Quantum computing offers several algorithms to compute the ground state of a problem Hamiltonian. The most desirable algorithms belong to the Fault Tolerant QuantumComputing (FTQC) regime, such as quantum algorithms with repetitive structure like Quantum Phase Estimation (QPE) and Quantum Signal Processing (QSP). However, in the Noisy In-termediate Scale Quantum (NISQ) regime, the most realistic approaches involve Variational Quantum Eigensolver (VQE) algorithms and their variants. VQE is an algorithm that searches for a parametrized unitary matrix called an ansatz whose purposeis to transform an easily prepared initial state into the groundstate of a given Hamiltonian. Adaptive Derivative-AssembledPseudo-Trotter (ADAPT)-VQE is a variant of VQE that im-proves this approach by constructing the ansatz iteratively so that the associated quantum circuit is as shallow as possible. A major difference between FTQC (i.e. not variational) algorithms and VQE is that FTQC algorithms do not construct a state transitiondirectly. Instead, they construct a projector that identifies the ground state using ancillary qubits that flag the good solution. The desired state is then obtained via amplitude amplification orpost-selection. In this work, we propose a VQE ansatz whose structure is more similar to that of an FTQC algorithm. Depending on its parametrization, this ansatz can be equivalent to either an Intermediate Scale Quantum (ISQ)-QSP or to an ADAPT-VQE quantum circuit structure. Our experimental results show that this first proposal of Projector Variational Ansatz (PVA) converges with a shallower ansatz than the usual ADAPT-VQE.
Quantum computing offers several algorithms to compute the ground state of a problem Hamiltonian. The most desirable algorithms belong to the Fault Tolerant QuantumComputing (FTQC) regime, such as quantum algorithms with repetitive structure like Quantum Phase Estimation (QPE) and Quantum Signal Processing (QSP). However, in the Noisy In-termediate Scale Quantum (NISQ) regime, the most realistic approaches involve Variational Quantum Eigensolver (VQE) algorithms and their variants. VQE is an algorithm that searches for a parametrized unitary matrix called an ansatz whose purposeis to transform an easily prepared initial state into the groundstate of a given Hamiltonian. Adaptive Derivative-AssembledPseudo-Trotter (ADAPT)-VQE is a variant of VQE that im-proves this approach by constructing the ansatz iteratively so that the associated quantum circuit is as shallow as possible. A major difference between FTQC (i.e. not variational) algorithms and VQE is that FTQC algorithms do not construct a state transitiondirectly. Instead, they construct a projector that identifies the ground state using ancillary qubits that flag the good solution. The desired state is then obtained via amplitude amplification orpost-selection. In this work, we propose a VQE ansatz whose structure is more similar to that of an FTQC algorithm. Depending on its parametrization, this ansatz can be equivalent to either an Intermediate Scale Quantum (ISQ)-QSP or to an ADAPT-VQE quantum circuit structure. Our experimental results show that this first proposal of Projector Variational Ansatz (PVA) converges with a shallower ansatz than the usual ADAPT-VQE.
In order to answer the problem of Quantum Phase Estimation Algorithm been not suitable for NISQ devices, and allows one to outperform classical computers, Variational Quantum Algorithms (VQAs) were designed. Our subject of interest is the so-called Variational Quantum Eigensolver (VQE) algorithm and was originally designed to simulate electronic structures and to compute the ground state of a given molecule. VQE is made of two main components : an ansatz and a classical optimizer. The ansatz runs on the quantum device and aims to simulate the wavefunction, the parameters of the ansatz will be optimized until the expectation value is minimum. The very first ansatz that has originally been used is called UCCSD and it is based on Coupled Cluster Theory. The main issue considering UCCSD is the large amount of parameters to optimize and this leads us to the introduction of Adaptive Derivative-Assembled Pseudo-Trotter ansatz VQE (ADAPT-VQE) which determines a quasi-optimal ansatz with a minimal number of parameters. The key point of ADAPT-VQE is to grow the ansatz at every step, by adding operators chosen from a pre-determined pool of operators one-at-a-time, assuring that the maximal amount of correlation energy is recovered at each step. There exists different kind of ADAPT-VQE depending on the starting pool of operators as the fermionic-ADAPT, the qubit-ADAPT or even the qubit excitation based (QEB). Our goal is to implement the different types of ADAPT-VQE mentioned before. After a quick review of the theoretical background under all of these concepts, we will implement each algorithm using quiskit. We will also compare all of these algorithms on different criterions such as the number of parameters, the accuracy or the number of CNOT gate used on H2 and LiH molecules. Then we will have a small discussion about the results we obtained.
Designing compact ansätze in Variational Quantum Eigensolver (VQE) is crucial for solving energetic problems of practical molecules on near-term quantum devices. However, existing Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT) ansätze face two challenges: improper operator selection and accumulation of degraded operators. In this paper, we propose the Hamiltonian-Aware (HA) ADAPT-VQE algorithm to address these issues. First, we establish a novel excitation operator selection criterion. It breaks the local constraint of existing criteria by incorporating Hamiltonian information, prioritizes physically meaningful excitation operators, and incurs no extra classical or quantum computational overhead. Furthermore, we develop a problem-adaptive method for discriminating and pruning redundant excitation operators stemming from improper selection and inevitable degradation. This method balances redundant operator pruning and convergence guarantee, and is applicable to ansätze with arbitrary scales. Systematic numerical experiments on typical strongly correlated molecular systems demonstrate that our HA-ADAPT-VQE avoids energy plateaus and outperforms baseline algorithms in terms of energy error, ansatz size, and measurement cost. This work offers an efficient, robust ansatz construction paradigm, facilitating the development and practical deployment of large-scale VQE in quantum chemistry.
The adaptive derivative-assembled pseudo-trotter (ADAPT) variational quantum eigensolver (VQE) is an appealing framework for iteratively constructing accurate representations of ground- and excited-state wave functions as products of exponentialized one- and two-body operators acting on a reference state. In this work, we propose a CNOT-efficient circuit for implementing exponentialized two-body qubit-excitation-based (QEB) operators, enabling the construction of the low-depth wave function Ansätze. Our proposed circuit architecture employs a 2-qubit-controlled rotation gate flanked by two layers of CNOT gates, requiring only 9 CNOT gates per two-body QEB operator while preserving essential symmetries, including particle number and the Z -component of spin. By integrating these optimized circuits into the ADAPT–VQE framework, we achieve a reduction in circuit depth of approximately 28% compared to the original QEB ADAPT–VQE method, without sacrificing accuracy in simulating ground- and excited-state properties of small molecules. Numerical simulations validate the improved performance of our approach, demonstrating its potential as an efficient tool for quantum simulations of electronic structures.
By design, the variational quantum eigensolver (VQE) strives to recover the lowest-energy eigenvalue of a given Hamiltonian by preparing quantum states guided by the variational principle. In practice, the prepared quantum state is indirectly assessed by the value of the associated energy. Novel adaptive derivative-assembled pseudo-trotter (ADAPT) ansatz approaches and recent formal advances now establish a clear connection between the theory of quantum chemistry and the quantum state ansatz used to solve the electronic structure problem. Here we benchmark the accuracy of VQE and ADAPT-VQE to calculate the electronic ground states and potential energy curves for a few selected diatomic molecules, namely H2, NaH, and KH. Using numerical simulation, we find both methods provide good estimates of the energy and ground state, but only ADAPT-VQE proves to be robust to particularities in optimization methods. Another relevant finding is that gradient-based optimization is overall more economical and delivers superior performance than analogous simulations carried out with gradient-free optimizers. The results also identify small errors in the prepared state fidelity which show an increasing trend with molecular size.
The transition from the Noisy Intermediate-Scale Quantum era toward fault tolerance will in-troduce an intermediate regime, Early Fault-Tolerant Quantum Computing. In this regime faulttolerance is first achieved but restricted by strict hardware limits, such as a small pool of logicalqubits or limited ancilla resources, and shallow logical circuits are guaranteed to be protected byerror correction protocols. This thesis investigates whether variational algorithms retain prac-tical utility in this regime, and whether anchoring them to a protocol with known analyticalcomplexity can supply the trainability guarantees that generic variational methods lack.We introduce a Variational Quantum Amplifier, a hybrid algorithm in which a parameter-ized quantum circuit prepares a trial state and a Quantum Phase Estimation block amplifies andreads out its overlap with a target eigenstate specified only by its eigenvalue. The central featureof this construction is that the cost function is the probability assigned to the target eigenvalue,ensuring the target eigenstate is addressed through its eigenvalue alone and never prepared as areference. We show that the trainable ansatz converts an otherwise exponential sampling costinto a polynomial one and for a local Hamiltonian, we derive a closed-form expression for thegradient variance, identifying in this manner conditions under which barren plateaus are avoided,and provide an analytical, hardware-estimable gradient via a modified Hadamard-test circuit.Building on this, we construct an adaptive variant of this Variational Quantum Amplifier, andbenchmark it on local autoencoder and entangled projector Hamiltonians. Finally, a rigorousresource estimation against a representative family of NISQ and fault-tolerant algorithms showsthat the proposed method occupies a distinctive niche, inheriting the circuit depth of an EFTQCmethod while actively reducing the sampling overhead through the trainable ansatz during execution.
Approximating the ground states of strongly interacting electron systems in quantum chemistry and condensed matter physics is expected to be one of the earliest applications of quantum computers. In this paper, we prepare highly accurate ground states of the Fermi-Hubbard model for small grids up to 6 sites (12 qubits) by using an interpretable, adaptive variational quantum eigensolver(VQE) called ADAPT-VQE. In contrast with non-adaptive VQE, this algorithm builds a system-specific ansatz by adding an optimal gate built from one-body or two-body fermionic operators at each step. We show this adaptive method outperforms the non-adaptive counterpart in terms of fewer variational parameters, short gate depth, and scaling with the system size. The fidelity and energy of the prepared state appear to improve asymptotically with ansatz depth. We also demonstrate the application of adaptive variational methods by preparing excited states and Green functions using a proposed ADAPT-SSVQE algorithm. Lower depth, asymptotic convergence, noise tolerance of a variational approach, and a highly controllable, system-specific ansatz make the adaptive variational methods particularly well-suited for NISQ devices.