# Quasi-Shor Algorithms for Global Benchmarking of Universal Quantum Processors
## 1 Introduction  
Building universal quantum computers is a goal in the field of quantum computing [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. Realizing various algorithms, such as the [PERSON] algorithm [12; 13], [PERSON] algorithm [14], Harrow-Hassidim-Lloyd algorithm [15; 16; 17; 18], [PERSON]'s algorithm [19; 20; 21; 22; 23; 24; 25], and other derivative results [26; 27; 28; 29; 30] on the universal quantum computers have led to tremendous challenges for both hardware and software. Being a paradigmatic example, [PERSON]'s algorithm provides a kind of method to find the factors of an odd composite integer of \(L\) bits using resources that are polynomial in the \(O(L)\) bits, and order-finding as the crux of [PERSON]'s algorithm is believed to be exponentially hard on classical computers [31; 32].  
En route to building scalable quantum processors to realize these algorithms, there have been a variety of benchmarking protocols, such as determining the fidelity of quantum gates [33; 34; 35; 36; 37], setting a standard benchmark between the expected and realized states based on different quantum devices [38; 39; 40; 41], characterizing quantum processors directly for scalability, and robustness [4; 42; 43; 44; 45; 46; 47; 48]. In addition, some relevant theories calculated for distillation and optimization in quantum sources are proposed [49; 50; 51; 52]. The architecture of superconducting quantum processors is the key to its performance, including the number of qubits, computational depth, and the connectivity of different qubits [53; 54; 55]. Single-qubit or even two-qubit gate operations could only focus on the properties of superconducting qubits and reduce the emphasis on the interconnection between these qubits [47; 53; 54; 56; 57]. Thus, it would be rewarding to carry out global benchmarking focused on the overall performances of small-scale circuits and then scale it to larger sizes. Quantum state tomography (QST) has generally been utilized for presenting the overall performances of quantum processors [58; 59]. However, the scalabilityof the existing QST protocol could be a major pitfall for the exponential growth in the resource with the scaling of the processor.  
To address this problem, we are attempting to cut down the measurement costs of QST by systematically selecting a subset of qubits in the quantum processor. This paper proposes a quasi-Shor algorithm, which replaces the modular exponentiation in [PERSON]'s algorithm with other unitary operations, preserves similar frames, and simplifies quantum operations. Taking quantum circuits for generating Bell states as specific examples, a so-called Bell-Shor algorithm was constructed. The Bell-Shor algorithm involves both single-qubit operations and two- or even three-qubit operations together. The evolution of quantum states is calculated during the whole Bell-Shor algorithm and the results of density matrices are visualized as two-dimensional chromatic graphs. The traits of these matrices include the location of the maximum and zero elements. Remarkably, the number of zero elements decreases to a few points in the density matrices after all the processes, which are the heralds of distinguishing features between different input states. Some suitable error metrics are introduced and benchmarking protocols are formulated. Due to the combination of both the single-qubit operations and two-qubit operations on any qubits, the Bell-Shor algorithm is a feasible setting for the global benchmarking of universal quantum processors.  
This paper is organized as follows. The circuits of the quasi-Shor and Bell-Shor are introduced in Section 2, which generalizes the structure of the algorithm. In Section 3, density matrices along the route of the circuit are shown, including both the real part and the imaginary part. Two metrics, \(\Gamma_{a}\) and \(\Gamma_{b}\), are discussed in Section 4, in which the locations of the special entries are displayed. Then the conclusion and summary are in Section 5.