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題名 以數位量子電腦模擬價鍵態動力學
Simulating the dynamics of a valence bond state with digital quantum computers
作者 黃靖泰
Huang, Ching-Tai
貢獻者 林瑜琤
Lin, Yu-Cheng
黃靖泰
Huang, Ching-Tai
關鍵詞 含噪聲中尺度量子計算
Hadamard 測試
隨機測量
經典影子
Rényi 糾纏熵
Loschmidt 回聲
Noisy Intermediate-Scale Quantum (NISQ) computation
Hadamard test
randomized measurement
classical shadow
Rényi entanglement entropy
Loschmidt echo
日期 2025
上傳時間 1-Sep-2025 16:52:09 (UTC+8)
摘要 本研究以 IBM Quantum 提供之量子電腦與模擬器，實作一維完全二聚化 XXZ 自旋-1/2反鐵磁鏈之量子淬火動力學，並透過「Hadamard 測試」(Hadamard test) 與「隨機測量」兩種方法計算隨時間演化之 Rényi 糾纏熵與 Loschmidt 回聲。我們以一價鍵固態為初始態，進行 XXZ 自旋鏈具各種非同向性參數之淬火實驗，比較兩種方法的效能及擴展性。 Hadamard 測試可直接估算由完全二聚化 XXZ 自旋鏈產生的任意時間點之量子態係數，能準確求得小系統於真實量子機器模擬之結果。然而當系統尺寸增加，因電路深度上升與係數絕對值遞減，在具噪聲的真實量子電腦上將導致顯著失真的結果，侷限其僅適用小系統模擬。為克服上述侷限，我們利用量子電路執行時間演化。因哈密頓量完全二聚化性質，我們得以在無 Trotter 誤差情形下以量子電路製備不同時間點之量子態，並進一步以隨機測量及其後處理法估算 Rényi 糾纏熵與 Loschmidt 回聲。在隨機測量選取量子閘方面，我們考慮兩種隨機基底轉換：Haar 隨機單量子閘以及隨機 Pauli 測量。隨機 Pauli 測量在真實量子硬體執行成本較低，可在現有資源下提供可靠的數據。在後處理部分，我們比較 Hamming 距離方法與經典影子 (classical shadow) 技術。我們實驗結果顯示，經典影子後處理法在高糾纏態下表現佳，具低誤差，而 Hamming 距離法在精確度上具整體穩定性。 綜合而言，Hadamard 測試適用於小系統高精確度交疊計算；而隨機測量法則更適用於含噪聲中尺度量子計算，具較高的實用性與擴展性。本研究展示兩類方法在不同情境下的優勢與限制，亦提供未來研究多體量子模擬與評估量子電腦效能之基礎。
This thesis investigates the quench dynamics of a one-dimensional fully dimerized spin-1/2 XXZ antiferromagnetic chain using IBM Quantum computers and simulators. Two approaches—the Hadamard test and randomized measurements—are employed to evaluate the time-evolved Rényi entanglement entropy and Loschmidt echo. We initiate the dynamics from a valence-bond solid and evolve it under the XXZ Hamiltonian with a range of anisotropy parameters, using this setting to benchmark the performance and scalability of these two methods. The Hadamard test provides direct amplitude estimation of the time-evolved quantum state generated by the fully dimerized Hamiltonian, yielding high-precision results for small systems even on real quantum devices. However, as system size increases, deeper circuits and smaller amplitude magnitudes significantly reduce accuracy on noisy hardware, restricting its use to small-scale simulations. To overcome this limitation, we execute Trotter-error-free time-evolution circuits for the dimerized XXZ chain and apply randomized measurement techniques to estimate observables. We compare Haar-random single-qubit unitaries and random Pauli measurements, with the latter offering lower hardware overhead and reliable performance on current hardware. For post-processing, we assess both the Hamming distance method and the classical shadow technique. Our results show that classical shadows provide lower estimation errors in highly entangled regimes, while the Hamming distance method provides robust performance across parameter ranges. In summary, the Hadamard test is well-suited for high-precision overlap estimation in small systems, whereas randomized measurement methods offer greater practicality and scalability for noisy intermediate-scale quantum (NISQ) computations. This study clarifies the strengths and limitations of both approaches under various scenarios and provides a foundation for future studies in many-body quantum simulation and quantum hardware benchmarking.
參考文獻 [1] P. W. Anderson. “The Resonating Valence Bond State in La2CuO4 and Supercon-ductivity.” Science 235 (1987), p. 1196. [2] H.-C. Chang, H.-C. Hsu, and Y.-C. Lin. “Probing entanglement dynamics and topo-logical transitions on noisy intermediate-scale quantum computers.” Phys. Rev. Res. 7 (2025), p. 013043. [3] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000. [4] S. Liang, B. Doucot, and P. W. Anderson. “Some New Variational Resonating-Valence-Bond-Type Wave Functions for the Spin-½ Antiferromagnetic Heisenberg Model on a Square Lattice.” Phys. Rev. Lett. 61 (1988), p. 365. [5] W. Marshall. “Antiferromagnetism.” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 232 (1955), p. 48. [6] B. Sutherland. “Systems with resonating-valence-bond ground states: Correlations and excitations.” Phys. Rev. B 37 (1988), p. 3786. [7] R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca. “Quantum algorithms revisited.” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454 (1998), p. 339. [8] H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf. “Quantum Fingerprinting.” Phys. Rev. Lett. 87 (2001), p. 167902. [9] L. C. Tazi, D. M. Ramo, and A. J. W. Thom. Shallow Quantum Scalar Products with Phase Information. 2024. arXiv: 2411.19072 [quant-ph]. [10] M. Heyl, A. Polkovnikov, and S. Kehrein. “Dynamical Quantum Phase Transitions in the Transverse-Field Ising Model.” Phys. Rev. Lett. 110 (2013), p. 135704. [11] M. Heyl. “Dynamical quantum phase transitions: a review.” Reports on Progress in Physics 81 (2018), p. 054001. [12] N. Hatano and M. Suzuki. “Finding exponential product formulas of higher orders.” Quantum annealing and other optimization methods. Springer, 2005, p. 37. [13] A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller. “The randomized measurement toolbox.” Nature Reviews Physics 5 (2023), p. 9. [14] T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos. “Probing Rényi entanglement entropy via randomized measurements.” Science 364 (2019), p. 260. [15] A. Elben, J. Yu, G. Zhu, M. Hafezi, F. Pollmann, P. Zoller, and B. Vermersch. “Many-body topological invariants from randomized measurements in synthetic quantum matter.” Science advances 6 (2020), eaaz3666. [16] H.-Y. Huang, R. Kueng, and J. Preskill. “Predicting Many Properties of a Quantum System from Very Few Measurements.” Nature Physics 16 (2020), p. 1050. [17] A. Elben, B. Vermersch, C. F. Roos, and P. Zoller. “Statistical correlations between locally randomized measurements: A toolbox for probing entanglement in many-body quantum states.” Phys. Rev. A 99 (2019), p. 052323. [18] J. Vovrosh, K. E. Khosla, S. Greenaway, C. Self, M. S. Kim, and J. Knolle. “Simple mitigation of global depolarizing errors in quantum simulations.” Physical Review E 104 (2021), p. 035309. [19] O. Kiss, M. Grossi, and A. Roggero. “Quantum error mitigation for Fourier moment computation.” Phys. Rev. D 111 (2025), p. 034504. [20] S. Chen, W. Yu, P. Zeng, and S. T. Flammia. “Robust Shadow Estimation.” PRX Quantum 2 (2021), p. 030348. [21] H. Jnane, J. Steinberg, Z. Cai, H. C. Nguyen, and B. Koczor. “Quantum Error Mitigated Classical Shadows.” PRX Quantum 5 (2024), p. 010324.
描述 碩士
國立政治大學
應用物理研究所
112755001
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0112755001
資料類型 thesis
dc.contributor.advisor 林瑜琤zh_TW
dc.contributor.advisor Lin, Yu-Chengen_US
dc.contributor.author (Authors) 黃靖泰zh_TW
dc.contributor.author (Authors) Huang, Ching-Taien_US
dc.creator (作者) 黃靖泰zh_TW
dc.creator (作者) Huang, Ching-Taien_US
dc.date (日期) 2025en_US
dc.date.accessioned 1-Sep-2025 16:52:09 (UTC+8)-
dc.date.available 1-Sep-2025 16:52:09 (UTC+8)-
dc.date.issued (上傳時間) 1-Sep-2025 16:52:09 (UTC+8)-
dc.identifier (Other Identifiers) G0112755001en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/159393-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用物理研究所zh_TW
dc.description (描述) 112755001zh_TW
dc.description.abstract (摘要) 本研究以 IBM Quantum 提供之量子電腦與模擬器，實作一維完全二聚化 XXZ 自旋-1/2反鐵磁鏈之量子淬火動力學，並透過「Hadamard 測試」(Hadamard test) 與「隨機測量」兩種方法計算隨時間演化之 Rényi 糾纏熵與 Loschmidt 回聲。我們以一價鍵固態為初始態，進行 XXZ 自旋鏈具各種非同向性參數之淬火實驗，比較兩種方法的效能及擴展性。 Hadamard 測試可直接估算由完全二聚化 XXZ 自旋鏈產生的任意時間點之量子態係數，能準確求得小系統於真實量子機器模擬之結果。然而當系統尺寸增加，因電路深度上升與係數絕對值遞減，在具噪聲的真實量子電腦上將導致顯著失真的結果，侷限其僅適用小系統模擬。為克服上述侷限，我們利用量子電路執行時間演化。因哈密頓量完全二聚化性質，我們得以在無 Trotter 誤差情形下以量子電路製備不同時間點之量子態，並進一步以隨機測量及其後處理法估算 Rényi 糾纏熵與 Loschmidt 回聲。在隨機測量選取量子閘方面，我們考慮兩種隨機基底轉換：Haar 隨機單量子閘以及隨機 Pauli 測量。隨機 Pauli 測量在真實量子硬體執行成本較低，可在現有資源下提供可靠的數據。在後處理部分，我們比較 Hamming 距離方法與經典影子 (classical shadow) 技術。我們實驗結果顯示，經典影子後處理法在高糾纏態下表現佳，具低誤差，而 Hamming 距離法在精確度上具整體穩定性。 綜合而言，Hadamard 測試適用於小系統高精確度交疊計算；而隨機測量法則更適用於含噪聲中尺度量子計算，具較高的實用性與擴展性。本研究展示兩類方法在不同情境下的優勢與限制，亦提供未來研究多體量子模擬與評估量子電腦效能之基礎。zh_TW
dc.description.abstract (摘要) This thesis investigates the quench dynamics of a one-dimensional fully dimerized spin-1/2 XXZ antiferromagnetic chain using IBM Quantum computers and simulators. Two approaches—the Hadamard test and randomized measurements—are employed to evaluate the time-evolved Rényi entanglement entropy and Loschmidt echo. We initiate the dynamics from a valence-bond solid and evolve it under the XXZ Hamiltonian with a range of anisotropy parameters, using this setting to benchmark the performance and scalability of these two methods. The Hadamard test provides direct amplitude estimation of the time-evolved quantum state generated by the fully dimerized Hamiltonian, yielding high-precision results for small systems even on real quantum devices. However, as system size increases, deeper circuits and smaller amplitude magnitudes significantly reduce accuracy on noisy hardware, restricting its use to small-scale simulations. To overcome this limitation, we execute Trotter-error-free time-evolution circuits for the dimerized XXZ chain and apply randomized measurement techniques to estimate observables. We compare Haar-random single-qubit unitaries and random Pauli measurements, with the latter offering lower hardware overhead and reliable performance on current hardware. For post-processing, we assess both the Hamming distance method and the classical shadow technique. Our results show that classical shadows provide lower estimation errors in highly entangled regimes, while the Hamming distance method provides robust performance across parameter ranges. In summary, the Hadamard test is well-suited for high-precision overlap estimation in small systems, whereas randomized measurement methods offer greater practicality and scalability for noisy intermediate-scale quantum (NISQ) computations. This study clarifies the strengths and limitations of both approaches under various scenarios and provides a foundation for future studies in many-body quantum simulation and quantum hardware benchmarking.en_US
dc.description.tableofcontents 致謝 i 摘要 iii Abstract v 目錄 vii 1 前言 1 2 模型與問題描述 3 2.1 模型概述 3 2.2 淬火時間演化 4 2.3 完全二聚化模型之時間演化 5 3 以 Hadamard 測試求量子態係數 11 3.1 Hadamard 測試基本電路 11 3.2 量子態係數之計算 13 3.3 Hadamard 測試之實作 14 3.4 Rényi 糾纏熵 24 3.5 Loschmidt 回聲 45 4 以量子電路執行時間演化 59 5 隨機測量 63 5.1 以二階交叉相關估算純度及交疊量 64 5.2 經典影子 65 5.3 測量次數 NM 及實驗次數 NU 的選取 67 5.4 隨機測量計算結果：Rényi 糾纏熵 74 5.5 誤差緩解 86 5.6 測量平均與集合平均之整合 93 5.7 隨機測量計算結果：Loschmidt 回聲 96 6 結論與討論 107 參考文獻 109 中英名詞對照 111zh_TW
dc.format.extent 5702840 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0112755001en_US
dc.subject (關鍵詞) 含噪聲中尺度量子計算zh_TW
dc.subject (關鍵詞) Hadamard 測試zh_TW
dc.subject (關鍵詞) 隨機測量zh_TW
dc.subject (關鍵詞) 經典影子zh_TW
dc.subject (關鍵詞) Rényi 糾纏熵zh_TW
dc.subject (關鍵詞) Loschmidt 回聲zh_TW
dc.subject (關鍵詞) Noisy Intermediate-Scale Quantum (NISQ) computationen_US
dc.subject (關鍵詞) Hadamard testen_US
dc.subject (關鍵詞) randomized measurementen_US
dc.subject (關鍵詞) classical shadowen_US
dc.subject (關鍵詞) Rényi entanglement entropyen_US
dc.subject (關鍵詞) Loschmidt echoen_US
dc.title (題名) 以數位量子電腦模擬價鍵態動力學zh_TW
dc.title (題名) Simulating the dynamics of a valence bond state with digital quantum computersen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] P. W. Anderson. “The Resonating Valence Bond State in La2CuO4 and Supercon-ductivity.” Science 235 (1987), p. 1196. [2] H.-C. Chang, H.-C. Hsu, and Y.-C. Lin. “Probing entanglement dynamics and topo-logical transitions on noisy intermediate-scale quantum computers.” Phys. Rev. Res. 7 (2025), p. 013043. [3] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000. [4] S. Liang, B. Doucot, and P. W. Anderson. “Some New Variational Resonating-Valence-Bond-Type Wave Functions for the Spin-½ Antiferromagnetic Heisenberg Model on a Square Lattice.” Phys. Rev. Lett. 61 (1988), p. 365. [5] W. Marshall. “Antiferromagnetism.” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 232 (1955), p. 48. [6] B. Sutherland. “Systems with resonating-valence-bond ground states: Correlations and excitations.” Phys. Rev. B 37 (1988), p. 3786. [7] R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca. “Quantum algorithms revisited.” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454 (1998), p. 339. [8] H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf. “Quantum Fingerprinting.” Phys. Rev. Lett. 87 (2001), p. 167902. [9] L. C. Tazi, D. M. Ramo, and A. J. W. Thom. Shallow Quantum Scalar Products with Phase Information. 2024. arXiv: 2411.19072 [quant-ph]. [10] M. Heyl, A. Polkovnikov, and S. Kehrein. “Dynamical Quantum Phase Transitions in the Transverse-Field Ising Model.” Phys. Rev. Lett. 110 (2013), p. 135704. [11] M. Heyl. “Dynamical quantum phase transitions: a review.” Reports on Progress in Physics 81 (2018), p. 054001. [12] N. Hatano and M. Suzuki. “Finding exponential product formulas of higher orders.” Quantum annealing and other optimization methods. Springer, 2005, p. 37. [13] A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller. “The randomized measurement toolbox.” Nature Reviews Physics 5 (2023), p. 9. [14] T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos. “Probing Rényi entanglement entropy via randomized measurements.” Science 364 (2019), p. 260. [15] A. Elben, J. Yu, G. Zhu, M. Hafezi, F. Pollmann, P. Zoller, and B. Vermersch. “Many-body topological invariants from randomized measurements in synthetic quantum matter.” Science advances 6 (2020), eaaz3666. [16] H.-Y. Huang, R. Kueng, and J. Preskill. “Predicting Many Properties of a Quantum System from Very Few Measurements.” Nature Physics 16 (2020), p. 1050. [17] A. Elben, B. Vermersch, C. F. Roos, and P. Zoller. “Statistical correlations between locally randomized measurements: A toolbox for probing entanglement in many-body quantum states.” Phys. Rev. A 99 (2019), p. 052323. [18] J. Vovrosh, K. E. Khosla, S. Greenaway, C. Self, M. S. Kim, and J. Knolle. “Simple mitigation of global depolarizing errors in quantum simulations.” Physical Review E 104 (2021), p. 035309. [19] O. Kiss, M. Grossi, and A. Roggero. “Quantum error mitigation for Fourier moment computation.” Phys. Rev. D 111 (2025), p. 034504. [20] S. Chen, W. Yu, P. Zeng, and S. T. Flammia. “Robust Shadow Estimation.” PRX Quantum 2 (2021), p. 030348. [21] H. Jnane, J. Steinberg, Z. Cai, H. C. Nguyen, and B. Koczor. “Quantum Error Mitigated Classical Shadows.” PRX Quantum 5 (2024), p. 010324.zh_TW