Improved upper bounds on the stabilizer rank of magic states

In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of m copies of the magic state |T⟩=2−1(|0⟩+eiπ/4|1⟩) in the limit of large m. In particular, we show that |T⟩⊗m can be exactly expressed as a superposition of at most O(2αm) stabilizer states, where α≤0.3963, improving on the best previously known bound α≤0.463. This furnishes, via known techniques, a classical algorithm which approximates output probabilities of an n-qubit Clifford + T circuit U with m uses of the T gate to within a given inverse polynomial relative error using a runtime poly(n,m)2αm. We also provide improved upper bounds on the stabilizer rank of symmetric product states |ψ⟩⊗m more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford gates and m instances of any (fixed) single-qubit Z-rotation gate with runtime poly(n,m)2m/2. We suggest a method to further improve the upper bounds by constructing linear codes with certain properties.

A policy iteration method for Mean Field Games

The policy iteration method is a classical algorithm for solving optimal control problems. In this paper, we introduce a policy iteration method for Mean Field Games systems, and we study the convergence of this procedure to a solution of the problem. We also introduce suitable discretizations to numerically solve both stationary and evolutive problems. We show the convergence of the policy iteration method for the discrete problem and we study the performance of the proposed algorithm on some examples in dimension one and two.

Comparison between classical ‘P&O’ algorithm and FLC of MPPT for GPV under partial shading

<p><span lang="EN-US">When the GPV is under partial shading, several peaks appear in the characteristic P-V, namely a GMP and one or more local maximums. The classical algorithm ‘P&amp;O’ MPPT cannot converge on the GMP for low irradiation values and is trapped by tracking down a LMP so making the algorithm ineffective making the algorithm ineffective, in this case under 200 W/m². An alternative objective function is developed to optimize the performance of the FLC by selecting the appropriate gains using PSO. In this simulation the GPV is composed of one hundred modules grouped parallel series (10x10) and subjected to partial shading. The proposed FLC provides better performance for GMP tracking for the chosen shade configuration selected.</span></p>

On completely factoring any integer efficiently in a single run of an order-finding algorithm

We show that given the order of a single element selected uniformly at random from $${\mathbb {Z}}_N^*$$ Z N ∗ , we can with very high probability, and for any integer N, efficiently find the complete factorization of N in polynomial time. This implies that a single run of the quantum part of Shor’s factoring algorithm is usually sufficient. All prime factors of N can then be recovered with negligible computational cost in a classical post-processing step. The classical algorithm required for this step is essentially due to Miller.

Implementation of Shor’s Algorithm in QISKIT by Quantum Computing using IBMQ

Quantum machine learning is the combination of quantum computing and classical machine learning. It helps in solving the problems of one field to another field. Shor’s algorithm is used for factoring the integers in polynomial time. Since the bestknown classical algorithm requires super polynomial time to factor the product of two primes, the widely used cryptosystem, RSA, relies on factoring being impossible for large enough integers. In this paper we will focus on the quantum part of Shor’s algorithm, which actually solves the problem of period finding. In polynomial time factoring problem can be turned into a period finding problem so an efficient period finding algorithm can be used to factor integers efficiently.

Optimal algorithm for distinguishing radio navigation signals by a set of “spaced” correlators

The classical algorithm for signal distinction, signal detecting and estimating signal parameters consists in analyzing discrete parameter values using a correlator. The value of the parameter with the maximum absolute value of the correlator is taken as an estimate. Obviously, this is accompanied by losses in sensitivity and noise immunity, since the specified discrete parameter values do not accurately correspond to the true parameter values of the real signal. In this case, the accuracy of the parameter estimation, even at large signal-to-noise ratios, is limited by the value of the correlators placement interval. Therefore, it is of interest to optimally use the entire set of correlators for parameter estimation and signal detection. The article presents the derivation of algorithm for distinguishing signals by a given parameter by a set of "spaced" correlators. Unlike the classical algorithm, it uses decisive statistics not by one, but by a pair of neighboring correlators, detuned by the correlation interval. In this case, at first, the number of the interval between correlators is estimated according to the maximum of the decisive statistics, and then the value of the parameter is refined within this interval. Additionally, the algorithm allows you to estimate the signal amplitude. The proposed algorithm is compared with the classical one. By means of simulation, the dependences on the energy potential of the average probability of signal distinction for both algorithms are plotted. It is shown that the proposed algorithm has a higher probability of correct distinction than the classical algorithm. It is also shown that the maximum and average energy losses of the distinction algorithm based on a set of "spaced" correlators are less than the losses of the classical algorithm. Thus, the proposed algorithm for distinction signals by a set of "spaced" correlators has greater noise immunity and accuracy of estimating the desired parameter than the classical distinction algorithm.

A systematic variational approach to band theory in a quantum computer

A quantum algorithm to calculate the band structure of any crystal, with efficiency comparable to the classical algorithm.