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Ground-state energy estimation of the water molecule on a trapped ion quantum computer

Yunseong Nam, Jwo-Sy Chen, Neal C. Pisenti, Kenneth Wright, Conor Delaney, Dmitri Maslov, Kenneth R. Brown, Stewart Allen, Jason M. Amini, Joel Apisdorf, Kristin M. Beck, Aleksey Blinov, Vandiver Chaplin, Mika Chmielewski, Coleman Collins, Shantanu Debnath, Andrew M. Ducore, Kai M. Hudek, Matthew Keesan, Sarah M. Kreikemeier, Jonathan Mizrahi, Phil Solomon, Mike Williams, Jaime David Wong-Campos, Christopher Monroe, Jungsang Kim

arXiv:1902.10171v2quant-phcs.ET

TL;DR

Useful quantum chemistry calculations remain resource-intensive for available noisy hardware. This paper co-designs a UCC-based VQE workflow with a trapped-ion computer and estimates H2O ground-state energies approaching chemical accuracy without error mitigation.

  • Problem

    Useful chemical simulations remain beyond current quantum-resource capabilities, despite quantum computing’s potential for quantum chemistry.

  • Method

    The paper develops a scalable co-design framework that optimizes UCC-VQE circuits, orbital selection, and gate implementations for a trapped-ion quantum computer.

  • Results

    The experiment obtains H2O energies for the first three post-Hartree-Fock correction terms in excellent agreement with theory at chemical-accuracy-level values and precision, without error mitigation.

  • Takeaways & Limitations

    The framework provides near-optimal circuits for existing NISQ hardware and establishes a path toward computations on more complex systems as trapped-ion quantum computers improve.

Abstract

from arXiv · show

Quantum computing leverages the quantum resources of superposition and entanglement to efficiently solve computational problems considered intractable for classical computers. Examples include calculating molecular and nuclear structure, simulating strongly-interacting electron systems, and modeling aspects of material function. While substantial theoretical advances have been made in mapping these problems to quantum algorithms, there remains a large gap between the resource requirements for solving such problems and the capabilities of currently available quantum hardware. Bridging this gap will require a co-design approach, where the expression of algorithms is developed in conjunction with the hardware itself to optimize execution. Here, we describe a scalable co-design framework for solving chemistry problems on a trapped ion quantum computer, and apply it to compute the ground-state energy of the water molecule. The robust operation of the trapped ion quantum computer yields energy estimates with errors approaching the chemical accuracy, which is the target threshold necessary for predicting the rates of chemical reaction dynamics.

I. TRAPPED-ION QUANTUM COMPUTER

The trapped-ion quantum computer uses individually controlled ytterbium-ion qubits, Raman-mediated gates, and all-to-all connectivity. Its small-angle XX gates offer an estimated per-gate error of at most 4 × 10^-3.

  • Hardware architecture: The register uses a linear chain of ^171Yb+ ions, with optical pumping for initialization and fluorescence-based simultaneous readout.High-NA optics enable individual addressing and detection of the ion qubits.
  • Hardware architecture: 355-nm Raman beams drive two-photon transitions, while independently controlled addressing beams manipulate individual qubits.The addressing beams provide control over amplitude, frequency, and phase.
  • Gate operations: Two-qubit operations use a Mølmer-Sørensen XX-Ising interaction mediated by shared motional modes.Because the modes involve the full ion chain, arbitrary ion pairs can be coupled with comparable speed and fidelity.
  • Gate operations: All-to-all two-qubit connectivity allows flexible qubit mappings and gate configurations optimized for circuit performance.The system maintains single-qubit gate fidelity ≳99.9% and maximally entangling XX(π/2) state fidelity ≳96%.
  • Gate performance: 4 × 10^-3: the estimated per-gate error for the small-angle XX(π/100) gate.The estimate comes from 50 consecutive small-angle XX gates, whose final state fidelity is approximately 78%.

II. MOLECULAR MODELING

The study models H2O with a Hartree-Fock-based molecular Hamiltonian, maps it to qubits, and constructs a progressively enriched UCC ansatz. Circuit and orbital-selection co-design reduces resources while retaining near-FCI energy accuracy.

  • II. MOLECULAR MODELING: H2O is used as a testbed because it is complex enough for scalable circuit-synthesis methods yet accessible to current trapped-ion quantum computers.Classical simulations provide verified reference solutions for assessing the quantum computation.
  • II. MOLECULAR MODELING: The molecular Hamiltonian is formulated under the Born-Oppenheimer approximation with nuclei fixed at their equilibrium geometry.Molecular spin orbitals are obtained from the minimal STO-3G basis using Hartree-Fock.
  • II. MOLECULAR MODELING: The Jordan-Wigner transformation maps creation and annihilation operators to Pauli operators, while UCC and one first-order Trotter step generate the ansatz.Hamiltonian expectations are estimated by measuring the Pauli bases associated with the transformed Hamiltonian terms.
  • II. MOLECULAR MODELING: The FCI ground-state energy is −75.0116 Ha, approximately 49.2 mHa below the Hartree-Fock result of −74.9624 Ha.Interaction terms are ranked by their contributions to the FCI diagonalization and added progressively to the UCC ansatz.
  • II. MOLECULAR MODELING: The in-silico VQE simulation estimates the lowest energy for each ansatz as its parameters are optimized, providing a reference for quantum-hardware results.The simulation examines ansätze formed by adding increasing numbers of significant interaction terms beyond Hartree-Fock.
  • II. MOLECULAR MODELING: Chemical accuracy is reached for the full Hamiltonian at HF+17 terms using 11 qubits and 143 entangling gates.The reduced Hamiltonian reaches within 2.1 mHa of FCI at HF+16 terms using 10 qubits and 140 entangling gates.

III. CIRCUIT OPTIMIZATION AND CO-DESIGN

The authors co-design generic UCC-VQE circuits with trapped-ion hardware, exploiting all-to-all connectivity, native XX gates, and encoding choices to reduce entangling and readout costs.

  • Circuit optimization: A modular software toolchain automatically generates optimized UCC ansatz circuits using hardware-specific but molecule-independent techniques.The framework is intended to scale to larger molecular systems and other target molecules.
  • Circuit optimization: All-to-all connectivity eliminates repeated SWAP operations for arbitrary two-electron interactions, improving VQE accuracy by avoiding their potentially dominant infidelity.This optimization directly leverages a trapped-ion hardware advantage.
  • Circuit optimization: Bosonic excitations simplify to pairwise arbitrary-angle XX(θ) gates, enabling efficient implementation of ansatz states containing only such excitations.The simplification represents paired-electron excitations with a single creation or annihilation operator.
  • Circuit optimization: Ordering UCC interaction terms and Jordan-Wigner strings cancels many CNOT gates; the eight-term main circuit requires 13 CNOT gates.The remaining Jordan-Wigner overhead is relatively low when terms are properly ordered.
  • Error-aware encoding: Encoding filled orbitals as |0⟩ exploits lower |0⟩ readout error and reduces systematic shifts, while its benefit diminishes as measurement errors become smaller or more symmetric.The encoding also requires fewer single-qubit initialization gates.
  • Integrated co-design: Combining these strategies produces a fully general, scalable, near-optimal framework with optimized total entangling-gate counts.The stated framework is designed for VQE simulations using a UCC ansatz.

IV. RESULTS

The experiment estimates H2O energies with calibrated trapped-ion gates, SPAM correction, and bootstrap uncertainty analysis. The first three correction-term energies agree well with simulation at approximately chemical accuracy without error mitigation.

  • Error control: Gate-angle calibration fits parity to sin(2θ), compensating for AOM nonlinearities and enabling interpolation for arbitrary XX(θ) angles.SPAM uncertainty can be reduced with sufficient measurement statistics.
  • Error control: For the demonstrated circuits, gate fidelity does not limit the computation, and Richardson-extrapolation error mitigation provides no observed benefit.Gate fidelity is expected to dominate as computation length increases.
  • Energy estimates: The first three experimentally determined H2O ground-state energies are −74.977(1) Ha, −74.979(2) Ha, and −74.985(5) Ha for HF+1, HF+2, and HF+3.Parenthetical uncertainties are 1σ values from bootstrapped distributions.
  • Energy estimates: The experimental energies match in-silico VQE theory closely, with absolute accuracy and precision comparable to chemical accuracy.The dominant experimental uncertainty arises from SPAM correction.

V. SUMMARY AND OUTLOOK

The work presents an end-to-end co-design framework for running quantum-chemistry computations on existing trapped-ion NISQ hardware. It estimates H2O energies near chemical accuracy without error mitigation, while the methodology is broader than this specific experiment.

  • Summary: The framework maps quantum-chemistry problems to trapped-ion hardware through end-to-end optimization and produces near-optimal circuits for existing NISQ devices.The authors identify improvements in both hardware and techniques as necessary for meaningful NISQ computations.
  • Summary: Without error mitigation, the first three H2O correction terms agree with theory at chemical-accuracy levels in both predicted values and precision.The calculation targets the post-Hartree-Fock ground-state energy on a trapped-ion quantum computer.
  • Outlook: The reported results are specific to one quantum-chemistry problem and trapped-ion hardware, although the computational methodology is presented as general for simulating quantum systems.The authors also anticipate applications to other variational optimization problems.

Methods

SPAM errors are monitored by preparing all-bright and all-dark ion-chain states, while relaxed confinement suppresses inter-ion crosstalk during detection.

  • SPAM characterization: Interleaved all-bright |11 · · · 11⟩ and all-dark |00 · · · 00⟩ preparations monitor SPAM error during computation.These reference states characterize state-preparation and measurement behavior.
  • SPAM characterization: Relaxing axial confinement reduces adjacent-ion crosstalk below detector dark-count error and enables rapid measurement of the full SPAM matrix.The matrix decomposes into a Kronecker product of individual-ion SPAM matrices.

B. Single-qubit operation.

Single-qubit states are prepared on the Bloch sphere using individually addressed Raman-driven Rabi oscillations and SK1 composite pulses.

  • Rabi oscillations between |0⟩ and |1⟩ implement single-qubit gates.The transition is driven by a two-photon Raman process using a pulsed 355 nm laser.
  • Individual addressing controls each ion through a tightly focused beam.The beam’s phase and pulse area determine the prepared state.
  • Arbitrary single-qubit states are prepared on the Bloch sphere.

C. Two-qubit operation.

Two-qubit operations use Mølmer–Sørenson-mediated XX gates, calibrated through parity oscillations and implemented with circuit optimizations tailored to trapped-ion connectivity.

  • Mølmer–Sørenson interactions implement entangling gates with tunable geometric phase θ.Amplitude-modulated laser pulses achieve spin-motion decoupling at the end of the gate.
  • θ = π/2 corresponds to a maximally entangling XX gate.
  • Parity Πij is fit to sin(2kΘ) to calibrate arbitrary-angle XX(θ) gates.The fitted factor k is applied to the AOM amplitude scale factor g.
  • Small-angle XX(π/100) gates have estimated fidelity ≳99.6%.The authors attribute this likely to smaller geometric-phase errors and weaker light shifts.
  • The molecular Hamiltonian is transformed from spin orbitals to qubits with Jordan–Wigner, then represented using a UCC ansatz and product formulas.The first-order Trotter formula with one stage is sufficient for the H2O ansatz to reach the FCI ground-state energy within chemical accuracy.
  • Qubit mappings and term orderings reduce Jordan–Wigner strings and simplify the circuit.Frequently interacting spin orbitals are mapped nearby, while adjacent ansatz terms are ordered to maximize cancellation.
  • A four-qubit two-electron interaction can be implemented with 2(m −1) CNOT gates.Here m is the number of non-identity Pauli factors, and the common target may be any of those qubits.

I. Circuit efficiency.

Circuit optimization reduces the entangling-gate cost of a four-qubit two-electron interaction while retaining a single basis-transformation gate.

  • 13 entangling gates implement the final four-qubit two-electron interaction template.The starting construction used eight circuits containing 48 CNOT gates.
  • The optimized design uses exactly one two-qubit gate for transforming between Pauli bases.

J. Higher fidelity two-qubit gates.

The trapped-ion implementation favors small-angle XX gates and selectively reduced spin-orbital representations to improve fidelity and conserve qubits, with savings that diminish for larger systems.

  • Small-angle XX gates perform better on this trapped-ion QC than CNOT gates requiring XX(π/2).
  • Replacing four of 13 CNOT gates with small-angle XX gates yields a ∼30% reduction in two-electron interaction infidelity.This estimate assumes small-angle XX infidelity is much lower than CNOT infidelity.
  • Bosonic excitations are executed first, followed by non-bosonic excitations using additional ancilla qubits.Each molecular-orbital occupation qubit is paired with a fresh ancilla through a CNOT gate.
  • The hybrid excitation savings diminish as problem size increases because non-pair excitations become more numerous.The reduction remains non-negligible for NISQ devices.
  • Restricting simulations to the most significant spin orbitals saves qubits at the cost of omitting less significant interactions.Freed resources can be allocated to interactions among more significant orbitals when qubits are limiting.

M. Bootstrap error analysis.

The study uses empirical bootstrapping to estimate energy uncertainties while characterizing circuit structure, gate fidelity, and interaction terms across supplementary analyses.

  • M. Bootstrap error analysis.: 500 bootstrap repetitions resample circuit and SPAM data to estimate uncertainties in computed energies.The procedure draws samples with replacement from both datasets for each circuit implementation.
  • M. Bootstrap error analysis.: HF+3 uses six XX(θ) gates with three independent ansatz parameters, and measurements generate expectation values for Pauli operators.The circuits also include measurement configurations for the required Pauli terms.
  • M. Bootstrap error analysis.: ⟨H0⟩= −74.985(5) Ha is obtained experimentally for HF+3, with uncertainty shown by 1σ error bars.The bootstrap distribution is compared with an in-silico result and the bounds of chemical accuracy.
  • M. Bootstrap error analysis.: 99.964(5)% is the extracted single-qubit gate fidelity from randomized benchmarking with an SK1 composite pulse sequence.The figure presents typical benchmarking data for single-qubit gates.
  • M. Bootstrap error analysis.: Extended Data Figure 7 lists the most significant interaction terms beyond the Hartree-Fock approximation.
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