Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

DOI: 10.1039/C9SC01313J
(Edge Article)
Chem. Sci., 2019, Advance Article

Sam McArdle^{a},
Alexander Mayorov^{ab},
Xiao Shan^{c},
Simon Benjamin^{a} and
Xiao Yuan*^{a}
^{a}Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK. E-mail: xiao.yuan.ph@gmail.com
^{b}Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK
^{c}Physical and Theoretical Chemical Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, UK

Received
17th March 2019
, Accepted 23rd April 2019

First published on 25th April 2019

Molecular vibrations underpin important phenomena such as spectral properties, energy transfer, and molecular bonding. However, obtaining a detailed understanding of the vibrational structure of even small molecules is computationally expensive. While several algorithms exist for efficiently solving the electronic structure problem on a quantum computer, there has been comparatively little attention devoted to solving the vibrational structure problem with quantum hardware. In this work, we discuss the use of quantum algorithms for investigating both the static and dynamic vibrational properties of molecules. We introduce a physically motivated unitary vibrational coupled cluster ansatz, which also makes our method accessible to noisy, near-term quantum hardware. We numerically test our proposals for the water and sulfur dioxide molecules.

In this work, we discuss a general method for efficiently simulating molecular vibrations on a universal quantum computer. Our method targets the eigenfunctions of a vibrational Hamiltonian with potential terms beyond quadratic order (‘anharmonic potentials’). These wavefunctions can then be used to efficiently calculate properties of interest, such as absorption spectra at finite temperatures. We can also use our method to perform simulations of vibrational dynamics, enabling the investigation of properties such as vibrational relaxation.

Despite their importance for accurate results, studying vibrations has proven difficult. There are several possible routes to obtaining an accurate description of vibrational behaviour. Real-space, grid based methods, which treat the electronic and nuclear degrees of freedom on an equal footing, are limited to systems of a few particles. While algorithms to efficiently solve this problem on a universal quantum computer exist,^{36,37} it will take many years to develop a quantum computer with the required number of qubits.^{38} Alternatively, one may separate the electronic and nuclear degrees of freedom. We can solve for the electronic energy levels of the system as a function of the nuclear positions, which enables us to map out potential energy surfaces for the system. A number of approximate classical methods have been developed to solve this problem,^{1,39} as well as several quantum algorithms.^{8,40–42} These electronic potential surfaces can then be viewed as the nuclear potential, determining the vibrational energy levels. This is known as the vibrational structure problem. The accuracy of the nuclear potential is determined by the accuracy of the electronic structure calculation, as well as the number of points obtained for the potential energy surface. Once this potential has been obtained, a number of classical methods can be used for solving both the time dependent and independent Schrödinger equations.

The most simple method uses the ‘harmonic approximation’. This treats the nuclear potential in the vicinity of the equilibrium geometry as a harmonic oscillator potential, resulting in energy eigenstates which are harmonic oscillator eigenfunctions.

Alternatively, one may consider higher order expansions of the nuclear potential, resulting in more accurate calculations.^{43} One common route towards obtaining the nuclear potential is to first carry out many electronic structure calculations on the system, in the vicinity of the minimum energy configuration. Each of these electronic structure calculations is approximate, and so the cost of each one scales polynomially with the system size. However, if one proceeds to obtain the nuclear potential using this simple grid based method, then a number of grid points scaling exponentially with the number of modes is required.^{44} In practice one can often instead construct an approximate nuclear potential by considering a reduced number of mode couplings, or using interpolation, or using adaptive methods. A review of these, and other state-of-the-art methods can be found in ref. 44. The requirement to first perform multiple electronic structure calculations to obtain the anharmonic nuclear potential makes calculating vibrational energy levels expensive,^{45} even if only mean-field vibrational calculations are then performed. If the correlation between different vibrational modes is included in the calculation, then the simulation becomes even more expensive. While most of the existing classical vibrational simulation methods scale polynomially with the number of modes in the system (e.g. vibrational self-consistent field methods,^{12} or vibrational coupled cluster theory^{46}), and are sufficiently accurate for some systems, they only provide approximations to the true full configuration interaction vibrational wavefunction, which can be exponentially costly to obtain. A similar hierarchy of accuracy also exists for dynamics simulations.

The computational difficulties described above make accurate vibrational calculations on large systems very challenging for classical computers. To overcome these challenges, quantum solutions have been suggested for the vibrational structure problem.^{13–19,47} To date, the majority of suggestions have focused on analog quantum simulation of vibrations. In analog simulations, the simulator emulates a specific system of interest, but cannot in general be programmed to perform simulations of other, different systems. Huh et al. proposed using boson sampling circuits to determine the absorption spectra of molecules.^{13} These boson sampling circuits consist of photons passing through an optical network. This initial proposal relied on the harmonic oscillator approximation at zero temperature, but does take into account bosonic mode mixing due to nuclear structural changes that result from electronic excitation. This method has since been experimentally demonstrated,^{15,16} and extended to finite temperature spectra.^{14,19} The main limitation of these simulations is the use of the harmonic oscillator approximation for the vibrational wavefunction. It is in general difficult to engineer ground states of anharmonic Hamiltonians using an optical network, as non-linear operations, such as squeezing, are required. Optical networks have also been used for simulating vibrational dynamics.^{17} These simulations investigated vibrational transport, adaptive feedback control, and anharmonic effects.

The aforementioned schemes make use of the analogy between the vibrational energy levels in molecules in the Harmonic oscillator approximation, and the bosonic energy levels accessible to photons and ions. One advantage of this is that the bosonic modes are in principle able to store an arbitrary number of excitations. As these analog simulators are relatively simple to construct (when compared to a universal, fault-tolerant quantum computer), they will likely prove useful for small calculations in the near-term. However, it is not yet known how to suppress errors to an arbitrarily low rate in analog simulators. As a result, if we are to simulate the vibrational behaviour of larger quantum systems, we will likely require error corrected universal quantum computers. This motivates our work on methods for vibrational simulation on universal quantum computers.

The rest of this paper is organised as follows. In Section III, we introduce the vibrational structure problem for molecules and show how this problem can be mapped onto a quantum computer. In Section IV, we show how to solve both static and dynamic problems of molecular vibrations. Finally, in Section V, we present the results of numerical simulations of the H_{2}O and SO_{2} molecules.

(1) |

(2) |

(3) |

(4) |

(5) |

In contrast, we show below that it is possible to efficiently encode the k^{th} order nuclear Hamiltonian into a Hamiltonian acting on qubits. We can then use quantum algorithms to efficiently calculate the static and dynamic properties of the nuclear Hamiltonian.

Focusing first on one harmonic oscillator, ĥ = ωa^{†}a, we consider the truncated eigenstates with the lowest d energies, |s〉 with s = 0,1,…,d − 1. A direct mapping of the space {|s〉} is to encode it with d qubits as

|s〉 = ⊗^{s−1}_{j=0}|0〉_{j}|1〉_{s}⊗^{d−1}_{j=s+1}|0〉_{j},
| (6) |

(7) |

The annihilation operator can be obtained by taking the Hermitian conjugate of a^{†}. As an alternative to the direct mapping, we can use a compact mapping, using K = [logd]′ qubits,

|s〉 = |b_{K−1}〉|b_{K−2}〉…|b_{0}〉,
| (8) |

(9) |

These binary projectors can then be mapped to Pauli operators;

(10) |

Fig. 1 Number of qubits required for the direct and compact mappings with d = 4 energy levels for each mode. |

As p and q can both be represented by a linear combination of creation and annihilation operators, we can thus map the nuclear vibrational Hamiltonian to a qubit Hamiltonian. If the molecule has n atoms, it has M = 3n − 6 vibrational modes for a nonlinear molecule, and M = 3n − 5 for a linear molecule. The vibrational wavefunction can then be represented with Md (direct mapping) or Mlog(d) (compact mapping) qubits. This can be contrasted with the exponentially scaling classical memory required to store the wavefunction. If the potential is expanded to k^{th} order (with M ≫ k), the Hamiltonian contains O(M^{k}d^{k}) (direct) or O(M^{k}d^{2k}) (compact) terms. These terms are strings of local Pauli matrices. In this work, we take d to be a small constant. This approximation constrains us to the low energy subspace of the Hamiltonian, which should be valid for calculations of ground and low-lying excited states. The applicability of this approximation to the simulation of dynamics is discussed in Section VI. We set k = 4 to investigate the Hamiltonian to quartic order. The resulting Hamiltonian has O(M^{4}) terms.

Another route to a state with a large overlap with the ground state, is to prepare the VSCF state, and then adiabatically evolve under a Hamiltonian that changes slowly from the VSCF Hamiltonian, to the full vibrational Hamiltonian H. This approach has received significant attention within quantum computing approaches to the electronic structure problem, since it was first proposed in the context of quantum computational chemistry in ref. 8. However, both adiabatic state preparation and phase estimation typically require long circuits, with a large number of gates. As a result, quantum error correction is required to suppress the effect of device imperfections. It is therefore helpful to introduce variational methods, which may make these calculations feasible for near-term, non-error corrected quantum computers. Variational methods replace the long gate sequences required by phase estimation with a polynomial number of shorter circuits.^{42,54} This dramatically reduces the coherence time required. As a result, quantum error correction may not be required, if the error rate is sufficiently low, in the context of the number of gates required. The circuits used consist of a number of parametrised gates which seek to create an accurate approximation of the desired state. The parameters are updated using a classical feedback loop, in order to produce better approximations of the desired state. The circuit used is known as the ‘ansatz’ circuit.

Inspired by classical methods for the vibrational structure problem, we introduce the unitary vibrational coupled cluster (UVCC) ansatz. This is a unitary analogue of the VCC ansatz introduced in ref. 12 and 46. We note that a similar pairing exists for the electronic structure problem, where the unitary coupled cluster (UCC) ansatz^{55,56} has been suggested as a quantum version of the classical coupled cluster method. The UVCC ansatz is given by

|Ψ()〉 = exp( − ^{†})|Ψ_{0}〉,
| (11) |

= _{1} + _{2} +…,
| (12) |

(13) |

The UVCC ansatz seeks to create a good approximation to the true ground state by considering excitations above a reference state. We note that the classical VCC ansatz is not a unitary operator. Correspondingly, the method is not variational, meaning that energies are not bounded from below. Moreover, we expect that the UVCC ansatz will deal better with problems of strong static correlation than the VCC ansatz, as the former can easily be used with multi-reference states. This echoes the way in which the UCC ansatz can be used with multi-reference states,^{56} while it is typically more difficult when using the canonical CC method.^{1}

Once we have obtained the energy levels of the vibrational Hamiltonian using the methods discussed above, we can calculate the infrared and Raman frequencies, using the difference between the excited and ground-state energies.^{57}

It is often also the case that one is interested in the properties of a system in thermal equilibrium, rather than a specific eigenstate. We can also use established quantum algorithms with the Hamiltonians described above to construct these thermal states. On error corrected quantum computers, we can use the heuristic algorithms presented in ref. 58 and 59 to construct these thermal states. Alternatively, we can use near-term devices to implement hybrid algorithms for imaginary time evolution.^{60–62}

H_{mol} = |i〉〈i|_{e}⊗H_{i} + |f〉〈f|_{e}⊗H_{f},
| (14) |

In practice, |ψ^{i}_{vib}〉 and |ψ^{f}_{vib}〉 are eigenstates of Hamiltonians with different harmonic oscillator normal modes q^{f} and q^{i}. These modes are related by the Duschinsky transform q^{f} = Uq^{i} + d.^{45,63} According to the Doktorov unitary representation of the Duschinsky transform, the harmonic oscillator eigenstates are related by^{13,14,64}

|s_{f}〉 = Û_{Dok}|s_{i}〉
| (15) |

If |Ψ^{i}_{vib}〉 and |Ψ^{f}_{vib}〉 are the qubit wavefunctions resulting from diagonalisation of H_{i} and H_{f} using a quantum computer, they will be obtained in different normal mode bases |s_{i}〉 and |s_{f}〉, respectively. We cannot directly calculate the Franck–Condon integrals using |〈Ψ^{f}_{vib}|Ψ^{i}_{vib}〉|^{2}, as this does not take into account the different bases. Instead, we must implement the Doktorov unitary to get the Franck–Condon integrals

|〈ψ^{f}_{vib}|ψ^{i}_{vib}〉|^{2} = |〈Ψ^{f}_{vib}|Û_{Dok}|Ψ^{i}_{vib}〉|^{2}.
| (16) |

|〈ψ^{f}_{vib}|(q)|ψ^{i}_{vib}〉^{2} = |〈Ψ^{f}_{vib}|(q^{f})Û_{Dok}|Ψ^{i}_{vib}〉|^{2}.
| (17) |

Alternatively, we can obtain the Franck–Condon integrals without realising the Doktorov transform. The qubit states |Ψ^{i}_{vib}〉 and |Ψ^{f}_{vib}〉 are obtained from H_{i}(q^{i}) and H_{f}(q^{f}) with normal mode coordinates q^{i} and q^{f}, respectively. Instead, we can focus on one set of normal mode coordinates q^{i} and represent the Hamiltonian H_{f} in q^{i}, H′_{f}(q^{i}). By solving the energy eigenstates of H′_{f}(q^{i}), we can directly get and calculate the Franck–Condon integrals without realising the Doktorov transform. However, as the Hamiltonian H′_{f}(q^{i}) is not encoded in the correct normal mode basis, the ground state of the harmonic oscillators or the VSCF state |Ψ_{VSCF}〉 may not be an ideal initial state to start with. However, this effect may be negligible if the overlap between |Ψ^{i}_{vib}〉 and is suitably large. In this case, the initial state |Ψ_{0}〉 for |Ψ^{i}_{vib}〉 should also be an ideal initial state for The aforementioned transformation can be implemented by transforming the normal mode coordinates q^{i} as described in ref. 45.

(18) |

(19) |

(20) |

Alternatively, the vibrational dynamics can be realised using a recently proposed variational algorithm.^{74} One could use either a UVCC ansatz, or a Trotterized ansatz.^{75,76}

k | H_{2}O |
SO_{2} |
---|---|---|

k_{1,1} |
0.275240 × 10^{−4} |
0.252559 × 10^{−5} |

k_{2,2} |
0.151618 × 10^{−3} |
0.125410 × 10^{−4} |

K_{3,3} |
0.161766 × 10^{−3} |
0.176908 × 10^{−4} |

K_{1,1,1} |
0.121631 × 10^{−6} |
0.316646 × 10^{−8} |

K_{1,1,2} |
0.698476 × 10^{−6} |
0.575325 × 10^{−8} |

K_{1,2,2} |
−0.266427 × 10^{−6} |
0.197771 × 10^{−7} |

k_{2,2,2} |
−0.312538 × 10^{−5} |
−0.668689 × 10^{−7} |

K_{1,3,3} |
−0.915428 × 10^{−6} |
−0.370850 × 10^{−9} |

k_{2,3,3} |
−0.964649 × 10^{−5} |
−0.284244 × 10^{−6} |

K_{1,1,1,1} |
−0.463748 × 10^{−9} |
0.330842 × 10^{−11} |

K_{1,1,2,2} |
−0.449480 × 10^{−7} |
−0.172869 × 10^{−9} |

K_{1,2,2,2} |
0.957558 × 10^{−8} |
−0.215928 × 10^{−9} |

k_{2,2,2,2} |
0.433267 × 10^{−7} |
0.225400 × 10^{−9} |

K_{1,1,3,3} |
−0.555026 × 10^{−7} |
−0.356155 × 10^{−9} |

K_{1,2,3,3} |
0.563566 × 10^{−7} |
−0.128135 × 10^{−9} |

k_{2,2,3,3} |
0.269239 × 10^{−6} |
−0.220168 × 10^{−8} |

K_{3,3,3,3} |
0.462143 × 10^{−7} |
0.458046 × 10^{−9} |

k_{2,3,3,3} |
0 | −0.720760 × 10^{−11} |

We first calculate the energy levels under the harmonic approximation. We compare this to the energy levels obtained with a fourth order expansion of the potential. The results for H_{2}O are shown in Fig. 2. We can see that although the ground state can be well approximated by the harmonic oscillators, the excited states deviate from the harmonic oscillators at higher energy levels. The results for SO_{2} can be found in the Appendix. These calculations highlight the importance of anharmonic terms in the potential for even small molecules.

Next, we implemented the UVCC ansatz to obtain the vibrational energy levels of H_{2}O using the variational quantum eigensolver.^{42} For simplicity, we considered two energy levels for each mode. To implement the UVCC ansatz, we first calculate the imaginary part of and encode it into a linear combination of local Pauli terms, i.e.,

(21) |

Using the UVCC ansatz, we can obtain the vibrational ground state with a variational procedure. As the ground state is close to the initial state |0〉^{⊗3}, we start with parameters slightly perturbed from zero. We then use gradient descent to find the minimum energy of the system. The results are shown in Fig. 4.

Fig. 4 Solving the vibrational ground state of H_{2}O with the UVCC ansatz. Here, we consider two energy levels for each mode. |

We have discussed ways to map between vibrational modes and qubit states. This is only possible when we consider a restricted number of harmonic oscillator energy levels. This approximation is appropriate when investigating the low energy properties of the Hamiltonian, such as the ground state energy. However, it may not always be possible to use this approximation when considering time evolution, as this requires the exponentiation of our truncated Hamiltonian. One possible route to overcome this challenge is to repeat simulations which consider an increasing number of energy levels, and then to extrapolate to the infinite energy level result. This technique was used in a similar context to this work in ref. 49.

Once the vibrational Hamiltonian has been mapped to a qubit Hamiltonian, much of the existing machinery for quantum simulation can be applied. Static properties, such as energy levels, can be calculated using phase estimation or variational approaches. To aid variational state preparation, we proposed a unitary version of the powerful VCC method used in classical vibrational simulations. The resulting energy eigenstates can be used as an input for SWAP-test circuits. These calculate the Franck–Condon factors for the molecules, which are related to the absorption spectra. Alternatively, one may investigate dynamic properties, using methods for Hamiltonian simulation to time evolve a specified state.

Compared with analog algorithms,^{13,14,16,17} our method can easily take into account anharmonic terms in the nuclear Hamiltonian. Moreover, it could be used to simulate large systems by protecting the quantum computer with error correction. As our technique is tailored for simulating vibrational states, it makes it simple to investigate interesting vibrational properties, such as the Franck–Condon factors. However, as our approach uses the Born–Oppenheimer approximation, it is not suitable for all problems in chemistry, such as problems including relativistic effects^{77} or conical intersections.^{78–80} Future work will address whether restricting our vibrational modes to low-lying energy levels poses a significant challenge for problems of practical interest.

(A1) |

(A2) |

(A3) |

Under the Born–Oppenheimer approximation, we assume the electrons and nuclei are in a product state,

|ψ〉 = |ψ〉_{n}|ψ〉_{e}.
| (A4) |

(A5) |

(A6) |

(A7) |

In general, considering a spectral decomposition of the molecular Hamiltonian is

(A8) |

(A9) |

(A10) |

Finding the spectra of H_{e}(R_{I}) is called the electronic structure problem, which can be efficiently solved using a quantum computer.^{8} One approach is to consider a subspace that the ground state lies in and transform the Hamiltonian H_{e}(R_{I}) into the second quantised formulation, with a basis determined by the subspace. As electrons are fermions, the obtained Hamiltonian is a fermionic Hamiltonian. By using the standard encoding methods, such as Jordan–Wigner and Bravyi–Kitaev,^{82} the fermionic Hamiltonian is converted into a qubit Hamiltonian, whose spectra can be efficiently computed.

Focusing on the ground state of the electronic structure Hamiltonian, we show how to redefine H_{0} in the mass-weighted basis and how to encode it with qubits. Denote then one can obtain the mass-weighted normal coordinates q_{i} by minimising the coupling between the rotational and vibrational degrees of freedom and diagonalising the Hessian matrix,

(A11) |

(A12) |

(A13) |

If we consider the higher order terms as a perturbation, one can get the normal modes by solving the harmonic oscillator

(A14) |

(A15) |

(A16) |

|VCC〉 = exp( − ^{†})|Φ_{0}〉
| (B1) |

= _{1} + _{2} +… |

The initial state can be the product of the ground-state of each mode

|Φ_{0}〉 = |ψ^{1}_{0}ψ^{2}_{0}…ψ^{M}_{0}〉.
| (B2) |

|Φ_{0}〉 = |ϕ^{1}ϕ^{2}…ϕ^{M}〉,
| (B3) |

(B4) |

H_{i}|ϕ_{i}〉 = E_{i}|ϕ_{i}〉,
| (B5) |

q_{1} = Uq_{2} + d,
| (C1) |

(C2) |

(C3) |

The transformation for p is p_{1} = U^{†}p_{2}. The transformation for the creation operators are

(C4) |

It was shown by Doktorov et al.^{64} that the Duschinsky transform can be implemented using a unitary transform inserted into the overlap integral

〈ν_{f}|ν_{i}〉 = 〈ν′_{f}|Û_{Dok}|ν_{i}〉
| (C5) |

Û_{Dok} = Û_{t}Û^{†}_{s′}Û_{s}Û_{r}
| (C6) |

(C7) |

(C8) |

(C9) |

(C10) |

These exponentials could be expanded into local qubit operators using Trotterization. It is important to note that these relations are only valid when the single-mode basis functions are chosen to be harmonic oscillator eigenstates.

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