Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDiscretized quantum adiabatic process for free fermions and comparison with the imaginary-time evolution
278
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Motivated by recent progress of quantum technologies, we study a discretized quantum adiabatic process for a one-dimensional free fermion system described by a variational wave function, i.e., a parametrized quantum circuit. The wave function is composed of $M$ layers of two elementary sets of time-evolution operators, each set being decomposed into commutable local operators. The evolution time of each time-evolution operator is treated as a variational parameter so as to minimize the expectation value of the energy. We show that the exact ground state is reached by applying the layers of time-evolution operators as many as a quarter of the system size. This is the minimum number $M_B$ of layers set by the limit of speed, i.e., the Lieb-Robinson bound, for propagating quantum entanglement via the local time-evolution operators. Quantities such as the energy $E$ and the entanglement entropy $S$ of the optimized variational wave function with $M < M_B$ are independent of the system size $L$ but fall into some universal functions of $M$. The development of the entanglement in these ansatz is further manifested in the progressive propagation of single-particle orbitals in the variational wave function. We also find that the optimized variational parameters show a systematic structure that provides the optimum scheduling function in the quantum adiabatic process. We also investigate the imaginary-time evolution of this variational wave function, where the causality relation is absent due to the non-unitarity of the imaginary-time evolution operators, thus the norm of the wave function being no longer conserved. We find that the convergence to the exact ground state is exponentially fast, despite that the system is at the critical point, suggesting that implementation of the non-unitary imaginary-time evolution in a quantum circuit is highly promising to further shallow the circuit depth.
rate research
Read More
We formulate an adiabatic approximation for the imaginary-time Schroedinger equation. The obtained adiabatic condition consists of two inequalities, one of which coincides with the conventional adiabatic condition for the real-time Schroedinger equation, but the other does not. We apply this adiabatic approximation to the analysis of Markovian dynamics of the classical Ising model, which can be formulated as the imaginary-time Schrodinger equation, to obtain an asymptotic formula for the probability that the system reaches the ground state in the limit of a long annealing time in simulated annealing. Using this form, we amend the theory of Somma, Batista, and Ortiz for a convergence condition for simulated annealing.
Tensor network methods are routinely used in approximating various equilibrium and non-equilibrium scenarios, with the algorithms requiring a small bond dimension at low enough time or inverse temperature. These approaches so far lacked a rigorous mathematical justification, since existing approximations to thermal states and time evolution demand a bond dimension growing with system size. To address this problem, we construct PEPOs that approximate, for all local observables, $i)$ their thermal expectation values and $ii)$ their Heisenberg time evolution. The bond dimension required does not depend on system size, but only on the temperature or time. We also show how these can be used to approximate thermal correlation functions and expectation values in quantum quenches.
The open-source library, irbasis, provides easy-to-use tools for two sets of orthogonal functions named intermediate representation (IR). The IR basis enables a compact representation of the Matsubara Greens function and efficient calculations of quantum models. The IR basis functions are defined as the solution of an integral equation whose analytical solution is not available for this moment. The library consists of a database of pre-computed high-precision numerical solutions and computational code for evaluating the functions from the database. This paper describes technical details and demonstrates how to use the library.
The phenomenon of localization usually happens due to the existence of disorder in a medium. Nevertheless, certain quantum systems allow dynamical localization solely due to the nature of internal interactions. We study a discrete time quantum walker which exhibits disorder free localization. The quantum walker moves on a one-dimensional lattice and interacts with on-site spins by coherently rotating them around a given axis at each step. Since the spins do not have dynamics of their own, the system poses the local spin components along the rotation axis as an extensive number of conserved moments. When the interaction is weak, the spread of the walker shows subdiffusive behaviour having downscaled ballistic tails in the evolving probability distribution at intermediate time scales. However, as the interaction gets stronger the walker gets exponentially localized in the complete absence of disorder in both lattice and initial state. Using a matrix-product-state ansatz, we investigate the relaxation and entanglement dynamics of the on-site spins due to their coupling with the quantum walker. Surprisingly, we find that even in the delocalized regime, entanglement growth and relaxation occur slowly unlike marjority of the other models displaying a localization transition.
A discretized version of the adiabatic theorem is described with the help of a rule relating a Hermitian operator to its expectation value and variance. The simple initial operator X with known ground state is transformed in a series of N small steps into a more complicated final operator Z with unknown ground state. Each operator along the discretised path in the space of Hermitian matrices is used to measure the state, initially the ground state of X. Measurements similar to the Zeno effect or Renningers negative measurements modify the state incrementally. This process eventually leads to an eigenstate combination of Z. In the limit of vanishing step size the state stays with overwhelming probability in the ground state of each of the N observables.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
