Ph.D. Thesis - Physics

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1996, right on the heels of Peter Shor’s 1994 factoring algorithm. Factoring and quantum simulation were the first two practical (and classically intractable) problems that quantum computers were shown to be able to efficiently solve. Since that time, numerous proposals have been set forth to do quantum simulation of physical systems. And, slowly but surely, experiments have started to catch up. Throughout this thesis, we will encounter several small-scale implementations of quantum simulation. There remain difficult problems to be solved, however, before a quantum simulator that can outperform classical computation is realized. In this thesis we address some of these problems. Our journey takes us across two different physical systems, nuclear spins and trapped ions, as we explore two of the most important problems facing quantum simulation: precision and scalability. The question of precision asks, for a given quantum simulation algorithm, how many digits of precision can be obtained, in principle, and how does this number scale with the space and time resources required? Also, in practice, how do the real-world effects of decoherence and control errors impact this precision? The exploration of this question in the context of solution-state nuclear magnetic resonance (NMR) forms the first part of this thesis. Quantum systems are generally hard for classical computers to simulate, although there do exist classical techniques for solving some large quantum systems either exactly or ap- proximately. To solve the exact dynamics of a general quantum system, however, requires a quantum simulator of the same number of interacting subsystems as the model being simulated. Moreover, each small quantum system is subject to the effects of decoherence and control errors. The question of scalability is: how do you build a reliable large-scale quantum simulator out of faulty small-scale quantum simulators? This is a highly nontrivial question, but trying to answer it for the case of trapped ions occupies the second and third parts of this work. This chapter is organized as follows. In Sec. 1.1, we present a brief overview of the con- cepts of quantum information processing, including motivations and experiments. Sec. 1.2 is an introduction to quantum simulation, including its history, most important literature, and most enticing applications. In Sec. 1.3, we discuss two different approaches to quantum simulation, termed digital and analog, both of which are of importance to this thesis. Some of the outstanding challenges in quantum simulation, focusing on the issues of precision and scaling mentioned above, are then discussed in Sec. 1.4, which in turn motivate the main results and content of the thesis, presented in Sec. 1.5. The final section, Sec. 1.6, enumerates the contributions of the author and coworkers to this thesis, as well as resulting publications. 22
1.1 Quantum information The concept of using the information contained in quantum states has led to the discovery of many new technologies, including computation, cryptography, communication, precision measurement, and simulation. The possibility of exponential speedups for tasks such as factoring and discrete loga- rithms [Sho94], in addition to the square-root (but nonetheless attractive) speed-up of the quantum search algorithm [Gro97], has spurred a huge effort from theoretical and experimental research groups toward developing a practical quantum information processor. Many experimental successes using NMR, trapped-ion, and other physical systems have demon- strated all the basic protocols of quantum computation and quantum communication. The theoretical breakthrough of the discovery of quantum error correction [CS96, Ste96], and with it fault-tolerant quantum computation [Sho96, DS96, Got97], has provided a way to overcome the unavoidable errors that will afflict the delicate information encoded in quantum states. With error correction, however, comes a large overhead in the number of qubits (quantum bits) that must be employed. The discovery of these algorithms has been accompanied by other interesting possibilities that make use of the tools of quantum information. One of these is quantum cryptography, which is a provably secure protocol for private key distribution [BB84]. A series of qubits is used in this scheme to transmit a key which can be used to decode a message; attempting to intercept the quantum message leads to a loss of the coherence of the message, and this can be detected and the key discarded in the event of eavesdropping. It is remarkable that the laws of physics guarantee privacy that no one may breech. This is to date the most technologically advanced application of quantum information, in that commercial quantum cryptography systems are already available 1 . Another exciting application of quantum information has been to precision measurement. There are two main approaches: the first has shown that phase measurement on a set of N entangled qubits has an error that scales as 1/N, whereas the best that is pos- sible classically is 1/ √ N. Another more recent technique uses a two-qubit quantum logic gate in an ion trap to probe the structure of an ion that itself has no “cycling transition” and thus cannot be directly probed [SRL + 05]. Such techniques have improved our measurement of time and frequency, and perhaps also enable more-precise measurements of physical constants such as the fine structure constant. The field of quantum communication is integral to many of the applications above, and is also interesting in its own right. The most noteworthy protocol for transmission of quantum information is quantum teleportation, in which a quantum state may be sent between two arbitrarily distant persons if they share a single entangled state in advance, and use two bits of classical communication per quantum bit [BBC + 93]. A related technique, superdense coding, permits the transmission of two classical bits using one qubit of a two-qubit entangled 1 For example, see MagiQ: http://www.magiqtech.com 23
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: Chapter 1 Introduction The growth o
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 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
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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
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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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