Rovibrational energy levels of H$_2$O by quantum computing

This paper demonstrates the calculation of low-lying rovibrational energy levels of H$_2$O with an accuracy of a few cm$^{-1}$ by deriving a qubit Hamiltonian from Watson's model and employing a hybrid quantum-classical quantum-selected configuration-interaction method on a trapped-ion quantum computer.

The Gist

Imagine a molecule like a tiny, chaotic dance troupe. The dancers are atoms, and they are constantly doing two things at once:

1. Vibrating: Jiggling back and forth like springs (stretching and bending).
2. Spinning: Twirling around like figure skaters.

In the world of chemistry, figuring out exactly how much energy this dance costs is called calculating rovibrational energy levels. If you know these levels, you can predict exactly what color of light the molecule will absorb or emit. This is crucial for everything from understanding the atmosphere to designing new drugs.

For decades, scientists have used supercomputers to simulate this dance. But as molecules get bigger, the dance gets so complex that even the fastest supercomputers get tired and give up.

This paper is about a new attempt to solve this dance using a Quantum Computer—a machine that doesn't just calculate numbers, but actually simulates the quantum nature of the dance itself.

Here is the story of how they did it, explained simply:

1. The Problem: The "Noise" in the Room

The quantum computers available today are like a brilliant but slightly drunk musician. They have incredible potential, but they make mistakes (called "noise"). If you ask them to play a long, complex symphony (like a standard calculation), the noise drowns out the music, and the result is garbage.

Most previous attempts to use quantum computers for chemistry focused only on the electrons (the "music" of the molecule), ignoring the heavy dancing of the atoms themselves. This paper decided to tackle the heavy dancing (the rotation and vibration) using a noisy machine.

2. The Strategy: The "Sample-and-Select" Game

Instead of trying to calculate the entire dance at once (which is too big and too noisy), the authors used a clever trick called Quantum-Selected Configuration Interaction (QSCI).

Think of it like this:

• The Old Way (VQE): Imagine trying to find the best route through a massive city by asking a noisy GPS for directions to every single street and then averaging the answers. The GPS gets confused, and you end up lost.
• The New Way (QSCI): Imagine you ask the noisy GPS to just drive for a little while and tell you which streets it passed through most often. You take that list of "popular streets," write them down on a piece of paper, and then use a regular, non-noisy calculator (a classical computer) to figure out the best route only among those specific streets.

In the paper's terms:

1. The Quantum Step: They let the quantum computer "evolve" a starting state (a simple dance move) for a short time. Because of the noise, the computer doesn't give a perfect answer, but it gives a probability distribution. It tells them: "Hey, the molecule is most likely to be found in these specific vibration/spin combinations."
2. The Selection: They pick the top few combinations (the "popular streets") that the quantum computer suggested.
3. The Classical Step: They take those few selected combinations and feed them into a normal supercomputer. Because the list is now short, the supercomputer can easily solve the math to find the exact energy levels.

3. The Water Molecule Test

They tested this method on Water (H₂O). Water is a small molecule, but it's complex enough to be a good test case.

• They mapped the water molecule's movements onto qubits (the quantum bits).
• They had to account for the fact that water spins and jiggles simultaneously. This is like trying to describe a spinning top that is also bouncing on a trampoline.
• They found that ignoring the "spin-jiggle" connection (called rovibrational coupling) leads to wrong answers. It's like trying to describe a dancer without mentioning that their arms move differently when they spin faster. Their method successfully included this connection.

4. The Results: "Good Enough" for Now

The quantum computer they used (Quantinuum's "Reimei") is noisy. It made mistakes.

• The Good News: Despite the noise, the method worked. They were able to calculate the energy levels of water with an accuracy of about 1 to 4 units (wavenumbers). In the world of spectroscopy, this is considered "spectroscopic accuracy"—it's close enough to match real-world experiments.
• The Analogy: It's like trying to hit a bullseye on a dartboard while standing on a shaking boat. You might miss the center by a few inches, but you're still hitting the board, and you're doing it with a machine that is much smaller and cheaper than the giant crane (supercomputer) usually required.

5. Why This Matters

This paper is a proof-of-concept. It shows that we don't need to wait for "perfect" (error-free) quantum computers to start doing useful chemistry.

• Hybrid Power: It proves that a "noisy" quantum computer can act as a smart filter to find the most important parts of a problem, while a classical computer does the heavy lifting on the final calculation.
• Future Potential: If this works for water, it could eventually work for much larger, more complex molecules (like those involved in climate change or drug discovery) that are currently impossible to simulate accurately.

Summary

The authors took a noisy, imperfect quantum computer, taught it to "sample" the most likely ways a water molecule dances, and then used a classical computer to calculate the exact energy of those dances. They successfully included the tricky connection between spinning and vibrating, proving that even with today's "noisy" technology, we can start solving complex molecular puzzles that were previously out of reach.

Technical Summary

Here is a detailed technical summary of the paper "Rovibrational energy levels of H2O by quantum computing" by Erik Lötstedt and Tamás Szidarovszky.

1. Problem Statement

The calculation of molecular energy levels typically focuses on electronic structure problems. However, a complete description of molecular spectra requires solving the rovibrational Schrödinger equation, which accounts for the coupled rotational and vibrational motions of nuclei.

• Challenge: The number of variables in the rovibrational wavefunction scales linearly with the number of atoms ($N_{var} = 3N_a - 3$), but the size of the basis set required for accurate solutions scales exponentially with the number of vibrational modes. This makes exact diagonalization on classical computers intractable for larger polyatomic molecules.
• Gap in Literature: While quantum computing has been applied to electronic structure and pure vibrational problems, few studies have addressed rovibrational coupling (the interaction between rotation and vibration) using current noisy intermediate-scale quantum (NISQ) devices. Previous quantum studies often neglected rotational degrees of freedom or relied on fault-tolerant algorithms not yet available.
• Goal: To demonstrate the calculation of rovibrational energy levels for a polyatomic molecule ($H_2O$) using a current trapped-ion quantum computer, specifically including the critical rovibrational coupling terms.

2. Methodology

A. Theoretical Framework: Watson's Hamiltonian

The authors derive the qubit form of the Watson Hamiltonian ($H_W$), which describes the rovibrational motion of a non-linear polyatomic molecule. The Hamiltonian is expanded up to fourth order in normal mode coordinates ($Q_k$) and grouped into five distinct terms:

1. $H_{RR}$: Rigid-rotor Hamiltonian.
2. $H_{HO}$: Harmonic oscillator Hamiltonian (vibrational).
3. $V_{anharm}$: Anharmonic potential energy surface.
4. $H_{vibCor}$: Vibrational Coriolis coupling (coupling between vibrational modes).
5. $H_{rovib}$: Rovibrational coupling (coupling between rotation and vibration).

Key Insight: The authors emphasize that neglecting $H_{vibCor}$ and $H_{rovib}$ leads to significant errors (tens of $cm^{-1}$), making their inclusion essential for "spectroscopic accuracy."

B. Qubit Encoding

• Binary Mapping: The vibrational quantum numbers ($v_1, v_2, v_3$) and the rotational projection quantum number ($K$) are mapped to qubits using a binary encoding scheme: $|v_1 v_2 v_3 K\rangle \rightarrow |bin(v_1) bin(v_2) bin(v_3) bin(K)\rangle$.
• Hamiltonian Expansion: The Hamiltonian is converted into a sum of Pauli strings ($H_q = \sum h_k P_k$). For $H_2O$, this results in a Hamiltonian with thousands of terms (e.g., ~200,000 terms for $J=31, v_{max}=3$), which is too large for standard Variational Quantum Eigensolver (VQE) methods due to the circuit depth and measurement overhead.

C. Algorithm: Quantum-Selected Configuration Interaction (QSCI)

To overcome the limitations of VQE, the authors employ a hybrid classical-quantum QSCI approach:

1. Trial Wavefunction Generation: A trial wavefunction $|\psi(\tau, N_{ST})\rangle$ is prepared on the quantum computer using Hamiltonian time evolution (Suzuki-Trotter decomposition) starting from an initial state (e.g., the ground state).
   • $|\psi\rangle = U^{N_{ST}} |\psi_0\rangle$, where $U$ includes only Pauli terms with coefficients $|h_k| > \lambda$ (a cutoff parameter).
2. Sampling: The quantum computer measures the trial wavefunction in the computational basis to estimate the probability distribution $p(x) = |\langle x | \psi \rangle|^2$.
3. Basis Set Selection: A compact basis set $\Omega$ is selected by sampling the most probable basis states from the measured distribution.
4. Classical Diagonalization: The Hamiltonian matrix is constructed and diagonalized classically using only the selected basis set $\Omega$.
5. Error Mitigation: A simple post-selection procedure is used to filter out states with incorrect parity (a known symmetry of the water molecule) caused by hardware noise.

3. Key Contributions

1. Derivation of Qubit Watson Hamiltonian: The paper provides a complete derivation of the Watson Hamiltonian in qubit form, explicitly including all rovibrational coupling terms ($H_{rovib}$) and Coriolis coupling, which were often omitted in previous quantum vibrational studies.
2. Demonstration on NISQ Hardware: The study successfully executes these calculations on the Quantinuum Reimei (20-qubit trapped-ion) quantum computer, a real-world noisy device.
3. Validation of QSCI for Rovibrational Problems: It demonstrates that the QSCI method is highly effective for rovibrational problems, generating compact basis sets that converge to accurate energies with relatively few qubits and shallow circuits.
4. Quantification of Coupling Importance: The work quantifies the impact of different Hamiltonian terms, showing that rovibrational coupling can shift energy levels by up to 30 $cm^{-1}$, significantly altering the spectral structure.

4. Results

• System: Water molecule ($H_2O$) with total angular momentum $J$ up to 6 and maximum vibrational quantum number $v_{max}$ up to 7.
• Hardware: Quantinuum Reimei (20 qubits, all-to-all connectivity, two-qubit gate error $\approx 10^{-3}$).
• Accuracy:
  • For low-lying states ($J=0, v_{max}=3$), the method achieved an accuracy of < 1 $cm^{-1}$ compared to exact classical diagonalization.
  • Even with noise, the QSCI results on Reimei converged to within a few $cm^{-1}$ of the exact values.
  • For $v_{max}=7$ (requiring 9 qubits), accuracies of < 1 $cm^{-1}$ were achieved for several states, with some states showing better convergence with noisy data than noise-free data (suggesting noise can occasionally aid in exploring the Hilbert space).
• Comparison:
  • The QSCI method outperformed simple first-order perturbation theory in generating compact basis sets.
  • It significantly surpassed random basis set sampling.
  • The inclusion of $H_{rovib}$ shifted energy levels by up to 30 $cm^{-1}$ compared to models ignoring rotation-vibration coupling.

5. Significance and Outlook

• Spectroscopic Relevance: The achieved accuracy (a few $cm^{-1}$) approaches "spectroscopic accuracy" (typically defined as 1 $cm^{-1}$), which is sufficient to interpret high-precision experimental data (e.g., from HITRAN databases) for molecules that are too complex for classical full-diagonalization.
• Quantum Advantage Pathway: This work suggests a viable pathway for verifiable quantum advantage in chemistry. Unlike electronic structure problems where the "exact" answer is often unknown for large systems, rovibrational spectra can be measured with extreme precision. A quantum computer could calculate transitions for large molecules (e.g., benzene, water clusters) that are classically intractable, and the results could be directly validated against experimental spectroscopy.
• Future Directions: The authors propose extending this method to larger molecules and molecular clusters. They also suggest investigating more efficient qubit mappings (e.g., grid-based/discrete variable representation) and optimizing the trial wavefunction generation for noisy hardware.

In summary, this paper represents a significant step forward in applying quantum computing to molecular dynamics, moving beyond static electronic structure to the complex, coupled motion of nuclei, and demonstrating that current noisy quantum hardware can already provide chemically relevant results for rovibrational spectra.