Quasi-Shor Algorithms for Global Benchmarking of Universal Quantum Processors

Abstract

This work generalizes Shor’s algorithm into quasi-Shor algorithms by replacing the modular exponentiation with alternative unitary operations. By using the quantum circuits to generate Bell states as the unitary operations, a specific example called the Bell–Shor algorithm was constructed. The system density matrices in the quantum circuits with four distinct input states were calculated in ideal conditions and illustrated through chromatic graphs to witness the evolution of quantum states in the quantum circuits. For the real part of the density matrices, it was revealed that the number of zero elements dramatically declined to only a few points after the operation of the inverse quantum Fourier transformation. Based on this property, a protocol constituting a pair of error metrics ${\mathsf{\Gamma }}_a$ and ${\mathsf{\Gamma }}_b$ is proposed for the global benchmarking of universal quantum processors by looking at the locations of the zero entries and normalized average values of non-zero entries. The protocol has polynomial resource requirements with the scale of the quantum processor. The Bell–Shor algorithm is capable of being a feasible setting for the 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 Deutsch algorithm [12,13], Grover algorithm [14], Harrow–Hassidim–Lloyd algorithm [15,16,17,18], Shor’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, Shor’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 $O\left(L\right)$ bits, and order-finding as the crux of Shor’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 scalability of 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 Shor’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, ${\mathsf{\Gamma }}_a$ and ${\mathsf{\Gamma }}_b$, are discussed in Section 4, in which the locations of the special entries are displayed. Then the conclusion and summary are in Section 5.

2. Quasi-Shor and Bell–Shor Algorithms

Shor’s factoring algorithm involves two registers in the quantum processors [19,59]. The first one contains t qubits set in $|0⟩$ state initially. The t relies on the number of digits of calculated accuracy and the probability of success of the phase estimation process. The second register starts with $|1⟩$ state, which should have enough qubits to store quantum information.

As shown in Figure 1a, quasi-Shor algorithms replace the modular exponentiation in Shor’s algorithm with unitary operations $M_i(i=1,2,3\dots )$. In the first register, the circuit contains a series of Hadamard gates, which are followed by controlled-$M_i$. Then inverse quantum Fourier transformation (${\mathrm{QFT}}^{†}$) is applied to the first register. It still holds the analog frame of the quantum circuit with Shor’s algorithm. The measurement of all these qubits after ${\mathrm{QFT}}^{†}$ could be exploited to benchmark the performance of the quantum processor.

Figure 1. Schematic quantum circuit for quasi-Shor algorithms (a) and Bell–Shor algorithm (b).

Figure 1b gives the schematic of the Bell–Shor algorithm based on five qubits, which are divided into two registers with three and two qubits, respectively. This combination keeps a certain complexity by retaining three computation periods with three controlling qubits in the first register while simplifying the varieties of entangled states through Bell states with two computing qubits. The quantum operation M is set as a combination of a Hadamard gate and a CNOT gate, which is denoted as a Controlled-Bell (C-Bell) quantum operation:

$$M=\mathrm{CNOT}\ast (H\otimes I).$$

(1)

As a case study, when $M_i(i=1,2,3)$ satisfy $M_i=M^{2^{i-1}}(i=1,2,3)$, the quasi-Shor algorithm could be utilized to complete phase estimation. For further simplification, $M_i(i=1,2,3)$ satisfy $M_i\equiv M(i=1,2,3)$, the brief quantum operations of Bell–Shor algorithm are displayed in Figure 1b. The quantum states along the quantum circuit are denoted as ${\varphi }_j(j=1,2,...,5)$. The first register is linked to the cycle index of the C-Bell operations and it is adjustable by changing the number of controlling qubits to alter the periodic operations in the second one. While for the computing qubits in the second register, $M_i$ is not limited by the quantum circuits for generating Bell states. It is also possible to increase the number to generate Greenberger-Horne-Zeilinger (GHZ) states [60] and W states [61] on more computing qubits. Furthermore, if $M_i$ is set as the Grover iteration G, the quasi-Shor algorithm would transform into an approximate quantum counting circuit [59].

3. Witness the Density Matrices

The initial states of the three controlling qubits and the two computing qubits are set as ${|0⟩}^{⨂3}|00⟩$. The quantum states ${\varphi }_1,{\varphi }_2,{\varphi }_3,{\varphi }_4$ are calculated analytically by matrices multiplications. To demonstrate the evolution of quantum states in Bell–Shor processing, Figure 2a–d display the density matrices ${\rho }_1,{\rho }_2,{\rho }_3,{\rho }_4$ corresponding to ${\varphi }_1,{\varphi }_2,{\varphi }_3,{\varphi }_4$, respectively. The elements marked with dark colors are zero (with accuracy $\le {10}^{-4}|E_{\max }|$, where $E_{\max }$ is the value of the element with the maximum absolute value). Running along the quantum circuit, it is noticeable that the states are becoming more complex and entangled. The zero elements are distributed as horizontal and vertical grids. The calculated results of the other three different input states $|01⟩,|10⟩,|11⟩$ of computing qubits (i.e., the second register qubits) are shown in Appendix A.

Figure 2. Density matrices for a series of quantum states during the processing ahead of ${\mathrm{QFT}}^{†}$. (The input state is ${|0⟩}^{⨂3}|00⟩$). (a) The density matrix ${\rho }_1$ of ${\varphi }_1$. (b) The density matrix ${\rho }_2$ of ${\varphi }_2$. (c) The density matrix ${\rho }_3$ of ${\varphi }_3$. (d) The density matrix ${\rho }_4$ of ${\varphi }_4$.

Applying the ${\mathrm{QFT}}^{†}$ operation to state ${\varphi }_4$, density matrices ${\rho }_5$ of ${\varphi }_5$ constructed with complex elements are shown in Figure 3. Here, the zero elements are marked with black color. In contrast to Figure 2, the translational symmetry of the grids is broken, whereas local rotational symmetry emerges. The real parts of the density matrices remain symmetric about the main diagonal just as the density matrices before ${\mathrm{QFT}}^{†}$, and the elements in the main diagonal are all positive. The zero elements of the imaginary parts of the density matrices are also symmetric about the main diagonal, whereas the other elements are all conjugate with each other about the main diagonal. Moreover, the elements in the main diagonal are all zero. The number of zero elements in the real and imaginary parts are counted for in each quantum state.

Figure 3. Density matrices ${\rho }_5$ for four distinct input states. The input state is ${|0⟩}^{⨂3}|00⟩$ (a), ${|0⟩}^{⨂3}|01⟩$ (b), ${|0⟩}^{⨂3}|10⟩$ (c), ${|0⟩}^{⨂3}|11⟩$ (d). Left panels are the real parts, while the right panels are the imaginary parts.

Denoting the number of zero elements as n, the ratio $n/1024$ as a function of the quantum circuit depth is illustrated in Figure 4. The ratio is decreasing during the whole process. At first, 93.75% of the 1024 elements are zero, then with the operations of C-Bell and ${\mathrm{QFT}}^{†}$, n decreases to 16–20 finally. The zero elements drop to just a few dark points in the real density matrices and generate imaginary parts after the operation of ${\mathrm{QFT}}^{†}$.

Figure 4. The ratio of the number of zero elements in the real part (solid lines) and that in the imaginary part (dashed lines) as a function of the quantum circuit depth. The inset presents the ratio of the final states after ${\mathrm{QFT}}^{†}$.

In Table 1, the coordinates of the maximum elements and zero elements in the real parts of density matrices ${\rho }_5$ are listed specifically. There are obvious distinctions among these four input states. Designating $|00000⟩⟨00000|$ as (1,1) and $|11111⟩⟨11111|$ as (32,32), the coordinate of the maximum is (1,1) for both ${|0⟩}^{⨂3}|00⟩$ and ${|0⟩}^{⨂3}|10⟩$, while (2,2) for ${|0⟩}^{⨂3}|01⟩$ and ${|0⟩}^{⨂3}|11⟩$. The landscapes of these density matrices are spectacular with remarkable features. For the zero entries, there are about 16–20 elements with unique distributions for different input cases. In contrast to the real parts, the imaginary parts have a lot of zero entries and most are degenerated for different input states, i.e., the patterns of zero entries are nearly identical. Appendix A presents calculations of the reduced density matrices and the coordinates of the maximum element and zero entries.

Table 1. Coordinates of elements with the maximum and zero values in the real part of density matrices for the four input states. Designate $|00000⟩⟨00000|$ as (1,1) and $|11111⟩⟨11111|$ as (32,32).

Input States	${\left|\mathit{E}\right|}_{\mathit{max}}$	Absolute Zero
${|0⟩}^{⨂3}|00⟩$	(1,1)	(6,24);(7,25);(7,26);(9,23);
		(9,31);(10,23);(10,31);(15,25);
		(15,26);(16,30);(23,9);(23,10);
		(24,6);(25,7);(25,15);(26,7);
		(26,15);(30,16);(31,9);(31,10)
${|0⟩}^{⨂3}|01⟩$	(2,2)	(5,23);(8,25);(8,26);(9,24);
		(9,32);(10,24);(10,32);(15,29);
		(16,25);(16,26);(23,5);(24,9);
		(24,10);(25,8);(25,16);(26,8);
		(26,16);(29,15);(32,9);(32,10)
${|0⟩}^{⨂3}|10⟩$	(1,1)	(6,27);(6,28);(11,22);(11,30);
		(12,22);(12,30);(14,27);(14,28);
		(22,11);(22,12);(27,6);(27,14);
		(28,6);(28,14);(30,11);(30,12)
${|0⟩}^{⨂3}|11⟩$	(2,2)	(5,27);(5,28);(11,21);(11,29);
		(12,21);(12,29);(13,27);(13,28);
		(27,5);(28,5);(21,12);(21,11);
		(29,11);(29,12);(27,13);(28,13)

4. Error Metrics for Benchmarking

Now, the coordinates of the maximum and the zero elements in the density matrices could be utilized as a set of traits for global benchmarking of universal quantum processors by comparing the measured result with calculated ones. Defining the measured value of the dark points as $P_i(i=1,2,...,n)$, the measured maximum as $P_m$ according to the coordinates obtained from theoretical values, an error metric ${\mathsf{\Gamma }}_a$ of benchmarking is tentatively written as

$${\mathsf{\Gamma }}_a=\left\{\begin{array}{cccc}\hfill & \mathrm{Fail}& & (P_m=0)\\ \hfill & & & \\ \hfill & 10\lg \left(\frac{{\sum }_{i=1}^n|P_i|}{n|P_m|}\right)& & (P_m\ne 0),\end{array}\right.$$

(2)

which provides information about the inaccuracy of operations on the quantum processor. Here, the traits of output states depend on the input states and the fidelity of operations.

In Figure 5a, four states, including ${|0⟩}^{⨂3}|00⟩$, ${|0⟩}^{⨂3}|01⟩$, ${|0⟩}^{⨂3}|10⟩$, ${|0⟩}^{⨂3}|11⟩$ are utilized as tools to benchmark those with different states (e.g., $|000⟩$, $|001⟩$, …, $|111⟩$) in the first register. It is obvious that the ${\mathsf{\Gamma }}_a$ is about –180 when using a few traits to benchmark themselves, while it becomes larger with the change in the first three states. Although ${\mathsf{\Gamma }}_a$ is reasonable and quite sensitive in most cases and has a complexity about $n+1$, the fluctuations at some states (e.g., $|100⟩|10⟩$ and $|100⟩|11⟩$) are too large to be acceptable.

Figure 5. The results of benchmarking with error metrics ${\mathsf{\Gamma }}_a$ and ${\mathsf{\Gamma }}_b$. (a) ${\mathsf{\Gamma }}_a$ uses ${|0⟩}^{⨂3}|00⟩$ (red line), ${|0⟩}^{⨂3}|01⟩$ (blue line), ${|0⟩}^{⨂3}|10⟩$ (green line), ${|0⟩}^{⨂3}|11⟩$ (purple line) as tools to benchmark these with different states in the first register, respectively. (b) ${\mathsf{\Gamma }}_b$ uses the state ${|0⟩}^{⨂3}|00⟩$, which combines the position of zero value in ${|0⟩}^{⨂3}|01⟩$, ${|0⟩}^{⨂3}|10⟩$ and ${|0⟩}^{⨂3}|11⟩$ together, and benchmark these states with the difference in the first register.

To enhance the robustness of the error metric, the completeness of the second register states is taken into consideration. Then another error metric ${\mathsf{\Gamma }}_b$ is constructed:

$${\mathsf{\Gamma }}_b=\left\{\begin{array}{cccc}\hfill & \mathrm{Fail}& & (P_m=0)\\ \hfill & & & \\ \hfill & 10\lg ({10}^3\frac{{\sum }_{j=1}^4{\sum }_{i=1}^{n_j}|P_i-P_{i_0}|}{n_{\mathrm{tot}}\left|P_m\right|}+1)& & (P_m\ne 0),\end{array}\right.$$

(3)

where $n_j$ is the number of zero elements in the jth calculated density matrix, $n_{\mathrm{tot}}$ is the total number of zero elements in the theoretical calculation, $P_i$ ($P_{i_0}$) is the measured (ideal) value in the position of the ith element, and $P_m$ is the measured value in the position of the maximum element. To enhance the sensitivity of ${\mathsf{\Gamma }}_b$, a scaling factor of ${10}^3$ is applied, and the addend constant 1 acts as a calibration term to ensure that the lower bound of ${\mathsf{\Gamma }}_b$ is zero.

Figure 5b gives the benchmarking results of a series of states via utilizing the first register state $|000⟩$, which has the complexity as $4n+1$. ${\mathsf{\Gamma }}_b$ improves the robustness of the error metric via the completeness of the second register. It employs the locations of zero entries in the four kinds of input states and mitigates the influence of state preparation errors to some extent. The fluctuations are suppressed efficiently in comparison with ${\mathsf{\Gamma }}_a$ in Figure 5a.

To turn an empirical value into an interpretation of the performance of a quantum processor, it is helpful to examine the formula of ${\mathsf{\Gamma }}_b$ in some detail. In an ideal condition, one has $P_m\ne 0$ and $|P_i-P_{i_0}|=0$, and this yields ${\mathsf{\Gamma }}_b$ = 0. While in practice, the value $P_m$ in the location of the maximum according to the calculated result $P_{m_0}$ is checked first. If $P_m=0$, it means the quantum processor is failed. For $P_m\ne 0$, it is usually less than the calculated value $P_{m_0}$, and $|P_i-P_{i_0}|$ is larger than zero, therefore ${\mathsf{\Gamma }}_b>0$. This indicates the monotonic increase of ${\mathsf{\Gamma }}_b$ with noise.

Based on the above calculations, a global benchmarking protocol for universal quantum processors is introduced as follows:

1. For an N-qubit quantum processor, choose five qubits randomly;

2. Run the Bell–Shor algorithm on the five qubits;

3. Carry out state tomography for the final state;

4. Construct density matrices of the final state;

5. Look up Table 1 for the coordinates of the required entries;

6. Calculate two metrics ${\mathsf{\Gamma }}_a$ and ${\mathsf{\Gamma }}_b$.

For the remaining $N-5$ qubits, there are at least three feasible treatments:

(1) Leave the remaining $N-5$ qubits in idle states when running the Bell–Shor algorithm on the five qubits. In this situation, it is important to move a certain number of frequencies of those $N-5$ qubits away from the work frequency to reduce the disturbance on working qubits.

(2) Initialize and keep the remaining $N-5$ qubits in the ground state. Because no other driving signal is applied to the $N-5$ qubits, less resonance or entanglement would be generated among the selected five qubits to other qubits.

(3) Try to conduct parallel operations on a multi-group of five (or other) qubits. The advantage of this method is the high benchmarking efficiency, nevertheless, more control and readout lines are needed.

For the state tomography [59], measure $⟨{\sigma }_{v_1}\otimes {\sigma }_{v_2}\otimes {\sigma }_{v_3}\otimes {\sigma }_{v_4}\otimes {\sigma }_{v_5}⟩$, where ${\sigma }_{v_i}(i=1,2,...,5)$ are chosen from the set of Pauli matrices ${\sigma }_0,{\sigma }_1,{\sigma }_2,{\sigma }_3$. The density matrices are reconstructed by

$$\rho =\sum _{\overrightarrow{v}}\frac{\mathrm{tr}({\sigma }_{v_1}\otimes {\sigma }_{v_2}\otimes ...\otimes {\sigma }_{v_5}\rho ){\sigma }_{v_1}\otimes {\sigma }_{v_2}\otimes ...\otimes {\sigma }_{v_5}}{2^5}.$$

(4)

An alternative method for state tomography and reconstructing density matrices has been demonstrated by Reed, et al [62,63].

The Bell–Shor algorithm can be extended to a reasonably larger number of qubits, e.g., five to eight qubits. Here, a subset of k qubits with $2^k\times 2^k$ density matrices is used to benchmark the N-qubit quantum processor. The total number of selections could be counted as:

$$C_N^k=\frac{N!}{k!(N-k)!}<\frac{N^k}{k!},(k=5,6,7,8).$$

(5)

Then the number of elements in the density matrices for the 5-qubit Bell–Shor algorithm is:

$$\frac{N!}{5!(N-5)!}\times {32}^2<\frac{N^5}{5!}\times {32}^2=\frac{N^5}{120}\times {32}^2=8.53N^5,$$

(6)

which has polynomial scalability. Comparing with benchmarking N-qubit directly by $2^N\times 2^N$ density matrices, the total number of elements is $4^N$, which scales exponentially.

As listed in Table 2, the number k of the calculating qubits is increased to study the change in the number (${\sum }_{j=1}^4{n}_j$) of zero elements. It could be seen that the total number of elements grows exponentially with k. However, the percentage of the number of zero elements keeps relatively steady except in the 6-qubit case, where an abnormal increase occurs. There is a trade-off between the choice of the qubit numbers k and the percentage of zero elements. The large sizes of elements would increase the calculation complexity, while too small qubit numbers would destroy the integrity of quasi-Shor algorithm.

Table 2. The change in the number of zero elements with an increase in the number of calculating qubits.

Number of Qubits k	Total Number of Elements $4\times 4^{\mathit{k}}$	Total Number of Zero Elements ${\sum }_{\mathit{j}=1}^4{\mathit{n}}_{\mathit{j}}$	Percentage (%)
5	4 × 1024	72	1.758
6	4 × 4096	1520	9.277
7	4 × 16,384	1096	1.672
8	4 × 65,536	1916	0.731

Based on the information mentioned in Table 2, the locations of these special entries are shown in Figure 6, including all the zero entries in the four final density matrices with the following different initial states ${|0⟩}^{\otimes k-2}{|00⟩,|0⟩}^{\otimes k-2}{|01⟩,|0⟩}^{\otimes k-2}|10⟩$, and ${|0⟩}^{\otimes k-2}|11⟩(k=5,$$6,7,8)$. It could be seen that though the number of special elements in the 6-qubit case is abnormally larger than the others, and the distribution of zero entries has a spectacular cross “+” feature. Moreover, all these special points keep some characteristics of symmetry. For $k=6,7,8$, the real part of the final density matrices (see Figure A3) are presented in Appendix B, where the application of ${\mathsf{\Gamma }}_b$ is also given (see Figure A4). A comparison of the number of elements among different k to the QST reveals that the present Bell–Shor global benchmarking protocol is polynomial to N (see Figure A5 in Appendix B).

Figure 6. The locations of special entries are marked in the density matrices. (a) Example of 5 qubits. (b) Example of 6 qubits. (c) Example of 7 qubits. (d) Example of 8 qubits.

At this point, a comparison between the present benchmarking protocol with other approaches is necessary. The trend of ${\mathsf{\Gamma }}_b$ is compared with 1-Fidelity, state-Fidelity, and trace distance by calculation [59] and simulation in Qiskit [64] in Appendix C. In fact, the existing benchmarking approaches are mainly for single-qubit gates and two-qubit gates [47,53,54,56,57], which focus on the properties of qubits and might lack an emphasis on the interconnection among all the qubits in the processors. So here comes up a moderately sized algorithm for the global benchmarking of the architecture of universal quantum processors, including the number and the interconnectivity of these qubits, as well as the computational depth of quantum circuits. The connection among qubits could be manifested through the entanglement generated among different qubits [65,66,67,68]. For more straightforward calculations, a quantum circuit for generating Bell states is taken as a specific example to construct the Bell–Shor algorithm on a subset of k qubits. This algorithm involves not only single-qubit operations on all these k qubits but also two-qubit operations between any two qubits, thus ensuring a benchmark of the universality of quantum processors.

5. Conclusions

In summary, this work proposes quasi-Shor algorithms, which share a similar framework with Shor’s algorithm and also demand moderate requirements for the performance of quantum processors. It simplifies quantum operations by replacing the modular exponentiation component with a series of controlled-unitary operations. One variant of the quasi-Shor algorithm involves utilizing the controlled Bell-state preparations and construct the Bell–Shor algorithm. Density matrices along the quantum circuit are calculated and visualized. Remarkably, it is revealed that the number of zero elements decreases from 960 to only a few points in the real part of the density matrices after the inverse quantum Fourier transformation. Based on this property, a pair of error metrics (${\mathsf{\Gamma }}_a$ and ${\mathsf{\Gamma }}_b$) with polynomial increases are defined and the robustness is analyzed for various instances. A systematical benchmarking protocol for quantum processors is formulated with the two error metrics playing key roles. The proposed benchmarking protocol uses quantum state tomography to estimate the resulting density matrices of the final quantum states. The protocol then calculates the benchmarking metric by looking at the locations of the zero entries and normalized average values of non-zero entries. Bell–Shor global benchmarking protocol has polynomial scalability, which cuts down the exponential costs of full-state tomography, therefore, addressing the need for benchmarking the global performance of universal quantum processors. The rich physics contained in the quasi-Shor algorithms may accompany the research and development of universal quantum computers.