Ground State Energy via Adiabatic Evolution and Phase Measurement for a Molecular Hamiltonian on an Ion-Trap Quantum Computer

Technical Summary: Ground State Energy via Adiabatic Evolution and Phase Measurement for a Molecular Hamiltonian on an Ion-Trap Quantum Computer

Problem Statement
Accurately estimating molecular ground-state energies (GSEs) is a primary application for quantum computing, yet practical realization is hindered by the difficulty of preparing accurate quantum states, estimating observables with high precision, and mitigating hardware noise. Classical methods like Hartree–Fock (HF) or coupled cluster often fail to capture essential electron correlation effects in strongly correlated systems, while more sophisticated classical methods scale exponentially. Quantum algorithms, specifically Adiabatic State Preparation (ASP) combined with Quantum Phase Estimation (QPE), offer a systematic improvement but face challenges regarding parameter optimization, barren plateaus, and the high circuit depth required for controlled-$U^{2^k}$ operations. Furthermore, distinguishing between errors that can be mitigated via software and those requiring hardware-level reduction is critical for advancing the field.

Methodology
The authors implemented a purely quantum, ab initio protocol on Quantinuum's H1-1 trapped-ion quantum computer to estimate the GSE of the $H_3^+$ molecule (encoded on six qubits). The workflow consists of three main components:

1. Adiabatic State Preparation (ASP): Instead of standard Trotter-Suzuki decomposition, which introduces discretization errors, the authors employed the TETRIS (Time Evolution Through Random Independent Sampling) algorithm. This method generates randomized unitaries that, on average, exactly reproduce the time-evolution operator without discretization error. The system evolves from the ground state of a simple starting Hamiltonian ($H_Z$) to the target molecular Hamiltonian ($H$). To improve efficiency, the authors optimized the interaction norm ($\mu_I$) of the Hamiltonian by adding conserved quantity terms (particle number operators) and applied classical pre-processing techniques, including parity-based and occupation-number-based operator reductions to shorten circuit depth.
2. Energy Estimation: The authors utilized a noise-resilient variant of Iterative Quantum Phase Estimation (iQPE). Rather than sampling expectation values directly, the algorithm measures the imaginary part of the overlap $\rho(s) = \text{Im} \langle \psi(T) | e^{is(E_{\text{trial}} - H)} | \psi(T) \rangle$ using a Hadamard test. By choosing two trial energies ($E_{\text{trial}}^\pm$) and measuring $\rho(s)$ at a fixed central time $s$, the GSE is extracted via an arctangent formula. This approach leverages amplitude information and is robust against depolarizing noise.
3. Error Mitigation and Post-Processing: The experiment employed two classical post-selection techniques:
   • Parity Post-selection: Discarding measurement outcomes where the total parity of the physical qubits deviated from the expected sector, indicating leakage or errors.
   • Hartree-Fock (HF) Projection: Setting the ancilla expectation value to zero for shots where the non-ancilla qubits did not return to the initial HF state, based on the theoretical expectation that such contributions average to zero in a noiseless setting.

Key Results

• Energy Estimation: The experiment yielded a final energy estimate with a deviation of $-25.5 \pm 5.3$ mHa from the exact GSE. This represents an improvement over the classical Hartree-Fock estimate, which deviated by $52.8$ mHa. The quantum result is statistically distinguishable from the HF energy by more than 5 standard deviations.
• Noise Analysis: The authors conducted extensive classical simulations to isolate the impact of different noise sources:
  • Coherent and Incoherent Noise: Simulations incorporating realistic rates of coherent (e.g., $Z$-rotations) and incoherent (depolarizing) errors predicted energy deviations close to chemical accuracy (within $\sim 2$ mHa). This suggests the algorithm is resilient to these standard noise types.
  • Leakage Errors: When leakage errors (transitions out of the computational subspace) were introduced in simulations, a significant systematic bias emerged. The magnitude of the bias in the simulations matched the experimental deviation ($\approx -25.5$ mHa) only when leakage rates were included.
• Leakage Signatures: Analysis of measurement statistics revealed a strong bias toward measuring the $|1\rangle$ state for both physical and ancilla qubits in the hardware data compared to noiseless simulations. This imbalance is consistent with leakage events, particularly noting that the ancilla qubit, which undergoes disproportionate shuttling and gating, is highly susceptible to leakage.

Significance and Claims
The paper demonstrates a practical, scalable quantum algorithm for ground-state energy estimation that does not rely on off-loading non-scalable tasks to classical computers. The primary contribution is the identification of leakage errors as the dominant obstacle to achieving chemical accuracy in current ion-trap hardware, rather than coherent or incoherent noise.

The authors claim that, absent leakage errors, their experimental settings and noise-resilient iQPE protocol would have achieved results close to chemical accuracy (within a factor of two), even with shot noise included. Consequently, the work underscores that future progress in quantum chemistry applications on ion-trap devices depends critically on targeting leakage suppression through improved hardware design (e.g., leakage detection codes for ancilla qubits) and algorithmic adaptations, rather than solely focusing on reducing standard gate infidelities.