Abstract

Inspired by photonic one-way quantum computation, we describe a microwave signal processing method for implementing unitary transforms based on measuring the cebits encoded in the “classical microwave graph state (CMGS).” Here the terms “cebit” and “CMGS” defined in our system are classical analogies of a qubit and certain target quantum graph states in quantum physics respectively, which can exhibit some similar behaviors and resultants. The constructions of 4- and 16-cebit CMGSs as examples are discussed in detail and specific tomography methods are introduced to characterize their qualities. By performing operations on these CMGSs, we implement some basic 2 × 2, 4 × 4, and specific generalized unitary transforms, and obtain output results with high fidelities. Furthermore, we also demonstrate that a simulation of an efficient Grover’s search algorithm, which has been executed in one-way quantum computing schemes, can be directly realized via a certain 4-cebit CMGS. Due to the excellent parallel efficiency and credible outcomes in the proposal, this quantum-inspired method may provide benefits for exploring new ways to microwave information processing, or in turn as an alternative tool for simulating particular quantum systems to some extent.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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2018 (1)

Z. Yang, J. Bao, H. Sun, and X. Zhang, “Experimental simulation of solution to boson sampling based on classical electronic circuits with exponential frequency bandwidth,” EPL 123(5), 50003 (2018).
[Crossref]

2017 (4)

L. Zhao, J. T. Sheridan, and J. J. Healy, “Unitary Algorithm for Nonseparable Linear Canonical Transforms Applied to Iterative Phase Retrieval,” IEEE Signal Process. Lett. 24(6), 814–817 (2017).
[Crossref]

S. Ouelha, S. Touati, and B. Boashash, “An efficient inverse short-time Fourier transform algorithm for improved signal reconstruction by time-frequency synthesis: Optimality and computational issues,” Digit. Signal Process. 65, 81–93 (2017).
[Crossref]

C. Figgatt, D. Maslov, K. A. Landsman, N. M. Linke, S. Debnath, and C. Monroe, “Complete 3-qubit Grover search on a programmable quantum computer,” Nat. Commun. 8(1), 1918 (2017).
[Crossref] [PubMed]

C. Li, Z. Peng, T.-Y. Huang, T. Fan, F.-K. Wang, T.-S. Horng, J.-M. Munoz-Ferreras, R. Gomez-Garcia, L. Ran, and J. Lin, “A review on recent progress of portable short-range noncontact microwave radar systems,” IEEE Trans. Microw. Theory Tech. 65(5), 1692–1706 (2017).
[Crossref]

2016 (4)

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116(25), 250404 (2016).
[Crossref] [PubMed]

C. Greganti, M.-C. Roehsner, S. Barz, T. Morimae, and P. Walther, “Demonstration of measurement-only blind quantum computing,” New J. Phys. 18(1), 013020 (2016).
[Crossref]

Z. Hou, H.-S. Zhong, Y. Tian, D. Dong, B. Qi, L. Li, Y. Wang, F. Nori, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Full reconstruction of a 14-qubit state within four hours,” New J. Phys. 18(8), 083036 (2016).
[Crossref]

B. R. La Cour, C. I. Ostrove, G. E. Ott, M. J. Starkey, and G. R. Wilson, “Classical emulation of a quantum computer,” Int. J. Quant. Inf. 14(04), 1640004 (2016).
[Crossref]

2015 (2)

B. R. L. Cour and G. E. Ott, “Signal-based classical emulation of a universal quantum computer,” New J. Phys. 17(5), 053017 (2015).
[Crossref]

T. Bäckström, “Decorrelating MVDR filterbanks using the non-uniform discrete Fourier transform,” IEEE Signal Process. Lett. 22(4), 479–483 (2015).
[Crossref]

2014 (1)

I. Amiri and J. Ali, “Simulation of the single ring resonator based on the Z-transform method theory,” Quantum Matter 3(6), 519–522 (2014).
[Crossref]

2013 (3)

B. Qi, Z. Hou, L. Li, D. Dong, G. Xiang, and G. Guo, “Quantum state tomography via linear regression estimation,” Sci. Rep. 3(1), 3496 (2013).
[Crossref] [PubMed]

B. A. Bell, M. S. Tame, A. S. Clark, R. W. Nock, W. J. Wadsworth, and J. G. Rarity, “Experimental characterization of universal one-way quantum computing,” New J. Phys. 15(5), 53030 (2013).
[Crossref]

J. Chen, L. Wang, E. Charbon, and B. Wang, “Programmable architecture for quantum computing,” Phys. Rev. A 88(2), 022311 (2013).
[Crossref]

2012 (5)

J. A. Smolin, J. M. Gambetta, and G. Smith, “Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise,” Phys. Rev. Lett. 108(7), 070502 (2012).
[Crossref] [PubMed]

X.-C. Yao, T.-X. Wang, H.-Z. Chen, W.-B. Gao, A. G. Fowler, R. Raussendorf, Z.-B. Chen, N.-L. Liu, C.-Y. Lu, Y.-J. Deng, Y.-A. Chen, and J.-W. Pan, “Experimental demonstration of topological error correction,” Nature 482(7386), 489–494 (2012).
[Crossref] [PubMed]

S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, “Programmable multimode quantum networks,” Nat. Commun. 3(1), 1026 (2012).
[Crossref] [PubMed]

X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, “Observation of eight-photon entanglement,” Nat. Photonics 6(4), 225–228 (2012).
[Crossref]

V. Gikas, “Ambient vibration monitoring of slender structures by microwave interferometer remote sensing,” J. Appl. Geod 6(3–4), 167–176 (2012).
[Crossref]

2011 (1)

A. Sehrawat, D. Zemann, and B.-G. Englert, “Hybrid Quantum Computation,” Phys. Rev. A 83(2), 022317 (2011).
[Crossref]

2010 (2)

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105(15), 150401 (2010).
[Crossref] [PubMed]

G. De Pasquale, G. Bernardini, P. P. Ricci, and C. Gentile, “Ambient Vibration Testing of Bridges by Non-Contact Microwave Interferometer,” IEEE Aerosp. Electron. Syst. Mag. 25(3), 19–26 (2010).
[Crossref]

2009 (2)

O. Gühne and G. Tóth, “Entanglement detection,” Phys. Rep. 474(1–6), 1–75 (2009).
[Crossref]

H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, “Measurement-based quantum computation,” Nat. Phys. 5(1), 19–26 (2009).
[Crossref]

2008 (1)

M. D. de Burgh, N. K. Langford, A. C. Doherty, and A. Gilchrist, “Choice of measurement sets in qubit tomography,” Phys. Rev. A 78(5), 052122 (2008).
[Crossref]

2007 (1)

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3(2), 91–95 (2007).
[Crossref]

2006 (1)

N. Yilmazer, Jinhwan Koh, and T. K. Sarkar, “Utilization of a unitary transform for efficient computation in the matrix pencil method to find the direction of arrival,” IEEE Trans. Antenn. Propag. 54(1), 175–181 (2006).
[Crossref]

2005 (4)

S. Samadi, M. O. Ahmad, and M. N. S. Swamy, “Z-transform of quantized ramp signal,” IEEE Trans. Signal Process. 53(1), 380–383 (2005).
[Crossref]

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434(7030), 169–176 (2005).
[Crossref] [PubMed]

G. Tóth and O. Gühne, “Detecting genuine multipartite entanglement with two local measurements,” Phys. Rev. Lett. 94(6), 060501 (2005).
[Crossref] [PubMed]

G. Tóth and O. Gühne, “Entanglement detection in the stabilizer formalism,” Phys. Rev. A 72(2), 22340 (2005).
[Crossref]

2004 (3)

K. Lee and J. Thomas, “Entanglement with classical fields,” Phys. Rev. A 69(5), 052311 (2004).
[Crossref]

M. Hein, J. Eisert, and H. J. Briegel, “Multiparty entanglement in graph states,” Phys. Rev. A 69(6), 062311 (2004).
[Crossref]

X. Lou and K. A. Loparo, “Bearing fault diagnosis based on wavelet transform and fuzzy inference,” Mech. Syst. Signal Process. 18(5), 1077–1095 (2004).
[Crossref]

2003 (2)

R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68(2), 022312 (2003).
[Crossref]

M. Fujishima, K. Saito, and K. Hoh, “16-qubit quantum-computing emulation based on high-speed hardware architecture,” Jpn. J. Appl. Phys. 42(4), 2182–2184 (2003).
[Crossref]

2002 (2)

K. F. Lee and J. E. Thomas, “Experimental simulation of two-particle quantum entanglement using classical fields,” Phys. Rev. Lett. 88(9), 097902 (2002).
[Crossref] [PubMed]

N. Bhattacharya, H. B. van Linden van den Heuvell, and R. J. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88(13), 137901 (2002).
[Crossref] [PubMed]

2001 (2)

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86(22), 5188–5191 (2001).
[Crossref] [PubMed]

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63(6), 062302 (2001).
[Crossref]

2000 (1)

T. M. Foltz, B. M. Welsh, and C. D. Holmberg, “Symmetric convolution using unitary transform matrices,” IEEE Trans. Signal Process. 48(9), 2691–2692 (2000).
[Crossref]

1998 (3)

H. J. Kim and C. C. Li, ““Lossless and lossy image compression using biorthogonal wavelet transforms with multiplierless operations,” IEEE Trans. Circuits Syst. II, Analog,” Digit. Signal Process. 45(8), 1113–1118 (1998).

R. J. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28(3), 361–374 (1998).
[Crossref]

I. L. Chuang, N. Gershenfeld, and M. Kubinec, “Experimental implementation of fast quantum searching,” Phys. Rev. Lett. 80(15), 3408–3411 (1998).
[Crossref]

1997 (1)

L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack,” Phys. Rev. Lett. 79(2), 325–328 (1997).
[Crossref]

1995 (3)

R. G. Baraniuk and D. L. Jones, “Unitary equivalence: a new twist on signal processing,” IEEE Trans. Signal Process. 43(10), 2269–2282 (1995).
[Crossref]

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52(5), 3457–3467 (1995).
[Crossref] [PubMed]

D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51(2), 1015–1022 (1995).
[Crossref] [PubMed]

1989 (1)

P. A. Regalia and S. K. Mitra, “Kronecker products, unitary matrices and signal processing applications,” SIAM Rev. 31(4), 586–613 (1989).
[Crossref]

1977 (1)

J. Allen, “Short term spectral analysis, synthesis, and modification by discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. 25(3), 235–238 (1977).
[Crossref]

1941 (1)

Afullo, T. J.

A. A. Alonge and T. J. Afullo, “Temporal characterization of rainfall time series analysis for wireless networks,” 3rd International Conference on Wireless Communications, Vehicular Technology, Information Theory and Aerospace & Electronic Systems (VITAE) (2013), pp. 1–5.
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 A designed classical microwave signal processing system to construct and characterize the 4-cebit CMGS, including transmit and receive dual-polarized antennas, an 8-channel receiver, an analog-to-digital converter (ADC) and a programmable DSP module. (Inset) A unit of the PROJ part including two multiplications and an adder, which can be tuned conveniently to perform desired measurement settings. The frequency of the microwave signal emitted from the dual-polarized antennas is taken as 17 GHZ, which would soon be down-converted to an intermediate frequency of 140MHz. Then all eight signals fed into the DSP are digitally down-converted to their own frequencies set in Eq. (2). After a measuring process through the PROJ part followed by several mixing and filtering processes for multiplex signals, the frequencies for 4 desired signal components A Ω 1 , A Ω 2 , A Ω 3 , and A Ω 4 selected by the FFT-based digital filter are 10.9MHz, 11.8MHz, 12.1MHz, and 13.0MHz, respectively, with a final composite signal A in a superposition of these four terms as defined in Eq. (6) collected.
Fig. 2
Fig. 2 (a) Density matrix of the 4-cebit classical cluster in Eq. (8) reconstructed from the experimental tomography data. (b) Theoretical results for the ideal quantum state in Eq. (9). The left column is the real part of the density matrix, and the right column is the imaginary part.
Fig. 3
Fig. 3 Scheme of the process flow in the DSP module to construct and measure the 16-cebit CMGS as described in Eq. (10). The whole system can be regarded as an extended version of the 4-cebit CMGS setup (Fig. 1) in channel number with an adjustment of the procedure (step-by-step digital filtering) in the programmable DSP, which shows good scalability and flexibility. The square ring enclosed by red dotted lines in each step represents a basic processing unit, which includes one multiplier, 2 time-domain filters and one adder. The two-star-shaped graph in the inset represents the 16-cebit graph state | G 16 cl ) under local unitary transforms.
Fig. 4
Fig. 4 Experimental results of the 16-cebit classical state | G 16 cl ). (a), (b), and (c) represent measured normalized intensities of composite signals in the Z 16 , Z 8 X 8 , and X 8 Z 8 basis, respectively, from which we can then compute the expectation value of the observable quantity, B, as defined in Eq. (34) in Appendix 5.2.
Fig. 5
Fig. 5 Processing circuits to implement unitary transforms involved in our measurement-based signal processing schemes. (a) A sequence of 2 × 2 unitary transforms, i.e., three rotations R z (α)(green square), R x (β)(yellow square), and R z (γ)equivalent to R z (γ) H ^ R z (β) H ^ R z (α)(the pink square indicates the Hadamard gate H ^ ) for the input state |+). (b) A 4 × 4 controlled-Z transform with two rotations R z (α)and R z (β); (c) a CZ gate, two rotations R z (α)and R z (β), a Hadamard gate with a controlled-NOT(‘CNOT’) gate, and another Hadamard gate with a SWAP gate for the input state |+)|+). (d) Generalized unitary transforms, such as two multi-cebit rotations R zzz 12m ( θ a )and R zzz m+1n ( θ b ) (green blocks) followed by n Hadamard gates and a (n-1)-controlled-Z gate C n1 Z(the blue square with ‘Z’ inside) for the 2n-length input state | ψ in (n)), which can be implemented by measuring cebits a and b in the displayed (n + 2)-cebit two-star CMGS.
Fig. 6
Fig. 6 (a) The output density matrix (real (Re) and imaginary (Im)) on cebits 1 and 4 from measurements on cebits 2 and 3 in the bases |0 ) 2 and |0 ) 3 on our 4-cebit CMGS. (b) The output density matrix (real (Re) and imaginary (Im)) on ccebits 2 and 3 from single-cebit measurements in H ^ |π ) 1 and H ^ |0 ) 4 on our 4-cebit CMGS.
Fig. 7
Fig. 7 Experimental results for the 14-cebit output classical state | O 14 cl ) . (a), (b), and (c) represent measured normalized intensities of composite signals in the Z 14 , Z 7 Y X 6 and Y X 6 Z 7 basis, respectively, from which we can then compute the lower bound of the fidelity (see Appendix 5.2 for details).
Fig. 8
Fig. 8 The experimentally measured outputs of the 2-qubit Grover’s algorithm in a CMGS scheme. The probability for successful identification of the function is almost 100% in all cases.
Fig. 9
Fig. 9 Measurement results of complex amplitudes and intensities of the 4 frequency-filtered signals A Ω j (j=1,2,3,4)under different bases in one experiment. (a) and (b) correspond to the magnitudes, and (c) and (d) to the phases. The basis for (a) and (c) is |+rh), while that for (b) and (d) is |ll). (e) shows intensities of final composite signals under various measurement bases.

Tables (1)

Tables Icon

Table 1 The Bell’s measure for 2-, 3-, and 4-dimensional maximum correlated states. The measured results match the corresponding theoretical predictions. The time-domain FIR custom filters with their center frequencies and the corresponding summed signals in three steps as shown in Fig. 3.

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

B i (t)=[ B hi , B vi ]=[ B hi exp(i ω hi t), B vi exp(i ω vi t)],
f h1 =1.0MHz, f v1 =1.6MHz, f h2 =2.0MHz, f v2 =2.3MHz, f h3 =3.2MHz, f v3 =4.0MHz, f h4 =4.7MHz, f v4 =5.1MHz.
S i (t)= c hi B hi (t)+ c vi B vi (t) c hi P hi exp(i ω hi t)+ c vi P vi exp(i ω vi t) = ( e ^ mi |h) i P hi exp(i ω hi t)+ ( e ^ mi |v) i P vi exp(i ω vi t),
Ω 1 = f h1 + f h2 + f h3 + f h4 =10.9MHz, Ω 2 = f v1 + f v2 + f h3 + f h4 =11.8MHz, Ω 3 = f h1 + f h2 + f v3 + f v4 =12.1MHz, Ω 4 = f v1 + f v2 + f v3 + f v4 =13.0MHz.
A Ω 1 ( e ^ m1 |h) 1 ( e ^ m2 |h) 2 ( e ^ m3 |h) 3 ( e ^ m4 |h) 4 P h1 P h2 P h3 P h4 A Ω 2 ( e ^ m1 |v) 1 ( e ^ m2 |v) 2 ( e ^ m3 |h) 3 ( e ^ m4 |h) 4 P v1 P v2 P h3 P h4 A Ω 3 ( e ^ m1 |h) 1 ( e ^ m2 |h) 2 ( e ^ m3 |v) 3 ( e ^ m4 |v) 4 P h1 P h2 P v3 P v4 A Ω 4 ( e ^ m1 |v) 1 ( e ^ m2 |v) 2 ( e ^ m3 |v) 3 ( e ^ m4 |v) 4 P v1 P v2 P v3 P v4
A= j=1,2,3,4 A Ω j P 1 P 2 P 3 P 4 ×[( e ^ m1 | h 1 )( e ^ m2 | h 2 )( e ^ m3 | h 3 ) ( e ^ m4 | h 4 )+( e ^ m1 | v 1 )( e ^ m2 | v 2 )( e ^ m3 | h 3 )( e ^ m4 | h 4 )+ ( e ^ m1 | h 1 )( e ^ m2 | h 2 )( e ^ m3 | v 3 )( e ^ m4 | v 4 ) ( e ^ m | v 1 )( e ^ m2 | v 2 )( e ^ m3 | v 3 )( e ^ m4 | v 4 )].
A( e ^ m1 |( e ^ m2 |( e ^ m3 |( e ^ m4 | G 4 cl ),
| G 4 cl )= 1 2 [ |h ) 1 |h ) 2 |h ) 3 |h ) 4 +|h ) 1 |h ) 2 |v ) 3 |v ) 4 + |v ) 1 |v ) 2 |h ) 3 |h ) 4 |v ) 1 |v ) 2 |v ) 3 |v ) 4 ]
| Φ cluster = 1 2 [ |H 1 |H 2 |H 3 |H 4 + |H 1 |H 2 |V 3 |V 4 + |V 1 |V 2 |H 3 |H 4 |V 1 |V 2 |V 3 |V 4 ].
| G 16 cl )= 1 2 (|h) 8 |h ) 8 +|h ) 8 |v ) 8 + |v ) 8 |h ) 8 |v ) 8 |v ) 8 ).
f h1 =1, f v1 =1.6, f h2 =2, f v2 =2.3, f h3 =3.2, f v3 =4, f h4 =4.7, f v4 =5.1, f h5 =1.12, f v5 =1.72, f h6 =2.43, f v6 =2.61, f h7 =3.24, f v7 =4.95, f h8 =5.05, f v8 =5.33, f h9 =1, f v9 =1.6, f h10 =2, f v10 =2.3, f h11 =3.2, f v11 =4, f h12 =4.7, f v12 =5.1, f h13 =1.12, f v13 =1.72, f h14 =2.43, f v14 =2.61, f h15 =3.24, f v15 =4.95, f h16 =5.05, f v16 =5.33.
R z (α)=exp(iαZ/2)=( e iα/2 0 0 e iα/2 ), R x (α)=exp(iαX/2)=( cos(α/2 ) isin(α/2 ) isin(α/2 ) cos(α/2 ) ),
|φ ) out = R z (γ) R x (β) R z (α)|+ ) in = R z (γ) H ^ R z (β) H ^ R z (α)|+ ) in = e i(α+γ)/2 2 ( 1 0 0 e iγ )( cos(β/2 ) isin(β/2 ) isin(β/2 ) cos(β/2 ) )( 1 0 0 e iα )( 1 1 ) = e i(α+γ)/2 2 ( cos(β/2 )+isin(β/2 ) e iα [isin(β/2 )+cos(β/2 ) e iα ] e iγ ).
( α 1 H ^ |( β 2 |( γ 3 | G 4 cl ) = ( β + 2 |( γ + 3 | 4 [(1+ e iα )|h) 2 (|h) 3 |h ) 4 +|v ) 3 |v ) 4 )+ (1 e iα )|v ) 2 (|h) 3 |h ) 4 |v ) 3 |v ) 4 )] = ( γ + 3 | 4 2 [(1+ e iα +e iβ e i(α+β) )|h) 3 |h ) 4 +(1+ e iα e iβ +e i(α+β) )|v ) 3 |v ) 4 ] = e iβ/2 4 [(cos(β/2 )+isin(β/2 ) e iα )|h) 4 +(isin(β/2 )+ cos(β/2 ) e iα ) e iγ |v ) 4 ].
CZ=( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ).
|φ ) out =[ R z (α) R z (β) ]CZ|+)|+) = e i(α+β)/2 2 ( 1 0 0 0 0 e iβ 0 0 0 0 e iα 0 0 0 0 e i(α+β) )( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 )( 1 1 1 1 ) = e i(α+β)/2 2 ( 1 e iβ e iα e i(α+β) ).
( α + 2 |( β + 3 | G 4 cl )= 1 4 [|h) 1 |h) 4 + e iβ |h ) 1 |v ) 4 + e iα |v ) 1 |h ) 4 e i(α+β) |v ) 1 |v ) 4 ].
CNOT=( 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ), SWAP=( 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ).
|φ ) out =SWAP[ HI ]CNOT[ HI ] [ R z (α) R z (β) ]CZ|+)|+) = e i(α+β)/2 4 ( 1 +e iβ +e iα +e i(α+β) 1 e iβ +e iα e i(α+β) 1 +e iβ e iα e i(α+β) 1 +e iβ +e iα e i(α+β) ),
( α 1 H ^ |( β 4 H ^ | G 4 cl )= 1 4 [(1+ e iα )(1+ e iβ )|h) 2 |h ) 3 +(1+ e iα )(1 e iβ )|h ) 2 |v ) 3 +(1 e iα )(1+ e iβ )|v ) 2 |h ) 3 (1 e iα )(1 e iβ )|v ) 1 |v ) 4 ].
| φ out (n))= (C n1 Z) H ^ n [ R zzz 12m ( θ a ) R zzz m+1n ( θ b )]| ψ in (n)) =cos( θ a /2 )|h ) 12m [cos( θ b /2 )|h) m+1n isin( θ b /2 )|v ) m+1n ]+sin( θ a /2 )|v ) 12m [icos( θ b /2 )|h) m+1n +sin( θ b /2 )|v ) m+1n ].
( θ a |( θ b | G n+2 cl ) = 1 2 { cos( θ a /2 )|h ) 12m [cos( θ b /2 )|h) m+1n isin( θ b /2 )|v ) m+1n ]+sin( θ a /2 )|v ) 12m [ icos( θ b /2 )|h ) m+1n +sin( θ b /2 )|v ) m+1n ] }.
| O 14 cl )= 1 2 (|h) 7 |h ) 7 i|h ) 7 |v ) 7 i|v ) 7 |h ) 7 +|v ) 7 |v ) 7 ).
ρ cl = i=0 255 λ i Ω i = I 16×16 16 + i=1 255 λ i Ω i ,
| E m )( E m | (j) = I 16×16 16 + i=1 255 e i (j) Ω i .
p j = |( E m (j) | Ψ cluster cl ) | 2 =Tr[| E m )( E m | (j) ρ cl ] = 1 16 + i=1 255 e i (j) λ i .
p ^ j = 1 16 + E (j) T Λ+ k j .
P= E ˜ Λ+K.
Λ ^ LS = ( E ˜ T E ˜ ) 1 E ˜ T P= ( E ˜ T E ˜ ) 1 j=1 M E (j) ( p ^ j 1/16) .
| C 16 = 1 2 ( |H 8 |H 8 + |H 8 |V 8 + |V 8 |H 8 |V 8 |V 8 ),
W 2 = 3 2 I i=2 9 g i +I 2 i=1,10 16 g i +I 2 ,
g i ={ σ x 1 σ x 2 σ x 8 σ z9 σ z1 σ zi σ z1 σ x 9 σ x 10 σ x 16 σ z9 σ zi i=1 2i8 i=9 10i16 .
Tr[( W 2 W 1 ) ρ exp ]= i ψ i | ρ exp | ψ i 0
B=I+ i=2 9 G i +I 2 + i=1,10 16 G i +I 2
G i ={ X 1 X 2 X 8 Z 9 Z 1 Z i Z 1 X 9 X 10 X 16 Z 9 Z i i=1 2i8 i=9 10i16 ,
| O 14 cl )= 1 2 (|h) 7 |h ) 7 i|h ) 7 |v ) 7 i|v ) 7 |h ) 7 +|v ) 7 |v ) 7 ),
| G 14 cl )= 1 2 (|h) 7 |h ) 7 +|h ) 7 |v ) 7 + |v ) 7 |h ) 7 |v ) 7 |v ) 7 ),
h i ={ Y 1 X 2 X 7 Z 8 Z 1 Z i Z 1 Y 8 X 9 X 14 Z 8 Z i i=1 2i7 i=8 9i14 ,

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