Quantum algorithms for fermionic simulations

Journal of Chemical Theory and Computation

The accurate computational determination of chemical, materials, biological, and atmospheric properties has critical impact on a wide range of health and environmental problems, but is deeply limited by the computational scaling of quantum-mechanical methods. The complexity of quantumchemical studies arises from the steep algebraic scaling of electron correlation methods, and the exponential scaling in studying nuclear dynamics and molecular flexibility. To date, efforts to apply quantum hardware to such quantum chemistry problems have focused primarily on electron correlation. Here, we provide a framework which allows for the solution of quantum chemical nuclear dynamics by mapping these to quantum spin-lattice simulators. Using the example case of a shortstrong hydrogen bonded system, we construct the Hamiltonian for the nuclear degrees of freedom on a single Born-Oppenheimer surface and show how it can be transformed to a generalized Ising model Hamiltonian. We then demonstrate a method to determine the local fields and spin-spin couplings needed to identically match the molecular and spin-lattice Hamiltonians. We describe a protocol to determine the on-site and inter-site coupling parameters of this Ising Hamiltonian from the Born-Oppenheimer potential and nuclear kinetic energy operator. Our approach represents a paradigm shift in the methods used to study quantum nuclear dynamics, opening the possibility to solve both electronic structure and nuclear dynamics problems using quantum computing systems.

Physical Review A, 2012

In principle a quantum system could be used to simulate another quantum system. The purpose of such a simulation would be to obtain information about problems which cannot be simulated with a classical computer due to the exponential increase of the Hilbert space with the size of the system and which cannot be measured or controlled in an actual experiment. The system will interact with the surrounding environment, with the other particles in the system and be implemented using imperfect controls making it subject to noise. It has been suggested that noise does not need to be controlled to the same extent as it must be for general quantum computing. However the effects of noise in quantum simulations and how to treat them are not completely understood. In this paper we study an existing quantum algorithm for the one-dimensional Fano-Anderson model to be simulated using a liquid-state NMR device. We calculate the evolution of different initial states in the original model, and then we add interacting spins to simulate a more realistic situation. We find that states which are entangled with their environment, and sometimes correlated but not necessarily entangled have an evolution which is described by maps which are not completely positive. We discuss the conditions for this to occur and also the implications.

Physical Review Letters, 2002

We propose a polynomial-time algorithm for simulation of the class of pairing Hamiltonians, e.g., the BCS Hamiltonian, on an NMR quantum computer. The algorithm adiabatically finds the lowlying spectrum in the vicinity of the gap between ground and first excited states, and provides a test of the applicability of the BCS Hamiltonian to mesoscopic superconducting systems, such as ultra-small metallic grains.

Quantum Information Processing, 2008

Only a few classes of quantum algorithms are known which provide a speed-up over classical algorithms. However, these and any new quantum algorithms provide important motivation for the development of quantum computers. In this article new quantum algorithms are given which are based on quantum state tomography. These include an algorithm for the calculation of several quantum mechanical expectation values and an algorithm for the determination of polynomial factors. These quantum algorithms are important in their own right. However, it is remarkable that these quantum algorithms are immune to a large class of errors. We describe these algorithms and provide conditions for immunity.

Physical Review A, 2019

The core concept of quantum simulation is the mapping of an inaccessible quantum system onto a controllable one by identifying analogous dynamics. We map the Dirac equation of relativistic quantum mechanics in 3+1 dimensions onto a multi-level superconducting Josephson circuit. Resonant drives determine the particle mass and momentum and the quantum state represents the internal spinor dynamics, which are cast in the language of multi-level quantum optics. The degeneracy of the Dirac spectrum corresponds to a degeneracy of bright/dark states within the system and particle spin and helicity are employed to interpret the multi-level dynamics. We simulate the Schwinger mechanism of electron-positron pair production by introducing an analogous electric field as a doubly degenerate Landau-Zener problem. All proposed measurements can be performed well within typical decoherence times. This work opens a new avenue for experimental study of the Dirac equation and provides a tool for control of complex dynamics in multi-level systems.

The European Physical Journal Special Topics, 2021

Quantum chemistry has been viewed as one of the potential early applications of quantum computing. Two techniques have been proposed for electronic structure calculations: (i) the variational quantum eigensolver and (ii) the phase-estimation algorithm. In both cases, the complexity of the problem increases for basis sets where either the Hamiltonian is not sparse, or it is sparse, but many orbitals are required to accurately describe the molecule of interest. In this work, we explore the possibility of mapping the molecular problem onto a sparse Hubbard-like Hamiltonian, which allows a Green's-function-based approach to electronic structure via a hybrid quantum-classical algorithm. We illustrate the time-evolution aspect of this methodology with a simple four-site hydrogen ring.

Physical Review Research, 2021

A central component of variational quantum algorithms (VQA) is the state-preparation circuit, also known as ansatz or variational form. This circuit is most commonly designed to respect the symmetries of the problem Hamiltonian and, in this way, constrain the variational search to a subspace of interest. Here, we show that this approach is not always advantageous by introducing ansätze that incorporate symmetry-breaking unitaries. This class of ansätze, that we call Quantum-Optimal-Control-inspired Ansätze (QOCA), is inspired by the theory of quantum optimal control and leads to an improved convergence of VQAs for some important problems. Indeed, we benchmark QOCA against popular ansätze applied to the Fermi-Hubbard model at half-filling and show that our variational circuits can approximate the ground state of this model with significantly higher accuracy and for larger systems. We also show how QOCA can be used to find the ground state of the water molecule and compare the performance of our ansatz against other common choices used for chemistry problems. This work constitutes a first step towards the development of a more general class of symmetry-breaking ansätze with applications to physics and chemistry problems.

Physical Review D, 2019

A general scheme is presented for simulating gauge theories, with matter fields, on a digital quantum computer. A Trotterized time-evolution operator that respects gauge symmetry is constructed, and a procedure for obtaining time-separated, gauge-invariant correlators is detailed. We demonstrate the procedure on small lattices, including the simulation of a 2+1D non-Abelian gauge theory.

International Journal of Quantum Chemistry, 2016

We present an efficient quantum algorithm for beyond-Born-Oppenheimer molecular energy computations. Our approach combines the quantum full configuration interaction method with the nuclear orbital plus molecular orbital (NOMO) method. We give the details of the algorithm and demonstrate its performance by classical simulations. Two isotopomers of the hydrogen molecule (H 2 , HT) were chosen as representative examples and calculations of the lowest rotationless vibrational transition energies were simulated.

Physical Review C

We use the Lipkin-Meshkov-Glick (LMG) model and the valence-space nuclear shell model to examine the likely performance of variational quantum eigensolvers in nuclear-structure theory. The LMG model exhibits both a phase transition and spontaneous symmetry breaking at the mean-field level in one of the phases, features that characterize collective dynamics in medium-mass and heavy nuclei. We show that with appropriate modifications, the ADAPT-VQE algorithm, a particularly flexible and accurate variational approach, is not troubled by these complications. We treat up to 12 particles and show that the number of quantum operations needed to approach the ground-state energy scales linearly with the number of qubits. We find similar scaling when the algorithm is applied to the nuclear shell model with realistic interactions in the sd and pf shells. Although most of these simulations contain no noise, we use a noise model from real IBM hardware to show that for the LMG model with four particles, weak noise has no effect on the efficiency of the algorithm.

Quantum Science and Technology, 2020

Relativistic fermionic field theories constitute the fundamental description of all observable matter. The simplest of the models provide a useful, classically verifiable benchmark for noisy intermediate scale quantum computers. We calculate the energy levels of the massive Thirring model-a model of Dirac fermions with four-fermion interactions-on a lattice in 1 + 1 space-time dimensions. We employ a hybrid classical-quantum computation scheme to obtain the mass gap in this model for three spatial sites. With error mitigation the results are in good agreement with exact classical calculations. Our calculations extend to the vicinity of the massless limit where chiral symmetry emerges, however relative errors for quantum computations in this regime are significant. We compare our results with an analytical calculation using perturbation theory.

Advances in Physics, 2004

We present an algebraic framework for interacting extended quantum systems to study complex phenomena characterized by the coexistence and competition of different states of matter. We start by showing how to connect different (spin-particle-gauge) languages by means of exact mappings (isomorphisms) that we name dictionaries and prove a fundamental theorem establishing when two arbitrary languages can be connected. These mappings serve to unravel symmetries which are hidden in one representation but become manifest in another. In addition, we establish a formal link between seemingly unrelated physical phenomena by changing the language of our model description. This link leads to the idea of universality or equivalence. Moreover, we introduce the novel concept of emergent symmetry as another symmetry guiding principle. By introducing the notion of hierarchical languages, we determine the quantum phase diagram of lattice models (previously unsolved) and unveil hidden order parameters to explore new states of matter. Hierarchical languages also constitute an essential tool to provide a unified description of phases which compete and coexist. Overall, our framework provides a simple and systematic methodology to predict and discover new kinds of orders. Another aspect exploited by the present formalism is the relation between condensed matter and lattice gauge theories through quantum link models. We conclude discussing applications of these dictionaries to the area of quantum information and computation with emphasis in building new models of computation and quantum programming languages.

npj Quantum Information, 2019

Modeling chemical reactions and complicated molecular systems has been proposed as the “killer application” of a future quantum computer. Accurate calculations of derivatives of molecular eigenenergies are essential toward this end, allowing for geometry optimization, transition state searches, predictions of the response to an applied electric or magnetic field, and molecular dynamics simulations. In this work, we survey methods to calculate energy derivatives, and present two new methods: one based on quantum phase estimation, the other on a low-order response approximation. We calculate asymptotic error bounds and approximate computational scalings for the methods presented. Implementing these methods, we perform geometry optimization on an experimental quantum processor, estimating the equilibrium bond length of the dihydrogen molecule to within $$0.014$$0.014 Å of the full configuration interaction value. Within the same experiment, we estimate the polarizability of the H$${}_{...

Physical Review B

Simulating the nonequilibrium dynamics of thermal states is a fundamental problem across scales from highenergy to condensed-matter physics. Quantum computers may provide a way to solve this problem efficiently. Preparing a thermal state on a quantum computer is challenging, but there exist methods to circumvent this by computing a weighted sum of time-dependent matrix elements in a convenient basis. While the number of basis states can be large, in this paper we show that it can be reduced by simulating only the largest density matrix elements by weight, capturing the density matrix to a specified precision. Leveraging Hamiltonian symmetries enables further reductions. This approach paves the way to more accurate thermal-state dynamics simulations on near-term quantum hardware.

npj Quantum Information

Modeling electronic systems is an important application for quantum computers. In the context of materials science, an important open problem is the computational description of chemical reactions on surfaces. In this work, we outline a workflow to model the adsorption and reaction of molecules on surfaces using quantum computing algorithms. We develop and compare two local embedding methods for the systematic determination of active spaces. These methods are automated and based on the physics of molecule-surface interactions and yield systematically improvable active spaces. Furthermore, to reduce the quantum resources required for the simulation of the selected active spaces using quantum algorithms, we introduce a technique for exact and automated circuit simplification. This technique is applicable to a broad class of quantum circuits and critical to enable demonstration on near-term quantum devices. We apply the proposed combination of active-space selection and circuit simplif...

Physical review, 2023

It has recently been argued that noisy intermediate-scale quantum computers may be used to optimize interpolating operator constructions for lattice quantum field theory (LQFT) calculations on classical computers. Here, two concrete realizations of the method are developed and implemented. The first approach is to maximize the overlap, or fidelity, of the state created by an interpolating operator acting on the vacuum state to the target eigenstate. The second is to instead minimize the energy expectation value of the interpolated state. These approaches are implemented in a proof-of-concept calculation in (1 þ 1)dimensions for a single-flavor massive Schwinger model to obtain quantum-optimized interpolating operator constructions for a vector meson state in the theory. Although fidelity maximization is preferable in the absence of noise due to quantum gate errors, it is found that energy minimization is more robust to these effects in the proof-of-concept calculation. This work serves as a concrete demonstration of how quantum computers in the intermediate term might be used to accelerate classical LQFT calculations.

Physical Review Letters, 2020

The only known way to study quantum field theories in non-perturbative regimes is using numerical calculations regulated on discrete space-time lattices. Such computations, however, are often faced with exponential signal-to-noise challenges that render key physics studies untenable even with next generation classical computing. Here, a method is presented by which the output of smallscale quantum computations on Noisy Intermediate-Scale Quantum era hardware can be used to accelerate larger-scale classical field theory calculations through the construction of optimized interpolating operators. The method is implemented and studied in the context of the 1+1-dimensional Schwinger model, a simple field theory which shares key features with the standard model of nuclear and particle physics.

Physics Reports, 2022

The variational quantum eigensolver (or VQE), first developed by Peruzzo et al. (Nature Comm.5, 4213 (2014)), has received significant attention from the research community in recent years. It uses the variational principle to compute the ground state energy of a Hamiltonian, a problem that is central to quantum chemistry and condensed matter physics. Conventional

Physical Review A, 2008

We consider the simulation of interacting high-dimensional systems using pairwise interacting qubits. The main tool in this context is the generation of effective many-body interactions, and we examine a number of different protocols for obtaining them. These methods include the usage of higher-order processes (commutator method), unitary conjugation or graph state encoding, as well as teleportation based approaches. We illustrate and compare these methods in detail and analyze the time cost for simulation. In the second part of the article, we investigate the influence of noise on the simulation process. We concentrate on errors in the interaction Hamiltonians and consider two generic noise models, (i) timing errors in pairwise interactions and (ii) noisy pairwise interactions described by Master equations of Lindblad form. We analyze and compare the effect of noise for the different simulation methods and propose a way to significantly reduce the influence of noise by making use of entanglement purification together with a teleportation based protocol.

Physical Review Letters, 2021

Variational quantum eigensolvers (VQEs) combine classical optimization with efficient cost function evaluations on quantum computers. We propose a new approach to VQEs using the principles of measurement-based quantum computation. This strategy uses entangled resource states and local measurements. We present two measurement-based VQE schemes. The first introduces a new approach for constructing variational families. The second provides a translation of circuitbased to measurement-based schemes. Both schemes offer problem-specific advantages in terms of the required resources and coherence times. VQEs with wide-ranging applications [14, 20-23, 26, 42]. The underlying idea is to use a tailored entangled state called 'custom state', that allows for exploring an appropriate corner of the system's Hilbert space [Fig. ]. This custom state includes auxiliary qubits which, once measured, modify the state of the output qubits. The choices of the measurement bases, and the corresponding variational changes to the state, are controlled by a classical optimization algorithm. This approach differs conceptually and practically from standard VQE schemes since MB-VQE * These authors contributed equally.