Quantum logic operations and algorithms in a single 25-level atomic qudit

This paper demonstrates the feasibility of high-dimensional quantum computing by experimentally realizing a 25-level $^{137}$Ba$^+$ ion qudit with high-fidelity state control, analyzing error scaling, and successfully executing complex multi-qubit algorithms like the Bernstein-Vazirani algorithm and Toffoli gate within a single ion.

The Gist

Imagine you are trying to build a super-computer, but instead of using tiny switches that can only be "on" or "off" (like a light switch), you want to use a single, magical dial that can point to 25 different positions at once. This is the core idea behind the research in this paper.

Most current quantum computers use qubits, which are like coins that can be Heads, Tails, or a wobbly mix of both. This team of researchers from the University of Waterloo decided to try something different: a qudit. Think of a qudit not as a coin, but as a 25-sided die. Instead of just 0 and 1, it can be in a state of 0, 1, 2... all the way up to 24, or any superposition of them.

Here is what they actually achieved, broken down into simple concepts:

1. The "Super-Atom" Dial

The researchers used a single atom of Barium-137. Inside this atom, electrons can sit in different energy "floors." Usually, scientists only use two floors (like a ground floor and a first floor) to make a qubit.

• The Achievement: They figured out how to access and control 25 different floors within that single atom simultaneously.
• The Analogy: Imagine a piano. Most quantum computers play only two keys at a time. This team learned how to play a chord using 25 specific keys on a single piano, and they can switch between them incredibly fast and accurately.

2. Setting the Stage (Preparation and Reading)

Before you can play a song on a piano, you need to make sure every key is in the right spot, and you need to be able to hear which keys were pressed at the end.

• The Challenge: Getting the atom to start in a specific "floor" and then reading it out without messing it up is very hard when you have 25 options. It's like trying to sort 25 different colored marbles into specific jars without dropping any.
• The Result: They developed a special "optical pumping" technique (using lasers like a vacuum cleaner and a funnel) to sort the atom into the right starting spot 98.6% of the time. When they read the result, they were correct 99.5% of the time. This is a very high score for such a complex system.

3. Keeping the "Spin" in Sync (Coherence)

Quantum magic relies on the atom being in a "superposition" (a mix of many states at once). However, if the environment is noisy (like a bumpy road), the atom gets confused and loses its mix, turning back into a simple state.

• The Test: They created a "Ramsey experiment," which is like spinning a top. They spun the atom into a mix of up to 24 different states at once and then tried to stop it perfectly back in its original spot.
• The Result: They successfully kept the atom coherent (in sync) even when mixing 24 states. However, as they added more states, it got harder to keep them all in sync, much like trying to balance more and more spinning plates on a single stick. They identified that magnetic field fluctuations and laser noise were the main things knocking the plates off.

4. Running Algorithms on One Atom

To prove this "25-sided die" could actually do math, they ran two famous quantum algorithms on a single atom:

• Bernstein-Vazirani Algorithm: This is a "secret code" finder. In a normal computer, you might need to ask a question multiple times to find a secret number. With their 25-level atom, they could find a 2-bit or 3-bit secret code in a single try. They succeeded 97.9% of the time for the 2-bit code and 83.8% for the 3-bit code.
• Toffoli Gate (CCCNOT): This is a complex logic gate that acts like a "triple-switch." They successfully implemented a version of this using 4 "virtual" bits encoded in their single atom, achieving a 99.5% success rate.

5. Why This Matters (According to the Paper)

The paper argues that using these high-dimensional "dials" (qudits) is a promising path forward.

• Efficiency: Instead of needing 4 separate atoms to hold 4 bits of information, you can hold that same amount of information in just one atom by using its 25 levels.
• Error Correction: Having more levels gives you more room to hide errors and fix them, similar to how a larger net catches more fish.
• Future Potential: They built a computer model showing that if they clean up the noise (like shielding the atom from magnetic fields), they could get these error rates down to extremely low levels, making this a viable way to build future quantum computers.

In Summary:
The researchers took a single atom, turned it into a 25-level quantum dial, taught it how to start and stop perfectly, and used it to solve math problems that usually require multiple atoms. They proved that using the full "richness" of an atom's energy levels is a powerful way to make quantum computers more efficient and compact.

Technical Summary

Technical Summary: Quantum Logic Operations and Algorithms in a Single 25-Level Atomic Qudit

Problem Statement
Scaling quantum computers remains a significant scientific and technological challenge. While current efforts largely focus on binary qubit architectures, trapped ion systems possess a richer energy-level structure than the two levels typically utilized. Leveraging the full range of intrinsic degrees of freedom in these systems offers a promising route toward enhanced algorithmic performance and hardware efficiency. However, previous demonstrations of trapped ion qudit-based quantum computing have been limited to smaller dimensions ($d \leq 7$), constrained by the number of metastable states available in chosen ion species. The challenge lies in achieving high-fidelity control, state preparation, and readout across a significantly larger Hilbert space (up to $d=25$) while understanding how errors scale with dimension and identifying dominant error sources.

Methodology
The authors experimentally investigate the use of single $^{137}\text{Ba}^+$ ions, which possess a large nuclear spin yielding numerous metastable states. The qudits are encoded in the $S_{1/2}$ and $D_{5/2}$ electronic manifolds.

• State Preparation and Measurement (SPAM): The team developed a novel extension to narrow-band optical pumping (NBOP) to initialize any of the 25 internal levels on demand. This involves flushing specific states, applying "shelving" pulses at 1762 nm to empty all but the target state, and repumping via 614 nm light. To achieve high-fidelity readout, they employ heralding techniques: for initialization, they check for fluorescence after shelving to discard failed attempts; for measurement, they sequentially de-shelve states from $D_{5/2}$ to $S_{1/2}$ and check for fluorescence, assigning the first bright outcome as the result.
• Coherence Probing: To demonstrate coherent control, the authors generalized qubit Ramsey interferometry to probe mutual coherence across up to $d=24$ states simultaneously. They utilize quadrupole transitions driven by a 1762 nm laser to create equal superpositions and re-phase populations, measuring the contrast of the $|0\rangle$ state population as a function of dimension.
• Algorithm Implementation: The researchers implemented quantum algorithms using "virtual qubits," where $n$ qubits are encoded in $d=2^n$ basis states. They executed the Bernstein-Vazirani algorithm (finding a secret key) for 2 and 3 virtual qubits and a 4-qubit Toffoli (CCCNOT) gate.
• Error Modeling: A full physical error model was constructed using shot-to-shot Monte Carlo sampling. This model incorporates independent measurements of magnetic field noise, laser frequency/power fluctuations, transition frequency miscalibration, and A/C line-induced magnetic field changes, with no free parameters.

Key Results

• High-Dimensional SPAM: The system achieved a heralded state preparation and measurement fidelity of 99.51(5)% averaged over all 25 levels. The primary error sources identified were off-resonant driving, spontaneous decay from $D_{5/2}$, and bright/dark discrimination errors. The authors project that with engineering improvements (e.g., optimized magnetic fields and photon collection), SPAM errors could be reduced to below $10^{-4}$ for $d \leq 25$.
• Coherence Scaling: Coherence was demonstrated for superpositions of up to 24 states. Contrast measurements showed a decrease as dimension $d$ increased, primarily due to the inclusion of states with differing magnetic field sensitivities and longer pulse sequences. For $d > 17$, where multi-pulse rotations are required, a sharp drop in contrast was observed. The error model, validated against experimental data, identified pulse angle errors as dominant at lower $d$, while laser frequency noise and A/C line-induced magnetic field changes dominated at higher $d$.
• Algorithmic Performance:
  • Bernstein-Vazirani: The algorithm successfully found a 2-bit secret key with 97.9(2)% probability and a 3-bit key with 83.8(8)% probability. The authors attribute the comparable fidelity to multi-ion implementations to the absence of multi-ion gate errors, despite the sequential nature of the virtual qubit gates.
  • Toffoli Gate: A 4-qubit CCCNOT gate was implemented with a truth-table fidelity of 99.5(2)%, limited mainly by SPAM errors.
• Error Projections: The physical error model predicts that for dimensions $d \leq 16$, combined errors from identified sources can be reduced to the $10^{-3}$ level or better using technologies already demonstrated in trapped-ion systems.

Significance
The paper demonstrates that high-dimensional qudit encoding in $^{137}\text{Ba}^+$ is a viable approach for quantum information processing. By achieving the largest digital encoding of a trapped ion qudit reported to date (25 levels), the work establishes that:

1. Hardware Efficiency: A single ion can host multiple virtual qubits, potentially reducing the need for complex multi-ion gate operations which are a primary source of error in current architectures.
2. Scalability: The error scaling with dimension is understood and manageable; the authors show that current noise levels allow for high-fidelity operations up to $d=16$, with projections for even higher fidelities using existing engineering improvements.
3. Algorithmic Potential: The successful implementation of non-trivial algorithms (Bernstein-Vazirani) and complex gates (Toffoli) on a single ion validates the utility of qudit-based approaches for quantum simulation and computation.

The findings suggest that quantum computing architectures based on large-dimensional qudits hold significant promise, offering a new region of the trade-off space in processor design where the complexity of single-qubit gates is balanced against the reduction in two-ion gate requirements.