Ph.D. Thesis - Physics

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To conclude our discussion of precision, we note that even as the number of classically simulable quantum systems grows through the discovery of new algorithms, it is generally believed that classical computation is not able to approximate any general quantum- mechanical system. If it could, then there would be no speedup of quantum computation over classical; a classical method would then exist for factoring, searching, and all other quantum algorithms, by classical simulation of a quantum simulator. For the quantum systems for which no classical approximation is known, quantum simulation is an attractive possibility, despite the inherent precision limitations. 1.4.3 Scalability Scalability is the study of how to take a small number of faulty systems and connect them to form one arbitrarily large, reliable system. This is an important subject in quantum computation; the solution of computationally hard problems requires such a network of qubits. To begin with, we note that a digital quantum simulator is a quantum computer that may not have a universal set of gates; that is, it may not require sufficient controls to simulate any Hamiltonian, but rather just the Hamiltonian of the problem of interest. A quantum simulator may not need the same degree of control, but the problem of scalability is the same. Analog systems will also be composed of many distinct subsystems, but the form of the scaled-up architecture may be different. For example, an optical lattice presents a framework for simulating the Bose-Hubbard model on a large scale, but one does not need the ability to control the internal states of individual atoms, or to implement deterministic two-atom gates, that would be necessary for digital quantum simulation. There is still the question of how large a quantum system will need to be to implement quantum dynamics that are intractable on a classical computer. A recent benchmark [RMR + 07] sets the record, to our knowledge, of exactly simulating 36 interacting spin-1/2 systems on a supercomputer. The importance of this number is that even if a given idea for scalability does not lead to arbitrarily large systems, it may still be of great use as a quantum simulator, and be able to exactly simulate quantum dynamics that classical computers cannot. To scale to this or a larger number of subsystems, however, requires a system that satisfies certain conditions. These are important for creating a system in which sufficient control is available to execute a given simulation, and to measure the result. DiVincenzo criteria To begin the discussion of scalability, we present the five basic DiVincenzo criteria [DiV00], which are the generally agreed-upon requirements for a scalable quantum computer. Al- though these were originally proposed for quantum computation, they still hold for less- general quantum simulations, albeit in a modified form, and we review them here. 1. A scalable physical system with well-defined qubits. 34
2. Initialization to a simple fiducial state such as |000...0〉. 3. Decoherence times much longer than the gate times. 4. A universal set of quantum gates. 5. High efficiency, qubit-specific measurements. We now briefly summarize how these criteria differ for quantum simulation. The first criterion certainly holds, although the form of the large-scale quantum simulator may be different for analog, as opposed to digital, quantum simulation (as discussed above). The second and third also hold, but the initial state will depend greatly on the problem being simulated. It may be necessary to create a state such as |000...0〉, which may then be transformed into the correct initial state through single-qubit rotations. An example of this is in Ref. [WBL02]. However, it is also possible that the relevant degrees of freedom may be described by continuous variables. In the example of cold atoms in an optical lattice, it is typically necessary for the translational degrees of freedom of the atoms to be cold; however, the simulation itself may not depend on the internal states. This is one way in which analog quantum simulation enables simulation with less control than digital; in the digital case, the momenta of the atoms would be discretized and then programmed into a set of qubits at the beginning of the simulation. The fourth criterion must also be modified for quantum simulation, and again varies between the digital and analog types. Depending on the problem being implemented, a set of controls less powerful than those of a universal quantum computer may be required. The optical lattices are again a good example. In this analog simulation, sufficient control is present to simulate a Bose-Hubbard model. By contrast, a universal quantum simulator is equally as powerful as a universal quantum computer, because for either any arbitrary Hamiltonian (and corresponding unitary gate) may be approximated to arbitrary accuracy. It is also true, however, that for certain digital quantum simulations, only a subset of a universal gate set will be required. Finally, high measurement fidelity is important in both analog and digital quantum simulation, but for the analog variety, it may be a measurement of a continuous variable, and “qubit-specific” may not have meaning. Measuring the density of atoms in space is an example of this. Scaling up quantum simulators is a major research question no matter what physical system forms the qubit. The two physical systems that are studied in this thesis are solution- state NMR and trapped ions; however, solution-state NMR is intrinsically unscalable. Ion traps, by contrast, do satisfy all the DiVincenzo criteria. The only caveat is that although ion traps are scalable in principle, we do not yet know which method for scaling them up will prove most effective. We therefore focus here on the scalability of ion traps. 35
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: Digital Analog Classical Quantum Sp
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 and 48: Chapter 2 Quantum simulation using
Page 49 and 50: µN the nuclear magneton, and B0 th
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
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GND RF r ENDCAP 0 z 0 ENDCAP Ring T
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where Q is the charge of the trappe
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angular momentum along the quantiza
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emission from the excited state |
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selection rule ∆m = ±1 is applie
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Figure 4-3: Schematic diagram of sp
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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
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[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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