Practical Quantum Chemistry on Quantum Computers via Overlap-Guided Compact Ansätze

An iterative procedure for generating compact ansatz wave-functions for adaptive variational quantum eigensolvers
Practical Quantum Chemistry on Quantum Computers via Overlap-Guided Compact Ansätze
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The development of programmable quantum devices able to solve many-body physics and chemistry problems which are otherwise computationally intractable on classical computing architectures has been an active field of research for several decades. While quantum computing has advanced tremendously since the pioneering proposal of Richard Feynman to ‘simulate physics with [quantum] computers’ [1], physically engineering a high-quality quantum computer has proven very challenging. The current stage of development of quantum hardware is known as the ‘noisy intermediate-scale quantum’ (NISQ) era [2]. It is characterized by relatively small quantum devices over which we have imperfect control. The question that we, as an interdisciplinary team of chemists and applied mathematicians, were confronted with is the following: can we contribute to the development of quantum algorithms– suitable for implementation on NISQ-era quantum devices– that can solve some problems of practical scientific interest?


Given our background and interest in the simulation of molecular quantum systems, a very natural field to focus our efforts on was the so-called variational quantum eigensolver (VQE) for estimating the ground state of quantum Hamiltonians [3]. The core idea of the VQE is to take advantage of the ability of quantum computers to encode wave-functions of exponentially scaling complexity using only a linearly scaling number of quantum bits (qubits). VQEs thus function by generating parameterised ‘ansatz’ wave-functions on a quantum device (using so-called quantum circuits) and then variationally tuning the parameters in the ansatz wave-function so as to minimize the expectation value of the molecular electronic Hamiltonian. Since the encoding of the ansatz wave-function on a quantum device is very efficient, the expectation value of the quantum Hamiltonian with respect to a given ansatz can be computed easily with a quartic (versus exponential) complexity.  This easy access to the expectation value of the quantum Hamiltonian, i.e., the objective function to be minimized, allows for the use of a traditional optimisation routine on classical computing architecture so as to achieve the necessary variational tuning. Thus, the VQE is a hybrid classical/quantum algorithm with the quantum device being used only to represent ansatz wave-functions and measure their expectation values.


The fundamental challenge in implementing the VQE methodology on NISQ devices is thus to construct an ansatz wave- function that can accurately describe the ground-state of the Hamiltonian under study and, at the same time, can be represented on shallow quantum circuits which are not dominated by noise. A promising approach to achieve these two requirements is to make use of adaptive algorithms which construct quantum circuits tailored to the specific Hamiltonian being considered using some type of iterative procedure, the hope being that these system-tailored quantum circuits allow for a very compact representation of the specific ground state wave-function being approximated. Among this class of algorithms, the Adaptive Derivative-Assembled Pseudo Trotter (ADAPT) VQE [4] has emerged as a gold-standard method thanks to its ability to generate ansatz wave-functions that are both highly accurate approximations to the ground state and also relatively compact in terms of the quantum circuits required to represent them on a quantum device.


Despite representing a significant advance over existing non-adaptive ‘fixed-ansatz’ VQEs, the ADAPT methodology has an unfortunate tendency to fall into local minima of the energy landscape. Thus, for certain types of strongly correlated Hamiltonians, we can encounter long stretches in the ADAPT-VQE iterative process during which the ansatz quantum circuit is continually grown across multiple iterations without meaningfully improving the quality of the resulting ansatz wave- function. As a consequence, very deep quantum circuits may be required to represent an accurate ansatz wave-function. Unfortunately, on present imperfect NISQ devices, the quantum circuit depth is directly correlated to the amount of noise and practical implementations of such quantum algorithms remain out of reach, as performing measurements on today’s quantum hardware introduce an overwhelming amount of noise. Consequently, adaptive algorithms such as ADAPT-VQE are thus only implementable on classical architectures such as high performance computing simulators (or personal workstations in the case of simple toy models).


The goal of our study was therefore to explore the possibility of further compactifying ansatz wave-functions generated by ADAPT-VQE-like procedures with the specific aim of mitigating the issue of energy plateaus. The outcome of our investigations is the Overlap-ADAPT-VQE, a new iterative procedure for generating ansatz wave-functions for variational quantumeigensolvers. The essential idea of Overlap-ADAPT-VQE is to iteratively generate a compact approximation of a target wave-function through a quasi-greedy procedure that maximises, at each iteration, the overlap of the current iterate with the target. Typically, the chosen target wave-function represents an intermediate state that approximates to a certain extent the sought-after ground state of the Hamiltonian under consideration. Once a compact and sufficiently accurate approximation of the target has been obtained through the Overlap-ADAPT procedure, the resulting approximate wave-function can be used as a high accuracy initialization for a new ADAPT-VQE-like procedure.

Figure 1. Comparison of the CIPSI-Overlap-ADAPT-VQE and ADAPT-VQE for the ground state energy of a linear H6 chain with an interatomic distance of 3 Angstrom. The plot represents the energy convergence as a function of the number of iterations in the ansatz. The CIPSI-Overlap ansatz is grown up to 20 parameters and then used as the initial state for an ADAPT-VQE process. This transition from Overlap-ADAPT-VQE to classical ADAPT-VQE is denoted by the top-pointing triangle. The horizontal black dotted line corresponds to the energy error of the initial CIPSI target wave-function. The light blue dotted line corresponds to the energy of the non-adaptive ‘fixed-ansatz’ tUCCSD method [5],  which consists in an 118 parameter ansatz wave-function. The pink area indicates the so-called chemical accuracy threshold of 103 Hartree, which is usually thought to be sufficient to accurately predict chemical bonding phenomenon
We subsequently applied the Overlap-ADAPT-VQE procedure to compute the ground state energies of a range of small benchmark molecules, and our results indicate significant savings in the quantum circuit depth required to achieve ‘chemically accurate’ simulations, with the benefits offered by overlap maximization increasing for strongly correlated molecules. As an example, the numerical results for a stretched linear H6 chain are displayed in Figure 1. For this particular example, the intermediate target wave-function for the Overlap-ADAPT procedure was produced using a selected Configuration Interaction (CI)  procedure (more specifically, the perturbatively selected configuration interaction scheme or CIPSI method [6]). Note also that the number of iterations of the adaptive procedure (indicated on the x-axis) can be viewed as a proxy for the quantum circuit depth. Let us also mention that, by comparison, the so-called Qubit-excitation-based (QEB) variant of ADAPT-VQE [7] (plotted in red and blue in Figure 1) requires more than 150 iterations to achieve eventually the same level of accuracy as the ansatz wave-function generated by Overlap-ADAPT-VQE after 50 iterations (among them only 30 of them need to be performed on a quantum architecture).
Of course the numerical simulations performed in the current study were all proof-of-concept te
sts carried out on classical computing architectures. The true test of our methodology is whether or not it allows for practical implementations on current generation NISQ devices. A first step in this direction has been taken in [8].

References
  1. Feynman, R.P. Simulating physics with computersInt. J. Theor. Phys. 21, 467–488 (1982)
  2. Preskill, J. Quantum Computing in the NISQ era and beyond.Quantum 2, 79 (2018)
  3.  Peruzzo, A., McClean, J., Shadbolt, P. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).
  4. Grimsley, H.R., Economou, S.E., Barnes, E.,  Mayhall, N. J. An adaptive variational algorithm for exact molecular simulations on a quantum computerNat Commun 10, 3007 (2019)
  5. Romero, J.,  Babbush, R.,  McClean, J. R.,  Hempel, C.,  Love, P.J.,  Aspuru-Guzik, A. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. Quantum Sci. Technol. 4, 014008 (2019)
  6. Huron, B., Malrieu, J. & Rancurel, P. Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions. J. Chem. Phys. 58, 5745 (1973)
  7. Yordanov, Y.S., Armaos, V., Barnes, C.H.W.,  Arvidsson-Shukur, D.R. Qubit-excitation-based adaptive variational quantum eigensolverCommun. Phys. 4, 228 (2021).
  8. Feniou, C., Claudon, B., Hassan, M., Courtat, A., Adjoua, O., Maday, Y., Piquemal, J.-P. Greedy Gradient-free Adaptive Variational Quantum Algorithms on a Noisy Intermediate Scale Quantum Computer. ArXiv Preprint (2023)

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