Abstract Gate-level quantum circuits are often derived manually from higher level algorithms. While this suffices for small implementations and demonstrations, ultimately automatic circuit design will be required to realise complex algorithms using hardware-specific operations and connectivity. Therefore, ab initio creation of circuits within a machine, either a classical computer or a hybrid quantum–classical device, is of key importance. We explore a range of established and novel techniques for the synthesis of new circuit structures, the optimisation of parameterised circuits, and the efficient removal of low-value gates via the quantum geometric tensor. Using these techniques we tackle the tasks of automatic encoding of unitary processes and translation (recompilation) of a circuit from one form to another. Using emulated quantum computers with various noise-free gate sets we provide simple examples involving up to 10 qubits, corresponding to 20 qubits in the augmented space we use. Further applications of specific relevance to chemistry modelling are considered in a sister paper, ‘Exploiting subspace constraints and ab initio variational methods for quantum chemistry’. The emulation environments used were QuEST, QuESTlink and pyQuEST. All resources will be made openly accessible and are currently available upon request.
Dynamical simulation of quantum many-body systems is a central task in quantum chemistry and requires efficient wavefunction representations. Although tensor-network and neural-network quantum states have achieved considerable success in ground-state calculations, entanglement growth hinders their application to quantum dynamics. Quantum computing may offer a route to quantum advantage, but algorithms such as Trotterization generally require fault-tolerant quantum computers, whereas near-term hardware supports only limited circuit sizes. Variational quantum simulation (VQS) instead represents the evolving state with a single shallow, fixed-size parameterized quantum circuit (PQC). Here, we introduce nonorthogonal variational quantum simulation (NOVQS), which applies linear combinations of parameterized quantum states to real- and imaginary-time evolutions. We design a shallow, hardware-friendly ansatz tailored to second-quantized electronic-structure Hamiltonians, together with resource-efficient protocols for measuring the matrices and vectors in the parameter equations of motion. Error analysis and resource estimation are also provided. Numerical simulations of hydrogen chains and the nitrogen molecule demonstrate that a collection of shallow, or even single-layer, PQCs can match or outperform a much deeper PQC in VQS. In particular, we identify a trade-off between circuit number and depth. NOVQS therefore provides a flexible route to enhancing wavefunction expressivity under circuit-depth constraints, making it a promising framework for quantum dynamics simulations on near-term quantum processors.
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.
In the noisy near-term quantum computing era variational quantum algorithms are promising to explore the boundaries of quantum advantage. One prevalent instance is the variational quantum eigensolver (VQE) employed in quantum chemistry to approximate ground-state energies of molecules. The performance of VQE depends on the design of the parametrized quantum circuit, as well as the optimization algorithm used to optimize those parameters. Circuit optimization methods have been developed to generate quantum circuits to reach accurate approximations to the ground-states. In this work, we conduct an empirical comparison between two prominent adaptive quantum circuit optimization algorithms, namely qubit-ADAPT-VQE and a reinforcement learning (RL) based method, on the problem of finding the ground state energy of Lithium Hydride (LiH). In our experiment, we focus on the circuit depth generated by each circuit optimization method and the number of two-qubit quantum gates in the best-found circuit.
Abstract Hybrid quantum-classical approaches offer potential solutions to quantum chemistry problems, yet they often manifest as constrained optimization problems. Here, we explore the interconnection between constrained optimization and generalized eigenvalue problems through the Unitary Coupled Cluster (UCC) excitation generators. Inspired by the generator coordinate method, we employ these UCC excitation generators to construct non-orthogonal, overcomplete many-body bases, projecting the system Hamiltonian into an effective Hamiltonian, which bypasses issues such as barren plateaus that heuristic numerical minimizers often encountered in standard variational quantum eigensolver (VQE). Diverging from conventional quantum subspace expansion methods, we introduce an adaptive scheme that robustly constructs the many-body basis sets from a pool of the UCC excitation generators. This scheme supports the development of a hierarchical ADAPT quantum-classical strategy, enabling a balanced interplay between subspace expansion and ansatz optimization to address complex, strongly correlated quantum chemical systems cost-effectively, setting the stage for more advanced quantum simulations in chemistry.
We present a self-consistent field (SCF) approach within the adaptive derivative-assembled problem-tailored ansatz variational quantum eigensolver (ADAPT-VQE) framework for efficient quantum simulations of chemical systems on near-term quantum computers. To this end, our ADAPT-VQE-SCF approach combines the idea of generating an ansatz with a small number of parameters, resulting in shallow-depth quantum circuits with a direct minimization of an energy expression that is correct to second order with respect to changes in the molecular orbital basis. Our numerical analysis, including calculations for the transition-metal complex ferrocene [Fe (C 5 H 5 ) 2 ], indicates that convergence in the self-consistent orbital optimization loop can be reached without a considerable increase in the number of two-qubit gates in the quantum circuit by comparison to a VQE optimization in the initial molecular orbital basis. Moreover, the orbital optimization can be carried out simultaneously within each iteration of the ADAPT-VQE cycle. ADAPT-VQE-SCF thus allows us to implement a routine analogous to the complete active space SCF, a cornerstone of state-of-the-art computational chemistry, in a hardware-efficient manner on near-term quantum computers. Hence, ADAPT-VQE-SCF paves the way toward a paradigm shift for quantitative quantum-chemistry simulations on quantum computers by requiring fewer qubits and opening up for the use of large and flexible atomic orbital basis sets in contrast to earlier methods that are predominantly based on the idea of full active spaces with minimal basis sets.