# Quantum Computing ⨷ Quantum Chemistry

## 24-25 September 2020

**Part of the Quantum Week of Fun**

### Invited talks

Kosuke Mitarai
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Extending the variational quantum eigensolver for practical applications
**Abstract: **
The variational quantum eigensolver (VQE) is a promising candidate among applications for the noisy intermediate scale quantum (NISQ) devices. The original formulation of the VQE has aimed to determine a ground state and its energy of a given Hamiltonian. However, for quantum chemistry applications, we must develop extended methods that can analyze excited states, its energy, derivatives of energy, and so on. These quantities play important roles in practical applications. For example, analysis of light absorption spectra of a molecule requires us to evaluate oscillator strengths between eigenstates, and vibrational spectra need second derivatives of the energy with respect to nuclear positions. Also, for practical NISQ applications, we also need sophisticated techniques to reduce quantum resources because of the limited capacity of devices.
In this talk, we discuss some of the techniques which we developed to tackle these problems. First, I give a brief review on extensions of the VQE for excited states including our work. A detailed analysis of its performance against other techniques will also be given. Next, we move on to a technique to extract the analytical derivatives of the VQE energy, which is based on the so-called parameter shift rule for calculating derivatives of parametrized quantum circuits. Lastly, we discuss techniques for reducing quantum resources.

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Panagiotis Barkoutsos
.
Exploring possibilities of near-term quantum computers for the simulation of the Hubbard model
**Abstract: **
Near term noisy quantum processors require the development of schemes that will allow us to get meaningful results without the overhead of error correction codes. One of the most established algorithms for noisy intermediate scale quantum computers is the Variational Quantum Eigensolver (VQE). In this talk I will present a plethora of approaches developed for the generation of a key component of the VQE algorithm, namely the variational form (Ansatz). Using the iconic Fermi-Hubbard model as a demonstrative example, I will discuss the different approaches varying from classically inspired to quantum native trial wavefunctions and give an overview of the existing research. In the last part I will present a novel error mitigation scheme, inspired from the classical Lanczos algorithm that when coupled with the VQE allow for the simulation of the Fermi Hubbard model in an IBM Quantum processor with significantly better results.

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Nicholas Rubin
.
First steps in intermediate scale quantum chemistry on superconducting quantum computers
**Abstract: **
Determining whether near-term quantum resources can aid in quantum chemical simulation is an important question for determining the utility of today’s quantum hardware. The work we present here takes steps towards describing the role of NISQ quantum computation in chemical simulation by focusing on the performance of an algorithmic primitive for simulating fermionic systems. In the process of this verification, we demonstrate the simulation of a chemistry model that is significantly larger than previous implementations on any quantum computing platform. Our experiments run on the Sycamore chip achieves chemical accuracy upon application of an error mitigation scheme based on pure-state representability. Most importantly we describe methods for going beyond this circuit primitive improving the model chemistry.

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Mario Motta
.
Quantum simulation of electronic structure with transcorrelated Hamiltonian: increasing accuracy without extra quantum resources
**Abstract: **
Quantum simulations of electronic structure with transformed ab initio Hamiltonians that include some electron correlation effects a priori are demonstrated. The transcorrelated Hamiltonians used in this work are efficiently constructed classically, at polynomial cost, by an approximate similarity transformation with an explicitly correlated two-body unitary operator; they are Hermitian, include up to two-particle interactions, and are free of electron-electron singularities. To investigate whether the use of such transformed Hamiltonians can reduce resource requirements for general quantum solvers for the Schrodinger equation, we explore the accuracy and the computational cost of the quantum variational eigensolver, based on the unitary coupled cluster with singles and doubles (q-UCCSD). Our results demonstrate that transcorrelated Hamiltonians, paired with extremely compact bases, produce explicitly correlated energies comparable to those from much larger bases. The use of transcorrelated Hamiltonians reduces the number of CNOT gates by up to two orders of magnitude, and the number of qubits by a factor of three.

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### List of contributed talks

Bruno Senjean,
Thomas O'Brien, Luuk Visscher, Xavier Bonet-Monroig, Ramiro Sagastizabal, Alicja Dutkiewicz, Francesco Buda and Leonardo DiCarlo
.
Calculating energy derivatives for quantum chemistry on a quantum computer
**Abstract: **
Quantum computers are at the verge of providing solutions for certain classes of problems that are intractable on a classical computer. As this threshold nears, an important next step is to investigate how these new possibilities can be translated into useful algorithms for specific scientific domains. Quantum chemistry has been identified as a key area where quantum computers can stop being science and start doing science. This observation has lead to an intense scientific effort towards developing and improving quantum algorithms to determine ground and excited state energies, but they are only one of the important targets for quantum chemistry calculations.
For many applications, molecular properties are of primary interest, however they have received relatively little focus. For example, the energy gradient (first-order derivative) and Hessian (second-order derivative) for nuclear displacements is used to search for minima, transition states, and reaction paths that characterize a molecular potential energy surface (PES).
In this talk, I will present our algorithms to compute the first- and second-order derivatives of the energy with respect to a change in the Hamiltonian, using both VQE- and QPE-based algorithms. We demonstrated our algorithms on a superconducting quantum processor by performing geometry optimization of the dihydrogen molecule, as well as its response to a small electric field (polarizability), and we found excellent agreement with the full configuration interaction (FCI) solution.

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David Wierichs, Christian Gogolin and Michael Kastoryano
.
Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer
**Abstract: **
We compare the BFGS optimizer, ADAM and Natural Gradient Descent (NatGrad) in the context of Variational Quantum Eigensolvers (VQEs). We systematically analyze their performance on the QAOA ansatz for the Transverse Field Ising Model as well as on overparametrized circuits with the ability to break the symmetry of the Hamiltonian. The BFGS algorithm is frequently unable to find a global minimum for systems beyond about 20 spins and ADAM easily gets trapped in local minima. On the other hand, NatGrad shows stable performance on all considered system sizes, albeit at a significantly higher cost per epoch. In sharp contrast to most classical gradient based learning, the performance of all optimizers is found to decrease upon seemingly benign symmetry-preserving and more complex symmetry-breaking overparametrization of the ansatz class, with BFGS and ADAM failing more often and more severely than NatGrad.
Additional tests for the Heisenberg XXZ model corroborate the accuracy problems of BFGS in high dimensions, but they reveal some shortcomings of NatGrad as well. Our results suggest that great care needs to be taken in the choice of gradient based optimizers and the parametrization for VQEs.

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Sumeet, V.S. Prasannaa, B.K. Sahoo and B.P. Das
.
Quantum Simulation using the Variational Quantum Eigensolver Algorithm: Hardware-efficient versus unitary coupled-cluster ansatz
**Abstract: **
The classical-quantum hybrid Variational Quantum Eigensolver (VQE) algorithm is considered well suited to obtain ground state energies of atoms and molecules in the Noisy Intermediate Scale Quantum computing era. An extremely important component of the VQE algorithm is the choice of a suitable ansatz for the trial wave function. In this expository work, we choose atomic systems, which also finds several applications like molecules but have received little attention until now in quantum simulation studies, as our test systems. We compare the atomic energies obtained from quantum simulation of the VQE algorithm that employs the unitary coupled-cluster wave function and a hardware-efficient ansatz, namely the RyRz variational form. The former is expected to set the bar from a quantum many-body theoretic point of view, whereas the latter is more practical due to the minimal resources that it consumes. We also carry out surveys on the influence of other modules of the VQE algorithm such as the choice of fermion to qubit mapping and atomic basis functions. Apart from that we also investigate expressibility and entanglement capacity of various ansatz used for VQE algorithm.

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David Zsolt Manrique, Irfan T. Khan, Kentaro Yamamoto, Vijja Wichitwechkarn and David Muñoz Ramo
.
Momentum-Space Unitary Couple Cluster and Translational Quantum Subspace Expansion for Periodic Systems on Quantum Computers
**Abstract: **
We demonstrate the use of the Variational Quantum Eigensolver (VQE) to simulate solid state materials. We adapt the Unitary Coupled Cluster ansatz to periodic boundary conditions in real space and momentum space representations. To reduce required quantum resources, such as the number of UCCSD amplitudes, circuit depth, required number of qubits and number of measurement circuits, we investigate a translational Quantum Subspace Expansion method for the localized representation of the periodic Hamiltonian. We compare accuracy and computational costs for a range of geometries for 1D chains of dimerized hydrogen, helium and lithium hydride and also demonstrate VQE calculations for 2D and 3D hydrogen and helium lattices.

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Hans Hon Sang Chan, Nathan Fitzpatrick, Javier Segarra-Marti and Mike Bearpark
.
Excited State Calculations of Small Molecules with Adaptive Wavefunctions on a Simulated Quantum Computer
**Abstract: **
Recently developed hybrid quantum-classical computing algorithms for ab initio molecular excited state calculations are gaining traction. They promise to address some of the prohibitive scaling challenges in expensive ab initio methods, still needed for quantitative study of excited states, by harnessing emerging quantum computers' ability to handle exponentially more information than their classical counterparts. They also boast to use short quantum circuits, the sequence of procedures needed to process information in a quantum computer, for calculations.
Consequently, they can potentially be executed on near-term quantum computers, which are susceptible to random errors due to noise, without fault correction. As such they are seen as one of the first quantum applications that will demonstrate supremacy over classical computing.
In practice, molecules with more than 6 light elements at anything other than the minimal orbital basis remain intractable on current high-noise quantum technology;
the quantum circuits required to encode molecular wavefunctions are still too deep to guarantee a noiseless calculation. In our work, we set out to shorten the quantum circuits needed for Variational Quantum Deflation (VQD), one of the most promising hybrid computing methods for calculating molecular excited states near-term. We invoked spin-constraints used in contemporary computational chemistry methods, and combined it with ADAPT-VQE to propose novel strategies that shorten the encoding of the excited state wavefunctions on a quantum computer for VQD. Emulating these calculations on the same molecules, we found that by using spin constraints to limit the wavefunction space explored, the quantum circuit needed to calculate excited states within chemical accuracy was half of the original proposed VQD scheme. ADAPT-VQE techniques combined with spin constraints also showed potential to further reduce circuit depth.
This work is a step towards improving the efficiency of hybrid quantum-classical algorithms for excited state quantum chemistry. We demonstrated that, through new scaling reduction techniques developed, we could in principle exploit the advantages of real quantum computers for electronic excited state calculations much sooner than previously anticipated.

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Ivan Rungger
.
Dynamical mean field theory algorithm and experiment on quantum computers
**Abstract: **
The developments of quantum computing algorithms and experiments for atomic scale simulations have largely focused on quantum chemistry for molecules, while their application in condensed matter systems is scarcely explored. Here we present a quantum algorithm to perform dynamical mean field theory (DMFT) calculations for condensed matter systems on currently available quantum computers, and demonstrate it on two quantum hardware platforms. DMFT is required to properly describe the large class of materials with strongly correlated electrons. The computationally challenging part arises from solving the effective problem of an interacting impurity coupled to a bath, which scales exponentially with system size on conventional computers. An exponential speedup is expected on quantum computers, but the algorithms proposed so far are based on real time evolution of the wavefunction, which requires high-depth circuits and hence very low noise levels in the quantum hardware. Here we propose an alternative approach, which uses the variational quantum eigensolver (VQE) method for ground and excited states to obtain the needed quantities as part of an exact diagonalization impurity solver. We present the algorithm for a two site DMFT system, which we benchmark using simulations on conventional computers as well as experiments on superconducting and trapped ion qubits, demonstrating that this method is suitable for running DMFT calculations on currently available quantum hardware.

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Saad Yalouz, Bruno Senjean, Jakob Gunther, Thomas E. O’brien, Francesco Buda and Lucas Visscher
.
Describing conical intersection in the NISQ era for photochemistryapplications: the Orbital-Optimized State-Averaged VQE method
**Abstract: **
In quantum chemistry, the study of photoisomerization processes is fundamental to understand many biological mechanisms. For example, the vision process is mediated by the photoisomerisation of the rhodopsin molecule. The absorption of a photon by the cis-conformer molecule promotes a wavepacket to its first electronic excited state and changes its geometric structure. The presence of a conical intersection (a singular point of degeneracy) between the ground and first excited state allows the molecule to relax to its trans-conformer ground state in a non-radiative way. This cycle of excitation/relaxation makes rhodopsin a genuine photo-receptor antenna responsible for sending signals from the eye to the brain.
Studying such complex processes requires a precise knowledge of the position of the conical intersection between the potential energy surfaces of the molecule. Although such knowledge could be obtained using a sufficiently large active space, NISQ-era quantum computers often do not have the qubits or the coherence time to achieve this.
To solve this problem, we have combined the classical technique of state-averaged orbital-optimization with the well-known variational quantum eigensolver (VQE). The resulting method, called "Orbital-Optimized State-Averaged VQE", is able to qualitatively and quantitatively reproduce conical intersections within a vastly reduced active space. We demonstrate the strength of this method on the formaldimine molecule. This is a minimal model for rhodopsin, featuring a similar cis-trans rotation with a conical intersection.

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Oinam Meitei, Bryan Gard, George Barron, Sophia Economou, Edwin Barnes and Nicholas Mayhall
.
Gate-free state preparation for fast variational quantum eigensolver simulations: ctrl-VQE
**Abstract: **
The variational quantum eigensolver (VQE) is currently the flagship
algorithm for solving electronic structure problems on near-term quantum
computers. This hybrid quantum/classical algorithm involves implementing a
sequence of parameterized gates on quantum hardware to generate a target
quantum state and measuring the expectation value of the molecular Hamiltonian. Due to finite coherence times and frequent gate errors, the number of gates that can be implemented remains limited on the current state of the art quantum devices, preventing application to systems with significant entanglement, such as strongly correlated molecules. In this work, we propose an alternative algorithm (which we refer to as ctrl-VQE) where the quantum circuit used for state preparation is removed entirely, being replaced by a quantum control routine which variationally shapes a pulse to drive the initial Hartree-Fock state to the full CI target state. As with VQE, the objective function optimized is the expectation value of the qubit-mapped molecular Hamiltonian. However, by removing the quantum circuit, the coherence times required for state preparation can be drastically reduced by directly optimizing the pulses. We demonstrate the potential of this method numerically by directly optimizing pulse shapes which accurately model the dissociation curves of the hydrogen molecule (covalent bond) and helium hydride ion (ionic bond).

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Dmitry Fedorov, Matthew Otten, Stephen Gray and Yuri Alexeev
.
Ab Initio Molecular Dynamics on Quantum Computers
**Abstract: **
Ab initio molecular dynamics (AIMD) is a valuable technique for studying molecular systems at finite temperatures where the nuclei evolve on potential energy surfaces obtained from accurate electronic structure calculations. In this work, a quantum computer-based AIMD method is presented. The electronic energies are calculated on a quantum computer using the variational quantum eigensolver (VQE) method. The energy gradients are computed numerically using the Hellmann-Feynman theorem, finite differences, and a correlated sampling technique. Our method only requires the calculation of electron integrals for each degree of freedom without any additional computations on a quantum computer. To achieve comparable accuracy, our gradient calculation method requires three to five orders of magnitude fewer measurements than the brute force methods without correlated sampling. As a proof of concept, AIMD dynamics simulations are demonstrated for the H2 molecule on IBM quantum devices. To the best of our knowledge, it is the first successful attempt to run AIMD on quantum devices for a chemical system. With the increasing quality of quantum hardware and noise mitigation techniques, our method can be utilized for studying larger molecular systems.

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Igor Sokolov, Panagiotis Barkoutsos, Lukas Moeller, Philippe Suchsland, Guglielmo Mazzola and Ivano Tavernelli
.
Microcanonical and finite temperature ab initio molecular dynamics simulations on quantum computers
**Abstract: **
Ab initio molecular dynamics (AIMD) is a powerful tool to predict properties of molecular and
condensed matter systems. The quality of this procedure is based on accurate electronic structure
calculations. The development of quantum processors has shown great potential for the efficient
evaluation of accurate ground and excited state energies of molecular systems, opening up new
avenues for molecular dynamics simulations. In this talk, we address the use of variational quantum
algorithms for the calculation of accurate atomic forces to be used in AIMD. In particular, we
provide solutions for the alleviation of the statistical noise associated to the measurements of the
expectation values of energies and forces, as well as schemes for the mitigation of the hardware noise
sources (in particular, gate infidelities, qubit decoherence and readout errors). Despite the relative
large error in the calculation of the potential energy, our results show that the proposed algorithms
can provide reliable MD trajectories in the microcanonical (constant energy) ensemble. Further,
exploiting the intrinsic noise arising from the quantum measurement process, we also propose a
Langevin dynamics algorithm for the simulation of canonical, i.e., constant temperature, dynamics.
Both algorithms (microcanonical and canonical) are applied to the simulation of simple molecular
systems such as the hydrogen molecule and the trihydrogen cation.
Finally, we also provide results for the dynamics of the hydrogen molecule obtained with IBM quantum computer ibmq_athens

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Mekena Metcalf, Nicholas Bauman, Karol Kowalski and Wibe De Jong
.
Resource Efficient Chemistry on Quantum Computers with the Variational Quantum Eigensolver and The Double Unitary Coupled-Cluster Approach
**Abstract: **
Applications of quantum simulation algorithms to obtain electronic energies of molecules on NISQ devices require careful consideration of resources for describing complex inter-electron correlation effects. In modeling of these problems, a big challenge is posed by the fact that the number of qubits scales linearly with the size of molecular basis, which significantly limits basis set size and the number of correlated electrons included in quantum simulations of chemical processes.
To address this issue and enable more realistic simulations on near-term quantum computers, several algorithms have been proposed to effectively downfold correlation effects into the reduced-size orbital space, commonly referred to as the active space.
Using downfolding techniques based on double unitary coupled-cluster (DUCC) Ansatz we demonstrate that dynamic correlation can be captured by small-size active spaces and by properly constructed active-space effective Hamiltonians. Combining the downfolding pre-processing technique with the Variational Quantum Eigensolver, we solve for the ground-state energy in the DUCC reduced active space and compare results to the configuration-interaction for $\text{Li}_2$, $\text{H}_2$ and $\text{BeH}_2$ using RHF and natural orbitals.

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