The use of the variational quantum eigensolver (VQE) for quantum chemistry is one of the most promising applications for noisy intermediate-scale quantum (NISQ) devices. A major limitation is represented by the need to build compact and shallow circuit ansatzes having the variational flexibility to catch the complexity of the electronic structure problem. To alleviate this drawback, we introduce a modified VQE scheme in which the form of the molecular Hamiltonian is adapted to the circuit ansatz through an optimization procedure. Exploiting the invariance of the Hamiltonian by molecular orbital rotations, we can optimize it using gradients that can be calculated without significant computational overload. The proposed method, named Wavefunction Adapted Hamiltonian Through Orbital Rotation (WAHTOR), has been applied to small molecules in numerical state vector simulations. The results demonstrate that, at variance with standard VQE, the method is less dependent on circuit topology and less prone to be trapped into high-energy local minima. It is able to recover a significant amount of electron correlation even with only empirical ansatzes with shallow circuit depth. Noisy calculations demonstrate the robustness and feasibility of the proposed methodology and indicate the hardware requirements to effectively apply the procedure using forthcoming NISQ devices.
Abstract By exploiting the invariance of the molecular Hamiltonian by a unitary transformation of the orbitals it is possible to significantly shorter the depth of the variational circuit in the variational quantum eigensolver (VQE) algorithm by using the wavefunction adapted Hamiltonian through orbital rotation (WAHTOR) algorithm. This work introduces a non-adiabatic version of the WAHTOR algorithm and compares its efficiency with three implementations by estimating quantum processing unit (QPU) resources in prototypical benchmarking systems. Calculating first and second-order derivatives of the Hamiltonian at fixed VQE parameters does not introduce a significant QPU overload, leading to results on small molecules that indicate the non-adiabatic Newton–Raphson method as the more convenient choice. On the contrary, we find out that in the case of Hubbard model systems the trust region non-adiabatic optimization is more efficient. The preset work therefore clearly indicates the best optimization strategies for empirical variational ansatzes, facilitating the optimization of larger variational wavefunctions for quantum computing.
State-of-the-art noisy digital quantum computers can only execute short-depth quantum circuits. Variational algorithms are a promising route to unlock the potential of noisy quantum computers since the depth of the corresponding circuits can be kept well below hardware-imposed limits. Typically, the variational parameters correspond to virtual
RZ gate angles, implemented by phase changes of calibrated pulses. By encoding the variational parameters directly as hardware pulse amplitudes and durations, we succeed in further shortening the pulse schedule and overall circuit duration. This decreases the impact of qubit decoherence and gate noise. As a demonstration, we apply our pulse-based variational algorithm to the calculation of the ground state of different hydrogen-based systems (
H2,H3, and
H4) using IBM cross-resonance-based hardware. We observe a reduction in schedule duration of up to $5\ifmmode\times\else\texttimes\fi{}$ compared to cnot-based Ans\"atze, while also reducing the measured energy. In particular, we observe a sizable improvement of the minimal energy configuration of
H3 compared to a cnot-based variational form. Finally, we discuss possible future developments including error-mitigation schemes and schedule optimizations, which will enable further improvements of our approach, paving the way towards the simulation of larger systems on noisy quantum devices.
Quantum computers promise a great computational advantage over classical computers, which might help solve various computational challenges such as the simulation of complicated quantum systems, finding optimum in large optimization problems, and solving large-scale linear algebra problems. Current available quantum devices have only a limited amount of qubits and a high level of noise, limiting the size of problems that can be solved accurately with those devices. Variational quantum algorithms (VQAs) have emerged as a leading strategy to address these limitations by optimizing cost function based on measurement results of shallow depth circuits. Recently, various pulse engineering methods were suggested in order to improve VQA results, including optimizing pulse parameters instead of gate angles as part of the VQA optimization process. In this paper, we suggest a novel pulse-based ansatz, which is parameterized mainly by pulses’ duration of pre-defined pulse structures. This ansatz structure provides relatively low amounts of optimization parameters while maintaining high expressibility, allowing fast convergence. In addition, the ansatz has structured adaptivity to the entanglement level required by the problem, allowing low noise and accurate results. We tested this ansatz against quantum chemistry problems. Specifically, finding the ground-state energy associated with the electron configuration problem, using the variational quantum eigensolver (VQE) algorithm for several different molecules. We manage to achieve chemical accuracy both in simulation for several molecules and on one of IBM’s NISQ devices for the H2 molecule in the STO-3G basis, without the need for extensive error mitigation. Our results are compared to a common gate-based ansatz and show better accuracy and significant latency reduction—up to 7× shorter ansatz schedules.
One of the first practical applications of quantum computers is expected to be molecular modelling. Performing this task would profoundly affect areas such as chemistry, materials science and drug synthesis. Modelling of molecules, which are classically intractable, can be achieved with just over
30 qubits, whereas state of the art quantum computers already have more than
50 qubits. The Variational Quantum Eigensolver (VQE) algorithm and VQE based protocols, are promising candidates to enable this task on emerging Noisy Intermediate-Scale Quantum (NISQ) computers. These protocols require short quantum circuits and short coherence times, and are particularly resilient to quantum errors. Nevertheless, there is still a significant gap between the accuracy and the coherence times of current NISQ computers, and the hardware requirements of VQE protocols to simulate practically interesting molecules. In this thesis, I present my contribution to narrowing this gap by developing VQE protocols for molecular modelling that are less demanding on quantum hardware. The VQE relies on the Rayleigh-Ritz variational principle to estimate the eigenvalues of a Hamiltonian operator, by minimizing its expectation value with respect to a trial quantum state, prepared by an ansatz. A major challenge for the practical realisation of VQE protocols on NISQ computers is to construct an ansatz that: (1) can accurately approximate the eigenstates of the Hamiltonian; (2) is easy to optimize; and (3) can be implemented by a shallow circuit, within the capabilities of a NISQ computer. The most widely used, unitary coupled cluster (UCC), type of ans\"atze mathematically correspond to a product of unitary evolutions of fermionic excitation operators. Owing to their fermionic structure, UCC ans\"atze preserve the symmetries of electronic wavefunctions, and thus are accurate and easy to optimize. Nevertheless, UCC ans\"atze are implemented by high depth circuits, which severely limit the size of the molecules that can be reliably simulated on NISQ computers. In this thesis, I begin by constructing efficient quantum circuits to perform evolutions of fermionic excitation operators. The circuits are optimized in the number of two-qubit entangling gates, which are the current bottleneck of NISQ computers. Compared to the standard circuits used to implement evolutions of fermionic excitation operators, the circuits derived in this thesis reduce the number of two-qubit entangling gates by more than
70% on average. As an intermediate result, I also derive efficient circuits to perform evolutions of qubit excitation operators (excitation operators that account for qubit, rather than fermionic commutation relations). Even with the fermionic-excitation-evolution circuits derived here, UCC ans\"atze still require very long circuits, with a particularly large number of two-qubit entangling gates. In this thesis, I consider the use of alternative VQE ans\"atze, based on evolutions of qubit excitation operators. Due to not accounting for fermionic anticommutation, evolutions of qubit excitation operators can be performed by circuits that require asymptotically fewer two-qubit entangling gates. Furthermore, qubit excitation operators preserve many of the physical properties of fermionic excitation operators. Performing a number of classical numerical VQE simulations for small molecules, I show that qubit-excitation-based ans\"atze can approximate molecular electronic wavefunctions almost as accurately as fermionic-excitation-based ans\"atze. Hence, I argue that evolutions of qubit excitation operators are more suitable to construct molecular ans\"atze than evolutions of fermionic excitation operators, especially in the era of NISQ computers. Motivated by the advantage of qubit-excitation-based ans\"atze, I introduce the qubit-excitation-based adaptive variational quantum eigensolver (QEB-ADAPT-VQE). The QEB-ADAPT-VQE belongs to a family of ADAPT-VQE protocols for molecular modelling that grow a problem-tailored ansatz by iteratively appending unitary operators sampled from a predefined finite-size pool of operators. The operator at each iteration is sampled based on an ansatz-growing strategy, which aims to achieve the lowest estimate for the Hamiltonian expectation value at each iteration. In this way, ADAPT-VQE protocols construct shallow-circuit, few-parameter ans\"atze tailored specifically to the molecular systems of interest. In the case of the QEB-ADAPT-VQE, the operator pool is defined by a set of evolutions of single and double qubit excitation operators. I benchmark the performance of the QEB-ADAPT-VQE, by performing classical numerical simulations. I demonstrate that it can construct ans\"atze that are several orders of magnitude more accurate, and require significantly shallower circuits, than standard UCC ans\"atze. I also compare the QEB-ADAPT-VQE against the original fermionic-ADAPT-VQE, which utilizes a pool of fermionic excitation evolutions, and the qubit-ADAPT-VQE, which utilizes a pool of Pauli-string evolutions. I demonstrate that, in terms of circuit efficiency and convergence speed, the QEB-ADAPT-VQE systematically outperforms the qubit-ADAPT-VQE, which to my knowledge was the previous most circuit-efficient, scalable VQE protocol for molecular modeling. The QEB-ADAPT-VQE protocol, therefore represents a significant improvement in the field of VQE protocols for molecular modelling and brings us closer to achieving practical quantum advantage. Lastly, I outline a modified version of the QEB-ADAPT-VQE, the excited-QEB-ADAPT-VQE, designed to estimate energies of excited molecular states. The excited-QEB-ADAPT-VQE is more robust to initial simulation conditions, at the expense of increased computational complexity.
Abstract The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectation values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose an analytical method to compute the energy gradient that reduces the sampling cost for gradient estimation by several orders of magnitude compared to numerical gradients. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the circuit depth of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.