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Chapter 3 Understanding Qubits Numerically 3.1 Solving the Schrödinger Equation The evolution of any quantum system is described by the Schrödinger equation [Schrödinger, 1926], named after Erwin Schrödinger who discovered it in 1926: i ∂ t ψ (r, t) = Ĥ ψ (r, t) (3.1) Here, ψ (r, t) is the state of the system expressed by its “wave-function” as a function of position r and time t. The wave-function is also called the “probability amplitude” as its square gives the probability of finding the system at position r 37
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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
Page 13 and 14: 99:187006 Lisenfeld, J., Lukashenko
Page 15 and 16: Abstract Benchmarking the Supercond
Page 17 and 18: Contents Contents List of Figures L
Page 19 and 20: 4.1.3 Readout Squid Parameters . .
Page 21 and 22: 7.5.5 Grapher . . . . . . . . . . .
Page 23 and 24: 11.5 Analysis and Verification . .
Page 25 and 26: List of Figures 2.1 Inductor-Capaci
Page 27 and 28: List of Tables 3.1 Transition Matri
Page 29 and 30: Chapter 1 Quantum Computation 1.1 M
Page 31 and 32: for such problems include factoring
Page 33 and 34: if present encrypted data will rema
Page 35 and 36: In terms of quantum bits, this mean
Page 37 and 38: 1.2.3 Implications - The EPR Parado
Page 39 and 40: proposed architecture of quantum bi
Page 41 and 42: “coherence times”, i.e. the tim
Page 43 and 44: Chapter 2 Superconducting Josephson
Page 45 and 46: of the glue-circuitry needed to con
Page 47 and 48: Figure 2.1: Inductor-Capacitor Osci
Page 49 and 50: • The circuit needs to be cooled
Page 51 and 52: Figure 2.2: Josephson Tunnel Juncti
Page 53 and 54: the trace along the V-axis, gives c
Page 55 and 56: Figure 2.4: Josephson Qubits: Sligh
Page 57 and 58: the cosine forms a local minimum al
Page 59 and 60: Figure 2.5: Example Qubit Coupling
Page 61 and 62: Readout schemes can further be cate
Page 63: states in the qubit’s inductor, t
Page 67 and 68: Equation 3.6, also called the “ti
Page 69 and 70: As the eigenstates are ortho-normal
Page 71 and 72: This can be expressed as a tri-diag
Page 73 and 74: pecially for the higher levels, for
Page 75 and 76: • The x-range over which the pote
Page 77 and 78: This will most likely take forbiddi
Page 79 and 80: Figure 3.3: Simulation of Mock-Qubi
Page 81 and 82: | 12 〉 and | 14 〉 (desired tran
Page 83 and 84: Figure 3.4: Bloch Sphere - a) Bloch
Page 85 and 86: safely ignored. This step is called
Page 87 and 88: 3.3.3 Single Qubit Operations in a
Page 89 and 90: To couple qubits 2 and 3 out of fiv
Page 91 and 92: P ′′ | 01 〉(t) = (1 − X 21
Page 93 and 94: sphere. Instead, it corresponds to
Page 95: [ ] 1 0 K ϕa (∆t) = 0 e −∆t/
Page 98 and 99: Figure 4.1: Qubit Circuit: The qubi
Page 100 and 101: obtain control electronics that can
Page 102 and 103: will focus only on the integrated c
Page 104 and 105: Figure 4.2: Readout Squid - a) Circ
Page 106 and 107: the qubit to dissipate energy and t
Page 108 and 109: Figure 4.4: Spice Coupler Design -
Page 110 and 111: 4.2.1 Qubit When designing the indu
Page 112 and 113: 4.2.2 Bias Coil The design of the b
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small scales of these barriers, it
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Chapter 5 Phase Qubit Fabrication T
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that will be used in the later fabr
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the sapphire wafer is positioned cl
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5.3.3 ICP Etch Next, the wafer is l
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quality of the film deposited in th
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During this step, argon ions are ac
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Figure 5.5: 3D View of Junction: Th
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5.9 Top Wiring Layer - Part III Sin
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Chapter 6 Device Testing Equipment
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6.1.2 Scanning Electron Microscopy
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6.2 Electrical Screening after Fabr
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down to temperatures around 4 K. Th
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happens continuously, the DR can th
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This melts the aluminum wire and bo
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a readout line. These two lines are
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described in Chapter 8 (specificall
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Filters selectively attenuate certa
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Chapter 7 Control Software - LabRAD
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Requirement. Second, as the control
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7.2.3 Efficiency To promote accepta
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dent modules, but it also provides
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Another benefit of this hard separa
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different angles simultaneously, if
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7.3.3 Cross-Language and Cross-Plat
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useful to us in the future. Beyond
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Table 7.1: Basic LabRAD Types Type
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Table 7.5: LabRAD Packet Structure
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7.4.2 LabRAD Manager The LabRAD Man
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process. The SI conversion factors
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languages like Python it should be
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structure kept in the Server that h
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This is especially useful if second
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Figure 7.1: Control Layout: The con
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The next layers correspond to the
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7.5.5 Grapher At the far left of th
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hard drive. Having a central author
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which it prepends with the current
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system that needs to be changed in
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are involved in the experiment and
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to the “Sampling Scope Server”,
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eception of packets, and return the
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7.5.18 IPython In addition to using
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Chapter 8 Single Qubit Bring-Up and
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Figure 8.1: Squid I/V: Axis scales
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Figure 8.2: Squid Steps - a) Bias s
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where the potential has two stable
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picking the “operating branch”,
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Figure 8.4: Step Edge - a) Bias seq
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Plotting P T unnel versus the qubit
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Figure 8.6: S-Curve - a) Bias seque
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8.6 Spectroscopy Using this prelimi
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wave pulse at the qubit resonance f
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Figure 8.9: Visibility - a) Bias se
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as a function of the delay between
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Figure 8.12: Spin Echo - a) Bias se
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Figure 8.13: Fine Spectroscopy - a)
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Chapter 9 Coupled Qubit Bringup The
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is of significant concern for relia
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Figure 9.1: Effect of Squid Crossta
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Figure 9.2: Measure Pulse Timing -
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Figure 9.3: Spectroscopy of Coupled
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Figure 9.5: Capacitive Coupling Res
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9.2.5 Phase Calibration The trickie
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9.3 Resonator Based Coupling The me
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9.3.1 2D-Spectroscopy To couple a s
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Figure 9.9: Swapping Photon out of
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Figure 9.10: Resonator Swaps - a) S
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a Ramsey-type experiment needs to b
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Chapter 10 Hidden Variable Theories
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10.1.3 Settling the Question Experi
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Table 10.1: Possible Populations fo
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This restriction is called the Bell
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This leads to the correlation value
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“choosing” between polarization
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croscopic quantum systems like atom
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Chapter 11 Implementing the Bell Te
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singlet state due to the extra fact
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process of the qubits in two ways:
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[Steffen et al., 2006] (the details
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11.2.1 Bell Rotations The first ste
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P | 11 〉 (a ′ b ′ )) estimate
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Table 11.3: Sequence Parameters - C
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Table 11.5: Sequence Parameters - C
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the data taking can lead to thermal
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and back towards the position of th
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Table 11.6: Qubit Sample Parameters
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Table 11.8: Optimization Results -
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this leads to a modification of the
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Table 11.11: Optimization Results -
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Figure 11.3: Standard Error Analysi
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20 minutes worth of data. Within ea
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Figure 11.4: Behavior of S: Examini
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Figure 11.5: Quantifying Measuremen
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11.5.4 Numerical Simulation To unde
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Table 11.13: Qubit Sample Parameter
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Figure 11.7: Visibility Analysis -
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pair, independent of the measuremen
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Chapter 12 Conclusion 12.1 Claim of
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information about the quality of th
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Bibliography J. Altepeter, E. Jeffr
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Max Hofheinz, E. M. Weig, M. Ansman
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Matthew Neeley, M. Ansmann, Radosla
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