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From transistor to trapped-ion computers for quantum chemistry

M. -H. Yung, J. Casanova, A. Mezzacapo, J. McClean, L. Lamata, A. Aspuru-Guzik, E. Solano

arXiv:1307.4326v1quant-ph

TL;DR

Large quantum-chemistry systems exceed the resource capabilities of classical digital methods, motivating controllable quantum simulations. The paper develops a trapped-ion toolkit using internal and motional degrees of freedom, and reports protocols for electronic, vibrational, and coupled simulations that may extend beyond classical capacity.

  • Problem

    Classical digital methods face exponentially growing resource requirements for large quantum systems.

  • Method

    The paper develops trapped-ion quantum-simulation methods combining spin qubits, phonon modes, variational optimization, phase estimation, and nonlocal gate constructions.

  • Results

    The toolkit supports efficient electronic simulation, interactions between electronic and vibrational degrees of freedom, and measurement of bosonic-internal correlations.

  • Takeaways & Limitations

    Trapped-ion systems provide a proposed route toward quantum-chemistry simulations combining classical computation with quantum processing.

Abstract

from arXiv · show

Over the last few decades, quantum chemistry has progressed through the development of computational methods based on modern digital computers. However, these methods can hardly fulfill the exponentially-growing resource requirements when applied to large quantum systems. As pointed out by Feynman, this restriction is intrinsic to all computational models based on classical physics. Recently, the rapid advancement of trapped-ion technologies has opened new possibilities for quantum control and quantum simulations. Here, we present an efficient toolkit that exploits both the internal and motional degrees of freedom of trapped ions for solving problems in quantum chemistry, including molecular electronic structure, molecular dynamics, and vibronic coupling. We focus on applications that go beyond the capacity of classical computers, but may be realizable on state-of-the-art trapped-ion systems. These results allow us to envision a new paradigm of quantum chemistry that shifts from the current transistor to a near-future trapped-ion-based technology.

RESULTS AND DISCUSSION

The paper maps quantum-chemistry Hamiltonians to trapped-ion spin and phonon operations, using internal and motional degrees of freedom to simulate electronic and vibrational problems. This mapping supports efficient implementation despite nonlocal electronic interactions.

  • Hamiltonian representation: Quantum chemistry is formulated as a many-body Hamiltonian whose electronic component can be expressed in a molecular-orbital basis with polynomially many terms.The electronic Hamiltonian contains O(M 4) terms, with M typically of the same order as the electron number N.
  • Hamiltonian representation: The Jordan-Wigner transformation maps fermionic occupation states to spin states while enforcing exchange symmetry for trapped-ion implementation.Unoccupied and occupied fermionic modes correspond to spin-down and spin-up states, respectively.
  • Quantum-assisted simulation: The quantum-assisted procedure measures Hamiltonian-term expectations, sums them classically, and repeats the calculation across nuclear configurations to probe potential-energy surfaces.Classical optimization selects variational parameters, while quantum phase estimation is applied after optimization to obtain energy-surface data.
  • Trapped-ion platform: Trapped ions use metastable internal levels as qubits and cooled collective motional modes as a quantum bus for multiqubit operations.The platform offers control over both individual-ion internal states and collective phonon degrees of freedom.
  • Trapped-ion platform: Adjusting laser detuning generates carrier, red-sideband, and blue-sideband interactions, whose combinations yield Mølmer-Sørensen gates.These gates provide the basic building blocks for simulating dynamics associated with arbitrary quantum-chemistry Hamiltonians.

Quantum-assisted optimization

Quantum-assisted optimization divides ground-state estimation between quantum and classical processors. Quantum states provide Hamiltonian expectation values, while classical optimization updates parameters before phase estimation generates potential-energy-surface data.

  • Quantum-assisted optimization: Quantum-assisted optimization strategically uses the quantum processor only for expensive tasks while a classical processor handles the optimization.The method is designed to reduce the quantum coherence required by the overall simulation.
  • Quantum-assisted optimization: The method prepares parameterized states and directly measures their Hamiltonian expectation values without intermediate quantum evolution.The Hamiltonian is divided into a polynomial number of smaller pieces for measurement.
  • Quantum-assisted optimization: Measurements of Hamiltonian terms can be performed in parallel without maintaining quantum coherence between measurements.Their expectation values are combined classically to obtain the total energy.
  • Quantum-assisted optimization: A classical optimization algorithm selects new variational parameters to minimize the measured energy until convergence.Each new parameter set defines another quantum state whose expectation value is evaluated.
  • Quantum-assisted optimization: After optimization, phase estimation produces electronic energy-surface points for nuclear configurations, enabling further study of vibronic coupling.The optimized state can be used to obtain ground- and excited-state potential-energy surfaces.

Unitary coupled-cluster (UCC) ansatz

The UCC ansatz represents correlated electronic states through particle-hole excitations, but classical implementation faces exponential resource growth. The trapped-ion protocol uses Suzuki-Trotter evolution and Mølmer-Sørensen gates to implement the required nonlocal operators efficiently.

  • Unitary coupled-cluster (UCC) ansatz: The UCC ansatz applies the unitary transformation e^(T−T†) to a reference state, such as a Hartree-Fock Slater determinant.The cluster operator is decomposed into particle-hole excitation ranks.
  • Unitary coupled-cluster (UCC) ansatz: The UCC energy is a variational upper bound on the exact ground-state energy.The expectation value is evaluated using the transformed reference state and the electronic Hamiltonian.
  • Unitary coupled-cluster (UCC) ansatz: Classical UCC implementation requires exponentially growing resources because the transformed Hamiltonian involves an infinite Baker-Campbell-Hausdorff expansion.Approximate truncation methods implicitly rely on the reference state being a good solution.
  • Implementation of UCC through time evolution: The trapped-ion protocol prepares UCC states by simulating pseudo-time evolution with a Suzuki-Trotter expansion from the Hartree-Fock reference state.The effective generator is decomposed into excitation-rank subgroups and individual fermionic terms.
  • evolution with trapped-ions: Jordan-Wigner mapping and trapped-ion gates simulate each nonlocal Pauli-product evolution, with local rotations completing the construction.The protocol exploits trapped ions’ ability to implement nonlocal controlled operations efficiently.
  • evolution with trapped-ions: 2 Mølmer-Sørensen gates replace as many as 2^N two-qubit gates for each N-mode Pauli-product evolution.The local rotation can also incorporate motional degrees of freedom to simulate fermionic Hamiltonians coupled linearly to bosonic operators.

Measurement of arbitrarily-nonlocal spin operators

The toolkit measures arbitrarily nonlocal spin correlations by converting them into single-qubit expectation values, with extensions to bosonic operators. This is well suited to trapped-ion measurements.

  • Arbitrarily nonlocal Pauli correlations can be encoded in the expectation value of a single qubit.The protocol applies a unitary generated by the corresponding Pauli product before measurement.
  • Measuring one qubit only is especially suitable for trapped ions, where single-qubit measurement fidelity is 99.99%.
  • The method extends to correlations involving bosonic operators by replacing local operations with spin-motion interactions.This produces expectation values containing (a + a†) and enables correlations between bosonic and internal degrees of freedom.
  • A derivative of the single-qubit measurement recovers the desired nonlocal spin correlation.The relation is evaluated for the transformed state as a function of θ.

Probing potential energy surfaces

The paper constructs potential energy surfaces by combining quantum-assisted electronic-state preparation with phase estimation across nuclear configurations. A hydrogen-molecule resource estimate places the digital dynamics within trapped-ion coherence times.

  • Probing potential energy surfaces: Potential energy surfaces are obtained by scanning electronic eigenvalues over nuclear coordinates and adding the nuclear potential.The electronic eigenvalues can be extracted with phase estimation using one extra ancilla qubit.
  • Probing potential energy surfaces: Phase estimation decomposes an input state into electronic eigenstates, encoding eigenvalue frequencies and eigenstate weights in the ancilla’s reduced density matrix.Repeating the circuit over times t and applying a classical Fourier transform extracts the frequencies and weights.
  • Probing potential energy surfaces: For H2 in a minimal STO-3G basis, the Hamiltonian has 12 terms for four spin-orbitals and requires 16 Mølmer-Sørensen gates per Trotter step.The estimate uses a 0.75 Å bond distance and two electrons.
  • Probing potential energy surfaces: 800 µs, 1.6 ms, and 2.4 ms are the estimated simulation times for n = 1, 2, and 3 protocols, respectively, within approximately 30 ms decoherence times.
  • Probing potential energy surfaces: The digitized hydrogen dynamics can recover the system energy with small error using a reduced number of digital steps.The analysis compares simulated-state loss with accumulated gate error and identifies regions dominated by digital approximation error.

Conclusions

The paper proposes a trapped-ion quantum-simulation toolkit that combines classical optimization with quantum-state preparation and measurement. Its stated advantages include electronic simulation, coupled electronic-vibrational dynamics, and trapped-ion scalability.

  • Conclusions: The toolkit targets quantum chemistry with trapped ions and combines classical and quantum computation.
  • Conclusions: Classical algorithms first provide approximate UCC solutions, after which an ion trap searches for the true minima.The procedure uses successive perturbed cluster operators and energy evaluations.
  • Conclusions: Fermionic operators are converted into spin operators for measurement, which can be implemented efficiently with trapped ions.
  • Conclusions: Gradient descent updates the cluster-operator parameters using energy gradients, with finite differences or the Hellmann-Feynman theorem available for gradient evaluation.The parametrization is assumed to define a smooth function in general cases.
  • Conclusions: The optimization converges to a minimum for large k and yields an optimized UCC quantum state.

1. Quantum simulation

Quantum simulation can use analog Hamiltonian engineering or digital decomposition into quantum gates. For quantum chemistry, this work mainly uses second quantization because it requires fewer qubits for low-energy properties.

  • 1. Quantum simulation: Analog simulation engineers a system Hamiltonian to mimic a target, whereas digital simulation decomposes evolution into quantum-logic submodules.Digital simulation does not necessarily require quantum logic gates in every implementation.
  • 1. Quantum simulation: The work mainly uses second quantization because it requires fewer qubits than first quantization for low-energy state properties.The described techniques can also apply to first-quantization approaches.
  • 1. Quantum simulation: Quantum computers do not have a rigorous proof of solving all quantum-chemistry ground-state problems, and some such problems are computationally hard.The paper cites N-representability as QMA-complete and the universal density-functional problem as QMA-hard.
  • 1. Quantum simulation: Molecular vibrations can be included after constructing the electronic potential surface by a local expansion near equilibrium.

a. Electronic transitions coupled with nuclear motion

The paper models vibronic states by combining electronic eigenstates with nuclear wavefunctions on electronic potential-energy surfaces. It then proposes trapped-ion procedures to probe energy surfaces, simulate two-level vibronic dynamics, and compute absorption-related correlations.

  • Within the Born-Oppenheimer approximation, vibronic states factor into electronic eigenfunctions at fixed nuclear coordinates and nuclear wavefunctions on corresponding potential surfaces.
  • Phase estimation can probe potential-energy surfaces for different electronic eigenstates, locate equilibrium minima, and support a second-order expansion around those configurations.
  • The two-level vibronic model uses separate nuclear Hamiltonians for ground and excited electronic states, with their normal modes related by rotation, translation, or both.
  • For a single normal mode, the ground and excited nuclear modes are related by a displacement b = a + λ, reducing the simulation to a trapped-ion Hamiltonian with spin-dependent bosonic terms.
  • Absorption under external perturbations is formulated through a dipole correlation function, with the final spectrum obtained by weighting expectation values and applying a Fourier transform.

b. Simulation of vibronic coupling with trapped ions

The trapped-ion implementation realizes vibronic-coupling dynamics through sideband interactions between internal states and motional modes. A related ancilla-assisted protocol extracts transition matrix elements, including electric dipole moments after fermion-to-qubit mapping.

  • The required single-mode vibronic Hamiltonian can be generated with two trapped ions by implementing dispersive and sideband interactions in an interaction-picture digital protocol.A detuned red-sideband excitation realizes the number-dependent evolution, while red and blue sidebands generate spin-boson terms.
  • Trapped-ion motional degrees of freedom provide bosonic modes, while internal states provide fermionic or electronic degrees of freedom in a mixed digital-analog simulator.
  • The transition matrix element ⟨e|A|g⟩ is measured by preparing an ancilla in |+⟩, applying controlled-UA, post-selecting the target state, and performing ancilla tomography.
  • For the electric dipole operator, the unitary UA = e−iλA can be simulated efficiently after replacing A with µ and applying the Jordan-Wigner transformation.

5. Derivation of the spin-boson coupling

The spin-boson formulation rewrites two electronic potential surfaces and their nuclear motions using a Pauli-matrix representation. Coordinate displacement and mode transformations produce effective bosonic terms whose frequencies depend on the electronic spin state.

  • The two-surface model assigns separate nuclear Hamiltonians to the electronic ground and excited states, including their respective zero-point energies.
  • For one normal mode per local minimum, the excited-state and ground-state modes are related by a real displacement λ.
  • Rewriting the electronic subspace with Pauli matrices separates identity and σz contributions associated with the two electronic energies.
  • The resulting spin-boson Hamiltonian contains a bosonic mode whose effective frequency is spin-dependent.

6. Multimode extension of simulating vibronic coupling

The multimode extension expresses excited-state normal modes through transformations of ground-state modes and uses the resulting parameters to determine absorption spectra for more complicated systems. The formulation also identifies a simplified case when mode rotations are negligible.

  • The multimode vibronic-coupling method expresses excited-state modes in terms of ground-state modes before constructing the transformed Hamiltonian.
  • Knowing the mode-transformation parameters s_ij and displacements λ_i allows the procedure to determine the absorption spectrum of a complicated system with a quantum computer.
  • The transformed Hamiltonian is rewritten in a form more familiar to quantum computation.
  • When Duschinsky rotations can be neglected, the mode transformation simplifies to s_ij = δ_ij.
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