Ph.D. Thesis - Physics

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Renormalization Group (DMRG), with which, for example, a pairing model with N ≈ 400 has been solved to five digits of precision [DS99]. However, DMRG is still a fundamentally approximate method. In this chapter we explore an alternative quantum algorithm that is, in principle, numerically exact. 3.2 The Wu-Byrd-Lidar proposal In 2002, Wu, Byrd, and Lidar (WBL) proposed a method for simulating the BCS Hamilto- nian on a NMR-type quantum computer [WBL02]. In this section we will, following their paper, explain how to use a digital NMR-type quantum simulator to calculate the low-lying spectrum of the pairing Hamiltonian. We are concerned with finding the energy gap ∆ between the ground and excited states for a specified Hamiltonian. Wu et al. prove that ∆ can be computed using O(N 4 ) steps, where N is the number of qubits, and also the number of quantum states included in the simulation, equivalent to the number of modes N above. They begin by mapping the Fermionic creation and annihilation operators to qubit operators. A computational |1〉 state signifies the existence of a Cooper pair in one of the possible modes, while |0〉 is the vacuum state for that mode. Thus, the total number of qubits in the |1〉 state signifies the number of Cooper pairs, and the total number of qubits translates into the total number of modes that might be occupied. Accordingly, a measurement of the number operator nm = (Zm + 1)/2 yields 1 if the mode m is excited, and 0 otherwise. The operator σ + m (σ − m) signifies the creation (annihilation) of a Cooper pair in mode m: σ + m ↦→ c−mcm and σ − m ↦→ c † mc † −m The above identification holds for pairing Hamiltonians that simulate Fermionic particle- particle interactions; thus, it is appropriate for the BCS Hamiltonian. Two other cases are given in their paper, but for simplicity, we will restrict our discussion to the case that we actually implemented experimentally (the BCS Hamiltonian). From the above point, it is simple to write HBCS in terms of qubit operators, using the identities σ + = (X +iY )/2 and σ − = (X − iY )/2. The final Hamiltonian that we use in our NMR implementation is HBCS = n m=1 νm 2 (−Zm) + m
up the evolution a bit, it is possible to produce excitation to |E〉 without placing significant population into the higher-energy excited states. 3. Apply the unitary transformation UBCS(tq) = exp (−iHBCStq/) for tq ranging from 0 to the total time tQ, where a total of Q steps are used. Each time, measure one qubit in the Z basis. Any operator M for which 〈G|M |E〉 = 0 may be measured. 4. Perform a (classical) discrete Fourier transform (DFT) on the measured spectra; frequency peaks will be present at ∆, the energy gap between |G〉 and |E〉. We now explain how each piece is implemented in an NMR system. Step 1 is a very common and well-understood operation in NMR quantum computing. Starting in the ground state |00...0〉, the spins may be initialized by simply applying single- qubit rotations Rx(π) to the desired qubits. Step 2, the quasi-adiabatic evolution, is implemented in the following way. The system is evolved under a Hamiltonian that gradually changes from the NMR Hamiltonian H0 + HI (c.f. Sec. 2.1) into HBCS. Specifically, the Hamiltonian used at each step is H = (1 − s/S)H0 + (s/S)HBCS. The unitary corresponding to each is U = exp(−iH(s)τ/), where the timestep τ and total number of steps S together control the rate of quasiadiabatic evolution. This procedure is discussed further in [SvDH + 03]. The construction of HBCS is discussed presently. Step 3, evolution under HBCS, is done by a pulse sequence designed as follows. We re- quire an approximation of the unitary evolution under HBCS, UBCS(qt0) = exp(−iHBCSqt0). An ideal NMR implementation accomplishes this by a repeatable pulse sequence VBCS(t0), where UBCS(qt0) ≈ (VBCS(t0)) q . The Hamiltonian HBCS (Eq. 3.2) contains three noncommuting parts: H0 = m νm 2 (−Zm), HXX = Vml m
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An Investigation of Precision and S
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Acknowledgments What a long, strang
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should totally start a band. I also
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2.4 Conclusions and further questio
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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 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 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 |↑〉
Page 99 and 100: the vibrational temperature is low.
Page 101 and 102: to the design of the trap itself. M
Page 103 and 104: to the quantum-mechanical ground st
Page 105 and 106: Chapter 5 Lattice ion traps for qua
Page 107 and 108: Figure 5-1: Schematic of the lattic
Page 109 and 110: Figure 5-3: Fit to Eq. 5.2 of the C
Page 111 and 112: Figure 5-4: The vacuum chamber for
Page 113 and 114: weeks. Low pressures depend on choo
Page 115 and 116: 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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