Ph.D. Thesis - Physics

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state [BW92]. Quantum communication also enables unconditional exponential gains in communication complexity for certain problems, e.g. as discussed in Refs. [HBW98, Raz99, BCWdW01, GKK + 08]. The application of quantum information processing that is of most interest to many physicists, and that motivates the work of this thesis, is quantum simulation. It is presented in its own section below. Quantum information primitives have been implemented in a number of different physical systems, including trapped ions ([LBMW03], and many others cited in this thesis), nuclear spins ([VSB + 01, VC05], and many others cited in this thesis), superconducting circuits [MNAU02, MOL + 99, NPT99, HSG + 07], neutral atoms [ALB + 07, BCJD99, MGW + 03], and quantum dots [LD98, KBT + 06, PJT + 05]. Each of these systems carries certain advantages and disadvantages. Solution-state NMR was the first system in which small algorithms, including the simplest nontrivial case of Shor’s algorithm, were implemented [VSB + 01]. The long coherence times of nuclear spins combined with the superb control techniques already developed (and developed further still by quantum information researchers [VC05]) enabled this system to get off the ground fairly quickly. However, NMR cannot scale to large numbers of qubits. Trapped ions have now emerged instead as the leading technology, because they combine coherence times even longer than nuclear spins in solution with a now highly-developed control apparatus, superior measurement accuracy, and several promising avenues to scalability. Of course, no technology has yet been scaled up to even more than ten qubits, meaning that a quantum computer or simulator that can outperform classical computation has not yet come close to being realized. 1.2 What is quantum simulation? We now begin to explore the overarching theme of this thesis, quantum simulation. Quan- tum simulation is the use of one controllable quantum system, referred to in this thesis as the model system, to calculate properties of some other quantum system that is more difficult to control, which we call the target system. For instance, one may map a Hamiltonian that describes a set of interacting fermions to a set of spin-1/2 systems which an experi- menter can control well, such as (in this thesis) nuclear spins in molecules or the electronic states of trapped ions. If controls are available to apply the same effective dynamics on the model system that is thought to govern the target system, then predictions regarding the target system can be inferred by performing experiments on the model system. Although deceptively simple as stated here, in practice it is difficult to discover good methods for performing quantum simulation. Of course, the person carrying out the quantum simulation also must make the assumption that the target system obeys a specific Hamiltonian. This could be stated in a positive light, in that simulation by model systems may inform 24
us whether the target system obeys a given Hamiltonian at all! The concept of quantum simulation was first conjectured by Feynman in 1982 [Fey82], and subsequently rigorously proven by Lloyd in 1996 [Llo96]. Following this, a number of significant papers presented in detail methods for simulating a variety of physical systems. Abrams and Lloyd discovered algorithms for calculating the eigenvalues and eigenvectors of a Hamiltonian on a quantum computer [AL97] and for simulating the dynamics of fermionic many-body systems [AL99]. Following this, a 2001 paper presented a method for efficiently simulating quantum chaos and localization [GS01]. In 2002, a paper of great importance to our work was published by Wu et al., which proposed a method for simulating pairing models on an NMR-type quantum computer [WBL02]. Jumping ahead a bit, in 2005 an interesting report was published, detailing classical simulations of a quantum computer calculating molecular energies [AGDLHG05]. This paper suggested that, using only tens of qubits, a quantum computer might indeed be able to solve certain problems in chemistry more efficiently than a classical computer can. This paper was followed-up by an article detailing a general polynomial-time algorithm for the simulation of chemical dynamics [KJL + 08]. The above partial listing of the important literature is meant to illustrate the breadth of interest in quantum simulations, but also an important caveat: it is not trivial to design a quantum simulation algorithm for a given quantum system. There is no “general purpose” quantum algorithm that can solve any problem in quantum mechanics efficiently. Although it may be efficient to implement a given simulated Hamiltonian, it is not, in general, straightforward to design an efficient measurement. It is fairly easy to see why: the length of the state vector for n qubits is of length 2 n . Even if the state could be measured without collapse, inquiring as to the state of the whole system is intractable. Part of the art of designing quantum simulations is asking the right questions. We would like to note a common theme throughout this thesis and a dominant one in the current effort of researchers worldwide. Some of the most interesting and classically- intractable models occur within the realm of solid-state physics, especially with regard to superconductivity. The BCS Hamiltonian is a model for which some good approximate methods (such as the density matrix renormalization group) exist, but which eludes a full quantum-mechanical description on a classical computer. The paper of Wu et al. [WBL02] provides a way for a quantum simulator to calculate the low-lying spectrum of this Hamil- tonian in polynomial time. This is only the beginning, though. BCS theory works well for Type-I superconductors, and the main advantage of quantum simulation of this model would be in calculating the spectra to greater precision. However, the mechanism behind Type-II superconductivity is still poorly understood. It has been conjectured that the phenomenon of spin frustration in a 2-D lattice of antiferromagnetically interacting spins may hold some insight into the dynamics of high-temperature superconductors [GP00, NGB92, BDZ08]. Research is active into using both neutral atoms and trapped ions to realize such spin Hamiltonians [DDL03, 25
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: 1.1 Quantum information The concept
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 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
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. Figure 3-4: The above probe is a
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Figure 3-5: The CHFBr2 molecule. Al
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Figure 3-6: Frequency-domain spectr
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used to extract the answer is irrel
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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
Page 205 and 206:
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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