Implementation of the Quantum Fourier Transform on a molecular qudit with full refocusing and state tomography

This paper demonstrates the successful implementation of the Quantum Fourier Transform on a 173Yb(trensal) molecular spin qudit by employing a full-refocusing protocol to mitigate inhomogeneous broadening and achieve high-fidelity state recovery, thereby validating the feasibility of complex quantum logic operations on this chemical platform.

The Gist

Imagine you have a tiny, magical spinning top made of a single molecule. In the world of quantum computing, this isn't just a toy; it's a potential computer. But unlike the standard "bits" (which are like light switches that are either ON or OFF) used in most quantum computers today, this molecule is a qudit. Think of a qudit not as a switch, but as a dial with many settings. In this specific experiment, the dial has 12 settings, but the researchers focused on using just three of them (a "qutrit") to do some math.

Here is the story of what they did, explained simply:

The Challenge: A Room Full of Confused Spinning Tops

The researchers wanted to perform a complex mathematical dance called the Quantum Fourier Transform (QFT). You can think of the QFT as a very specific recipe for rearranging information. In a perfect world, if you tell a single spinning top to dance this recipe, it moves perfectly.

However, the researchers didn't use just one molecule; they used a crystal containing millions of these molecules. This is like asking a stadium full of people to perform a synchronized dance.

• The Problem: In a real stadium, not everyone hears the music at the exact same time. Some people are slightly out of sync. In the molecular world, this is called inhomogeneous broadening. Because of tiny differences in their environment, the "spinning tops" in the crystal started to drift out of step with each other very quickly.
• The Consequence: If you tried to run the full dance routine (the QFT) without fixing this, the molecules would get so confused by the time the dance ended that the final result would be a mess. The information would be lost.

The Solution: The "Reset Button" (Refocusing)

To solve this, the team invented a special technique called full refocusing.

Imagine you are leading a group of runners. They start running, but because they have different shoe sizes, they start to spread out and lose their formation.

• The Trick: Instead of letting them run until they are lost, you stop them halfway, tell them to turn around and run back exactly the way they came, and then tell them to turn around again to finish the race.
• The Result: Even though they ran at different speeds, the act of turning around and retracing their steps cancels out their mistakes. When they reach the finish line, they are all perfectly synchronized again, just as if they had never been out of step.

The researchers embedded this "turn around and run back" trick directly into the middle of their quantum dance routine. They used a series of radio pulses (like whistles) to flip the molecules' states back and forth, effectively erasing the confusion caused by the crystal's imperfections.

The Performance: A Perfect Dance

The team tested this on a molecule called 173Yb(trensal).

1. The Setup: They cooled the crystal to near absolute zero (colder than outer space) to keep the molecules calm.
2. The Test: They asked the molecules to perform the QFT dance.
   • Without the trick: The dance was sloppy. The molecules got confused, and the final result was only about 85-90% accurate.
   • With the trick (Refocusing): The molecules stayed in perfect sync. The final result was 96% to 98% accurate.

The Proof: Taking a Snapshot

How do you know the dance was perfect? You can't just look at a spinning top and see its quantum state. The researchers had to take a "snapshot" of the entire system, a process called state tomography.

Think of this like trying to figure out the shape of a spinning top by taking photos of it from every possible angle. By combining all these photos, they reconstructed the exact state of the molecules. The photos proved that the molecules had indeed performed the complex math correctly and that the "refocusing" trick had saved the information from being lost.

Why This Matters (According to the Paper)

The paper claims this is a major step forward because:

• It proves you can run complex algorithms (like the QFT) on molecular "dials" (qudits), not just simple switches.
• It shows that even with a "noisy" crowd of millions of molecules, you can keep them perfectly synchronized using this refocusing technique.
• It demonstrates that molecular spin qudits are a viable candidate for future quantum technologies, offering a way to store more information in a single physical object than current methods allow.

In short, the researchers taught a crowded room of confused molecular spinning tops to perform a complex, synchronized dance by teaching them how to "reset" their steps mid-performance, resulting in a near-perfect execution of a difficult quantum algorithm.

Technical Summary

Technical Summary: Implementation of the Quantum Fourier Transform on a Molecular Qudit with Full Refocusing and State Tomography

Problem Statement
Molecular spin qudits (MSQs), particularly those based on lanthanide complexes like $^{173}$Yb(trensal), offer a promising platform for quantum technologies due to their chemical tunability and ability to encode high-dimensional Hilbert spaces ($d > 2$) within a single physical system. However, experimental demonstrations of complex quantum algorithms on these platforms have been limited. A primary obstacle is the difficulty of maintaining precise control over coherences between qudit states during long pulse sequences. In molecular ensembles, inhomogeneous broadening leads to rapid dephasing ($T_2^* \sim 500$ ns), which is comparable to or shorter than the duration of necessary pulse sequences. This effect causes the loss of quantum information stored in the relative phases of coherences, hindering the implementation of algorithms that require high-fidelity manipulation of the full density matrix, such as the Quantum Fourier Transform (QFT).

Methodology
The authors implemented the QFT on a 3-level subspace (qutrit) of the hyperfine manifold of a single crystal of $^{173}$Yb(trensal), diluted to 0.05% in a diamagnetic Lu(trensal) matrix to minimize intermolecular interactions. The system operates at 1.4 K with a static magnetic field of 0.2 T, defining a qutrit subspace with states $|0\rangle, |1\rangle, |2\rangle$ and transitions at 333.0 MHz and 359.9 MHz.

To address the challenge of inhomogeneous broadening, the authors embedded a full-refocusing protocol directly into the QFT pulse sequence. Unlike standard echo techniques applied post-computation, this scheme interleaves the algorithmic operations with a specific sequence of $\pi$-pulses.

• Refocusing Mechanism: The protocol utilizes a block of five $\pi$-pulses alternating between the two addressable transitions. This sequence swaps the amplitudes of the basis states ($|0\rangle, |1\rangle, |2\rangle$) such that each amplitude traverses all basis states during the free evolution periods. Consequently, the differential phases accumulated due to inhomogeneous broadening are converted into a global phase, effectively refocusing the state.
• Pulse Sequence: The standard QFT decomposition for a qutrit (9 planar rotations) was modified into a 19-pulse sequence where refocusing blocks are inserted every six pulses.
• State Preparation and Readout: Since thermal initialization at 1.4 K does not yield a pure state, a pseudo-pure state was generated by equipopulating states $|1\rangle$ and $|2\rangle$ and allowing coherence to decay. The output of the algorithm was characterized using complete quantum state tomography, measuring the populations and all off-diagonal coherences (including two-quanta coherences) via Hahn-echo detection.

Key Contributions

1. First Full Refocusing in MSQs: The paper presents the first experimental implementation of a full refocusing scheme tailored for molecular spin qudits within a complex quantum algorithm.
2. Algorithmic Control on Qudits: It demonstrates the execution of the QFT, a fundamental primitive for algorithms like Shor's and phase estimation, on a molecular qutrit with high fidelity.
3. Mitigation of Inhomogeneous Broadening: The work provides a practical strategy to recover coherences in molecular ensembles where $T_2^*$ is significantly shorter than the sequence duration, effectively extending the usable coherence time to the intrinsic pure dephasing limit ($T_2 > 0.1$ ms).
4. Microscopic Insight: Through simulations and experimental data, the authors identify strain in the hyperfine coupling constants (rather than the electronic Zeeman term) as the dominant source of inhomogeneous broadening in their system.

Results
The performance of the algorithm was evaluated by comparing the experimental density matrix ($\rho_{exp}$) against the ideal theoretical output ($\rho_{ideal}$) using state fidelity ($F$).

• Non-Refocused QFT: Without the refocusing scheme, the implementation suffered from significant loss of coherence, particularly for two-quanta coherences. Fidelities were limited to $F \leq 0.90$ (e.g., $F = 0.85 \pm 0.01$ for initial state $|0\rangle$).
• Refocused QFT: The implementation with the embedded refocusing protocol achieved near-complete recovery of all coherences across the density matrix. Fidelities improved significantly, reaching $F \geq 0.96$ (e.g., $F = 0.98 \pm 0.02$ for initial state $|0\rangle$ and $F = 0.98 \pm 0.02$ for the superposition state $\frac{1}{\sqrt{2}}(|0\rangle - i|1\rangle)$).
• Simulation: Numerical simulations incorporating the measured inhomogeneous broadening confirmed that the non-refocused sequence leads to coherence attenuation, while the refocused sequence theoretically recovers the ideal operation in the absence of other error sources.

Significance
The authors claim this work represents a key step toward practical quantum logic operations on molecular spin qudits. By demonstrating that complex algorithms can be executed with high fidelity despite the inherent inhomogeneous broadening of molecular ensembles, the study validates the potential of MSQs as viable candidates for qudit-based quantum technologies. The successful integration of refocusing into the algorithmic sequence suggests a pathway to scaling these systems, potentially enabling more efficient error correction and deeper circuits in the Noisy Intermediate-Scale Quantum (NISQ) era. The authors note that while their results are specific to the $^{173}$Yb(trensal) system, the refocusing strategy is generalizable to other qudit dimensions and algorithms, provided the connectivity allows for the necessary pulse sequences.