尋求創建非高斯量子狀態的探索，這構成了連續變數量子計算領域的一個艱鉅挑戰。許多方法，包括光子的增加和減少，已經被開發出來，試圖創建這些難以捉摸的狀態。我們的理論研究在這一環境中探索了光子催化的新領域，重點放在實現類似貓的狀態，特別是擠壓真空（SSV）狀態上。這一發現對於發展GKP狀態具有重大意義，GKP狀態是量子計算的重要組成部分。

在光束分束器的幫助下，我們的研究表明，結合擠壓真空狀態和Fock狀態（|1⟩ ⟨1|）可以提高類似貓的狀態的振幅。我們還使用光子數解析檢測器研究了串級光子催化，展示了增加SSV狀態振幅的有趣可能性。這項研究為通過創建一種主要依賴於操縱擠壓真空狀態的創新方法來可靠地進行量子處理打開了一條有希望的新途徑，以便創建非高斯狀態。

The search for creating non-Gaussian quantum states, which poses a difficult challenge, dominates the field of continuous variable quantum computation. Many approaches, including photon addition and subtraction, have been developed to attempt to create these elusive states. Our theoretical research explores the novel field of photon catalysis in this environment, focusing on the realisation of cat-like states, especially the Superposition of Squeezed Vacuum (SSV) state. The GKP state, a crucial element of quantum computation, is being developed, and this revelation has significant implications for that process.

With the aid of a beam splitter, our research shows that by mixing a squeezed vacuum state with a Fock state (|1⟩ ⟨1|), it is feasible to increase the amplitudes of cat-like states. We also investigate cascade photon catalysis using Photon Number Resolving Detectors, demonstrating an interesting possibility of increasing the amplitude of the SSV state. This research opens up a promising new path towards reliable quantum processing through the creation of an innovative approach for creating non-Gaussian states that rely primarily on manipulating squeezed vacuum states.

Abstract (Chinese) I
Abstract II
Acknowledgements III
Contents IV
List of Figures VI
List of Tables VIII
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Continuous Variable Quantum Computation . . . . . . . . . . . 1
1.2 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Quantum Optics 3
2.1 Quantized Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . 3
2.2 Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Number State . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.2 Coherent State . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.3 Squeezed Vacuum State . . . . . . . . . . . . . . . . . . . . . 6
2.2.4 Superposition of Squeezed Vacuum (SSV) State . . . . . . . . . 7
2.3 Quantum Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Displacement Operator . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Phase Shift Operator . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Wigner Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Quantum Photon Catalysis 11
3.1 Squeezed Vacuum State Optical Catalysis . . . . . . . . . . . . . . . . 12
3.2 Single Photon Detection Noise Analysis . . . . . . . . . . . . . . . . . 15
3.2.1 Loss due to Vacuum in Fock state |1⟩ ⟨1| . . . . . . . . . . . . . 15
3.2.2 Loss effect of Dark Count . . . . . . . . . . . . . . . . . . . . 17
3.3 Losses modelled using Beam Splitter . . . . . . . . . . . . . . . . . . . 18
3.4 Higher Photon Number Measurement . . . . . . . . . . . . . . . . . . 21
3.4.1 Odd photon Measurement Analysis . . . . . . . . . . . . . . . 21
3.4.2 Even photon measurement analysis . . . . . . . . . . . . . . . 22
4 Cascade Photon Catalysis 23
4.1 Machine Learning Techniques to Quantum Circuit . . . . . . . . . . . . 24
5 Conclusions 26
A Quantum mechanical description of Beam Splitter 28
B Modelling Vacuum & Dark Count effect for SPD 30
B.1 Squeezed state interference . . . . . . . . . . . . . . . . . . . . . . . . 30
B.1.1 Interference with Vacuum . . . . . . . . . . . . . . . . . . . . 30
B.1.2 Interference with |1⟩ ⟨1| . . . . . . . . . . . . . . . . . . . . . 31
C Optimization method analysis 33
D Optimized Parameters 37
Bibliography 40

[1] David Deutsch. Quantum theory, the church–turing principle and the universal quantum computer. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 400:97–117, 1985. doi: 10.1098/rspa. 1985.0070. URL https://royalsocietypublishing.org/doi/10.1098/rspa.1985.0070.
[2] Stephen D. Bartlett and Barry C. Sanders. Universal continuous-variable quantum computation: Requirement of optical nonlinearity for photon counting. Phys. Rev.A, 65:042304, Mar 2002. doi: 10.1103/PhysRevA.65.042304. URL https://link.aps.org/doi/10.1103/PhysRevA.65.042304.
[3] Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. Nature, 614:676–681, 2023. doi: 10.1038/s41586-022-05434-1. URL https://doi.org/10.1038/s41586-022-05434-1.
[4] Christian Weedbrook, Stefano Pirandola, Raul Garcıa-Patron, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd. Gaussian quantum information. Rev. Mod. Phys., 84:621–669, May 2012. doi: 10.1103/RevModPhys.84.621. URL https://link.aps.org/doi/10.1103/RevModPhys.84.621.
[5] Samuel L. Braunstein and Peter van Loock. Quantum information with continuous variables. Rev. Mod. Phys., 77:513–577, Jun 2005. doi:10.1103/RevModPhys.77.513. URL https://link.aps.org/doi/10.1103/RevModPhys.77.513.
[6] Seth Lloyd and Samuel L. Braunstein. Quantum computation over continuous variables. Phys. Rev. Lett., 82:1784–1787, Feb 1999. doi: 10.1103/PhysRevLett.82.1784. URL https://link.aps.org/doi/10.1103/PhysRevLett.82.1784.
[7] Yunong Shi, Christopher Chamberland, and Andrew Cross. Fault-tolerant preparation of approximate gkp states. New Journal of Physics, 21(9):093007, sep 2019.doi: 10.1088/1367-2630/ab3a62. URL https://dx.doi.org/10.1088/1367-2630/ab3a62.
[8] Nicolas C. Menicucci. Fault-tolerant measurement-based quantum computing with continuous-variable cluster states. Phys. Rev. Lett., 112:120504, Mar 2014.doi: 10.1103/PhysRevLett.112.120504. URL https://link.aps.org/doi/10.1103/PhysRevLett.112.120504.
[9] Kadir Gum¨ us¸, Tobias A. Eriksson, Masahiro Takeoka, Mikio Fujiwara, Masahide ¨Sasaki, Laurent Schmalen, and Alex Alvarado. A novel error correction protocol for continuous variable quantum key distribution. Scientific Reports, 11(1):10465,5 2021. ISSN 2045-2322. doi: 10.1038/s41598-021-90055-3. URL https://www.nature.com/articles/s41598-021-90055-3.
[10] Kosuke Fukui, Akihisa Tomita, Atsushi Okamoto, and Keisuke Fujii. Highthreshold fault-tolerant quantum computation with analog quantum error correction. Phys. Rev. X, 8:021054, May 2018. doi: 10.1103/PhysRevX.8.021054. URL https://link.aps.org/doi/10.1103/PhysRevX.8.021054.
[11] Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. Phys. Rev. A, 64:012310, Jun 2001. doi:10.1103/PhysRevA.64.012310. URL https://link.aps.org/doi/10.1103/PhysRevA.64.012310.
[12] H. M. Vasconcelos, L. Sanz, and S. Glancy. All-optical generation of states for “encoding a qubit in an oscillator”. Opt. Lett., 35(19):3261–3263, Oct 2010. doi: 10.1364/OL.35.003261. URL https://opg.optica.org/ol/abstract.cfm?URI=ol-35-19-3261.
[13] P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L.Frunzio, M. Mirrahimi, and M. H. Devoret. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature, 584(7821):368–372, 8 2020. ISSN 1476-4687. doi: 10.1038/s41586-020-2603-3. URL https://www.nature.com/articles/s41586-020-2603-3.
[14] Miller Eaton, Rajveer Nehra, and Olivier Pfister. Non-gaussian and gottesman–kitaev–preskill state preparation by photon catalysis. New Journal of Physics, 21(11):113034, nov 2019. doi: 10.1088/1367-2630/ab5330. URL https://dx.doi.org/10.1088/1367-2630/ab5330.
[15] A. I. Lvovsky and J. Mlynek. Quantum-optical catalysis: Generating nonclassical states of light by means of linear optics. Phys. Rev. Lett., 88:250401, Jun 2002.doi: 10.1103/PhysRevLett.88.250401. URL https://link.aps.org/doi/10.1103/PhysRevLett.88.250401.
[16] Richard J. Birrittella, M. El Baz, and Christopher C. Gerry. Photon catalysis and quantum state engineering. J. Opt. Soc. Am. B, 35(7):1514–1524, Jul
2018. doi: 10.1364/JOSAB.35.001514. URL https://opg.optica.org/josab/abstract.cfm?URI=josab-35-7-1514.
[17] T. Kiss, U. Herzog, and U. Leonhardt. Compensation of losses in photodetection and in quantum-state measurements. Phys. Rev. A, 52:2433–2435, Sep 1995.
doi: 10.1103/PhysRevA.52.2433. URL https://link.aps.org/doi/10.1103/PhysRevA.52.2433.
[18] D.F. Walls and G.J. Milburn. Quantum Optics. Springer Berlin Heidelberg, 2008.ISBN 9783540285731. URL https://books.google.com.tw/books? id=LiWsc3Nlf0kC.
[19] Roy J. Glauber. Photon correlations. Phys. Rev. Lett., 10:84–86, Feb 1963. doi: 10.1103/PhysRevLett.10.84. URL https://link.aps.org/doi/10.1103/PhysRevLett.10.84.
[20] Roy J. Glauber. Coherent and incoherent states of the radiation field. Phys. Rev.,131:2766–2788, Sep 1963. doi: 10.1103/PhysRev.131.2766. URL https://
link.aps.org/doi/10.1103/PhysRev.131.2766.
[21] Erwin Schrodinger. Die gegenw ¨ artige situation in der quantenmechanik. ¨ Naturwissenschaften, 23(48):807–812, 1935. ISSN 1432-1904. doi: 10.1007/
BF01491891. URL https://doi.org/10.1007/BF01491891.
[22] Pierre Meystre and Marlan O Scully. Quantum optics. Springer, 2021.
[23] Bernard Yurke, Samuel L. McCall, and John R. Klauder. Su(2) and su(1,1) interferometers. Phys. Rev. A, 33:4033–4054, Jun 1986. doi:10.1103/PhysRevA.33.4033.URL https://link.aps.org/doi/10.1103/PhysRevA.33.4033.
[24] Leonard Mandel and Emil Wolf. Optical coherence and quantum optics. Cambridge university press, 1995.
[25] R. Loudon. The Quantum Theory of Light. Clarendon Press, Oxford, second edition, 1983.
[26] Matteo G.A. Paris. Displacement operator by beam splitter. Physics Letters A, 217(2):78–80, 1996. ISSN 0375-9601. doi: https://doi.org/10.1016/0375-9601(96)00339-8. URL https://www.sciencedirect.com/science/article/pii/0375960196003398.
[27] E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev.,40:749–759, Jun 1932. doi: 10.1103/PhysRev.40.749. URL https://link.
aps.org/doi/10.1103/PhysRev.40.749.
[28] U. Leonhardt and H. Paul. Measuring the quantum state of light. Progress in Quantum Electronics, 19(2):89–130, 1995. ISSN 0079-6727. doi: https://doi.org/10.1016/0079-6727(94)00007-L. URL https://www.sciencedirect.com/science/article/pii/007967279400007L.
[29] Ingrid Strandberg, Yong Lu, Fernando Quijandr´ıa, and Goran Johansson. Numerical study of wigner negativity in one-dimensional steady-state resonance
fluorescence. Phys. Rev. A, 100:063808, Dec 2019. doi: 10.1103/PhysRevA.100.063808. URL https://link.aps.org/doi/10.1103/PhysRevA.100.063808.
[30] David T. Pegg, Lee S. Phillips, and Stephen M. Barnett. Optical state truncation by projection synthesis. Phys. Rev. Lett., 81:1604–1606, Aug 1998. doi:10.1103/PhysRevLett.81.1604. URL https://link.aps.org/doi/10.1103/PhysRevLett.81.1604.
[31] S¸ahin Kaya Ozdemir, Adam Miranowicz, Masato Koashi, and Nobuyuki Imoto. ¨Quantum-scissors device for optical state truncation: A proposal for practical realization. Phys. Rev. A, 64:063818, Nov 2001. doi: 10.1103/PhysRevA.64.063818. URL https://link.aps.org/doi/10.1103/PhysRevA.64.
063818.
[32] Thomas Gerrits, Scott Glancy, Tracy S. Clement, Brice Calkins, Adriana E. Lita, Aaron J. Miller, Alan L. Migdall, Sae Woo Nam, Richard P. Mirin, and Emanuel Knill. Generation of optical coherent-state superpositions by number-resolved photon subtraction from the squeezed vacuum. Phys. Rev. A, 82:031802, Sep 2010. doi: 10.1103/PhysRevA.82.031802. URL https://link.aps.org/doi/10.1103/PhysRevA.82.031802.
[33] Ching Tsung Lee. External photodetection of cavity radiation. Phys. Rev. A,48:2285–2291, Sep 1993. doi: 10.1103/PhysRevA.48.2285. URL https:
//link.aps.org/doi/10.1103/PhysRevA.48.2285.
[34] Rajveer Nehra, Miller Eaton, Carlos Gonzalez-Arciniegas, M. S. Kim, Thomas Gerrits, Adriana Lita, Sae Woo Nam, and Olivier Pfister. Generalized overlap quantum state tomography1. Phys. Rev. Res., 2:042002, Oct 2020. doi:10.1103/PhysRevResearch.2.042002. URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.042002.
[35] Adriana E. Lita, Aaron J. Miller, and Sae Woo Nam. Counting near-infrared single-photons with 95% efficiency. Opt. Express, 16(5):3032–3040, Mar
2008. doi: 10.1364/OE.16.003032. URL https://opg.optica.org/oe/abstract.cfm?URI=oe-16-5-3032.
[36] Rajveer Nehra, Kevin Valson Jacob, Miller Eaton, and Olivier Pfister. Characterizing quantum detectors by wigner functions. In OSA Quantum 2.0 Conference, page QTu8A.6. Optica Publishing Group, 2020. doi: 10.1364/QUANTUM.2020.QTu8A.6. URL https://opg.optica.org/abstract.cfm?URI=
QUANTUM-2020-QTu8A.6.
[37] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii. Quantum circuit learning. Phys. Rev. A, 98:032309, Sep 2018. doi: 10.1103/PhysRevA.98.032309. URL
https://link.aps.org/doi/10.1103/PhysRevA.98.032309.
[38] Juan Miguel Arrazola, Thomas R Bromley, Josh Izaac, Casey R Myers, Kamil Bradler, and Nathan Killoran. Machine learning method for state preparation and gate synthesis on photonic quantum computers. Quantum Science and Technology, 4(2):024004, jan 2019. doi: 10.1088/2058-9565/aaf59e. URL https://dx.
doi.org/10.1088/2058-9565/aaf59e.
[39] Nathan Killoran, Thomas R. Bromley, Juan Miguel Arrazola, Maria Schuld, Nicolas Quesada, and Seth Lloyd. Continuous-variable quantum neural networks. Phys. Rev. Res., 1:033063, Oct 2019. doi: 10.1103/PhysRevResearch.1.033063. URL https://link.aps.org/doi/10.1103/PhysRevResearch.1.033063.
[40] Krishna Kumar Sabapathy, Haoyu Qi, Josh Izaac, and Christian Weedbrook.Production of photonic universal quantum gates enhanced by machine learning.
Phys. Rev. A, 100:012326, Jul 2019. doi: 10.1103/PhysRevA.100.012326. URL https://link.aps.org/doi/10.1103/PhysRevA.100.012326.
[41] SciPy Developers. Scipy documentation. https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html, 2023.