Left inset: The reachable time t * under the condition F ⩾ 1 − 10 − 4 as a function of N q. The blue markers show the results for l-USC, and the line shows the linear fit. The black crosses represent the standard iTEBD approach with the black dashed line showing an exponential fit. (b) The number of parameters required to reach time t * with the fidelity density at least F = 1 − 10 − 4.

Inset: The difference in the single-site density matrices between the quasi-exact iTEBD simulation and optimization of l-USC. (a) Expectation value of 〈 σ ̂ z ( t ) 〉 using l-USC with different values of N q. Simulations of time evolution using l-USC with M U = 1 and exact environment for the Hamiltonian in Eq. ( 8) with g / J = 1.0, h / J = 0 (upper row) and h / J = 1 with M U = 1 and 4 (lower row). With more efficient tensor contraction schemes, this algorithm may also offer improvements as a classical numerical algorithm. After benchmarking the evolution algorithm on a classical computer, we demonstrate the measurement of observables of this uniform state using a finite number of qubits on a cloud-based quantum processing unit. All steps of the hybrid optimization are designed with near-term digital quantum computers in mind.

Furthermore, this favorable scaling of the Ansatz is maintained during our variational evolution algorithm. We show numerically that this Ansatz requires a number of parameters polynomial in the simulation time for a given accuracy. This algorithm uses a layered uniform sequential quantum circuit as a variational Ansatz to represent infinite translation-invariant quantum states. In this work we present a polynomially scaling hybrid quantum-classical algorithm for time evolving a one-dimensional uniform system in the thermodynamic limit. If we know arr is sorted, binary search is a better choice.Simulating time evolution of generic quantum many-body systems using classical numerical approaches has an exponentially growing cost either with evolution time or with the system size. This variant can be written in numerous ways, including with a while loop.Īs with the other variant, this is not something that would generally be done in practice. If arr is sorted (in increasing/non-decreasing/smallest to largest order) and a value greater than key is found before key, key is not in arr. This variant stops when it finds a value in arr that is greater than key. With extra variable public static int search(int arr, int key) search(int, int) calls the helper method, with 0 as startIndex, and returns the result. We use recursive helper methods when the parameters for the recursive method differ from those we want for the public method. Search(int, int, int) is a private recursive helper method. Public static int search(int arr, int key) Iterative implementations int return public static int search(int arr, int key) If we get to the end of the array and we have not found the key, we return the value that indicates that the key was not found (typically -1 or false). If we find an element that matches the key, we stop and return the desired value. We check each element, starting with the first, against the given key. If nothing is known about the order of the elements in the array, we use a sequential/linear search. In the case of an element that appears in arr more than once (such as 86 in the example) some implementations explicitly return the position of the first occurrence while others may return the position of any occurrence. Return true if key is in arr otherwise, return false.Return the position of key in arr or -1 if key is not in arr.Sequential/linear search for the AP CS A Exam ExampleĬonsider the problem of finding a key in an array.

Algorithm for sequential search