Direct Fidelity Estimation of Quantum States using Machine Learning
Xiaoqian Zhang, Maolin Luo, Zhaodi Wen, Qin Feng, Shengshi Pang, Weiqi Luo, Xiaoqi Zhou
aa r X i v : . [ qu a n t - ph ] F e b Direct Fidelity Estimation of Quantum States using Machine Learning
Xiaoqian Zhang, ∗ Maolin Luo, ∗ Zhaodi Wen, Qin Feng, Shengshi Pang, Weiqi Luo, and Xiaoqi Zhou † School of Physics and State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510000, China College of Information Science and Technology, College of Cyber Security, Jinan University, Guangzhou 510632, China (Dated: February 5, 2021)In almost all quantum applications, one of the key steps is to verify that the fidelity of the prepared quantumstate meets the expectations. In this paper, we propose a new approach to solve this problem using machinelearning techniques. Compared to other fidelity estimation methods, our method is applicable to arbitrary quan-tum states, the number of required measurement settings is small, and this number does not increase with the sizeof the system. For example, for a general five-qubit quantum state, only four measurement settings are requiredto predict its fidelity with ±
1% precision in a non-adversarial scenario. This machine learning-based approachfor estimating quantum state fidelity has the potential to be widely used in the field of quantum information.
In the field of quantum information, almost all quantum ap-plications require the generation and manipulation of quantumstates. However, due to the imperfections of equipment andoperation, the prepared quantum state is always di ff erent fromthe ideal state. Therefore, it is a key step to evaluate the devi-ation of the prepared state from the ideal one in the quantumapplications. Quantum state tomography (QST) [1, 2, 4–17]is the standard method for reconstructing a quantum state toobtain its density matrix, which can be used to calculate thefidelity of the quantum state with respect to the ideal one. Inrecently years, researchers have proposed compressed sens-ing methods [5–10] to improve the e ffi ciency of QST for thepure quantum states. Despite the fact that compressed sensinggreatly reduces the measurement resources, the measurementsettings for QST still grow exponentially with the size of thesystem.However, to evaluate the fidelity of a quantum state, fullreconstruction of its density matrix is not needed. Recently,schemes [18–37] for directly estimating the fidelity of quan-tum states, including the quantum state verification (QSV)method [26–37] and the direct fidelity estimation (DFE)method [20], have been proposed. The QSV method can de-termine whether a quantum state is the target state with fewmeasurement resources, but this method is only applicable tospecial quantum states, such as the stabilizer states or the Wstates, and is not applicable to general quantum states. Com-pared with the QSV method, the DFE method [20] is appli-cable to general quantum pure states but requires more mea-surement settings. In most practical experiments, the numberof measurement settings has a significant impact on the to-tal measurement time (changing measurement setting is time-consuming). Both the QSV and the DFE methods assume thatthe measured quantum state may be prepared or manipulatedby an adversary, which is valid for the case of quantum net-works. For most local experiments in which the quantum de-vices are trusted, the imperfections of the quantum state arecaused by noise and device defects, not by the adversary. As aresult, our aim is to devise a direct fidelity estimation protocolfor this scenario, further reducing the number of measurement ∗ These authors contributed equally † [email protected] settings required.In this work, we use machine learning methods [1, 38–45, 47] to tackle this problem. So far, machine learning meth-ods have been used for classification problems [38–43] in thefield of quantum information to detect the non-locality [38],steerability [40], entanglement [39] and coherence [47] ofquantum states. In these previous works, the classificationof quantum states can be performed with high accuracy us-ing fewer measurement settings by using artificial neural net-works (ANNs) to learn the potential information between theinternal structures of the quantum state space. In this work,we transform the quantum state fidelity estimation probleminto a classification problem, by dividing the quantum statespace into di ff erent subspaces according to the value of fi-delity, and then using a neural network to predict which sub-space the quantum state is in to obtain an estimate of the quan-tum state fidelity. Compared with previous methods for directestimation of fidelity, this method not only works for arbitraryquantum states, but also greatly reduces the number of mea-surement settings required. Representing fidelity using Pauli observables. —The fidelity[48] of an arbitrary quantum state ρ with respect to the desiredpure state ρ can be written as F ( ρ , ρ ) = tr q ρ / ρ ρ / = p tr ( ρρ ) , (1)where ρ = n n − X j = a j W j , ρ = n n − X j = β j W j . (2)Here W j represents Pauli operators which are n-fold tensorproducts of I , σ x , σ y , σ z . The fidelity in Eq.(1) can be ex-panded in terms of the Pauli operators’ expectation values a j and β j , F ( ρ , ρ ) = vut n n − X j = β j a j . (3) The artificial neural networks. —For an n -qubit quantumstate, there exist 3 n Pauli operators consisting of a tensor prod-uct of σ x , σ y and σ z , from which k Pauli operators are selectedfor measurement. Here we have chosen the k Pauli operatorswith the largest absolute value of the expectation value of the M Input Layer Hidden Layers
M M
Output Layer M FIG. 1. (Color online) The artificial neural network for quantumstate fidelity evaluation. The input layer neurons are loaded withthe measurements of the Pauli operators, the output layer neuronscorrespond to di ff erent fidelity intervals, and the input and outputlayers are fully connected by several hidden layers. After hundredsof training sessions, a neural network model that can evaluate thefidelity of quantum states is obtained. desired quantum state. For each of these k Pauli operators,there are 2 n possible outcomes and the probability of eachoutcome occurring will be used as an input. Thus the inputlayer has a total of k × n neurons.The input layer neurons arefully connected to the hidden layer, i.e., each neuron of theinput layer is connected to each neuron of the hidden layer.The hidden layer is also fully connected to the output layer.The output layer has a total of 122 neurons corresponding todi ff erent fidelity intervals of the quantum states. We generatedvarious quantum states of di ff erent fidelity using the programwe designed (see Supplementary Information) and calculatedthe k × n probability values corresponding to each quantumstate and its fidelity label which ranges from 1 to 122. Foreach fidelity interval, we generated 20,000 quantum states,16,000 of which were used for neural network training and4,000 for neural network validation. After several hundredrounds of training, the prediction accuracy of the neural net-work saturates, resulting in a neural network model that canpredict the fidelity of the quantum states with high confidence(See Supplementary Information).Taking a general five-qubit quantum state | ψ i as an exam-ple [49], by setting k =
2, 3, 4, and 5 and four di ff erent neuralnetwork models are generated respectively using the methodsdescribed above to predict the fidelity of the input quantumstate with respect to | ψ i . When predicting fidelity with a neu-ral network model, the accuracy of the prediction is inverselyrelated to the confidence level, and here we set the confidencelevel fixed at 95%. Figure 2 shows that the higher the num-ber of Pauli operators used, the higher the prediction accuracyof the neural network model is. For the same neural networkmodel, the higher the value of the predicted fidelity, the higherthe accuracy of that prediction is.Now we look at how to use a neural network model fora specific problem—to determine whether the fidelity of theinput quantum states | ψ i and | ψ i (See Supplementary Infor-mation) with respect to | ψ i exceeds 96%. For | ψ i , we choosethe top three Pauli operators in the absolute value of the ex-pectation value for measurement, and input the measurementresults into the neural network model with k =
3, and obtain -2-1012 P r e c i s i on ( % ) ++ -2-1012 ++ -2-1012 ++ -2-1012 ++ FIG. 2. (Color Online) A plot of the prediction accuracy of the neu-ral network versus the quantum state fidelity when measurements aremade using three, four, five, and six Pauli operator measurement set-tings. The higher the number of Pauli operator measurement settingsused, the higher the prediction accuracy is. The higher the fidelity ofthe predicted quantum states is, the higher the accuracy is. a fidelity prediction of (97 . ± . | ψ i exceeds 96%. For | ψ i , repeating the above op-erations, the fidelity result obtained is (95 . ± . | ψ i exceeds 96% fornow. Then the Pauli operator with the fourth largest absolutevalue of expectation value is measured, and the measurementresult was input into the k = . ± . | ψ i does not exceed 96%.Figure 3 illustrates the cases of n -qubit quantum states ( n ranges from 2 to 6), indicating that the number of Pauli opera-tors that need to be measured does not increase as the numberof qubits of the quantum state increases. For example, fora general six-qubit quantum state, only three Pauli operatorsare needed for measurement to make predictions about the fi-delity. This phenomenon seems a bit counterintuitive, and itcan be understood as follows: when the fidelity is 1, β j , the ex-pectation value of Pauli operators for the input quantum state,will be equal to a j , the expectation value of Pauli operatorsfor the desired quantum state. When the fidelity is less than1, β j deviates with respect to a j , and the smaller the fidelity,the larger this deviation will be. Since we consider the non-adversarial scenario, the deviation of β j with respect to a j willnot be particularly preferred to specific Pauli operators. Afraction of β j , which can be derived from the measurementresults of the k Pauli bases, is equivalent to a sampling of the T he nu m be r o f P au li c o m b i na t i on s FIG. 3. (Color Online) For general quantum states (fidelity between0.95 and 1) with two, three, four, five, and six qubits, the number ofPauli operator measurement settings required to predict fidelity with ±
1% accuracy using neural network models, are 7, 5, 4, 4, and 3,respectively. entire set of β j . Therefore, by comparing the deviation of thisfraction of β j with the corresponding fraction of a j , the devi-ation of the entire set of β j with respect to the entire set of a j can be estimated, and thus the fidelity of the input state with respect to the ideal state can be predicted.To summarize, we present in this paper a method for pre-dicting the fidelity of quantum states using neural networkmodels. Compared with previous methods for quantum statefidelity estimation, our method uses fewer measurement set-tings and works for arbitrary quantum states. Here our methodis applicable to non-adversarial scenarios. It has the poten-tial to be used in a wide variety of local quantum informationapplications, such as quantum computation, quantum simula-tion, and quantum metrology. A future research direction is todesign machine-learning-based quantum state fidelity estima-tion schemes in adversarial scenarios.This work was supported by the National Key Re-search and Development Program (2017YFA0305200 and2016YFA0301700), the Key Research and DevelopmentProgram of Guangdong Province of China (2018B030329001and 2018B030325001), the National Natural Science Founda-tion of China (Grant No. 61974168). X. Zhou acknowledgessupport from the National Young 1000 Talents Plan. W. Luoacknowledges support from the National Natural ScienceFoundation of China (Grant No. 61877029). X. Zhangacknowledges support from the National Natural ScienceFoundation of China (Grant No. 62005321). S. Pangacknowledges support from the National Natural ScienceFoundation of China (Grant No. 12075323). [1] A. Chantasri, S.-S. Pang, T. Chalermpusitarak, A. N. Jor-dan, Quantum state tomography with time-continuous mea-surements: reconstruction with resource limitations, QuantumStud.: Math. Found 7 (2020) 23–47.[2] G. Toth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer,H. Weinfurter, Permutationally invariant quantum tomography,Phys. Rev. Lett. 105 (2010) 250403.[3] M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross,S. D. Bartlett, O. Landon-Cardinal, D. Poulin, Y. Liu, E ffi cientquantum state tomography, Nature Communications 1 (2010)149.[4] J. Renes, R. Blume-Kohout, A. Scott, C. Caves, Symmetric in-formationally complete quantum measurements, J. Math. Phys.45(6) (2004) 2171–2180.[5] D. Gross, Y. Liu, S. T. Flammia, S. Becker, J. Eisert, Quantumstate tomography via compressed sensing, Phys. Rev. Lett. 105(2010) 150401.[6] S. T. Flammia, D. Gross, Y. Liu, J. Eisert, Quantum tomographyvia compressed sensing: error bounds, sample complexity ande ffi cient estimators, New J. Phys. 14 (2012) 095022.[7] A. Smith, C. A. Riofr ´ I o, B. E. Anderson, H. Sosa-Martinez,I. H. Deutsch, P. S. Jessen, Quantum state tomography by con-tinuous measurement and compressed sensing, Phys. Rev. A 87(2013) 030102(R).[8] A. Kalev, R. L. Kosut, I. H. Deutsch, Quantum tomographyprotocols with positivity are compressed sensing protocols, npjQuantum Information 1 (2015) 15018.[9] C. Riofr ´ I o, D. Gross, S. Flammia, T. Monz, D. Nigg, R. Blatt,J. Eisert, Experimental quantum compressed sensing for aseven-qubit system, Nat. Commun. 8 (2017) 15305. [10] A. Kyrillidis, A. Kalev, D. Park, S. Bhojanapalli, C. Caramanis,S. Sanghavi, Provable compressed sensing quantum state to-mography via non-convex methods, npj Quantum Information4 (2018) 36.[11] J. Shang, Z. Zhang, H. K. Ng, Superfast maximum-likelihoodreconstruction for quantum tomography, Phys. Rev. A 95(2017) 062336.[12] G. Silva, S. Glancy, H. Vasconcelos, Investigating bias inmaximum-likelihood quantum-state tomography, Phys. Rev. A95 (2017) 022107.[13] C. Oh, Y. Teo, H. Jeong, E ffi cient bayesian credible regioncertification for quantum-state tomography, Phys. Rev. A 100(2019) 012345.[14] V. Siddhu, Maximum a posteriori probability estimates forquantum tomography, Phys. Rev. A 99 (2019) 012342.[15] X. Ma, H. Z. T. Jackson, J.-X. Chen, D.-W. Lu, M. D. Mazurek,K. A. G. Fisher, X.-H. Peng, D. Kribs, K. J. Resch, Z.-F. Ji,B. Zeng, R. Laflamme, Pure-state tomography with the expec-tation value of pauli operators, Phys. Rev. A 93 (2016) 032140.[16] D. Martnez, M. Sol ´ I -Prosser, G. Ca˜ n as, O. Jim´ e nez, A. Del-gado, G. Lima, Experimental quantum tomography assisted bymultiply symmetric states in higher dimensions, Phys. Rev. A99 (2019) 012336.[17] H. Sosa-Martinez, N. Lysne, C. Baldwin, A. Kalev, I. Deutsch,P. Jessen, Experimental study of optimal measurements forquantum state tomography, Phys. Rev. A 119 (2017) 150401.[18] C. Y. L. O. G¨ u hne, W. B. Gao, J. W. Pan, Toolbox for entangle-ment detection and fidelity estimation, Phys. Rev. A 76 (2007)030305(R).[19] Y. Tokunaga, T. Yamamoto, M. Koashi, N. Imoto, Fidelity estimation and entanglement verification for experimentallyproduced four-qubit cluster states, Phys. Rev. A 74 (2006)020301(R).[20] S. T. Flammia, Y. K. Liu, Direct fidelity estimation from fewpauli measurements, Phys. Rev. Lett. 106 (2011) 230501.[21] da S. Marcus P., L. C. Olivier and P. David, Practical Charac-terization of Quantum Devices without Tomography, Phys. Rev.Lett. 107 (2011) 210404.[22] H. J. Zhu, M. Hayashi, Optimal verification and fidelity esti-mation of maximally entangled states, Phys. Rev. A 99 (2019)052346.[23] M. Cerezo, A. Poremba, L. Cincio, P. J. Coles, Variationalquantum fidelity estimation, Quantum 4(248) (2020) 1-16.[24] D. S. Rolando, J. Chiaverini, D. J. Berkeland, Lower boundsfor the fidelity of entangled-state preparation, Phys. Rev. A 74(2006) 052302.[25] J. Wang, Z. Han, S. Wang, Z. Li, L. Mu, H. Fan, L. Wang,Scalable quantum tomography with fidelity estimation, Phys.Rev. A 101 (2020) 032321.[26] D. Mahler, L. A. Rozema, A. Darabi, C. Ferrie, R. Blume-Kohouz, A. Steinberg, Adaptive quantum state tomographyimproves accuracy quadratically, Phys. Rev. Lett. 111 (2013)183601.[27] Z.-H. Li, Y.-G. Han, H.-J. Zhu, E ffi cient verification of bipartitepure states, Phys. Rev. A 100 (2019) 032316.[28] X.-D. Yu, J.-W. Shang, O. G¨ u hne, Optimal verification of gen-eral bipartite pure states, npj Quantum Information 5 (2019)112.[29] H.-J. Zhu, M. Hayashi, E ffi cient verification of pure quantumstates in the adversarial scenario, Phys. Rev. Lett. 123 (2019)260504.[30] Y.-C. Liu, X.-D. Yu, J.-W. Shang, H.-J. Zhu, X.-D. Zhang, Ef-ficient verification of dicke states, Phys. Rev. APPL. 12 (2019)044020.[31] H.-J. Zhu, M. Hayashi, E ffi cient verification of hypergraphstates, Phys. Rev. APPL. 12 (2019) 054047.[32] K. Wang, M. Hayashi, Optimal verification of two-qubit purestates, Phys. Rev. A 100 (2019) 032315.[33] H.-J. Zhu, M. Hayashi, General framework for verifying purequantum states in the adversarial scenario, Phys. Rev. A 100(2019) 062335.[34] Y.-G. H. Z.-H. Li, H.-J. Zhu, Optimal verification ofgreenberger-horne-zeilinger states, Phys. Rev. APPL. 13 (2020)054002.[35] S. Pallister, N. Linden, A. Montanaro, Optimal verification ofentangled states with local measurements, Phys. Rev. Lett. 120(2018) 170502.[36] W.-H. Zhang, C. Zhang, Z. Chen, X.-X. Peng, X.-Y. Xu, P. Yin,S. Yu, X.-J. Ye, Y.-J. Han, J.-S. Xu, G. Chen, C.-F. Li, G.-C.Guo, Experimental optimal verification of entangled states us-ing local measurements, Phys. Rev. Lett. 125.[37] X. H. Jiang, K. Wang, K. Y. Qian, Z. Z. Chen, Z. Y. Chen, L. L.Lu, L. J. Xia, F. M. Song, S. N. Zhu and X. S. Ma, Towardsthe standardization of quantum state verification using optimalstrategies, npj Quantum Information 6(2020) 90.[38] M. Yang, C. Ren, Y. Ma, Y. Xiao, X. Ye, Experimental simulta-neous learning of multiple nonclassical correlations, Phys. Rev.Lett. 123 (2019) 190401.[39] Y.-C. Ma, M.-H. Yung, Transforming bell’s inequalities intostate classifiers with machine learning, npj Quantum Informa-tion 4 (2018) 34.[40] S.-R. Lu, S.-L. Huang, K.-R. Li, J. Li, J.-X. Chen, D.-W. Lu, Z.-F. Ji, Y. Shen, D.-L. Zhou, B. Zeng, Separability-entanglementclassifier via machine learning, Phys. Rev. A 98 (2018) 012315. [41] J. Gao, L.-F. Qiao, Z.-Q. Jiao, Y.-C. Ma, C.-Q. Hu, R.-J. Ren,A.-L. Yang, H. Tang, M.-H. Yung, X.-M. Jin, Experimental ma-chine learning of quantum states, Phys. Rev. Lett. 120 (2018)240501.[42] D.-L. Deng, Machine learning detection of bell nonlocalityin quantum many-body systems, Phys. Rev. Lett. 120 (2018)240402.[43] C.-L. Ren, C.-B. Chen, Steerability detection of an arbitrarytwo-qubit state via machine learning, Phys. Rev. A 100 (2019)022314.[44] T. Xin, S.-R. Lu, N.-P. Cao, G. Anikeeva, D.-W. Lu, J. Li,G.-L. Long, B. Zeng, Local-measurement-based quantum statetomography via neural networks, npj Quantum Information 5(2019) 109.[45] A. Ling, K.-P. Soh, A. Lamas-Linares, C. Kurtsiefer, Experi-mental polarization state tomography using optimal polarime-ters, Phys. Rev. A 74 (2006) 022309.[46] J. A. Miszczak, Generating and using truly random quantumstates in mathematica, Computer Physics Communications 183(2012) 118–124.[47] Q. M. Ding, X. X. Fang, X. Yuan, T. Zhang, H.Lu, E ffi cient estimation of multipartite quantum coherence,arXiv:2010.02612v2 [quant-ph] 15 Oct 2020.[48] F. Tacchino, C. Macchiavello, D. Gerace, D. Bajoni, Measure-ment of qubits, Phys. Rev. A 64 (2001) 052312.[49] The five-qubit state can be represented by the followingvector [0 . + . i ; 0 . − . i ; − . − . i ; − . − . i ; − . − . i ; 0 . − . i ; − . + . i ; 0 . + . i ; 0 . − . i ; 0 . + . i ; 0 . − . i ; 0 . − . i ; − . − . i ; − . − . i ; 0 . − . i ; − . − . i ; − . − . i ; 0 . − . i ; 0 . + . i ; − . − . i ; 0 . − . i ; 0 . + . i ; 0 . + . i ; 0 . + . i ; − . − . i ; 0 . + . i ; 0 . − . i ; − . + . i ; − . + . i ; − . − . i ; 0 . − . i ; − . − . i ]. SUPPLEMENTARY MATERIALS
In this supplementary material, we discuss more results.Sec.(I) describes the basic structure of artificial neural net-works (ANNs) and our artificial neural network. Sec.(II) givesthe method to generate quantum states with specified fidelity.It also analyzes the uniformity of pure state fidelity, mixedstate fidelity The purity distribution of mixed states, the gen-erality of our method, and Poisson noise are also presentedhere. The method for selecting the Pauli operators is givenin Sec.(III). Sec.(IV) compares the accuracy of the neuralnetwork at two di ff erent inputs, i.e., the probability of eachoutcome and the expectation values of the Pauli operators.Sec.(V) analyzes the accuracy of the quantum state fidelity es-timation. Sec.(VI) gives the verification accuracy of the neu-ral network. Sec.(VII) presents a practical application of ourneural network. I. THE STRUCTURE OF ANNS AND OUR ANN
The basic structure of ANNs
ANNs consists of an input layer, hidden layers and an out-put layer (See FIG.1). The input layer consists of k × n neurons corresponding to the probability of each outcome, inwhich n represents the number of qubits and k represents thenumber of Pauli combinations. We set the inputs x and theintermediate vector x in the hidden layer generated by thenon-linear relation x = σ RL ( W x + ω ) , (4)where σ RL is the ReLU function for each neuron in the hiddenlayer, defined as σ RL ( z i ) = max ( z i , i = , , , ... ). The ma-trix W is the initialized weight and the vector ω is the biasbetween the input layer and the hidden layer. The optimaloutput vector denoted as x is generated using the function x = σ s ( W x + ω ) , (5)where σ s is the Softmax function defined by σ ( z i ) = e zi P k = e zk ( i = , ..., W is the initialized weightbetween the hidden layer and the output layer, while the vector ω is the bias. The loss function is categorical cross-entropyand is written as − n [ y s loga s + (1 − y s ) log (1 − a s )]. The sub-script s denotes the sequence number of the training sample,and the notation y represents the labels defined by the cri-terion, a means the output labels of the ANN and n is thetraining set number, respectively. During the machine learn-ing process, W , ω , W , ω are continuously optimized untilthe confidence level reaches saturation, and then the trainingis stopped. M Input Layer Hidden Layer Output Layer MM & (cid:20) w W x x x & (cid:21) W w FIG. 4. (Color online) Artificial neural networks with hidden layers.The objective of machine learning is to optimize σ s ( W σ RL ( W x + ω ) + ω ), where σ RL is ReLU function and σ s ( z ) i = e zi P kk = e zk ( i = , , ..., K ) is softmax function. The input data x = { a , a , . . . , a k } are the measurements of Pauli operators. Matrix W and the vector ω are initialized uniformly and optimized during the learning pro-cess. The number of neurons in the hidden layer can be varied foroptimal performance. The output data x are the predicted fidelity. The ANN for fidelity estimation
We use four 2080Ti GPUs. We choose the optimizer thathas the best performance in our task among almost all thebuilt-in optimizers in TensorFlow: NadamOptimizer (adap-tive moment estimation). This neural network contains 122 labels, using 1,952,000 data for training and 488,000 data forvalidation, i.e. each label contains 16,000 training data and4,000 validation data. By tuning the batch size of inputs,the number of neurons, and the number of training rounds,the performance of the neural network is continuously opti-mized. Eventually, the parameters required for training thisneural network from two-qubit to seven-qubit quantum statesare shown in Table S1. Hid-neu represents the number of thehidden neurons.
Table S1. Parameters of our ANNtwo-qubit states three-qubit statesstate epoch Batch size Hid-neu state epoch Batch size Hid-neuBell 200 2048 1000 GHZ 400 4096 1000 W | ϕ i | ϕ i | ϕ i Wsix-qubit states | ϕ i state epoch Batch size Hid-neu seven-qubit states C
500 16384 state epoch Batch size Hid-neuDicke 500-500 C
500 32768GHZ 1000-1000 300-600-300W 1500-1500 500-1000-500 | ϕ i Next, we show the specific forms of the special numberstates that appear in Table S1 (See FIG.2). Moreover, | ϕ i , | ϕ i , | ϕ i , | ϕ i , | ϕ i are general quantum states [5]. The two-qubit Bell state and the W state are | φ + i Bell = √ | i + | i ) , | W i = √ | i + | i + | i ) . (6)The three-qubit GHZ state and the W state are | GHZ i = √ | i + | i ) , | W i = √ | i + | i + | i ) . (7)The four-qubit Cluster state, the Dicke state, the GHZ stateand the W state are | Cluster i =
12 ( | i + | i + | i − | i ) , | Dicke i = √ | i + | i + | i + | i + | i + | i ) , | GHZ i = √ | i + | i ) , | W i =
12 ( | i + | i + | i + | i ) . (8)The five-qubit Cluster state, the C-ring state, the Dicke state,the GHZ state and the W state are | Cluster i =
12 ( | + + + i + | + − −i + | − − + i + | − + −i ) , | C - ring i = √ | + + i + | − + i + | + − i − | − − i + | − − i + | + − i + | − + i − | + + i ) , | Dicke i = √
10 ( | i ) + | i + | i + | i + | i + | i + | i + | i + | i + | i ) , | GHZ i = √ | i + | i ) , | W i = √ | i + | i + | i + | i + | i ) . (9)The six-qubit Cluster state, the Dicke state, the GHZ state andthe W state are | C i =
12 ( | + + + + i + | + + − −i + | − − + + i − | − − − −i ) , | Dicke i = √
15 ( | i + | i + | i + | i + | i + | i + | i + | i + | i + | i + | i + | i + | i + | i + | i ) , | GHZ i = √ | i + | i ) , | W i = √ | i + | i + | i + | i + | i . (10)The seven-qubit Cluster state is | C i = √ | + + + + i + | + + − −i + | + − + + i + | + − − −i + | − + + + i − | − + − −i−| − − + + i + | − − − −i ) . (11) II. GENERATION AND ANALYSIS OF QUANTUM STATES
A. Generation of quantum pure states with specified fidelity
Preparing quantum states has an important role in realiz-ing quantum information and quantum computing, but oftenthe imperfection of devices and the influence of noise resultin obtaining quantum states that are all mixed states. Here,we use neural network techniques to evaluate whether the fi-delity between this quantum state and the ideal state satisfiesthe requirements. Our neural network inputs are derived frommixed states with specified fidelity. We first introduce themethod for generating a pure state.Step 1. Generating an arbitrary pure stateIn Mathematics, we create a pure state of arbitrary dimen-sion with the help of the function
RandomKet(D) [1]. Specifi-cally, The
RandomKet(D) function calls
RandomSimplex(D) and
Randomreal(D) to generate a D-dimensional arbitrarypure state. The Mathematic code of generating an arbitrarypure state is shown below. RandomSimplex [ d ] : = Blo ck [ { r , r 1 , r 2 } ,r = S o r t [ T a b l e [ RandomReal [ { } ] , { i , 1 , d − } ] ] ;r 1 = Append [ r , 1 ] ;r 2 = P r e p e n d [ r , 0 ] ; r 1 − r 2] ;RandomKet [ n ] : = Blo ck [ { p , ph } ,p = S q r t [ RandomSimplex [ n ] ] ;ph = Exp [ I RandomReal [ { \ [ P i ] } , n − = P r e p e n d [ ph , 1 ] ;p * ph] ;Step 2. Generation of a pure state with specified fidelitycorresponding to the state | i ⊗ n An arbitrary pure state can be expanded as | ϕ i = n − X i = α i | i i , (12)where | i i ( i = , , ..., n −
1) is basis vector of calculations.For convenience, we rewrite Eq.(9) as follows. | ϕ i = f | i ⊗ n + n − X i = α i | i i = f | i + q − f | φ i n − , (13)where α = f . The state | φ i n − is a (2 n −
1) dimensionalarbitrary pure state. Therefore, the fidelity f between | i ⊗ n and | ϕ i is given as f = F ( | i ⊗ n , | ϕ i ) = |h | ⊗ n | ϕ i| (14)Hence, we can generate an n -qubit pure state dataset with thetarget state | i ⊗ n using Matlab.Step 3. Generation of a pure state with a specified fidelitycorresponding to an arbitrary pure stateFor convenience, we set | i = | i ⊗ n . Then we rewrite theEq.(11) as follows. f = F ( | i , σ ) = p h | σ | i , (15)where σ can be viewed as a state in the dataset S with a targetstate | i . If we choose a new target pure state ρ , there is aunitary matrix transformation U from | ih | to ρ . This unitary U can be calculated by the code Findunitary.m . As soon assuch unitary is found [2, 3], the relative state of σ is directlyobtained as σ ′ = U σ U † , (16)where the state σ ′ belongs to the database S ′ of the target state ρ . The fidelity f ′ between the target state ρ and the state σ ′ can be calculated, that is, f ′ = F ( ρ, σ ′ ) = p tr ( ρ, σ ′ ) = p h | U † σ ′ U | i = p h | σ | i = f . (17)Step 4. Projective measurementsEach measurement setting k is characterized by W k , and eachspecific result p ( k j ) is associated with a projection operator P k j = | v k j ih v k j | , j = , , ..., n , (18) (a) (b) (c) (d) (e) FIG. 5. The structure of Cluster states. (a) The four-qubit line Cluster state | Cluster i . (b) The five-qubit line Cluster state | Cluster i . (c) Thefive-qubit ring Cluster state | C - ring i . (d) The six-qubit grid Cluster state | C i . (e) The seven-qubit Cluster state | C i . where | v k j i is the j th eigenvector of the W k , p ( k j ) is equal to tr ( ρ P k j ). B. Generation of quantum mixed states with specified fidelity
Here we give the method for generating mixed states withspecified fidelity. In a Ginibre matrix G , each element is thestandard complex normal distribution CN (0 , ρ = GG † tr ( GG † ) . (19) Inspired by the Ginibre matrix G , we propose a method toprepare specified fidelity states with the target state | i , i.e.,the density matrix of desired N-qubit mixed state is ρ n = G n G † n tr ( G n G † n ) . (20)where the matrix G n can be expressed as G n = √ m x e − π i ∗ rand q − x e − π i ∗ rand | ϕ i , ..., √ m n x n e − π i ∗ rand n p − x n e − π i ∗ rand n | ϕ n i ! (21) The notations rand b and rand b ( b = , , ..., n ) represent ran-dom numbers. The set {| ϕ i , ..., | ϕ n i} is a collection of 2 n − RandomKet . { m , .., m n } is a set of real numbers of 2 N dimension normal-ized standard normal distribution. { x , ..., x n } is a set of un-defined real numbers that are closely related to the expectedfidelity.The steps for preparing the mixed state are similar to thosefor preparing the pure state. As can be seen from Eq.(18), thedensity matrix of a mixed state with specified fidelity needsto determine the values of { m , ..., m n } and { x , ..., x n } , wherethe former is generated randomly using the Matlab code and the latter is determined according to the corresponding con-straints. An example. –For a two-qubit state, suppose the desired fi-delity is f for the target state | i . The values { x , x , x , x } in the matrix G n can be determined from Eqs. (19-22) us-ing Matlab. The value range of x can first be determined inEq.(19) by the given fidelity f . When we fix the x randomlyand uniformly, the value range of x can also be determinedin Eq.(20). Then we fix the x randomly and uniformly, andthe value range of x can also be defined in Eq.(21). Once thevalue of x is fixed randomly and uniformly, the value of x isalso fixed in Eq.(22). Therefore, we obtain the matrix G n . x ∈ ( min ( max ( f − P i = m i m , , , min ( max ( f m , , , (22) x ∈ ( min ( max ( f − m x P i = m i m , , , min ( max ( f − m x m , , , (23) x ∈ ( min ( max ( f − P i = m i x i − m m , , , min ( max ( f − P i = m i x i m , , , (24) x = f − P i = m i x i m . (25) C. Uniformity analysis of pure state fidelity
We use a computer to generate 500 single-qubit pure states. The fidelity between each of 500 pure states and the state | i is √ .
5. We also give a geometric representation of the Blochball in FIG.3(a). It is obvious that the distribution of 500 statesis uniform.In addition, we verify the uniformity of four-qubit purestates fidelities. We generate 10,000 four-qubit pure states,in which the fidelity between each of these states and the state | i is 0.25. We select 20 states out of the 10,000 states, andthe fidelity between each selected state and 9,999 other statescan be calculated. We get 20 similar distributions, includingthe distribution of random state 6 and the distribution of therandom state 20 in FIG.3(b). We conclude that the distributionof 10,000 states is uniform. D. Uniformity analysis of mixed state fidelity
Here, we verify the uniformity of four-qubit mixed statesdatasets in FIG.3(c). A total of 1,220 states are generated, inwhich the fidelity between 1,220 states and the state | i is0.25. We also select 20 states out of 1,220 states. The fidelity,between each selected state and the other 1,219 states, can becalculated. We get 20 similar distribution patterns, includingthe distribution of random state 6, random state 15 and thedistribution of random state 20 in FIG.3(c). We found that thedistribution of the 1,220 states is uniform. E. Distribution of di ff erent purities of mixed states The purity of a quantum state ρ is defined as tr ( ρ ), where tr ( ρ ) = m can control thepurity of quantum states. In FIG.4(a-b) the fidelity f betweenthe prepared state and | i is 0.25, and the fidelity f is 0.8in FIG.4(c-d). The controller m is equal to 1, 0.9, 0.6, 0.2 or0.01. Moveover, m can also be a uniform distribution U (0 , | ϕ i , in which eight datasets belong to mixedstates and the rest one to pure states. In eight mixed datasets,the purity distribution of m is changed for seven datasets andthe remaining one does not change the distribution of m .By comparing the distribution from Table S3, it can beseen that the distribution m = − rand is better than oth-ers because this distribution is e ff ective for mixed states andpure states. It is worth note that the states in this distribu-tion m = − rand are closer to pure states than those inother distributions. Here we only show the validation resultsof k = , , F. Universality of our method
We will prove the universality of our methods for prepar-ing quantum states. To better demonstrate this inference, weshow three dataset classes with three six-qubit general targetpure states | ϕ i , | ϕ i , | ϕ i , respectively. By analyzing the re-sults from Table S4, we conclude that our method of prepar-ing database is universal, corresponding to an arbitrary targetpure state. G. Poisson noise analysis
In an experiment, noise is inevitable. In our work, we as-sume that the noise follows the Poisson law. Using the random function in Matlab, we can obtain one value from the Poissondistribution with NP , in which P is the basis measurements,and NP is the ideal coincidence count.We give a comparison of di ff erent number of samples fora four-qubit general state (See FIG.5). The noise model usesthe number of samples of N = N = N = III. SELECTION OF PAULI COMBINATIONS
Let W k ( k = , , ..., n ) denotes all possible Pauli operators( n -fold tensor products of ( σ x , σ y and σ z ). Then tr ( ρσ ) canbe rewritten in Ref.[1] as follows. tr ( ρσ ) = X k d χ ρ ( k ) χ σ ( k ) , (26)where the characteristic function is defined as χ ρ ( k ) = tr ( ρ W k ). For a target pure state ρ , we define the weight Pr ( k )corresponding to the Pauli operators W k . Pr ( k ) = [ χ ρ ( k )] , (27)If we want to know the fidelity between a target pure state andan arbitrary state, we need to get χ σ ( k ) in Eq.(23) related tothe weight Pr ( k ) ,
0. Obviously, the general state requiresmore Pauli combinations than a special state. Here we havechosen the k Pauli operators with the largest absolute value ofthe expectation value of the desired quantum state.Here we show the accuracy of the neural network for thePauli operators with a large absolute value of expectation andfor the Pauli operators with a small absolute value of expecta-tion, respectively (See Table S5). Take the four-qubit clusterstate and the general state as an example, the ANN model forthe Pauli operator with a large absolute value of expectationcan get more information than that for the Pauli operator witha small absolute value of expectation.We then show the selected Pauli operators from two-qubitto seven-qubit states in Table S6.
IV. COMPARISON OF THE ACCURACY OF NEURALNETWORK WITH TWO CLASSES OF INPUTS
In Table S7, we present the results of the artificial neu-ral network trained with the probability of the measurementoutcomes of the expectation and the artificial neural networktrained with the expectation. The bases ANN represents thatthe inputs that are the probabilities of measurement. The ex-pectation ANN represents that the inputs are the expectationvalues. It can be seen that the accuracy of the two kinds ofneural network models is almost equivalent. (a)
Random state 6
Fidelity N u m Random state 20
Fidelity N u m (b) Randomstate6
Fidelity N u m Randomstate15
Fidelity N u m Randomstate20
Fidelity N u m (c) FIG. 6. (Color Online) (a) Distribution of 500 single-qubit states in the Bloch ball; (b)The distribution of 10,000 four-qubit pure statescontaining state 6 and state 20; (c) The distribution of 1,220 four-qubit mixed states including state 6, state 15 and state 20.Table S2. Nine distributions of m ANN models datasets m distributionA 1 m = − rand ∗ rand B 2 m = − √ rand ∗ rand C 3 m belongs to a uniform distribution U (0 , m belongs to a random standard normal distribution N (0 , m = − rand ∗ rand ∗ rand G 7 m = − rand ∗ rand ∗ rand ∗ rand H 8 m = − rand I 9 m = − rand ∗ rand V. PRECISION OF FIDELITY ESTIMATION
FIG.6 presents some important information about the neu-ral network with di ff erent number of labels. In FIG.6(a-c),in the case of a fixed number of labels, the higher the num-ber of Pauli operator measurement settings used, the higherthe prediction accuracy is. Meanwhile, the higher the fidelityof the predicted quantum states is, the higher the accuracy is.In FIG.6(d), the more quantum state fidelity intervals are di-vided, the higher the accuracy is. However, the accuracy ofthe neural network model with 234 labels is not much higherthan that of the neural network model with 122 labels. More-over, the higher the number of labels are, the more resourcesand time are consumed, so the neural network with 122 labelsis selected as the most appropriate. The fidelity interval of thequantum state is respectively divided into 66 labels, 122 labelsand 234 labels for the specific interval in Ref.[4]. VI. ACCURACIES OF ANN MODELS FOR N-QUBITSTATES
In Table S8, we show all the accuracies of the ANN modelsfrom two-qubit to seven-qubit states with the precision ± VII. APPLICATIONS OF OUR NEURAL NETWORKS
How can we use a trained neural network to determinewhether the fidelity of an input quantum state is higher than96%. The fidelity of this quantum state is first predicted us-ing a neural network with k =
2. If the upper bound of thepredicted fidelity range given does not exceed 96%, the neu-ral network gives the prediction that determines that the fi-delity of this state does not exceed 96%. If the lower boundof the predicted fidelity range is more than 96%, the neuralnetwork will give a prediction that the fidelity of this state is0
Table S3. Accuracy of nine ANN models for nine datasetsdatasets 1 2 3 4 5 6 7 8 9 k model A3 79.78% 80.57% 81.11% 77.19% 72.60% 78.09% 76.41% 78.32% 77.12%4 86.99% 87.33% 87.42% 83.37% 82.62% 86.34% 85.45% 86.10% 85.63%5 89.82% 90.26% 89.94% 85.96% 87.26% 90.01% 89.48% 89.43% 89.59% k model B3 78.84% 80.24% 81.21% 77.61% 67.65% 76.47% 73.82% 77.04% 74.98%4 86.34% 87.03% 87.36% 83.75% 78.13% 84.70% 82.97% 84.84% 83.57%5 89.58% 90% 89.70% 86.49% 83.56% 88.89% 87.93% 88.65% 88.17% k model C3 77.53% 79.07% 80.29% 77.44% 66.83% 74.62% 71.98% 75.59% 75.59%4 85.36% 86.32% 87.11% 83.61% 76.39% 83.14% 81.01% 83.69% 83.69%5 88.67% 89.38% 89.72% 86.24% 81.36% 87.02% 85.34% 87.29% 87.29% k model D3 75.29% 77.62% 79.37% 76.67% 59.93% 70.81% 67.10% 72.51% 69.18%4 82.65% 84.39% 85.60% 82.80% 69.73% 79.13% 76.01% 80.27% 77.61%5 85.72% 86.91% 87.74% 85.26% 76.09% 83.25% 80.90% 83.89% 82.05% k model E3 22.93% 15.73% 11.10% 10.96% 85.05% 38.62% 54.07% 34.86% 46.63%4 26.66% 18.21% 12.64% 12.63% 91.83% 44.57% 61.47% 39.29% 52.52%5 27.60% 18.59% 12.49% 12.24% 95.07% 46.41% 64.04% 40.60% 54.55% k model F3 80.02% 80.13% 80.18% 76.15% 75.52% 79.66% 78.62% 79.19% 78.86%4 87.47% 86.94% 86.49% 82.64% 85.10% 87.66% 87.48% 86.93% 87.10%5 90.45% 90.24% 89.55% 85.58% 90.70% 91.27% 91.37% 90.46% 91.04% k model G3 79.13% 78.84% 78.65% 74.30% 78.77% 79.58% 79.67% 78.96% 78.39%4 86.63% 85.97% 85.35% 81.16% 86.86% 87.63% 88.16% 86.78% 87.17%5 90.43% 89.57% 88.58% 84.44% 91.87% 91.67% 92.25% 90.50% 90.92% k model H3 79.09% 79.51% 80.78% 76.93% 76.03% 78.37% 77.45% 78.65% 78%4 87.26% 87.15% 86.80% 83.11% 84.64% 87.09% 86.27% 86.75% 86.58%5 90.65% 90.32% 89.24% 85.77% 89.30% 90.94% 90.38% 90.28% 90.59% k model I3 79.73% 79.88% 79.15% 75.84% 76.82% 79.62% 79.18% 79.33% 78.65%4 86.82% 86.36% 86.39% 82.18% 86.92% 87.30% 87.53% 86.79% 87.39%5 90.08% 89.46% 88.81% 85.18% 91.23% 91% 91.46% 90.26% 91.08%Table S4. Datasets comparison of three six-qubit general statessix-qubit (Hid-neu:1500-1500, precision with ± k states | ϕ i | ϕ i | ϕ i m = 0.01 purity N u m m = 0.2 purity N u m m = 0.6 purity N u m (a) f = m = 0.9 purity N u m m = 1 purity N u m m U(0,1) purity N u m (b) f = m = 0.01 purity N u m m = 0.2 purity N u m m = 0.6 purity N u m (c) f = m = 0.9 purity N u m m = 1 purity N u m m U(0,1) purity N u m (d) f = FIG. 7. (Color Online) Comparison of mixed-state distributions for di ff erent purity, where m = , . , . , . , . , U (0 , more than 96%. If the range of prediction fidelity is given in-cluding 96%, it is necessary to determine whether the predic-tion accuracy reaches ± ± ± = [1] J. A. Miszczak, Generating and using truly random quantumstates in mathematica, Computer Physics Communications 183 (2012) 118–124. Number of Samples (N) A cc u r a cy k=2k=3k=4k=5k=6k=7 FIG. 8. (Color online) Comparison of the accuracy of neural network models with di ff erent number of samples. Taking a four-qubit generalstate as an example, when k = N is, the higher the accuracy is.Table S5. Comparison of two types of weightsfour-qubit states (Precision ± k states | ϕ i (largest weight, | ϕ i (smallest weight, Cluster (weight =
1, Cluster (weight = ffi cientquantum state tomography, Nature Communications 1 (2010)149.[3] F. Tacchino, C. Macchiavello, D. Gerace, D. Bajoni, An artifi-cial neuron implemented on an actual quantum processor, npjQuantum Information 5 (2019) 26.[4] For 66 labels, the division of the fidelity is [0 : 0 . . , .
61 :0 .
02 : 0 . , − . : . : 1]. For 122 labels, the division of the fidelity is [0 : 0 .
05 : 0 . , .
61 : 0 .
01 : 0 . , − . : . : 1].For 234 labels, the division of fidelity is [0 : 0 .
025 : 0 . , .
605 :0 .
005 : 0 . , − . : . : 1]. A selection of 66 labels, 122labels and 234 labels is convenient for comparing results as theerror bar can be set to ± | ϕ i , | ϕ i , | ϕ i , | ϕ i , | ϕ i asfollows. Table S6. Pauli operatorstwo-qubit three-qubitstates Pauli operators states Pauli operatorsBell XX;YZ;ZY;YY;ZX;XZ;XY GHZ ZZZ;XXX;XYY;YXY;YYX; YYY;XXZ W XX;YZ;ZY;XZ;YY;ZX;ZZ W ZZZ;ZXX;ZYY;XZX;XXZ;YZY;YYZ | ϕ i XY;ZX;YZ;XX;YY;ZZ;XZ | ϕ i YYY;XXY;ZZY;XYZ;YXX;YXZ;YXYfour-qubitCluster ZZXX;ZZYY;XXZZ;YYZZ;XYXY;XYYX;YXXYDicke XXXX;YYYY;ZZZZ;XXZZ;ZZYY;ZZXX;YYZZGHZ XXXX;YYYY;ZZZZ;XXYY;XYXY;YXYX;YYXXW ZZZZ;ZZXX;ZZYY;XXZZ;YYZZ;XZXZ;YZYZ | ϕ i YXXZ;XYZX;ZZYY;YZXX;XYZY;YXZZ;ZZYXfive-qubitCluster XZZXZ;ZYXYZ;ZXZZX;YXXXY;YYZZX;XZZYY;ZYXXYC-ring XXXXX;ZYXYZ;ZZYXY;XYZZY;YXYZZ;YZZYX;XZZYXDicke ZZZZZ;XXXXZ;YYYZY;ZZZXX;ZYYYY;YZZYZ;XXXZXGHZ XXXXX;ZZZZZ;XYXXY;XYXYX;XYYXX;YXYYY;YYXYYW ZZZZZ;XXZZZ;XZZXZ;YYZZZ;YZZYZ;ZXZXZ;ZYZYZ | ϕ i YYXZX;XZZYZ;ZYYYY;XZYXX;XXZZY;YZXXZ;ZYXZYsix-qubitC23 XZXYXY;XZXYXY;ZXYZYZ;ZYZYXZ;YXYXZX;XZXZYY;ZYZZXYDicke ZZZZZZ;XXZZZZ;ZZZZYY;ZZYYZZ;ZZXZZX;ZZZXXZ;YYZZZZGHZ XXXXXX;YYYYYY;ZZZZZZ;XXXYYY;XYYYYX;YXYYXY;YYYYXXW ZZZZZZ;XZXZZZ;XZZZZX;YYZZZZ;YZZZYZ;ZXZXZZ;ZYZZYZ | ϕ i XXYXYX;ZZXYZZ;YYZYXZ;ZZXZXY;ZXYXYY;ZYZZZX;YZZYZYseven-qubitC34 XZXZYXY;ZYXYXZX;ZXYZZYZ;XZYYYYZ;YYYZZXY;YXXYXZX;XZXZZYYTable S7. Comparison of two classes of ANN inputsA four-qubit general state(Hid-neu:5000, precision with ± k Bases ANN Expectation ANN2 62.15% 61.47%3 76.28% 77.06%4 84.22% 84.84%5 87.98% 88.47%6 91.22% 91.86%7 94.03% 94.22% -2-1012 + P r e c i s i on ( % ) +++++ -2-1012 -2-1012 -2-1012 ++ (a) the state | ϕ i with 66 labels -2-1012 P r e c i s i on ( % ) ++ -2-1012 ++ -2-1012 ++ -2-1012 ++ (b) the state | ϕ i with 122 labels -2-1012 + P r e c i s i on ( % ) +++++ -2-1012 -2-1012 -2-1012 ++ (c) the state | ϕ i with 234 labels P r e c i s i on ( % ) Fidelity (d) the state | ϕ i with 66 labels, 122 labels and 234 labels FIG. 9. (Color Online) A plot of the prediction accuracy of the neural network versus the quantum state fidelity when measurements are madeusing three, four, five, and six Pauli operator measurement settings. Here we choose a 5-qubit general state | ϕ i as the target state. The higherthe number of Pauli operator measurement settings used, the higher the prediction accuracy is. The higher the fidelity of the predicted quantumstates is, the higher the accuracy is. The more quantum state fidelity intervals are divided, the higher the accuracy is. However, the higherthe number of labels are, the more resources and time are consumed, so the neural network model with 122 labels is selected as the mostappropriate. Table S8. Accuracies of ANN models k states two-qubit (Hid-neu:2000) three-qubit (Hid-neu:2000)Bell W | ϕ i GHZ W | ϕ i k states four-qubit (Hid-neu:2000)Cluster GHZ W Dicke | ϕ i k states five-qubit (Hid-neu:1500-1500)Cluster C-ring Dicke GHZ W | ϕ i k states six-qubit (Hid-neu:1500-1500) seven-qubit (Hid-neu:1500-1500)GHZ C23 W Dicke | ϕ i C342 91.24% 98.23% 99.03% 96.12% 78.61% 99.81%3 99.89% 98.59% 99.17% 97.25% 86.84% 99.82%4 99.95% 98.77% 99.57% 98% 91.39% 99.94%5 99.94% 99.76% 99.67% 98.30% 93.86% 99.96%6 99.95% 99.84% 99.66% 98.96% 96.18% 99.97%7 99.98% 99.92% 99.75% 99.07% 97.20% 99.99% | ϕ i = [0 . − . − . i ; − . + . i ; − . + . i ] | ϕ i = [0 . . + . i − . + . i ; 0 . − . i ; − . + . i ;0 . + . i ; − . − . i ; 0 . + . i ] , | ϕ i = [0 . . + . i ; − . − . i ; 0 . − . i ; 0 . − . i ; 0 . + . i ;0 . − . i ; − . + . i ; − . − . i ; − . + . i ; − . − . i ;0 . + . i ; − . + . i ; 0 . + . i ; − . − . i ; − . + . i ] , | ϕ i = [0 . . − . i ; − . − . i ; − . − . i ; − . − . i ; 0 . − . i ; − . + . i ; 0 . + . i ; 0 . − . i ; 0 . + . i ; 0 . − . i ; 0 . − . i ; − . − . i ; − . − . i ; 0 . − . i ; − . − . i ; − . − . i ; 0 . − . i ;0 . + . i ; − . − . i ; 0 . − . i ; 0 . + . i ; 0 . + . i ; 0 . + . i ; − . − . i ; 0 . + . i ; 0 . − . i ; − . + . i ; − . + . i ; − . − . i ;0 . − . i ; − . − . i ] , | ϕ i = [0 . − . − . i ; − . + . i ; − . − . i ; − . + . i ; 0 . + . i ;0 . + . i ; − . − . i ; − . − . i ; 0 . − . i ; 0 . − . i ; 0 . + . i ;0 . − . i ; − . − . i ; 0 . + . i ; 0 . − . i ; − . − . i ; − . + . i ; − . + . i ; − . + . i ; 0 . − . i ; 0 . − . i ; − . + . i ; 0 . + . i ; − . − . i ; 0 . + . i ; − . − . i ; − . − . i ; 0 . − . i ; 0 . − . i ;0 . + . i ; 0 . + . i ; − . − . i ; 0 . − . i ; 0 . − . i ; 0 . + . i ;0 . − . i ; − . + . i ; − . + . i ; − . − . i ; 0 . + . i ; − . − . i ; − . − . i ; − . + . i ; 0 . − . i ; − . + . i ; 0 . + . i ; 0 . + . i ; − . + . i ; − . − . i ; 0 . − . i ; 0 . − . i ; − . − . i ; − . + . i ; − . + . i ; 0 . + . i ; − . − . i ; − . − . i ; − . + . i ; − . + . i ;0 . + . i ; 0 . + . i ; − . + . i ; − . − . ii