The transverse-field Ising model has attracted a lot of attention in recent years, especially in the quantum simulation and quantum computation literature. This interest is driven by many platforms for analog quantum computation, which implement the transverse-field Ising model for solving optimization problems, such as quantum annealing. However, it has remained an open question whether the Ising model with a global transverse field is equivalent to the gate model of quantum computation. Here we answer this question affirmatively for the case of a non-monotonic time-dependent transverse field. Building on a recent result by Cesa and Pichler on global control of Rydberg atoms, we provide a construction that allows simulating arbitrary quantum circuits using the Ising model with global transverse field with polynomial overhead in time, qubit number, and energy scale. Although the polynomial overheads we establish here are large relative to what is feasible on real-world quantum hardware, our result motivates the development of more sophisticated methods for simulating quantum circuits using the Ising model with a global transverse field. Additionally, under the assumption that quantum computing is strictly more powerful than classical computing, our result serves as a no-go theorem for efficient classical simulation of the transverse-field Ising model with a time-dependent global transverse field. Therefore, our finding is relevant for multiple communities, from analog quantum simulation and quantum optimization on various platforms to complexity and control theory.
We study the transverse field Ising model, defined by the Hamiltonian
H=21∑i,j∈[n]JijZiZj+∑i=1nhizZi+η∑iXi where $J $ is the symmetric interaction matrix, and
η is the transverse field strength. Let
Δ(J)=λmax(J)−λmin(J) be the spectral width of
J. When the inverse temperature
β≥0 satisfies
Δ(J)⋅ηtanh(βη)≤1, we give a randomized classical algorithm that approximates the partition function
Z(β)=Tr(e−βH) to a given relative error
ε∈(0,1) in time polynomial in
n,
β, the model parameters, and
ε−1. When $ Δ(J) \cdot \frac{\tanh(βη)}η > 1 ,$ we show that approximating $ Z(β)$ within an
exp(o(n))-multiplicative factor is
NP-hard, and thus unlikely to admit an efficient classical or quantum algorithms under standard complexity theoretic assumptions. Furthermore, in the regime
Δ(J)⋅ηtanh(βη)≤1, we provide an efficient randomized classical algorithm that approximates Pauli string observables of the Gibbs state $ ρ_β= \frac{e^{-βH}}{\operatorname{Tr}(e^{-βH})}$ within an arbitrarily small additive error. In the special case when the observable is also diagonal in the
X-basis, i.e.
P∈{I,X}⊗n, the algorithm further achieves arbitrarily small relative error.
We present an exact simulation of a one-dimensional transverse Ising spin chain with a quantum computer. We construct an efficient quantum circuit that diagonalizes the Ising Hamiltonian and allows to obtain all eigenstates of the model by just preparing the computational basis states. With an explicit example of that circuit for n = 4 spins, we compute the expected value of the ground state transverse magnetization, the time evolution simulation and provide a method to also simulate thermal evolution. All circuits are run in IBM and Rigetti quantum devices to test and compare them qualitatively.
The field of quantum computing has grown fast in recent years, both in theoretical advancements and the practical construction of quantum computers. These computers were initially proposed, among other reasons, to efficiently simulate and comprehend the complexities of quantum physics. In this paper, we present a comprehensive scheme for the exact simulation of the 1-D XY model on a quantum computer. We successfully diagonalize the proposed Hamiltonian, enabling access to the complete energy spectrum. Furthermore, we propose a novel approach to design a quantum circuit to perform exact time evolution. Among all the possibilities this opens, we compute the ground and excited state energies for the symmetric XY model with spin chains of n=4 n = 4 and n=8 n = 8 spins. Further, we calculate the expected value of transverse magnetization for the ground state in the transverse Ising model. Both studies allow the observation of a quantum phase transition from an antiferromagnetic to a paramagnetic state. Additionally, we have simulated the time evolution of the state all spins up in the transverse Ising model. The scalability and high performance of our algorithm make it an ideal candidate for benchmarking purposes, while also laying the foundation for simulating other integrable models on quantum computers.
The classical simulation of many body quantum systems has long been of interest to both condensed matter and quantum information theorists. The use of matrix product state representations of the wavefunction allows suitable approximations to be made that permit quantum simulations to be performed in polynomial time rather than exponential in the size of the system. The numerical methods are based on a minimization algorithm that can be used to nd the ground state and rst excited state energies and their respective matrix product states. Using the matrix product states, other quantities such as magnetization and correlation can also be computed. In this thesis, the theory and numerical techniques necessary for these simulations are studied and applied to the one dimesional Ising model in a transverse eld. The results of the model are compared to the analytical solution and analyzed in depth.
We study classical polynomial-time approximation algorithms for the transverse-field Ising model (TFIM) Hamiltonian, allowing a mixture of ferromagnetic and anti-ferromagnetic interactions between pairs of qbits, alongside transverse field terms with arbitrary non-negative weights.
Our main results are a series of approximation algorithms (all approximation ratios with respect to the true quantum optimum): (i) a simple maximum of two product state rounding algorithm achieving an approximation ratio
γ≈0.71 , (ii) a strengthened rounding, inspired by the anticommutation property of the two
Xi,ZiZj observables achieving ratio
γ≈0.7860, and (iii) a further improvement by interpolation achieving ratio
γ≈0.8156. We also give an explicit (purely ferromagnetic) TFIM instance on three qbits for which every product state achieves at most
169/180≈0.9389 of the true optimum, yielding an upper bound for all algorithms producing product state approximations, even in the purely ferromagnetic case.