The advantage of error correction significantly increases with larger numbers of qubits.

Depolarizing channel and Erasure Channel, Image Credit: [1]

Importantly, the ability to correct twice as many errors does not merely result from the presence of twice as many erasure errors compared to depolarizing errors. The advantage is a characteristic not limited to this specific example but applicable to both quantum and classical codes.

Depolarizing channel correct (d-1)/2 errors. Erasure channel correct d-1 errors. Thus, any code, classical or quantum, can correct twice as many erasures as depolarizations. Image Credit: [1]

Implementing Erasure Errors in Neutral Atom Qubits

Inspired by the advantages of erasure errors, exploration has been conducted on neutral atom qubits, where the qubit is stored in the spin sublevel of the ground state. In these neutral atom qubits, the primary source of errors occurs when transitioning to the Rydberg states during two-qubit gates. During this process, the atoms must remain in the Rydberg state long enough for the gate operation to take place. However, the Rydberg state has a finite lifetime and carries the risk of decay. By utilizing metastable states in ytterbium, we can ensure that decay predominantly occurs either back to the true ground state (which can be detected through fluorescence) or becomes trapped in other Rydberg states (detectable by converting the remaining atoms into ions and observing fluorescence). Approximately 2% of the decays during gate operations lead to errors remaining in the qubit subspace, resulting in a significant majority of errors being detectable and converted into erasure errors.

FIG. 1 provides an overview of the neutral atom quantum computer setup. (a) A schematic representation is shown. (b) The physical qubits are individual 171Yb atoms, with qubit states encoded in the metastable 6s6p 3P0 F = 1/2 level (subspace Q). Two-qubit gates are performed using the Rydberg state |ri accessed through a single-photon transition. Errors during gates mainly result from decays from |ri, with a total rate Γ = ΓB + ΓR + ΓQ. These decays can be detected and converted into erasure errors by observing fluorescence from ground state atoms (subspace R) or ionizing remaining Rydberg population and collecting fluorescence (subspace B). © A patch of the XZZX surface code is depicted, showing data qubits, ancilla qubits, and stabilizer operations. (d) The quantum circuit represents a single stabilizer measurement in the XZZX surface code with erasure conversion, where erasure detection occurs after each gate and erased atoms are replenished using a movable optical tweezer. Image Credit: [3]

Quantitative simulation studies of the XZZX surface code, where 98% of the errors are erasure errors, have demonstrated a substantial increase in the threshold from the typical surface code value of around 0.9% to 4.1%. This four-fold increase in the threshold greatly facilitates error correction and reduces the logical error rate by orders of magnitude, even when operating far below the threshold. Thus, incorporating erasure errors can considerably enhance the effectiveness and ease of error correction.

The XZZX surface code. Image Credit: [4]

The investigation of error correction techniques and error models, including biased noise and erasure errors, holds significant promise for advancing quantum computing capabilities.

These endeavors offer the potential to increase thresholds, reduce logical error rates, and pave the way for practical and robust quantum information processing.

The figure shows the scaling of the logical error rate with the physical qubit error rate p in two scenarios: pure computational errors (Re = 0) and a high conversion to erasure errors (Re = 0.98). The error thresholds are determined from the crossing of d = 11 and d = 15, with pth = 0.937(4)% and pth = 4.15(2)%, respectively. The error bars indicate the 95% confidence interval in the logical error rate pL, estimated from the number of trials in the Monte Carlo simulation. Image Credits: [3]

Metastable Yb: A Novel Neutral Atom Qubit

The use of neutral atom qubits, specifically metastable Ytterbium (Yb) qubits, represents a significant advancement in quantum computing. The process begins with a Ytterbium metal chunk placed in an oven. Through successive stages of laser cooling, the atoms are trapped in a magneto-optical trap (MOT) inside a vacuum chamber. Optical tweezers, created using a microscope objective on top of the chamber, allow for the manipulation and measurement of the qubits. An array of 400 optical tweezers is generated, with approximately 200 atoms occupying half of them (±40 atoms).

Metastable Qubit Overview

The metastable Yb qubits rely on the up and down nuclear spin states in the 3p zero state. Single-qubit gates are achieved by driving the rotation of the nuclear spin, yielding high-fidelity results with typical gate values of 99.9%. Two-qubit gates are performed by selectively exciting one nuclear spin level to the Rydberg state, achieving gate fidelities of approximately 98%. Crucially, the ability to detect population in the ground state without disturbing the qubits in the metastable state is essential. Two types of images are captured: a high-fidelity, slow green inter-combination transition image and a fast image with a duration of 20μs, allowing for erasure error detection.

(a) Pertinent energy levels of 174Yb are displayed, with indicated transition wavelengths (λ) and linewidths (Γ). (b) The experimental setup diagram highlights the configuration of cooling, imaging, and trapping beams. Two of the 3D MOT beams lie in the xy plane, while the third beam propagates along the z axis through the objective lens. The angled imaging beam lies in the xz plane. Additional details can be found in the text. Average images and (d) single-shot images of atoms in a 4x4 tweezer array are shown, with a spacing of 6 μm (35 ms exposure time). The color bar represents the number of detected photons per pixel. Image Credits: [5]

Preparing and Measuring 3p0 State

To prepare the qubit in the metastable state, a laser-driven transition from the ground state is utilized. Optical pumping techniques are employed, exciting the atoms to a higher state using allowed transitions and allowing them to decay into the desired metastable state. The reverse process is applied for qubit measurement, where the atoms are optically pumped back to the ground state. The combined fidelity of the initialization and readout processes is approximately 90%, limited by certain effects during the de-pumping process.

The metastable 171Yb qubit setup includes an array of optical tweezers and cameras for nondestructive and fast imaging. The qubit encoding is based on nuclear spin sublevels within the metastable 3P0 state. Single-qubit operations use an RF magnetic field, while entangling gates involve coupling the qubit state to a Rydberg state with a UV laser. Rabi oscillations are observed between qubit states, with different pulse times for nuclear spin and Rydberg state transitions. Image Credits: [6]

Metastable State Lifetime and Coherence

Coherence is a crucial factor in the performance of qubits. The metastable state of Yb atoms has a lifetime of about three seconds in optical tweezers, compared to 20 seconds in free space. This slightly shorter lifetime in optical tweezers is due to processes like Raman scattering and photoionization caused by the intense light. The Raman scattering process is mainly linearly dependent on trap power. The coherence time (T2) for superposition states is approximately 1 second, significantly longer than the metastable state lifetime. Notably, errors occurring within the metastable state are highly suppressed, with T1 errors practically non-existent.

Figure showcases the performance of single-qubit gates with mid-circuit erasure conversion. (a) The lifetime of the 171Yb metastable qubit in an optical tweezer is measured, with population decay times of Γm = 2.96(12) s and an average spin-flip time of T1 = 23(14) s. (b) The decay rate Γm of the metastable state is shown as a function of trap power, following a quadratic model. The probability of recovering an atom in the ground state after a decay from the metastable state is demonstrated. (d) Fast images on the 1S0 − 1P1 transition are used for discrimination, achieving a fidelity of 0.986. (e,f) Randomized benchmarking (RB) of single-qubit gates is performed, with erasure errors converted into population errors by probing the ground state. The total error rate is  = 1.0(1) × 10−3 (green), which decreases to c = 4.5(3) × 10−4 after conditioning on not detecting a ground state atom. (g) The threshold for detecting a ground state atom affects erasure conversion performance, with the analysis using a threshold near 700. Image Credits: [6]

Converting 3p0 Decays into Erasures

To detect errors and convert them into erasure errors, the decay rate out of the metastable state is characterized during a sequence of single-qubit gates. Randomized benchmarking techniques are employed, involving the preparation of a qubit in a specific state followed by numerous single-qubit rotations. By analyzing the exponential decay of the probability to find the qubit in the correct state, the error probability per gate (or fidelity) can be extracted. Mid-circuit erasure detection is performed by periodically checking for errors during the circuit. This detection does not disturb the circuit if no errors have occurred. The conditional probability of finding the qubit in the correct state significantly improves, reaching 0.9996%. This demonstrates the successful conversion of errors into erasure errors, enhancing the qubit fidelity.

Rydberg Gates on Nuclear Spin

Two-qubit gates in the metastable Yb qubits are accomplished by exciting one atom to the Rydberg state. A specific type of gate, known as a time-optimal pulse, is employed. This pulse induces a phase difference of π depending on the initial state, effectively implementing the CZ gate. The experimental implementation of this pulse involves laser excitation at the appropriate frequency with phase modulation. Initial characterization of the gate involves the preparation of entangled states, followed by fidelity measurement using parity oscillations. The intrinsic fidelity of the gate is estimated to be around 99% ± 2%.

Time-optimal two-qubit gates are described as follows: (a) The laser amplitude and phase profiles for the time-optimal CZ gate are shown. (b) Bloch sphere trajectories during the gate are depicted for the {|01⟩, |0r⟩} subspace (red) and the {|11⟩, |W⟩} subspace (grey), where |W⟩ represents (|1r⟩ + |r1⟩)/√2. The gate sequence used to prepare and characterize the Bell state |ψ_B⟩ = (|00⟩ + |11⟩)/√2 is presented. (d) The population of |00⟩ and |11⟩ in the Bell state is illustrated, with the dashed line indicating the probability of preparing and measuring the bright state |00⟩ without the CZ gate. (e) Parity oscillations demonstrating the coherence of the Bell state are shown, measured using only the bright state population. The off-diagonal part of the Bell state density matrix is denoted as Pc = 4A, where A represents the oscillation amplitude cos(2θ). The Bell state fidelity is calculated as (P00 + P11 + Pc)/2. (f) The two-qubit gate is characterized using a randomized circuit, with an error rate of  = 2.0(1) × 10^−2 per gate. Image Credits: [6]

Benchmarking with Random Circuit and Interleaved Benchmarking

To obtain a more precise measurement of the gate fidelity, randomized benchmarking techniques are employed. Random circuits consisting of alternating single-qubit and two-qubit gates are implemented, with the single-qubit gates randomized to rotate the state in a random manner. The probability of correctly ending in the desired state as a function of the circuit depth provides insights into the intrinsic error of the two-qubit gate. Additionally, interleaved benchmarking techniques are applied to evaluate the fidelity of the CZ gate in the presence of mid-circuit erasure detection. By removing errors detected during the circuit, the fidelity is enhanced from 98% to 98.7%.

Erasure conversion for two-qubit gates is described as follows: (a, b) Any population that leaks into the |r⟩ state or other Rydberg states can be recovered by allowing it to decay. Randomized circuit characterization of CZ gates is performed with interleaved erasure detection after every two gates. The total error rate of  = 2.0(1) × 10−2 per gate (green) is reduced to c = 1.3(1) × 10−2 per gate (blue) after conditioning on not detecting an atom in the ground state before the end of the circuit. It is worth noting that the green curve is reproduced in Fig. 3f. (d) The erasure detection fidelity during two-qubit gates is analyzed, following the approach in Fig. 2g. (e) The erasure detection probability for different initial states under repetitive CZ gates is examined. Linear fits are overlaid, with the shaded area representing one standard deviation. The erasure probability for the |00⟩ state is determined to be 4(6) × 10−4 per gate, which is consistent with zero. Image Credits: [6]

Future Improvements and Generalizability

The concept of erasure conversion opens up new avenues for improving logical error rates in quantum computing. By engineering qubits to have a higher fraction of erasure errors, the threshold for error correction can be increased. This approach applies not only to metastable Yb qubits but also to other quantum computing platforms. The exchange of ideas between different platforms, such as quantum dots, superconductors, trapped ions, and neutral atoms, allows for the exploration and implementation of erasure-dominated qubits. This concept holds promise for detecting and correcting errors in various quantum computing and quantum simulation applications.

References

Photo by Sigmund on Unsplash

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[3] Y. Wu, S. Kolkowitz, S. Puri, et al., “Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays,” Nat Commun, vol. 13, no. 1, p. 4657, Aug. 2022. DOI: 10.1038/s41467–022–32094–6. J.P.

[4]Bonilla Ataides, D.K. Tuckett, S.D. Bartlett, et al., “The XZZX surface code,” Nat Commun, vol. 12, no. 1, p. 2172, Apr. 2021. DOI: 10.1038/s41467–021–22274–1.

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[6] S. Ma et al., “High-fidelity gates with mid-circuit erasure conversion in a metastable neutral atom qubit,” arXiv:2305.05493 [quant-ph], May 2023.