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From the resulting states, the 16 probabilities P ab (| 00 〉), P ab (| 01 〉), . . . are extracted and combined to yield S. If the simulation is run as described, i.e. without including imperfections, it yields the expected value of S = 2.828. Using the Kraus operators K 1a and K 1b as described in Section 3.4.5, energy decay can be added to the simulation using the values for T 1 from Table 11.13 and the pulse lengths from Table 11.11. This lowers the resulting S-value to S = 2.443. If dephasing is added as well, using K ϕa and K ϕb , the value further decreases to S = 2.247. The final error mechanism to include are the errors caused by non-ideal measurement fidelities, which yields S = 1.984. Since this simulation corresponds to an experiment where the Bell rotations and measurement happen simultaneously on both qubits, this S-Value needs to be compared to the one shown in Figure 11.4b at t = −1 ns: S = 1.986. The agreement is remarkably good. If the simulation is modified to move the Bell rotations and measurement on the second qubit forward as done in the actual experiment, it yields S = 2.064. The corresponding error budget is shown in Table 11.14. The observation that the experimental result is even slightly better than the prediction by simulation is most likely rooted in that fact that the theoretical rotation angles (α = −135 ◦ , α ′ = 135 ◦ , β = 0 ◦ , and β ′ = −90 ◦ ) used in the simulation are not actually optimal in the presence of decoherence and imperfect measurement. Also, the 278
Figure 11.7: Visibility Analysis – Resonator Coupled Sample: Composite of several datasets. Blue dots represent data, red lines are fits through the data, and green lines are fits through the extrema of the red lines. The upper parabolas correspond to Rabi oscillations driven with pulses at fixed length and increasing amplitude (Power Rabis) around the point where they yield a π-pulse. The bottom parabolas are Power Rabis driven around the point where they yield a 2π pulse. The horizontal dataset at the bottom corresponds to no drive on the qubit. The green fits through the parabolas’ extrema (optimal π or 2π pulses) give the measurement visibility when extrapolated to t = 0, i.e. to an optimal, instantaneous pulse. The horizontal line checks the method by providing a direct measurement of the | 0 〉 state visibility. Since the measurements obtained agree to high precision, the method can be trusted to extract a | 1 〉 state fidelity for which no direct measurement is available. – a) Qubit A: F | 0 〉, Rabis = 96.86%, F | 0 〉, Direct = 97.04%, F | 1 〉 = 96.32%. b) Qubit B: F | 0 〉, Rabis = 96.06%, F | 0 〉, Direct = 96.18%, F | 1 〉 = 98.42%. 279
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UNIVERSITY of CALIFORNIA Santa Barb
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Benchmarking the Superconducting Jo
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viii
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Curriculum Vitæ Markus Ansmann Edu
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99:187006 Lisenfeld, J., Lukashenko
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Abstract Benchmarking the Supercond
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Contents Contents List of Figures L
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4.1.3 Readout Squid Parameters . .
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7.5.5 Grapher . . . . . . . . . . .
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11.5 Analysis and Verification . .
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List of Figures 2.1 Inductor-Capaci
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List of Tables 3.1 Transition Matri
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Chapter 1 Quantum Computation 1.1 M
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for such problems include factoring
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if present encrypted data will rema
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In terms of quantum bits, this mean
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1.2.3 Implications - The EPR Parado
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proposed architecture of quantum bi
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“coherence times”, i.e. the tim
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Chapter 2 Superconducting Josephson
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of the glue-circuitry needed to con
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Figure 2.1: Inductor-Capacitor Osci
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• The circuit needs to be cooled
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Figure 2.2: Josephson Tunnel Juncti
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the trace along the V-axis, gives c
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Figure 2.4: Josephson Qubits: Sligh
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the cosine forms a local minimum al
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Figure 2.5: Example Qubit Coupling
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Readout schemes can further be cate
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states in the qubit’s inductor, t
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at time t. r is not restricted to b
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3.1.2 Effects of a Time Dependent P
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In some cases, it is possible to so
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Figure 3.1: Examples of Numerical S
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• The energy difference between t
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like this: V = ( V (−1, −1), V
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Figure 3.2: Simulation of LC Oscill
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Table 3.1: Transition Matrix Elemen
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with ω mn = Em−En . Multiplying
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α, it can be ignored. Thus, the in
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e solved exactly: A(t + ∆t) = e
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qubits would be simulated using: A(
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This calculation assumes that the s
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Decoherence consists of two parts:
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Note the difference in signs of the
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Chapter 4 Designing the Phase Qubit
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mutual inductance between the qubit
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During the measurement, the | 1 〉
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the right impedance transformation
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excitations. Since these are a pote
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Figure 4.3: Squid I/V Traces - a) L
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Figure 4.5: Qubit Integrated Circui
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The geometry of the qubit junction
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squid loop. Thus, this tool can be
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now, amorphous silicon seems to pro
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Figure 5.1: L-Edit Mask Layout Tool
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Figure 5.2: Fabrication Building Bl
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Figure 5.3: Photolithography and Et
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times the removal can be a bit tric
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Figure 5.4: Clearing Vias from Nati
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5.6 Junction Layers 5.6.1 Oxidation
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top wiring layer to protect all low
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104
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6.1 Physical Quality Control during
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6.1.3 Atomic Force Microscopy To re
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Figure 6.1: 4-Wire Measurement - a)
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6.3 Quantum Measurements at 25 mK 6
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seems to be a box machined out of s
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Figure 6.2: Dilution Refrigerator W
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cessing data. This protects the vol
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6.3.9 Anritsu Microwave Source The
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122
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ment, the scalability requirements,
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people without any formal training
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7.2.4 Performance Last, but certain
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or a Client Module. Client Modules
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second Module talks to all these an
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puters to talk to each other. Usual
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7.3.4 Performance Addressing the Pe
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is designed such that the LabRAD Ma
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Table 7.3: LabRAD Type Annotations
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listed in Table 7.3. For transmissi
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Architecture to manage network conn
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Manager. In fact, in our lab, the o
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waiting for their completion. The C
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mentation of pipelining and certain
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Since the API guarantees that all R
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one microwave line for X/Y-rotation
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7.5.3 DC Rack Server The DC Rack Se
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data taking on the lab servers and
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keys and the ability to set Context
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a certain time. 7.5.9 Optimizer Cli
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ters read from different sub-direct
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efore the execution of the sequence
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can achieve very-close-to hardware
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type to provide a one-stop location
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172
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8.1 Squid I/V Response As explained
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a digital signal via the use of a c
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energy landscape (see Chapter 2.2.3
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Figure 8.3: Squid Steps Failure Mod
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At this point, the squid ramp can b
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starts to tunnel to the neighboring
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Figure 8.5: General Bias Sequence -
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Figure 8.7: Spectroscopy - a) Bias
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Figure 8.8: Rabi Oscillation - a) B
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ensemble with respect to each other
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Figure 8.10: T 1 - a) Bias sequence
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Figure 8.11: Ramsey - a) Bias seque
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phase shift into the middle of the
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photon excitation behaves similarly
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is desirable to modularize the cont
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sequence was run. This allows for t
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possible to read out all qubits cor
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simultaneous application of such me
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Figure 9.4: Capacitive Coupling Swa
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Figure 9.6: Capacitive Coupling Pha
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a phase difference of 0 ◦ rather
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Figure 9.7: Fine Spectroscopy of Re
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Figure 9.8: Swapping Photon into Re
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esonator. The latter calibration is
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Figure 9.11: Resonator T 1 - a) Seq
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224
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He does not throw dice” [Einstein
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Page 268 and 269: 11.1 State Preparation Since the ab
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Page 272 and 273: Table 11.1: Entangled State Density
Page 274 and 275: Figure 11.2: Bell Measurements - a)
Page 276 and 277: 11.2.3 Statistical Analysis For the
Page 278 and 279: 11.3 Calibration The most difficult
Page 280 and 281: Table 11.4: Sequence Parameters - R
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Page 288 and 289: Table 11.7: Bell Violation Results
Page 290 and 291: Table 11.9: Error Budget - Capaciti
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Page 298 and 299: 11.5.2 Dependence of S on Sequence
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Page 304 and 305: Figure 11.6: Resonator Coupled Samp
Page 308 and 309: Table 11.14: Error Budget - Resonat
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Page 314 and 315: educe energy decay and dephasing an
Page 316 and 317: John F. Clauser, Michael A. Horne,
Page 318 and 319: J. Majer, J. M. Chow, J. M. Gambett
Page 320: Matthias Steffen, M. Ansmann, Rados
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