Ph.D. Thesis - Physics

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precision of a quantum simulation is then discussed, in light of these experiments, in Sec. 2.3. In Sec. 2.4, we summarize the chapter and then enumerate some outstanding questions that remain to be addressed. 2.1 Hamiltonian and control techniques In this section we present a simple description of the NMR system, and with it the Hamilto- nian that governs the nuclear spins. We then show how this Hamiltonian enables quantum control of the nuclear spins. In so doing, we explain how a NMR system satisfies most of the DiVincenzo criteria for quantum computation (Sec. 1.4.3). We begin with a comment on the first criterion. Although suitable for small systems, NMR is not a scalable architecture for quantum computation (or universal quantum simulation) because there is no way to do operations fault-tolerantly. It is not possible to perform operations on the system conditioned on fast measurements, as would be required in the diagnosis and correction of errors. This stems from the nature of measurement in solution-state NMR, which is based on a slow recording of the bulk magnetization of the system (see Sec. 2.1.6). The other criteria are state initialization, a universal set of gates, high-efficiency measurement, and a sufficiently long coherence time; the satisfaction of these by NMR will be explained in the remainder of this section. Throughout we will use “nucleus,” “spin,” and “qubit” interchangeably, since the exclusive topic of this section is spin-1/2 nuclei. 2.1.1 The static Hamiltonian The NMR system consists of a sample of dissolved molecules, the nuclear spins of which, in thermal equilibrium, are aligned (or antialigned) with a strong external magnetic field of magnitude B0, which by convention points along the ˆz direction. Oscillating magnetic fields of peak magnitude B1, and oriented along ˆx and ˆy, are applied to the nuclei to rotate their spin states. Finally, the bulk magnetization of the sample may be read out inductively, giving an ensemble measurement of the spin states. Details of this setup for a real experimental system are given in Sec. 3.4. In this chapter, we focus on the Hamiltonian and control techniques. The static portion of the NMR Hamiltonian is composed of two terms: one describes rotations of the nuclei about the ˆz axis due to the the static magnetic field, and the other describes the spin-spin coupling between nuclei. Using i to index the individual nuclear spins, the term due to B0 is written H0 = ω0Zi , (2.1) i where the Larmor frequency ω0 is given by ω0 = gNµnB0/, and gN is the nuclear g-factor, 48
µN the nuclear magneton, and B0 the magnitude of the magnetic field along ˆz. Zi is the Pauli Z matrix operating on qubit i. Here, and throughout, we use the most common phase convention for the Pauli matrices: X = 0 1 1 0 ; Y = 0 −i i 0 ; Z = 1 0 0 −1 . (2.2) where the spin states, unless otherwise noted, are written in the ˆz basis, with vector repre- sentations |↑〉 = 1 0 and |↓〉 = 0 1 . (2.3) The other important term in the static Hamiltonian arises from couplings between individual nuclei that are mediated by the electron clouds surrounding them, which form the chemical bond in the molecule. Although, in general, there are also direct dipole-dipole couplings, in a dilute solution this interaction averages to zero. The through-bond coupling is called scalar coupling, or just J-coupling. This interaction takes the form of a ZZ interaction: HI = 4 N JijZiZj , (2.4) i,j
Page 1: An Investigation of Precision and S
Page 5 and 6: Acknowledgments What a long, strang
Page 7: should totally start a band. I also
Page 10 and 11: 2.4 Conclusions and further questio
Page 12 and 13: 7.6.1 Regular array of stationary q
Page 15 and 16: List of Figures 2-1 The CNOT gate i
Page 17: 7-12 Magnetic fields due to a “tr
Page 21 and 22: Chapter 1 Introduction The growth o
Page 23 and 24: 1.1 Quantum information The concept
Page 25 and 26: us whether the target system obeys
Page 27 and 28: Nevertheless, there are a number of
Page 29 and 30: Digital Analog Classical Quantum Sp
Page 31 and 32: electrodes [TKK + 99]. This heating
Page 33 and 34: Digital Analog Classical Quantum Sp
Page 35 and 36: 2. Initialization to a simple fiduc
Page 37 and 38: qubits, and in Ref. [SGA + 05] in t
Page 39 and 40: the lattice architecture, we discov
Page 41 and 42: 1.6 Contributions to this work In t
Page 43: 5) Ref. [DLC + 09b] Wiring up trapp
Page 47: Chapter 2 Quantum simulation using
Page 51 and 52: ⎡ ⎤ a 0 0 0 ⎢ ⎥ ⎢ ρ2 =
Page 53 and 54: The same effect as number 2 above i
Page 55 and 56: where V0 is the maximum signal stre
Page 57 and 58: actions that occur between nuclei i
Page 59 and 60: Figure 2-2: The 2,3-dibromothiophen
Page 61: detail how this was done with a sim
Page 64 and 65: and thus requires invocation of the
Page 66 and 67: Renormalization Group (DMRG), with
Page 68 and 69: 3.3 The bounds on precision In this
Page 70 and 71: Figure 3-1: A schematic of a generi
Page 72 and 73: Figure 3-3: The 11.7 T, 500 MHz sup
Page 74 and 75: . Figure 3-4: The above probe is a
Page 76 and 77: Figure 3-5: The CHFBr2 molecule. Al
Page 78 and 79: Figure 3-6: Frequency-domain spectr
Page 80 and 81: used to extract the answer is irrel
Page 83 and 84: Chapter 4 Theory and history of qua
Page 85 and 86: GND RF r ENDCAP 0 z 0 ENDCAP Ring T
Page 87 and 88: where Q is the charge of the trappe
Page 89 and 90: angular momentum along the quantiza
Page 91 and 92: emission from the excited state |
Page 93 and 94: selection rule ∆m = ±1 is applie
Page 95 and 96: Figure 4-3: Schematic diagram of sp
Page 97 and 98: to address an ion in state |↑〉
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the vibrational temperature is low.
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to the design of the trap itself. M
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to the quantum-mechanical ground st
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Chapter 5 Lattice ion traps for qua
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Figure 5-1: Schematic of the lattic
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Figure 5-3: Fit to Eq. 5.2 of the C
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Figure 5-4: The vacuum chamber for
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weeks. Low pressures depend on choo
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Figure 5-5: Left: Level diagram for
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Figure 5-7: (a) Schematic of the cr
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(a) Lattice Trap Schematic TOP PLAT
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mix the microspheres evenly in the
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Using the expression for ωˆr deri
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Figure 5-15: Plot of the motional c
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Figure 5-17: Dependence of the trap
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Chapter 6 Surface-electrode PCB ion
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Figure 6-1: Schematic of a linear i
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Figure 6-2: Layout of the trap elec
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Figure 6-3: Above are the cross sec
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Figure 6-6: Photograph of the trap
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Figure 6-8: CCD image of a cloud of
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Figure 6-9: Measurement results sho
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Figure 6-11: Bastille mounted in th
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Figure 6-13: A diagram of the setup
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Figure 6-14: A plot of the trapped
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allows us to upper-bound the amount
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Chapter 7 Quantum simulation in sur
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Figure 7-1: Left: Uraniborg 1, as r
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Figure 7-3: Calculated secular freq
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Figure 7-5: Left: Structure of 2-D
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Figure 7-7: Scaling of the order of
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involved. Because ions near the cen
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Results Before presenting the numer
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Figure 7-10: Calculation results fo
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7.3 Magnetic gradient forces Now th
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Figure 7-13: Scaling of the J-coupl
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how are single-ion operations to be
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250 psi, which increases to around
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Figure 7-18: At the center of this
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Figure 7-19: The plastic socket tha
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Figure 7-21: The two internal lense
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Figure 7-22: Measured secular frequ
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Along ˆy, the spacing is dˆy = 28
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of O2 to 2 × 10 −12 torr. Decohe
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Part III Toward ion-ion coupling ov
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Chapter 8 Motivation for and theory
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Figure 8-1: Schematic of the experi
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are in close agreement in the case
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8.2.3 Simulated coupling rates We n
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Johnson noise We begin our discussi
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should suffice. Note that the above
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Chapter 9 Measuring the interaction
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Figure 9-2: Left: Level diagram for
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Figure 9-5: Photograph of the micro
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A fit of the resulting curve return
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Figure 9-9: Linear fit of the ωˆy
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Figure 9-12: Heating rate as determ
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Chapter 10 Conclusions and outlook
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traps, linking ions using photons,
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Bibliography [ADM05] Paul M. Alsing
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[CGB + 94] J. I. Cirac, L. J. Garay
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[GME + 02] M. Greiner, O. Mandel, T
Page 223 and 224:
[LCL + 07] D. R. Leibrandt, R. J. C
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[PC04a] D. Porras and J. I. Cirac.
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[TBZ05] L. Tian, R. Blatt, and P. Z
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Appendix A Matlab code for Ising mo
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err_result = mean(theerr) 231
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J = J0; Jt(1) = J; Bt(1) = B; for k
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A.3 Simulation with constant force
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Appendix B Mathematica code for ion
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2 crystal_shape_6uraniborg.nb Out[6
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Appendix C How to trap ions in a cl
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Finally, we note that the chilled w
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C.4 Laser alignment and imaging As
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