Ab initio quantum chemistry simulations are crucially useful for the accurate prediction andunderstanding of chemical phenomena. The quality of such simulations rely heavily on precise solutions to the electronic structure problem. To facilitate the approximate yet efficient solving of the electronic structure problem, many methods rely on a set of problem assumptions, such as the validity of truncations of the fermionic excitation hierarchy, or active space selections. In this work, we describe black-box algorithms for the electronic structure problem, namely the iterative qubit coupled cluster (iQCC) algorithm along with a few of its variants and extensions. The iQCC method is a variational adaptive technique, where an iteratively updated effective Hamiltonian directly guides the selection of transformation generators of the next iteration. We describe the algorithm in detail, and introduce variations of the algorithm to alleviate the proliferation of terms in the effective Hamiltonians. Finally, we describe how the algorithm can be modified to perform simultaneuous determination of ground and excited states. These methods are amenable to implementation using near-term quantum computing devices, and with certain restrictions on meta-parameters, can be implemented as fully classical “quantum-inspired” algorithms. ii
Ab initio quantum chemistry simulations are crucially useful for the accurate prediction andunderstanding of chemical phenomena. The quality of such simulations rely heavily on precise solutions to the electronic structure problem. To facilitate the approximate yet efficient solving of the electronic structure problem, many methods rely on a set of problem assumptions, such as the validity of truncations of the fermionic excitation hierarchy, or active space selections. In this work, we describe black-box algorithms for the electronic structure problem, namely the iterative qubit coupled cluster (iQCC) algorithm along with a few of its variants and extensions. The iQCC method is a variational adaptive technique, where an iteratively updated effective Hamiltonian directly guides the selection of transformation generators of the next iteration. We describe the algorithm in detail, and introduce variations of the algorithm to alleviate the proliferation of terms in the effective Hamiltonians. Finally, we describe how the algorithm can be modified to perform simultaneuous determination of ground and excited states. These methods are amenable to implementation using near-term quantum computing devices, and with certain restrictions on meta-parameters, can be implemented as fully classical “quantum-inspired” algorithms. ii
We introduce the multistate iterative qubit coupled cluster (MS-iQCC) method, a quantum-inspired algorithm that runs efficiently on classical hardware and is designed to predict both ground and excited electronic states of molecules. Accurate excited-state energetics are essential for interpreting spectroscopy and chemical reactivity, but standard electronic structure methods are either too computationally expensive for larger systems or lose reliability in the presence of strong electron correlation. MS-iQCC addresses this challenge by simultaneously optimizing multiple electronic states in a single, state-averaged procedure that treats ground and excited states on equal footing. This removes the energetic bias that is introduced when excited states are computed one at a time and constrained to remain orthogonal to previously optimized states. The approach supports multireference electronic structure by allowing multideterminantal initial guesses and by adaptively building a compact exponential ansatz from a pool of qubit excitation generators. We apply MS-iQCC to H4, H2O, N2, and C2, including strongly correlated geometries, and observe robust convergence of all targeted state energies to chemically meaningful accuracy across their potential energy surfaces.
Abstract The iterative qubit coupled cluster (iQCC) method is a systematic variational approach to solve the electronic structure problem on universal quantum computers. It is able to use arbitrarily shallow quantum circuits at expense of iterative canonical transformation of the Hamiltonian and rebuilding a circuit. Here we present a variety of a posteriori corrections to the iQCC energies to reduce the number of iterations to achieve the desired accuracy. Our energy corrections are based on a low-order perturbation theory series that can be efficiently evaluated on a classical computer. Moreover, capturing a part of the total energy perturbatively, allows us to formulate the qubit active-space concept, in which only a subset of all qubits is treated variationally. As a result, further reduction of quantum resource requirements is achieved. We demonstrate the utility and efficiency of our approach numerically on the examples of 10-qubit N 2 molecule dissociation, the 24-qubit H 2 O symmetric stretch, and 56-qubit singlet-triplet gap calculations for the technologically important complex, tris-(2-phenylpyridine)iridium(III) Ir(ppy) 3 .
Noisy intermediate-scale quantum (NISQ) devices is the current realm of quantum computing. To exploit their limited powers to full extent it is imperative to develop suitable algorithms. We present the iterative qubit coupled cluster (iQCC) method aimed to perform realistic quantum chemistry calculations on such devices. The iQCC method can use as few as a single generator of entanglement (a single Pauli word) yet is able to reach the full configurational interaction (FCI) limit. The iQCC method is also endowed with a few techniques that allow for relatively large problems to be addressed. We show the capacity of our method on the electronic structure calculations for the LiH (4 qubits), H2O (6 qubits) and N2 (14 qubits) molecules
Immense interest in quantum computing has prompted the development of electronic structure methods that are suitable for quantum hardware. However, the slow pace at which quantum hardware progresses forces researchers to implement their ideas on classical computers despite the obvious loss of any “quantum advantage.” As a result, the so-called q uantum-inspired methods emerge. They allow one to look at the electronic structure problem from a different angle, yet to fully exploit their capacity, efficient implementations are highly desirable. Here, we report two schemes for improving the amplitude optimization in the iterative qubit coupled cluster (iQCC) method: a variational quantum eigensolver-type approach, which is based on the qubit coupled cluster (QCC) Ansatz. Our first scheme approximates the QCC unitary as a sum of symmetrical polynomials of generators up to a given order. The resulting energy expression allows for flexible control of computational complexity via the order parameter. It also guarantees smoothness of trial energies and their derivatives, which is important for gradient-based optimization strategies. The second scheme limits the size of the expansion space in which the QCC unitary is generated. It provides better control of memory requirements but, in general, may lead to the nonsmooth variation of energy estimates upon changes in amplitudes. It can be used, however, to extrapolate energies for a given set of amplitudes toward the exact QCC value. Both schemes allow for a larger number of generators to be included in the QCC form compared to the exact formulation. This reduces the number of iterations in the iQCC method and/or leads to higher accuracy. We assess the capabilities of the new schemes to perform QCC amplitudes optimization for a few molecular systems: dinitrogen (N 2, 16 qubits), water (H 2 O, 36 qubits), and tris(2-(2,4-difluorophenyl)pyridine) iridium(III), (Ir(F 2 ppy) 3, 80 qubits).
In ab-initio quantum chemistry, solving the electronic structure problem of a molecular system of interest to obtain an approximation of the ground and, in some cases, the excited state energies is a core and central task in various quantum chemistry processes. These include, but certainly not limited to, the calculation of the activation energy–that is the minimal energy required to launch a given chemical reaction–, the dipole and multipole moments which describe the polarity of the molecule, performing the optimization of the geometry of the molecule and searching for transition states. Furthermore, many quantum chemistry applications are predicated, not merely on the approximation of the ground state energy, but also on the corresponding first order derivatives. The body of literature of classical algorithms approximating the ground state energy and its first order derivative is extensively rich with a plethora of methods such as self-consistent mean-field methods, coupled-cluster methods, density matrix renormalization group and many more, each of which has intrinsic set of properties and characteristics that makes it suitable for specific tasks and cases. However, classical methods face a practically insurmountable challenge: the efficient computation of high accuracy approximation of ground state energies of large system sizes. This is fundamentally because the electronic structure problem provably belongs to the complexity class of QMA-hard problems. Which also means that it is out of reach for quantum algorithms as well in general. However, with certain assumptions that are expected to hold molecules of interest–that so far proved to be classically challenging because the required accuracy set by the standard of chemical accuracy threshold (generally held in quantum chemistry)– quantum algorithms can efficiently treat such cases. While originally theorized by Richard Feynmann as a potentially efficient way of simulating some classically intractable many-body systems, quantum computing has since proven to have a stronger and richer potential paving the way for the inception of new algorithms such as Shor’s algorithm for integer factorization, HHL for solving linear system of equation and Grover’s algorithm for constrained binary search, just to name a few. These algorithms provably exhibit an improvement rate that goes respectively exponentially and quadratically in the time complexity. Quantum chemistry is considered a prominent candidate to benefit from quantum computing given the variety of quantum algorithms put forth to solve the electronic structure problem of particular systems of interest. These quantum algorithm proposals can be broadly categorized in three main categories that span the spectrum: from noisy intermediate scale quantum (NISQ) computers to early and full scale fault tolerant quantum ((E)FTQC) computers. Firstly, the variational quantum eigensolver (VQE) family of quantum algorithms which variationally search for the ground state energy through the optimization of the energy of a parametrized state prepared in a quantum computer using an ansatz. Secondly, the quantum Krylov subspace diagonalization (QKSD) methods, which generate a subspace of the original Hilbert space and efficiently produce the lowest energy in said subspace as its output. Lastly, we have the quantum phase estimation (QPE) algorithms, which provably produce the ground state energy in Heisenberg scaling presenting hence a provable efficient solution. In this thesis, we firstly set to work on the problem of efficient Hamiltonian linear combination of unitaries (LCU) representation. This is mainly motivated by the wide-range impact an efficient compression of the Hamiltonian can potentially have on several quantum algorithms such as VQE, QPE...etc. We designed a method that efficiently compress the Hamiltonian representation in a way that is beneficially impactful across the set of the aforementioned quantum algorithms. Namely, we motivate and present in Sections 1.3 and 1.4 respectively, the regularized compressed double-factorization (RC-DF) and show how it reduces the energy measurement complexity both in terms of number of distinct observables and number of samples. Moreover, it also reduces the total depth of the quantum phase estimation circuit thanks to its low normalization factor. Secondly, given the importance of the energy derivatives, in Sections 2.1 and 2.2, we introduce and present an efficient quantum algorithm for the calculation of the analytic gradient of VQE energies of double-factorized Hamiltonians. Furthermore, in Section 2.1, we expand on the utility of RC-DF by presenting a quantum algorithm that determines the analytic gradient of energies measured with respect to RC-DF Hamiltonains. Third, in Section 3.1, we summarize the theoretical results of purification error-mitigation techniques, namely echo verification and virtual distillation. In Section 3.2, we report on the results of the experimental bench mark, of said techniques, performed on on Google’s Sycamore device to validate the theoretical scaling of error suppression and extrapolate the resources needed for classically intractable system sizes. Fourth, in Section 4 we establish the challenges of computing energy gradients in a quantum Krylov frame work. To overcome the problem, we provide, in Section 4.3, an efficient quantum algorithm for calculating energy derivative obtained through QKSD procedures. Our approach shows how the QKSD ground state can be prepared using quantum signal processing (QSP) and reduces the number of observables to O(1) per reduced density matrix (RDM). Lastly, in light of the importance of quantum RAMs due to their utility in a variety of quantum algorithms that require efficient initial state preparation, as explained in Section 5.5, we address the problematic exponential depth of the bucket-brigade QRAM (when decomposed). We present in Section 5.6 a very efficient Clifford+T decomposition of the bucket-brigade QRAM, which preserves the linear scaling of the depth of the circuit without introducing any additional ancilla.
With quantum computers on the horizon, it is essential to understand what problems they can be used to solve. One potential application is simulating quantum systems, in particular, systems with many interacting electrons. However, there is a disparity between the computational resources required to study these problems and the capabilities of modern experimental quantum devices. In this thesis, we make progress toward closing this gap by reducing the cost of quantum simulation of quantum chemistry on a number of fronts. In the first half, we develop three improved quantum simulation algorithms by approximating time evolution under the Hamiltonian using a truncated Taylor series rather than the Trotter-Suzuki decomposition. We do this first for the molecular electronic structure Hamiltonian in two representations, obtaining asymptotic improvements in terms of both the number of orbitals and the required precision. We also apply the truncated Taylor series paradigm to quantum simulation in real space, where we consider simulation of general wave functions discretized on a uniform grid. In the second half, by working with Hamiltonians tailored specifically to quantum algorithms, we demonstrate that Trotter steps of the electronic structure Hamiltonian can be simulated in exactly N depth and with only linear connectivity, where N is the number of spin-orbitals. This resolves the problem of non-locality in fermion-to-qubit mappings, and has wide-ranging applications for variational and phase estimation-based quantum simulation of quantum chemistry. We further study this algorithm on an error-corrected quantum computer using the surface code. Benchmarking it alongside other product formula approaches, we show that quantum simulation of scientifically important instances of the electronic structure problem that are classically intractable can be performed with only a few hundred thousand physical qubits and fewer than one billion costly fault-tolerant gates. Finally, we study the effect of using randomized Hamiltonians in iterative phase estimation, and how this allows us to reduce the costs of simulation as well as to exploit knowledge of the ground state in how we design our simulation algorithm.