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End-to-End Simulation of Chemical Dynamics on a Quantum Computer

Elliot C. Eklund, Arkin Tikku, Patrick Sinnott, William J. Huggins, Guang Hao Low, Dominic W. Berry, Ivan Kassal

Abstract

Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.

End-to-End Simulation of Chemical Dynamics on a Quantum Computer

Abstract

Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.
Paper Structure (83 sections, 13 theorems, 589 equations, 8 figures, 3 tables)

This paper contains 83 sections, 13 theorems, 589 equations, 8 figures, 3 tables.

Table of Contents

  1. Algorithm overview
  2. Input data
  3. Electronic-state data
  4. Nuclear-state data
  5. Non-separable state data
  6. Classical preprocessing
  7. MO preprocessing
  8. SM preprocessing
  9. Initial-state preparation
  10. Common grid
  11. Electronic ISP
  12. Arbitrary state preparation
  13. ONB preparation
  14. ONB to MOB transformation
  15. Antisymmetrization
  16. ...and 68 more sections

Key Result

Lemma 1

Let $\hat{\Phi}_{i \mu}$ be an SM defined in eq:project_pwb, $\delta^{(n,t)}_{i \mu}\in(0,1)$, and $K_{i \mu}^{(n)}\in\mathbb{R}$ such that Then there exists an approximation $\tilde{\hat{\Phi}}_{i \mu}$ to $\hat{\Phi}_{i \mu}$ given by replacing $\mathbb{K}$ with $\mathbb{K}_{\mathrm{cut}}$ as defined in eq:mom-cut-set, where $K_{i \mu}^{(n)}$ is the momentum cutoff, such that the error $\epsilo

Figures (8)

  • Figure 1: Chemical dynamics algorithm. Given a specification of the initial state in the occupation-number basis (ONB), the algorithm returns an estimate of an observable. Each step is followed by an illustration of the wavefunction at that point. Step 1: Classical preprocessing projects the initial state onto a plane-wave basis (PWB) and finds a matrix-product-state (MPS) encoding it. Step 2: Initial-state preparation converts the MPS into $|\hat{\Psi}(0)\rangle$ on the quantum computer in a PWB representation, and a normal-mode transformation $\textsc{nct}$ is applied to the nuclear states. Step 3: Time evolution evolves $|\hat{\Psi}(0)\rangle$ under the molecular Hamiltonian into $|\hat{\Psi}(t)\rangle$. $\textsc{qft}^{\dagger}$ is applied if the observable of interest is most easily measured in position space, yielding $|\Psi(t)\rangle$. Step 4: Measurement of observable $O$ on $|\Psi(t)\rangle$ gives outcomes that can be averaged to yield an expectation value. Not shown is quantum amplitude estimation, used to reduce the number of measurements.
  • Figure 2: Wavefunction representations (bases and coordinate systems) used in the algorithm. Top row: example nuclear and electronic wavefunctions in the occupation number (ON) basis. Bottom row: the same wavefunctions in first-quantized representations, including the molecular orbital (MO) and single-modal (SM) bases. The grid bases are related by the normal-mode transformation $\textsc{nct}$ or quantum Fourier transform QFT. Hats indicate functions in plane-wave bases.
  • Figure 3: Eliminating modular-addition error by padding the grid $\mathcal{B}_{\mathrm{ISP}}$. (a) Computational basis states $|\mathbf{n}\rangle$ are represented by grid points. Wavefunction $|g\rangle$ (blue contours) is supported by grid points in $\mathcal{B}_{\mathrm{ISP}}$ (blue square). Padding qubits expand the grid to $\bar{\mathcal{B}}_{\mathrm{ISP}}$. The padded state $|g\rangle$ has non-zero amplitude within $\bar{\mathcal{B}}_{\mathrm{ISP}}$ and is zero outside it. (b) Applying the shear $U_{\mathbf{S}}$ to $|g\rangle$, with transformed grid points shown in red. We choose $n_{\mathrm{pad}}$ sufficiently large that grid points in $\mathcal{B}_{\mathrm{ISP}}$ remain within $\bar{\mathcal{B}}_{\mathrm{ISP}}$ after the transformation. (c) Performing $\mathrm{cmod}\, \bar{N}_{\mathrm{ISP}}/2$ leaves grid points in the blue rhombus unchanged while those outside it are wrapped around to the opposite edge of the grid (purple regions). Because these grid points have zero amplitude, their wrapping around does not contribute an error.
  • Figure 4: Circuit for the propagator $\tilde{U}_{\mathrm{prop}}$ via bidirectional QSP.
  • Figure 5: Circuit for the controlled iterate $\widetilde{W}_{\Pi_{S}}$ on which we perform QAE to estimate a quantum yield with respect to the real-space final state $\ket{\tilde{\Psi}(t)}$. The controls on $\tilde{U}_{\mathrm{prop}}$, $\tilde{U}_{\mathrm{ISP}}$, $\widetilde{\textsc{qft}}$ and $U_{\Pi_{S}}$ are dropped as depicted. Ancilla qubits are not shown, except the QAE ancilla "flag" and the exterior grid ancillas "ext", which continue throughout the circuit.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 16 more