High-Resolution Dynamical Eigenspectroscopy via Variational Trotter Compression on a Superconducting Qubit Processor

AI summary

Overview

Research area: Quantum simulation / digital quantum dynamics on superconducting hardware — specifically, extracting energy spectra (eigenspectroscopy) from real-time quantum evolution, applied to a molecular system (H₂) and a strongly correlated lattice model (Fermi-Hubbard).

Technical level: Advanced. The paper assumes familiarity with Trotterization, variational ansätze, quantum state tomography, and exact diagonalization.

Scope (one sentence): The authors demonstrate that Variational Trotter Compression (VTC) can extend coherent on-chip time evolution on a 9-qubit superconducting processor long enough to resolve fine-grained eigenspectra of H₂ at five bond lengths and of a four-qubit Fermi-Hubbard impurity model across weak-to-strong correlation regimes.

What This Paper Is About

Extracting precise energy levels from a quantum simulation requires letting a system evolve for a long time, but real hardware loses coherence quickly, and standard Trotterized evolution grows deeper (and noisier) as evolution time increases. The authors solve this by periodically "compressing" the ideal but deep Trotter circuit into a shallow variational circuit, so that long-time dynamics can be run on today's noisy devices. They then read out the survival amplitude at many time points, fit it to a sum of sine waves, and Fourier-transform it to recover the eigenenergies.

Key Contributions

1. Experimental dynamical eigenspectroscopy using genuine on-chip evolution. Unlike the authors' prior work on the same L9 processor, where the time series used for Fourier analysis came from classical post-processing of numerically computed evolution, here the time-domain signal is reconstructed from actual quantum dynamics executed on hardware, via quantum state tomography.
2. An ancilla-free VTC workflow at the device level. The authors replace the Hadamard-test/controlled-unitary approach (which the paper notes introduces significant overhead and amplifies CNOT counts) with a protocol that uses computational-basis initial states, reconstructs the evolved state by QST, purifies it by eigen-decomposition, and classically aligns its global phase to a simulated reference trajectory.
3. A modified VTC loss function plus warm-starting. Instead of the ℓ² (Euclidean) norm used in the original Berthusen et al. formulation, the loss is the ℓ¹ (Manhattan) norm of the deviation from the initial state, which the authors argue aligns more naturally with experimentally reconstructed probability amplitudes. Each step's optimization is initialized from the previous step's optimal parameters.
4. A unified demonstration bridging quantum chemistry and condensed matter. The same VTC framework is applied to H₂ (mapped to a two-site Hubbard dimer, weak-to-strong coupling tuned by bond length) and to a four-qubit DMFT impurity representation of the single-band Hubbard model spanning metallic, intermediate, and Mott-insulating parameters.

Main Findings

• H₂ spectral accuracy: Fourier analysis of the fitted survival amplitudes at R = 2.25 Å yields eigenenergy pairs (−0.9399, −0.3729) E_h for initial state |00⟩ and (−0.9294, −0.3866) E_h for |01⟩, deviating from exact diagonalization of H_eff by less than 2×10⁻⁴ E_h. Across all five bond distances, the maximum absolute deviation between ED, numerical VTC, and experiment is 2×10⁻³ E_h.
• State-preparation and evolution fidelity: Averaged over sampled evolution times, the fidelity between experimentally evolved and numerically simulated states exceeds 99.6% for H₂ at the five bond lengths; for R = 2.25 Å specifically, the state fidelity stays above 99.4% across all measured time steps for both |00⟩ and |01⟩.
• Ansatz depth threshold: In noiseless numerical benchmarks, a single layer (L = 1) cannot hold the ℓ¹ loss below 10⁻⁴, and fidelity degrades with evolution time. For L ≥ 2 the loss is stably suppressed below 10⁻⁴ and fidelity stays above 0.99, which the authors use as the criterion for choosing depth: L = 2 for H₂ (8 parameters) and L = 2 or 3 for the Hubbard model (16 parameters at L = 2).
• Optimizer convergence: For H₂ at R = 2.25 Å the ℓ¹ loss plateaus at ~10⁻⁶ within 200 steps, with fidelity > 99.9% and state distance collapsing from ~0.18 to 10⁻⁶; for the intermediate Hubbard point (U = 1, V = 0.121) a comparable ~10⁻⁶ loss is reached within 100 steps with fidelity > 99.9%. Cold-starting at t = τ = 0.5 converges slowly (shown in an inset); warm-started later steps converge much faster.
• Hubbard error suppression: For the single-band Hubbard model, the same VTC ansatz with L ≥ 2 suppresses the evolution error below 10⁻⁴ across extended time scales.
• Adaptive sampling: To satisfy the Nyquist–Shannon criterion, the shortest bond length (R = 0.25 Å, largest level spacing) uses N = 100 steps of τ = 0.1 for a total T = 10 (a.u.), while R = 0.75–2.25 Å uses N = 100–400 with fixed τ = 0.5, giving T = 50–200 (a.u.), where near-degeneracies demand finer resolution.
• Near-degenerate levels resolved: The abstract states agreement with exact diagonalization to within 2×10⁻³ including near-degenerate levels in the strongly interacting regime; Fig. 3(d) highlights near-degeneracy of excited states around R = 1.75 Å and R = 2.25 Å.
• Hubbard regimes probed: Three parameter points were emulated — metallic (U = 0, V = 0.171), intermediate correlated (U = 1, V = 0.121), and Mott insulating (U = 2, V = 0.082). The specific experimentally extracted Hubbard eigenvalues are not available in the provided (truncated) content.

Methodology in Plain English

The recipe works in five stages.

1. Encode the problem in a few qubits. H₂ in the STO-3G minimal basis becomes a two-qubit Hamiltonian of the form α₀I + α₁Z₁ + α₂Z₂ + α₃X₁X₂ + α₄Z₁Z₂, with coefficients that depend on the internuclear distance R. The Fermi-Hubbard model is reduced, via two-site dynamical mean-field theory, to a half-filled Anderson impurity model on four qubits with Hamiltonian (U/4)Z₁Z₃ + (V/2)(X₁X₂ + Y₁Y₂ + X₃X₄ + Y₃Y₄).

2. Compress the evolution. Exact time evolution e^(−iHt) would need an ever-deeper Trotter circuit, whose error bound scales with t²/N (commutator-weighted). VTC instead advances time in small steps τ: apply the shallow parameterized circuit optimized for time t, apply one short exact Trotter step e^(−iHτ), then apply the adjoint of the parameterized circuit for the next time. Minimizing the ℓ¹ distance between the resulting state and the original initial state gives the parameters for step t + τ. Because adjacent steps differ only infinitesimally, starting each optimization from the previous solution (warm start) converges quickly; optimization uses the L-BFGS-B quasi-Newton algorithm with numerical gradients.

3. Run it on hardware. The L9 device is a flip-chip-integrated processor with nine frequency-tunable transmons and 24 tunable couplers, arranged with a central qubit linked to eight peripherals. Two disjoint pairs (Q1,Q2) and (Q8,Q9) were used for H₂; four-qubit subsets (Q1,Q2,Q8,Q9) and (Q1,Q3,Q7,Q8) for the Hubbard model. Device metrics: qubit frequencies 4–5 GHz, anharmonicity ≈ −200 MHz, average T₁ = 57.3 μs (37.9–85.6 μs), average T₂^echo = 26.8 μs (10.3–47.3 μs), single-qubit Clifford RB fidelity 99.93% (99.89%–99.95%), two-qubit CZ fidelity 99.46% (98.36%–99.94%), gate times ≈40 ns (single-qubit) and 30 ns (two-qubit).

4. Read out the survival amplitude. At each time point, full quantum state tomography reconstructs the density matrix; readout errors are corrected using the inverse of a pre-characterized assignment matrix; eigen-decomposition selects the dominant eigenvector to purify the state. Because tomography is insensitive to global phase, the phase is fixed by classical post-processing that aligns the reconstructed state to a numerically simulated reference trajectory — avoiding ancilla qubits and controlled unitaries. The imaginary part Im[S(t)] is used for the Fourier analysis to remove sign ambiguity in the energies.

5. Extract the spectrum. The time series is fit to f(t) = C + Σ_k A_k sin(ω_k t + φ_k), initialized roughly from a coarse Fourier analysis, and then Fourier-transformed to locate the eigenenergies. Since the amplitude of each frequency component is |c_i|², only eigenstates with nonzero overlap with the chosen initial state appear.

Why This Matters

Impact on research. The work closes a gap the authors explicitly identify: earlier spectroscopy on this processor relied on classically computed dynamics, whereas this work extracts spectra from genuine quantum time evolution. It shows that VTC-supplied evolution windows — not the depth-limited Trotter window — set the achievable frequency resolution, offering a route to high-resolution spectroscopy on near-term devices without deep multi-controlled gates or ancilla overhead.

Real-world applications (as framed or implied by the paper):

• Computational chemistry: reconstructing potential energy surfaces and excited-state structure of small molecules, here H₂ from R = 0.25 Å to 2.25 Å.
• Strongly correlated materials: probing the competition between kinetic delocalization and interaction-driven localization relevant to Mott physics.
• Benchmarking digital quantum simulation workflows against exact diagonalization on small, classically verifiable systems.
• Establishing reusable hardware-efficient compilation of long-time dynamics for future devices.

Industry relevance. The work is a collaboration between Zhejiang University and Tencent Quantum Laboratory, and the results concern practical device-level engineering (tunable-coupler Z̄Z̄ suppression, flip-chip fabrication, high-fidelity CZ gates), which is directly relevant to groups building superconducting quantum processors and to quantum-chemistry software efforts that need shallow-circuit dynamical primitives.

Future Directions

1. On-hardware parameter optimization. The authors used classically pre-optimized parameters because gradient-based optimization from experimental measurements was judged computationally prohibitive due to measurement overhead and sensitivity to accumulated noise; closing that loop is an open problem.
2. Combining parameterized state preparation with VTC evolution. The authors state that this combination imposes prohibitive circuit depth, making practical demonstration contingent on future hardware advances.
3. Scaling beyond finite-size impurity models. The four-qubit DMFT impurity mapping is explicitly described as limited by finite-size effects and insufficient to reproduce a true metal-insulator transition; larger lattices and true thermodynamic-limit behavior are left open.
4. Removing the classical phase-alignment step. The paper mentions an alternative ancilla-free route based on the invariance of Hermitian observable expectation values under global phase rotations (Appendix B), which would avoid alignment to a numerical reference.

Target Audience

Quantum-computing researchers and graduate students working on digital quantum simulation, variational compilation, and superconducting hardware; quantum-chemistry method developers interested in spectral extraction from small molecular Hamiltonians; and condensed-matter physicists interested in quantum-simulation approaches to Mott and impurity physics. Readers should be comfortable with Trotter error analysis, variational ansätze, and tomography; the paper is not suitable as an introductory read.

Authors’ abstract

The pursuit of high-resolution eigenspectroscopy on noisy intermediate-scale quantum devices is often hindered by the trade-off between circuit depth and coherence time. In this work, we introduce and experimentally demonstrate a dynamical eigenspectroscopy protocol that extracts fine-grained energy structures from time-dependent survival amplitudes. To overcome the finite coherence window of current superconducting processors, we employ Variational Trotter Compression (VTC) as a practical means to extend the duration of high-fidelity unitary evolution. Using a multi-connected 9-qubit superconducting processor, we reconstruct the time-domain autocorrelation signal via quantum state tomography for the H2 molecule at different bond lengths and for the Fermi-Hubbard model across different correlation regimes. Through multi-frequency fitting and Fourier analysis, the extracted eigenenergies agree with the exact-diagonalization values to within 2x10^-3, including the near-degenerate levels in the strongly interacting regime. Our results establish experimental dynamical spectroscopy as a robust and generalizable framework for simulating both quantum chemistry and strongly correlated lattice systems, bridging weak- and strong-coupling regimes on near-term quantum hardware.