Abstract This chapter presents some quantum algorithms to simulate the dynamics of a quantum system, one of the most important applications of a quantum computer. We introduce the conditions for efficient simulation of Hamiltonian dynamics. We then present some Trotterization methods, in which the time evolution operator is approximated as a product of terms, each of which can be implemented by a quantum circuit at polynomial cost. We show high-order decompositions and illustrate the use of a Trotterization method to simulate sparse Hamiltonians. We discuss the Taylor series method and the qubitization method. The latter achieves optimal cost in simulation time and target accuracy. We conclude the chapter with a proof that simulating Hamiltonian dynamics is BQP-complete
Abstract This chapter presents some quantum algorithms to simulate the dynamics of a quantum system, one of the most important applications of a quantum computer. We introduce the conditions for efficient simulation of Hamiltonian dynamics. We then present some Trotterization methods, in which the time evolution operator is approximated as a product of terms, each of which can be implemented by a quantum circuit at polynomial cost. We show high-order decompositions and illustrate the use of a Trotterization method to simulate sparse Hamiltonians. We discuss the Taylor series method and the qubitization method. The latter achieves optimal cost in simulation time and target accuracy. We conclude the chapter with a proof that simulating Hamiltonian dynamics is BQP-complete
In this work we propose an approach for implementing time-evolution of a quantum system using product formulas. The quantum algorithms we develop have provably better scaling (in terms of gate complexity and circuit depth) than a naive application of well-known Trotter formulas, for systems where the evolution is determined by a Hamiltonian with different energy scales (i.e., one part is "large" and another part is "small"). Our algorithms generate a decomposition of the evolution operator into a product of simple unitaries that are directly implementable on a quantum computer. Although the theoretical scaling is suboptimal compared with state-of-the-art algorithms (e.g., quantum signal processing), the performance of the algorithms we propose is highly competitive in practice. We illustrate this via extensive numerical simulations for several models. For instance, in the strong-field regime of the 1D transverse-field Ising model, our algorithms achieve an improvement of one order of magnitude in both the system size and evolution time that can be simulated with a fixed budget of 1000 arbitrary 2-qubit gates, compared with standard Trotter formulas.
Abstract In this chapter, we present quantum algorithms to approximate Hamiltonian eigenstates. We start discussing the quantum phase estimation algorithms, an important subroutine that allows us to approximate eigenpairs of a Hamiltonian operator given an initial guess and a quantum circuit implementing the time evolution operator. We then present the adiabatic state preparation, a technique to transform the ground state of a Hamiltonian into the ground state of another Hamiltonian. We describe two important heuristic methods, the quantum approximate optimization algorithm, and the variational quantum eigensolver. The second part of the chapter discusses concrete applications in the field of quantum chemistry. After introducing the Born-Oppenheimer Hamiltonian in second quantization and the Hartree-Fock method, we illustrate how a chemical problem can be mapped onto a quantum computer. We conclude the chapter with the simulation of the hydrogen molecule in a minimal basis with the iterative phase estimation algorithm.
We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of physical systems. As in another recent algorithm, the cost of our method depends only logarithmically on the inverse of the desired precision, which is optimal. However, we simplify the algorithm and its analysis by using a method for implementing linear combinations of unitary operations together with a robust form of oblivious amplitude amplification.
Abstract We present a quantum algorithm for simulating the dynamics of a first-quantized Hamiltonian in real space based on the truncated Taylor series algorithm. We avoid the possibility of singularities by applying various cutoffs to the system and using a high-order finite difference approximation to the kinetic energy operator. We find that our algorithm can simulate η interacting particles using a number of calculations of the pairwise interactions that scales, for a fixed spatial grid spacing, as O ~ ( η 2 ) , versus the O ~ ( η 5 ) time required by previous methods (assuming the number of orbitals is proportional to η ), and scales super-polynomially better with the error tolerance than algorithms based on the Lie–Trotter–Suzuki product formula. Finally, we analyze discretization errors that arise from the spatial grid and show that under some circumstances these errors can remove the exponential speedups typically afforded by quantum simulation.
(Suzuki-)Trotter decompositions, splitting methods, (Lie) product formulae... The most common numerical methods for the time evolution of quantum systems come with many names. And they are used practically everywhere with applications ranging from the solution of classical equations of motion and various Monte Carlo simulations to the real and imaginary time evolution on classical as well as quantum computers. Here we review the state of the art of said methods, focussing especially on the progress made over the last few years. We highlight recently discovered efficient time evolution algorithms and explain how best to use them in practice. A central part of this work is the estimation of error bounds that has improved greatly within the past decade. The relevance of time evolution methods for quantum computing is discussed with a focus on noisy hardware. Finally, a comprehensive overview of generalisations, related methods and alternatives to Trotterization is provided. This includes time-dependent Hamiltonian dynamics, processed methods, multi-product formulae, symplectic integrators, TDVP for tensor networks, quantum signal processing, Crouch-Grossman methods and more. The overall perspective in this work is that of a theoretical physicist. All mathematical proofs as well as some technical details are omitted for easier readability. Instead, this review serves as a hands-on guide and, of course, as a starting point for references that provide further details.
In this thesis, we present three innovations that can be used to simulate quantum dynamics on both quantum and classical computers. Our primary innovation is an efficient quantum algorithm for simulating time-dependent Hamiltonian evolution of general input states on a quantum computer. Given conditions on the smoothness of the Hamiltonian, the complexity of the algorithm is close to linear in the evolution time, and therefore is comparable to algorithms for time-independent Hamiltonians. In addition, we show how the complexity can be reduced by optimizing the time steps. The complexity of the algorithm is quantified by calls to an oracle, which yields information about the Hamiltonian. In contrast to previous work, which allowed an oracle query to yield an arbitrary number of bits or qubits, we assign a cost for each bit or qubit accessed, revealing hitherto unnoticed simulation costs. We also account for discretization errors in the time and the representation of the Hamiltonian. This work is enabled by our development of a decomposition scheme based on Lie-Trotter- Suzuki product formulae for ordered operator exponentials. We show, by counterexample, that Lie-Trotter-Suzuki approximations may be lower order than expected when applied to problems that have singularities or discontinuous derivatives of appropriate order. We address this problem by presenting a set of criteria that is sufficient for the validity of these approximations, prove convergence and provide upper bounds for the approximation error. We also provide a new analysis of the evolution of quantum adiabatic systems in this thesis. Our work contains the tightest upper bounds that are currently known, and the first tight lower bounds, for the error in the adiabatic approximation applied to fixed Hamiltonians in the limit of slow evolution. We use our results to address the Marzlin- Sanders counterexample Hamiltonian. Specifically, we provide sufficiency conditions for the validity of the adiabatic approximation and demonstrate that the Marzlin-Sanders counterexample fails to satisfy them. We also find a new class of time-dependent Hamiltonians that are related to the Marzlin0Sanders counterexample Hamiltonian, but nonetheless obey the adiabatic approximation for sufficiently slow evolutions.