Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPreparing exact eigenstates of the open XXZ chain on a quantum computer
101
0
0.0
(
0
)
Added by
Rafael I. Nepomechie
Publication date
2021
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The open spin-1/2 XXZ spin chain with diagonal boundary magnetic fields is the paradigmatic example of a quantum integrable model with open boundary conditions. We formulate a quantum algorithm for preparing Bethe states of this model, corresponding to real solutions of the Bethe equations. The algorithm is probabilistic, with a success probability that decreases with the number of down spins. For a Bethe state of $L$ spins with $M$ down spins, which contains a total of $binom{L}{M}, 2^{M}, M!$ terms, the algorithm requires $L+M^2+2M$ qubits.
rate research
Read More
Several quantum many-body models in one dimension possess exact solutions via the Bethe ansatz method, which has been highly successful for understanding their behavior. Nevertheless, there remain physical properties of such models for which analytic results are unavailable, and which are also not well-described by approximate numerical methods. Preparing Bethe ansatz eigenstates directly on a quantum computer would allow straightforward extraction of these quantities via measurement. We present a quantum algorithm for preparing Bethe ansatz eigenstates of the XXZ spin chain that correspond to real-valued solutions of the Bethe equations. The algorithm is polynomial in the number of T gates and circuit depth, with modest constant prefactors. Although the algorithm is probabilistic, with a success rate that decreases with increasing eigenstate energy, we employ amplitude amplification to boost the success probability. The resource requirements for our approach are lower than other state-of-the-art quantum simulation algorithms for small error-corrected devices, and thus may offer an alternative and computationally less-demanding demonstration of quantum advantage for physically relevant problems.
A key requirement to perform simulations of large quantum systems on near-term quantum hardware is the design of quantum algorithms with short circuit depth that finish within the available coherence time. A way to stay within the limits of coherence is to reduce the number of gates by implementing a gate set that matches the requirements of the specific algorithm of interest directly in hardware. Here, we show that exchange-type gates are a promising choice for simulating molecular eigenstates on near-term quantum devices since these gates preserve the number of excitations in the system. Complementing the theoretical work by Barkoutsos et al. [PRA 98, 022322 (2018)], we report on the experimental implementation of a variational algorithm on a superconducting qubit platform to compute the eigenstate energies of molecular hydrogen. We utilize a parametrically driven tunable coupler to realize exchange-type gates that are configurable in amplitude and phase on two fixed-frequency superconducting qubits. With gate fidelities around 95% we are able to compute the eigenstates within an accuracy of 50 mHartree on average, a limit set by the coherence time of the tunable coupler.
We obtain analytically close forms of benchmark quantum dynamics of the collapse and revival (CR), reduced density matrix, Von Neumann entropy, and fidelity for the XXZ central spin problem. These quantities characterize the quantum decoherence and entanglement of the system with few to many bath spins, and for a short to infinitely long time evolution. For the homogeneous central spin problem, the effective magnetic field $B$, coupling constant $A$ and longitudinal interaction $Delta$ significantly influence the time scales of the quantum dynamics of the central spin and the bath, providing a tunable resource for quantum metrology. Under the resonance condition $B=Delta=A$, the location of the $m$-th revival peak in time reaches a simple relation $t_{r} simeqfrac{pi N}{A} m$ for a large $N$. For $Delta =0$, $Nto infty$ and a small polarization in the initial spin coherent state, our analytical result for the CR recovers the known expression found in the Jaynes-Cummings model, thus building up an exact dynamical connection between the central spin problems and the light-matter interacting systems in quantum nonlinear optics. In addition, the CR dynamics is robust to a moderate inhomogeneity of the coupling amplitudes, while disappearing at strong inhomogeneity.
We construct the states that are invariant under the action of the generalized squeezing operator $exp{(z{a^{dagger k}}-z^*a^k)}$ for arbitrary positive integer $k$. The states are given explicitly in the number representation. We find that for a given value of $k$ there are $k$ such states. We show that the states behave as $n^{-k/4}$ when occupation number $ntoinfty$. This implies that for any $kgeq3$ the states are normalizable. For a given $k$, the expectation values of operators of the form $(a^{dagger} a)^j$ are finite for positive integer $j < (k/2-1)$ but diverge for integer $jgeq (k/2-1)$. For $k=3$ we also give an explicit form of these states in the momentum representation in terms of Bessel functions.
We present quantum algorithms, for Hamiltonians of linear combinations of local unitary operators, for Hamiltonian matrix-vector products and for preconditioning with the inverse of shifted reduced Hamiltonian operator that contributes to the diagonal matrix elements only. The algorithms implement a convergent series of approximations towards the exact solution of the full CI (configuration interaction) problem. The algorithm scales with O(m^5 ), with m the number of one-electron orbitals in the case of molecular electronic structure calculations. Full CI results can be obtained with a scaling of O(nm^5 ), with n the number of electrons and a prefactor on the order of 10 to 20. With low orders of Hamiltonian matrix-vector products, a whole repertoire of approximations widely used in modern electronic structure theory, including various orders of perturbation theory and/or truncated CI at different orders of excitations can be implemented for quantum computing for both routine and benchmark results at chemical accuracy. The lowest order matrix-vector product with preconditioning, basically the second-order perturbation theory, is expected to be a leading algorithm for demonstrating quantum supremacy for Ab Initio simulations, one of the most anticipated real world applications. The algorithm is also applicable for the hybrid variational quantum eigensolver.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
