臺灣博碩士論文加值系統

我們探討在矽半導體上參雜磷原子架構下的量子電腦內，如何使用磷的電子自旋來當量子位元並做邏輯計算。我們採用脈衝序列控制來實作CNOT量子邏輯閘，並透過gradient ascent pulse engineering (GRAPE) 演算法來尋找量子邏輯閘的時間最佳化解。
首先我們使用reduced Hamiltonian來尋找脈衝式控制序列之時間最佳解，其可控制的變數為電子與核子自旋的交互作用力(hyperfine A interactions) 和兩相鄰電子之間的交互作用力(exchange J interactions). 我們嘗試不同的控制步數，並數值計算其錯誤率是否低於我們的要求。我們發現，在時間為100ns, 控制步數為30個分段時，其錯誤率大約為〖1.11×10〗^(-16). 接下來我們用完整的Hamiltonian來模擬先前找到的時間最佳化之操作序列，我們發現錯誤率約為〖10〗^(-6), 小於量子計算中容許的錯誤率閥值(〖10〗^(-4)).
使用GRAPE實作此CNOT邏輯閘的運作時間為100ns. 在同樣的硬體上使用globally controlled electron spin scheme來實作CNOT邏輯閘需要297ns. 使用GRAPE比先前的方法約快了三倍。在我們的構想中還有一大優勢，我們不需要很強的電子和電子交互作用力。在我們的理論計算中最大的交互作用能量只需要J/h=20MHz,傳統上該作用力需要10.2GHz. 先前若要達到如此強的作用力，兩個量子位元約需間隔10nm, 但因為在我們的架構上不需要如此強的交互作用力，我們可以把兩個量子位元的距離拉開到30nm, 如此有可能可以解決當前製造技術上的難題。

We investigate how pulse-sequences and operation times of elementary quantum gates can be optimized for silicon-based donor electron spin quantum computer architecture, complementary to the original Kane''s nuclear spin proposal. This gate-sequence-optimal or time-optimal quantum gate control in a quantum circuit is in addition to the more conventional concept of optimality in terms of the number of elementary gates needed in a quantum transformation.
The optimal control method we use is the so-called gradient ascent pulse engineering (GRAPE) scheme. We focus on the high fidelity controlled-NOT (CNOT) gate and explicitly find the digitized control sequences by optimizing the effective, reduced donor electron spin Hamiltonian, with external controls over the hyperfine A and exchange J interactions. We first try different piecewise constant control steps and numerically calculate the fidelity (error) against the time needed to implement a CNOT gate with stopping criteria of error in the optimizer set to 〖10〗^(-9) in order to economize the simulation time. Here, the error is defined as 1-F, where F is fidelity. The error is less than 〖10〗^(-8) for times longer than 100ns, and it is found that 30 piecewise constant control steps for the CNOT gate operation will be sufficient to meet the required fidelity (error), and the performance would not be improved further with more steps.
With operation time t=100ns and stopping criteria of error set to 〖10〗^(-16), we can find that the near time-optimal, high-fidelity CNOT gate control sequence has an error of 〖1.11×10〗^(-16). We then simulate the control sequences of the CNOT gate, obtained from reduced Hamiltonian simulations, with the full spin Hamiltonian. We find the error of about 〖10〗^(-6) which is below the error threshold required for fault-tolerant (〖10〗^(-4)) quantum computation. The CNOT gate operation time of 100ns is 3 times faster than the globally controlled electron spin scheme of 297ns. One of the great advantages of this near optimal-time high fidelity CNOT gate is that the exchange interaction is not required to be strong (the maximum value is J/h=20MHz compared to the typical value of 10.2GHz. This relaxes significantly the stringent distance constraint of two neighboring donor atoms of about 10nm as reported in the original Kane''s proposal to be about 30nm which is within the reach of the current fabrication technology.

Table of Contents v
List of Figures vii
1 Introduction 1
2 The Si based Quantum Computer and Quantum Computing 5
2.1 The architecture and the Hamiltonian . . . . . . . . . . . . . . . . . . 5
2.2 The reduced Hamiltonian for a single qubit . . . . . . . . . . . . . . . 11
2.3 Two-qubit system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 GRAPE algorithm 15
3.1 Density Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Optimal Unitary transformations . . . . . . . . . . . . . . . . . . . . 16
3.3 Optimal Transfer between Hermitian density operators . . . . . . . . 19
4 Variational Principle Approach of Time-Optimal Evolution 21
4.1 Time-optimal evolution between a given set of initial and final states 23
4.2 Time-optimal realization of unitary operators . . . . . . . . . . . . . 30
4.3 Compare the Controlled Z gate implemented by the time-optimal approach
and the canonical decomposition respectively . . . . . . . . . 33
4.3.1 The time-optimal of the controlled Z gate using variational principle
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3.2 The canonical decomposition of the controlled Z gate . . . . . 38
4.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5 Optimal CNOT Gate 44
5.1 Control sequence obtained from reduced Hamiltonian . . . . . . . . . 44
5.2 Full Hamiltonian simulation . . . . . . . . . . . . . . . . . . . . . . . 46
5.3 Reinitialize the nuclear spin . . . . . . . . . . . . . . . . . . . . . . . 51
5.4 The robust control over the AWGN channel . . . . . . . . . . . . . . 53
6 Conclusions 56
v
CONTENTS vi
Bibliography 59
A Computing Matrix Exponentials 63
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A.2 Computed by Taylor Series Expansion . . . . . . . . . . . . . . . . . 64
A.3 Computed by Diagonalization of the Matrix . . . . . . . . . . . . . . 64
A.4 Computed by Pad′e Approximation . . . . . . . . . . . . . . . . . . . 65
A.5 Matrix Exponential Source Code . . . . . . . . . . . . . . . . . . . . 69

[1] P. W. Shor, “Algorithms for quantum computation: discrete logarithms and
factoring,” In 35th Annual Symposium on Foundations of Computer Science.
IEEE Press, Los Alamos (1994).
[2] P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete
logarithms on a quantum computer,” SIAM J. Computing 26 (1997).
[3] L. K. Grover, “A fast quantum mechanical algorithm for database search,” In
STOC ’96: Proceedings of the twenty-eighth annual ACM symposium on Theory
of computing, pp. 212–219 (ACM, New York, NY, USA, 1996).
[4] L. K. Grover, “Quantum Mechanics Helps in Searching for a Needle in a
Haystack,” Phys. Rev. Lett. 79, 325–328 (1997).
[5] L. K. Grover, “Quantum Computers Can Search Rapidly by Using Almost Any
Transformation,” Phys. Rev. Lett. 80, 4329–4332 (1998).
[6] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Informa-
tion, 2 ed. (Cambridge University Press, 2001).
[7] B. E. Kane, “A silicon-based nuclear spin quantum computer,” Nature 393,
133–137 (1998).
[8] L. C. L. Hollenberg, A. D. Greentree, A. G. Fowler, and C. J. Wellard, “Twodimensional
architectures for donor-based quantum computing,” Phys. Rev. B
74, 045311 (2006).
[9] A. Carlini, A. Hosoya, T. Koike, and Y. Okudaira, “Time-Optimal Quantum
Evolution,” Phys. Rev. Lett. 96, 060503 (2006).
[10] A. Carlini, A. Hosoya, T. Koike, and Y. Okudaira, “Time-optimal unitary operations,”
Phys. Rev. A 75, 042308 (2007).
[11] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr‥uggen, and S. J. Glaser, “Optimal
control of coupled spin dynamics: design of NMR pulse sequences by gradient
ascent algorithms,” J. Magn. Reson. 172 (2005).
[12] A. G. Fowler, C. J. Wellard, and L. C. L. Hollenberg, “Error rate of the Kane
quantum computer controlled-NOT gate in the presence of dephasing,” Phys.
Rev. A 67, 012301 (2003).
[13] C. D. Hill and H.-S. Goan, “Fast nonadiabatic two-qubit gates for the Kane
quantum computer,” Phys. Rev. A 68, 012321 (2003).
[14] C. D. Hill and H.-S. Goan, “Gates for the Kane quantum computer in the presence
of dephasing,” Phys. Rev. A 70, 022310 (2004).
[15] A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring, “Electron
spin relaxation times of phosphorus donors in silicon,” Phys. Rev. B 68, 193207
(2003).
[16] C. D. Hill, L. C. L. Hollenberg, A. G. Fowler, C. J. Wellard, A. D. Greentree,
and H.-S. Goan, “Global control and fast solid-state donor electron spin quantum
computing,” Phys. Rev. B 72, 045350 (2005).
[17] T. Yamamoto, Y. A. Pashkin, O. Astafiev, Y. Nakamura, and J. S. Tsai, “Demonstration
of conditional gate operation using superconducting charge qubits,” Nature
425, 941–944 (2003).
[18] A. Sp‥orl, T. Schulte-Herbr‥uggen, S. J. Glaser, V. Bergholm, M. J. Storcz, J.
Ferber, and F. K. Wilhelm, “Optimal control of coupled Josephson qubits,”
Phys. Rev. A 75, 012302 (2007).
[19] L. Kettle, H.-S. Goan, S. C. Smith, L. C. L. Hollenberg, C. I. Pakers, and C.
Wellard, (unpublished).
[20] C. D. Hill, Ph.D thesis (University of Queensland, Brisbane, Australia, 2006).
[21] C. Herring and M. Flicker, “Asymptotic Exchange Coupling of Two Hydrogen
Atoms,” Phys. Rev. 134, A362–A366 (1964).
[22] M. J. Testolin, C. D. Hill, C. J. Wellard, and L. C. L. Hollenberg, “Robust
controlled-NOT gate in the presence of large fabrication-induced variations of
the exchange interaction strength,” Phys. Rev. A 76, 012302 (2007).
[23] R. B. Sidje, “Expokit. A Software Package for Computing Matrix Exponentials,”
ACM Trans. Math. Softw. 24 (1998).
[24] C. Moler and C. V. Loan, “Nineteen Dubious Ways to Compute the Exponential
of a Matrix, Twenty-Five Years Later,” Society for Industrial and Applied
Mathematics 45 (2003).
[25] E. G. Birgin, J. M. Martinez, and M. Raydan, “Nonmonotone Spectral Projected
Gradient Methods on Convex Sets,” SIAM J. on Optim. 10 (2000).
[26] B. Kraus and J. I. Cirac, “Optimal creation of entanglement using a two-qubit
gate,” Phys. Rev. A 63, 062309 (2001).
[27] G. Vidal, K. Hammerer, and J. I. Cirac, “Interaction Cost of Nonlocal Gates,”
Phys. Rev. Lett. 88, 237902 (2002).
[28] M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer,
M. A. Nielsen, and T. J. Osborne, “Practical Scheme for Quantum Computation
with Any Two-Qubit Entangling Gate,” Phys. Rev. Lett. 89, 247902
(2002).
[29] M. L. Liou, “A novel method of evaluating transient response,” Proc. IEEEE 54
(1966).
[30] R. B. Sidje, Parallel Algorithms for Large Sparse Matrix Exponentials: applica-
tion to numerical transient analysis of Markov processes. (Ph. D. thesis, Univ.
of Rennes 1., 1994).