Andrey E. Korolkov

Factoring integers is considered as a computationally-hard problem for classical methods, whereas there exists polynomial-time Shor's quantum algorithm for solving this task. However, requirements for running the Shor's algorithm for realistic tasks, which are beyond the capabilities of existing and upcoming generations of quantum computing devices, motivates to search for alternative approaches. In this work, we experimentally demonstrate factoring of the integer with a trapped ion quantum processor using the Schnorr approach and a modified version of quantum approximate optimization algorithm (QAOA). The key difference of our approach in comparison with the recently proposed QAOA-based factoring method is the use of the fixed-point feature, which relies on the use of universal parameters. We present experimental results on factoring $1591=37\times43$ using 6 qubits as well as simulation results for $74425657=9521\times7817$ with 10 qubits and $35183361263263=4194191\times8388593$ with 15 qubits. Alongside, we present all the necessary details for reproducing our results and analysis of the performance of the factoring method, the scalability of this approach both in classical and quantum domain still requires further studies.

An efficient implementation of the Toffoli gate is of conceptual importance for running various quantum algorithms, including Grover's search and Shor's integer factorization. However, direct implementation of the Toffoli gate either entails a prohibitive increase in the number of two-qubit gates or requires ancilla qubits, whereas both of these resources are limited in the current generation of noisy intermediate-scale quantum devices. Here, we experimentally demonstrate a scalable $N$-qubit Toffoli gate improvement using $^{171}$Yb$^{+}$ trapped-ion-based optical-metastable-ground encoded qutrits for the cases of up to $N$=10. With the use of the Molmer-Sorensen gate as a basic entangling operation, we compare the standard qubit decomposition with the qutrit approach, where upper levels are used as ancillas.The presented decomposition requires only global control of the ancilla levels, which simplifies experimental implementation of the proposed approach. Using the example of a three-qubit Grover's search, we also demonstrate an increase in the algorithm's accuracy by monitoring the leakage from the qubit subspace during a qutrit-based Toffoli gate implementation.

Scalable quantum computers hold the promise to solve hard computational problems, such as prime factorization, combinatorial optimization, simulation of many-body physics, and quantum chemistry. While being key to understanding many real-world phenomena, simulation of non-conservative quantum dynamics presents a challenge for unitary quantum computation. In this work, we focus on simulating non-unitary parity-time symmetric systems, which exhibit a distinctive symmetry-breaking phase transition as well as other unique features that have no counterpart in closed systems. We show that a qutrit, a three-level quantum system, is capable of realizing this non-equilibrium phase transition. By using two physical platforms -- an array of trapped ions and a superconducting transmon -- and by controlling their three energy levels in a digital manner, we experimentally simulate the parity-time symmetry-breaking phase transition. Our results indicate the potential advantage of multi-level (qudit) processors in simulating physical effects, where additional accessible levels can play the role of a controlled environment.