Abstract
Quantum computers utilize the fundamentals of quantum mechanics to solve computational problems more efficiently than traditional computers. Gate-model quantum computers are fundamental to implement near-term quantum computer architectures and quantum devices. Here, a quantum algorithm is defined for the circuit depth reduction of gate-model quantum computers. The proposed solution evaluates the reduced time complexity equivalent of a reference quantum circuit. We prove the complexity of the quantum algorithm and the achievable reduction in circuit depth. The method provides a tractable solution to reduce the time complexity and physical layer costs of quantum computers.
Subject terms: Mathematics and computing, Computer science, Pure mathematics
Introduction
Gate-model quantum computers are realized by unitary operators (quantum gates) and quantum states1–29. As the technological limits of current semiconductor technologies will be reached within the next few years30–40, fundamentally different solutions provided by quantum technologies will be significant in the experimental realization of future computations15–18,31,32,41–72. A quantum computer is set up with a quantum gate structure, that is, via a set of unitary operators. These quantum gates can realize different quantum operations and can be defined on different numbers of input quantum states15–18,41–43,45,52,53. In a quantum computer environment, the depth of the quantum gate structure refers to the number of time steps (time complexity) required for the quantum operations making up the circuit to run on the quantum hardware15–18,41–43,45,52–59. A crucial problem here is the time complexity reduction of the quantum gate structure of the quantum computer. Practically, this problem is such that an equivalent quantum state of the output quantum state of the original the reference quantum circuit (e.g., non-reduced time complexity circuit) can be obtained using a reduced time complexity quantum gate structure. Particularly, currently there exists no plausible and implementable solution for the time complexity reduction of quantum computers. Gate-model quantum computer implementations are affected by the problem of high time complexities and a universal (i.e., platform independent) and tractable solution for the time complexity reduction is essential. Relevant implication of this problem is the high economic cost of the physical apparatuses required for experimentally implementing practical quantum computation: specifically, the high economic cost of the high-precision quantum hardware elements required in the implementation of high-performance quantum circuits.
The quantum circuit of the quantum computer is modeled as an arbitrary quantum circuit with an arbitrary circuit depth formulated via a unitary sequence of L unitary operators. Each unitary is set via a particular Pauli operator and gate parameter (see also Section 2 for a detailed description). The input problem fed into the quantum computer is an arbitrary computational problem with an objective function C. The C objective function is a subject of maximization via the quantum computer, i.e., via the unitaries of the circuit structure of the quantum computer. The objective function can model arbitrary combinatorial optimization problems9,10,42, large-scale programming problems10 such as the graph coloring problem, molecular conformation problem, job-shop scheduling problem, manufacturing cell formation problem, or the vehicle routing problem10. For a detailed description of input problems, we suggest2–4,8–10,42–45.
Another important application of gate-model quantum computations is the near-term quantum devices of the quantum Internet20,30,36–39,46–49,59,61,62,73–93.
Here, we define a quantum algorithm for the time complexity reduction of any quantum circuit of quantum computers set up with an arbitrary number of unitary gates. The aim of the proposed framework is to reduce the time complexity of an arbitrary reference quantum circuit and a maximization of the objective function of the computational problem fed into the quantum computer. The method defines the reduced time complexity equivalent of the reference quantum circuit and recovers the reference output quantum state via the reduced time complexity quantum circuit (Note: the terminology of quantum state refers to an input or output quantum system, while the terminology of quantum gate refers to a unitary operator.). The reduced structures are determined via a pre-processing phase in the logical layer, and only the reduced time complexity quantum circuit and reduced quantum state are implemented in the physical layer. The pre-rocessing phase integrates a machine learning94–97 unit for the parameter optimization. The high complexity reference quantum circuit and reference quantum input are characterized only in the pre-processing phase without any physical level implementation. The framework applies a quantum algorithm on the output of the reduced quantum gate structure to recover the equivalent quantum state of the output quantum state of the non-reduced reference structure. In particular, the proposed framework and the defined quantum algorithm are universal since they have no requirements for the structure of the reference (e.g., non-reduced) quantum circuit subject to be reduced, for the number of unitaries in the reference structure, for the size of the input quantum state of the reference quantum circuit, nor for the dimensions of the actual quantum state. The quantum algorithm is defined as a fixed, auxiliary hardware component for an arbitrary quantum computer environment, with a pre-determined constant computational complexity as an auxiliary cost of the application of the algorithm. Specifically, we prove that the auxiliary cost of the proposed quantum algorithm is orders lower than the reachable amount of the reduction in time complexity, and the computational cost of the quantum algorithm becomes negligible in practice. The method also allows significantly reducing the economic cost of physical layer implementations, since the required elements and high-cost hardware components can be reduced in an experimental setting.
The novel contributions of our manuscript are as follows:
We define a quantum algorithm for circuit depth reduction of quantum circuits of gate-model quantum computers.
We define the computational cost of the proposed quantum algorithm and prove that it is significantly lower than the gainable reduction in time complexity.
The algorithm provides a tractable solution to reduce circuit depth and the economic cost of implementing the physical layer quantum computer by reducing quantum hardware elements.
The results are useful for experimental gate-model quantum computations and near-term quantum devices of the quantum Internet.
This paper is organized as follows. Section 2 defines the system model. Section 3 proposes the quantum algorithm and proves the computational complexity. Section 4 discusses the performance of the algorithm. Finally, Section 5 concludes the results. Supplemental material is included in the Appendix.
System Model
Let QG0 be a reference quantum circuit (quantum gate structure) with a sequence of L unitaries42, defined as
1 |
where is the L-dimensional vector of the gate parameters of the unitaries (gate parameter vector),
2 |
and an i-th unitary gate Ui(θi) is evaluated as
3 |
where Pi is a generalized Pauli operator acting on a few quantum states (qubits in an experimental setting) formulated by the tensor product of Pauli operators {σx, σy, σz}42. Note, that in (1) identifies a unitary resulted from the serial application of the L unitary operators UL(θL)UL−1(θL−1) … U1(θ1), and for an input quantum state |φ〉,
4 |
A qubit system example with a sequence of L unitaries is as follows. Let assume that the QG0 structure of the quantum computer consists of g single-qubit and m two-qubit unitaries, L = g + m, such that a j-th single-qubit gate realizes an operator, while a two-qubit gate realizes a operator (see also42). Then, at a particular objective function C of an arbitrary computational problem subject of a maximization via the quantum computer, the sequence from (1) can be rewritten as
5 |
where
6 |
where is the gate parameter vector of the g single-qubit unitaries,
7 |
while B is defined as
8 |
with
9 |
and
10 |
where Bj = Xj, while the two-qubit unitaries are defined as
11 |
where 〈jk〉 ∈ QG0 is a physical connection between qubits j and k in the hardware-level of the QG0 structure of the quantum computer, is the gate parameter vector of the m two-qubit unitaries
12 |
while
13 |
where Cjk is a component of the objective function, while unitary U(Cjk, γjk) for a given 〈jk〉 is defined as
14 |
where
15 |
At a particular physical connectivity of QG0, the objective function C therefore can be written as
16 |
where Cjk(z) is the objective function component evaluated for a given 〈jk〉, as
17 |
while z is an N-length input bitstring,
18 |
where zi identifies an i-th bit, zi ∈ {−1, 1}.
For a given z, a |z〉 computational basis state is defined as
19 |
and the |ϕ〉 output system of QG0 is as
20 |
that can be evaluated further via (6) and (11), as
21 |
To achieve the quantum parallelism, the input system |φ〉 = |X〉 of the quantum computer is set as an N-length d-dimensional (d = 2 for a qubit system) quantum state in the superposition of all possible dN states. For a qubit system, it means that input |X〉 is an N-qubit quantum state in a superposition of all possible 2N states |0〉 to |2N − 1〉, and the computations are performed on 2N states in parallel in the quantum computer.
Let |X〉 be a superposed input system of the non-reduced QG0 gate structure:
22 |
where |xi〉 is an i-th input state (represented as an N-length bit string for a qubit system), i = 1, …, n, n = dN, of the QG0 structure of the quantum computer.
To describe the parallel processing of the n input vectors of |X〉 (see (22)), {|x1〉, …, |xn〉} of |X〉 (see (22)) in the quantum computer, let be the gate parameter vector associated with a given |xi〉 of |X〉:
23 |
Let X be the classical representation of |X〉 in (22) to get
24 |
where xi is the classical representation of |xi〉. (Note, that X and xi are not accessible in the quantum computer, since the quantum algorithm operates in the quantum regime on quantum states. The classical representation is used only as an abstracted auxiliary representation to describe the steps of the algorithm in a plausible manner).
Then, let be the non-reduced gate structure matrix of QG0:
25 |
where
26 |
and is the unitary sequence associated with |xi〉 in QG0, defined as
27 |
At an n-dimensional output vector
28 |
and the |Y〉 output quantum state of the non-reduced QG0 structure is
29 |
To define the reduced gate structure, QG*, it is necessary to find a reduced with a reduced input , for all i.
Then, let be the classical representation of the reduced quantum state fed into QG*, as
30 |
and
31 |
where N* is the number of d-dimensional (physical) quantum states that formulate , n* = dN*, while the unitaries of QG* are
32 |
where
33 |
and is the reduced unitary sequence associated with , defined as
34 |
The pre-processing phase determines output Z of QG* as a classical representation
35 |
and the output quantum state |Z〉 of QG* therefore yields
36 |
The notations of the system model are also summarized in Table A.1 of the Supplemental Information.
Problem statement
Problems 1–3 summarize the problems to be solved.
Problem 1
Find a classical pre-processing for calculating the reduced input system and the gate parameters of the QG* reduced time complexity gate structure.
Problem 2
Find a universal (independent of the number L of the unitaries in QG0) unitary operator UR with a set of quantum registers to recover output |Y〉 of the non-reduced QG0 structure from output |Z〉 of the QG* reduced time complexity gate structure.
Problem 3
Determine the time complexity of UR and the reduction in the overall time complexity of QG*.
Theorems 1–3 give the resolutions of Problems 1–3.
The non-reduced time complexity quantum circuit QG0 (reference circuit) with an input quantum state |X〉 is showed in Fig. 1(a). Figure 1(b) depicts the system model for the problem resolution. The method is realized via unitary UR and pre-processing, such that UR is implemented in the physical layer, while is only a logical-layer process. Only the reduced input quantum state and the reduced quantum gate structure QG* must be built up in the physical layer to yield the reference output system |Y〉 of the reference circuit QG0 via |YR〉. In both cases, the output states are measured via a measurement M to get a classical bitring for the objective function evaluation. As a next step, the gate parameter values of the unitaries of the circuits are calibrated until an optimal objective function value is not reached. The calibration of the gate parameters is a separate optimization procedure and its aim is fundamentally differ from the aim of , and therefore it is not part of the circuit depth reduction method. Note that existing algorithms can be utilized for this task (such as a the algorithms proposed in19 and20, or some gradient independent methods98).
Figure 1.
(a) The non-reduced time complexity quantum circuit QG0 (reference circuit) with an input quantum state |X〉. The output of QG0 is |Y〉. The state |Y〉 is measured via a measurement M to get the classical string z to evaluate the objective function C(z). (b) System model of the time complexity reduction scheme. Pre-processing phase : the Y classical representation of output |Y〉 of QG0 is pre-processed by the pre-processing unit . Unit contains a computational block that outputs a vector κ, fed into an machine learning control unit for the Δ error feedback. Unit outputs and the gate parameters of the reduced structure that defines QG*. Quantum phase: from and the gate parameters, and QG* are set up. System is fed into the reduced quantum circuit QG*. The output of QG* is |Z〉, which is fed into the UR recovery quantum algorithm. The UR quantum operation outputs the |YR〉 system, which is the reference output |Y〉 of the reference circuit QG0. The state |YR〉 is measured via a measurement M to get the classical string zR to evaluate objective function C(zR).
Pre-processing
Theorem 1
There exists a pre-processing to determine the input system and the gate parameters, i = , …, n, for the reduced QG* gate structure for an arbitrary non-reduced QG0 structure with and input |X〉.
Proof. The pre-processing phase can be decomposed as , where is a computational block, while is a machine learning control block to calibrate the results of . We first define block , then discuss . The pre-processing is a procedure to stabilize the output of the reduced quantum circuit. is defined between the components and to evaluate and to set the gate parameters of the reduced quantum circuit structure QG* using the reference output |Y〉 of QG0. Note, as the output |YR〉 is fed into an M measurement block, the measurement results provide a feedback to calibrate in every subroutine of the protocol to produce a final saturated output. The Δ output of the machine learning control unit is used as a feedback in unit . For the definition of Δ, see (116) in Algorithm 1.
In the computational block, the reduced and are determined for ∀i, in the following manner. Note, since outputs the parameters of the reduced quantum gate structure, the extra complexity of a quantum structure can be replaced with classical complexity in the form of machine learning in the proposed framework.
Operation sets a one-dimensional discrete cosine transform99 in the reduction method, thus a matrix G is defined as a generator matrix to evaluate the output coefficients of , see later (45). The definition of (see later in (40)) comes from the fact that any U unitary operator can be rewritten via the cos and sin functions, and using cosine functions rather than sine functions is critical for a compression99. In our setting, this is because fewer cosine functions are needed to approximate a particular U unitary operator.
Let xi be the classical representation of |xi〉, and be the classical representation of |yi〉. Using the sequences of the L unitaries in (29), define a matrix G with n coefficients ai, i = 1, …, n, as
37 |
where
38 |
where θi,j identifies the gate parameter of a j-th unitary Ui,j(θ) associated to an i-th input xi, while unitary sequence to an i-th input xi is
39 |
where Pi is a generalized Pauli operator.
First, the operation (one-dimensional discrete cosine transform99) is applied to the input matrix G from (37),
40 |
where cp is the p-th coefficient of ,
41 |
where
42 |
and Ap is
43 |
The coefficients of defines matrix γ as
44 |
where · is the inner product,
45 |
where coefficients ak-s are given in (37), and χ is
46 |
where ςi is an n-length vector
47 |
Then, the n-length output vector κ of is defined as
48 |
where Y is given in (28), while Ωi is as
49 |
Then, using the coefficients (41), (42) and (43) of , of the reduced state from (31) can be evaluated via the components of of (30). A p-th input for QG* is defined via (49) as
50 |
and the reduced quantum gate sequence of in QG*, as
51 |
where Pp is a generalized Pauli operator, and is as
52 |
Therefore, the quantum state |Z〉 of QG* is
53 |
The description of the machine learning control unit is as follows. Unit uses the results of to provide feedback for the pre-processing via supervised machine learning control.
The machine learning algorithm for the pre-processing control is defined in Algorithm 1.
The steps of the pre-processing method is given in Procedure 1.
■
Quantum Algorithm
Theorem 2
The |Y〉 output of the non-reduced QG0 structure can be recovered from the output |Z〉 of the reduced structure QG* via a unitary operator UR.
Proof. Let be the input quantum state fed into the reduced structure QG*, and let |Z〉 (see (53)) be the output of the reduced gate structure. The task here is therefore to recover from |Z〉. The problem is solved via a unitary UR, as follows.
Without loss of generality, in an i-th step, i = 1, …, n, the goal of the UR operation is to calculate the quantum state as
54 |
where κ is as in (48), while ωi = (ωi,1, …, ωi,n)T is an n-length vector defined for a given j, as
55 |
where ∑θi is as given in
56 |
where is given in (52).
Then let
57 |
such that
58 |
Applying UR for all i, yields the recovered quantum state |YR〉 as
59 |
where an i-th |xi〉 is as
60 |
where i ≤ n − 1, and p ≥ 0, and is as given in (50); while the gate parameters (see (39)) of the L unitaries for a given i are evaluated as
61 |
The unitary UR is defined via a set of quantum registers as
62 |
where |Ri〉 is the i-th quantum register. The registers are initialized via set as
63 |
where κ is given in (48), while ∂ and η are initial parameters defined as
64 |
and
65 |
where
66 |
where , and
67 |
where .
Then, unitary UR is defined as
68 |
where
69 |
and US is a unitary defined as
70 |
with eigenstate
71 |
where U0 is an initial unitary operator that prepares state |R5〉 = |ωi〉 for a given index state |R4〉 = |i〉, where ωi is given in (55); from an initial |R4〉|R5〉 = |i〉|0〉 as
72 |
in the register set (see (63)), where is the CNOT operation, while is an oracle applied on to compute Φi (54), defined as
73 |
where is the resulting register set, while is an oracle that outputs function fi, as
74 |
Specifically, note that (70) changes only the phase of the state as , where fi is given in (74), while
75 |
Applying (74) on (63) yields a register state as
76 |
where is the eigenvalue of US in (70).
Then, using the register set (63), let |ϕ0〉 be the input state for UR as
77 |
Applying (68) k-times on (77) yields
78 |
The k iteration number in (78) is a random number, k < c, where , and m is initialized as m = 199.
Then let OZ be an oracle defined on as
79 |
Applying OZU0 on (78), outputs system state
80 |
In particular, in system state (80), the state of register |R6〉 is
81 |
therefore yields (59), such that
82 |
holds for all i of |YR〉, due to the conditions set in the pre-processing procedure (see (67)).
Assuming that the input system (77) for UR is prepared for R-times and the output register (81) is measured for R-times, i.e., UR is applied for R times in overall, in an r-th repetition, r = 1, …, R, the parameters of the procedure can be valuated as
83 |
where
84 |
where is the measured value of |Φi〉 in the r-th repetition of UR, while q(r) is the number of coefficients have been already found99.
The actual value of r requires no increment if the relation
85 |
holds, where τ(r) is a threshold value in the r-th iteration. Otherwise, the value of r can be increased, r = r + 1, as r < R.
The steps of the quantum algorithm UR are given in Algorithm 2.
Distortion measure
As (81) is prepared in Step 4 of UR, the state |YR〉 can be measured to get the classical string zR to evaluate objective function C(zR), as follows. Measure register |R6〉 of via a measurement operator M to evaluate objective function C(zR), where zR is a classical string resulted from the measurement of |YR〉, while C is an objective function of an arbitrary computational problem fed into the quantum computer.
The distortion coefficient associated with the |YR〉 recovered quantum state (59) can be evaluated at a particular objective function C, associated to the computational problem fed into the quantum computer as
86 |
where z is a classical string resulting from the M measurement of |Y〉, while zR is a classical string resulting from the M measurement of |YR〉.
Precisely, assuming R measurement rounds, an average of distortion yields
87 |
where C(r)(z) and C(r)(zR) are the objective function values respectively associated with z and zR in the r-th round, r = 1, …, R.
Computational Complexity
Theorem 3
Quantum algorithm UR can be implemented with time complexity for the time complexity reduction of any non-reduced QG0 with an arbitrary number of L unitaries.
Proof. Let
88 |
be a global space spanned by |i〉, an n-dimensional vector |b〉, and by |c〉, which represents the inner product state.
Particularly, the UR unitary in (68) applied on an input |φ〉 formulated via set of quantum registers gives
89 |
where
90 |
thus UR can be interpreted as a rotation on an n-dimensional subspace , 0 ≤ i < n, i.e., on a span of all |i〉.
Let ∏ be the solution set with conditions (82) for all i of ∏,
91 |
and let be the superposition of all solutions:
92 |
The operation UR on |φ〉 yields the state (see (80)):
93 |
thus, UR is a rotation on the subspace by angle towards (92), as
94 |
where |∏| is the number of solutions (cardinality of solution set ∏).
UR can be implemented as a rotation of on subspace (instead of a rotation on global space (88)) via a generalized quantum searching100 that yields time complexity for an arbitrarily large quantum circuit QG0. ■
Performance Evaluation
Assuming that the initial time complexity of the QG0 non-reduced gate structure is
95 |
where N is the number of d-dimensional (physical) quantum states in the superposed input system, and L is the number of unitaries in QG0, the time complexity of the reduced QG* structure is
96 |
where N* is the number of d-dimensional (physical) quantum states in the reduced superposed input system, and L* is the number of unitaries in the reduced gate structure QG*.
Since the complexity of the proposed scheme is
97 |
the result of (96) is a reduced time complexity with respect to (95), as the relation
98 |
holds; thus
99 |
The overall complexity of the QG* reduced structure at the application of UR is therefore
100 |
Figure 2 depicts the resulting time complexities for a qubit implementation (N-qubit superposed input system, and qubit gate structure with L unitaries).
Figure 2.
The time complexities (number of operations) for an N-qubit system, d = 2, n = 2N, with an initial non-reduced gate structure QG0 with L unitaries, L = {10, 100, 1000, 10000}. The time complexity of QG0 is , while is an upper a bound on of QG*, .
To achieve time complexity reduction using and QG* instead of |X〉 and QG0, the relation must be straightforwardly satisfied, i.e., the initial complexity has to be reduced by more than . Since the complexity of the procedure is independent from the actual size of the gate structure, the cost remains constant for an arbitrarily large L.
Conclusions
Gate-model quantum computers are equipped with a collection of quantum states and unitary quantum gates. The realization of the quantum circuit of a quantum computer requires high fidelity, high precision, and high-level control. Since both the timecomplexity (depth of the circuits) and the economic costs of these implementations are high in practical scenarios, a reduction of these costs is essential. Here, we defined a quantum algorithm for reducing the circuit depth of gate-model quantum computers. The method achieves a reduction in the physical layer allowing significantly reducing implementation costs. The framework is flexible and can be used for arbitrary circuit depths.
Submission note
Parts of this work were presented in conference proceedings101.
Ethics statement
This work did not involve any active collection of human data.
Supplementary information
Acknowledgements
Open access funding provided by Budapest University of Technology and Economics (BME). The research reported in this paper has been supported by the Hungarian Academy of Sciences (MTA Premium Postdoctoral Research Program 2019), by the National Research, Development and Innovation Fund (TUDFO/51757/2019-ITM, Thematic Excellence Program), by the National Research Development and Innovation Office of Hungary (Project No. 2017-1.2.1-NKP-2017-00001), by the Hungarian Scientific Research Fund - OTKA K-112125 and in part by the BME Artificial Intelligence FIKP grant of EMMI (Budapest University of Technology, BME FIKP-MI/SC).
Appendix
Algorithm 1
supervised machine learning control of
Input: The Ωi coefficients from block .
Output: Classification of the Ωi coefficients, error Θ of ; error Δ of , updated Ωi coefficients.
Step 1. Let be the training set of as , where κi is an i-th instance and an n-length vector (see (48)) drawn from the input space , while is the class label associated with κi, where is the set of labels .
Step 2. For i = 1, …, m, set the initial D0(i) distribution of the i-th instance of as .
Step 3. Set the R iteration number. For an r-th iteration, r = 1, …, R, let Dr be distribution. Define hypothesis (implementable via a weak learning algorithm102–105) using the input distribution Dr as
101 |
such that the εr training error (error of hr) of is
102 |
is minimized with respect to the distribution Dr.
Step 4. If εr ≤ 0.5 goto Step 5, otherwise set R = r − 1, and stop.
Step 5. If hr(κi) = li, set Dr+1(i) as
103 |
where χr is a normalization term, while Wr is a weighting coefficient, as
104 |
If hr(κi) ≠ li, set
105 |
Step 6. Repeat steps 3–5 for R times.
Step 7. From the R hypotheses h1, …, hR, set the final hypothesis H to classify the Ωi coefficients and the input system Y as
106 |
and calculate the ε(H) error of H as
107 |
where DH is the distribution associated to H.
Step 8. Initialize and parameters as
108 |
where
109 |
where Y is defined in (28), , and
110 |
where .
Step 9. For an i-th coefficient Ωi, set
111 |
Step 10. Set σi threshold for an i-th coefficient as
112 |
Repeat step 9, until
113 |
where ς, ς > 0 is defined via the relation
114 |
where B is the number of non-zero elements Ωi of κ, B ≤ n. If σi < ς, goto step 11.
Step 11. Output the Θ error of as
115 |
where ρ is a constant, while ε(H) is given in (107).
Step 12. Output the error Δ of as
116 |
Step 13. According to (116), calibrate the Ωi coefficients (defined in (49)) of output κ in (48) of , to yield in (115), where is a threshold on the error of .
Procedure 1 Pre-processing
Input: Output Y (28) of the non-reduced QG0 reference circuit.
Output: Input (31) of reduced quantum circuit QG*.
Step 0. Characterize the (25) unitaries of the QG0 reference circuit via gate parameters (33), and the reference input |X〉 (22) via X (24). Evaluate |Y〉 (29) via Y (28).
Step 1. Fed Y into block to determine κ (48), and control the results of via .
Algorithm 2
Quantum algorithm UR
Input: Output state |Z〉 (36) of QG*.
Output: Quantum state |YR〉 (59).
Step 1. Prepare the reduced quantum state (30) using (31) of .
Step 2. Prepare the unitaries of (32) via (33) of QG*.
Step 3. Fed into QG* to yield output system |Z〉 (36).
Step 4. Utilize the quantum register set (63). Apply UR on to get |YR〉 (59) in register |R6〉 of .
Author contributions
L.GY. designed the protocol and wrote the manuscript. L.GY. and S.I. analyzed the results. All authors reviewed the manuscript.
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
is available for this paper at 10.1038/s41598-020-67014-5.
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