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Table 3.1: Transition Matrix Elements for Qubit-Like Potential m T | 12 〉,m T | 14 〉,m Left Index Right Index | 0 〉 0.000 0.000 0 . . . . | 11 〉 0.000 0.000 11 | 12 〉 -3.180 -0.250 0 | 13 〉 0.000 0.000 12 | 14 〉 -0.250 0.311 1 | 15 〉 0.000 0.000 13 | 16 〉 -0.016 -0.359 2 | 17 〉 0.000 0.000 14 | 18 〉 0.000 -0.032 3 | 19 〉 0.000 0.000 15 | 20 〉 0.000 0.002 4 | 21 〉 0.000 0.000 16 . . . minimum, i.e. next-to-nearest neighbors in energy. • The top-right corner corresponds to transitions between states that span both minima. In the region where the barrier is disappearing (around level 20), the potential looks less and less like a harmonic oscillator. This makes transitions between non-neighboring states progressively easier. Since the qubit will be formed by the lowest two energy levels in the shallow (left) minimum, the transitions of interest here are the ones between states 52
| 12 〉 and | 14 〉 (desired transitions) and transitions from states | 12 〉 or | 14 〉 to all other states (undesired transitions). The transition amplitudes T nm are shown in Table 3.1. Luckily, it is already virtually impossible to accidentally drive transitions between the desired states and states in the deep minimum. The main concern therefore is to avoid driving unwanted transitions to states | 16 〉, | 18 〉, or | 20 〉. Specifically, the transition from level | 14 〉 to level | 16 〉 needs to be watched closely. To understand how this can be achieved, let’s examine how a drive of the form ∆I dc (t) = X cos ωt + Y sin ωt + Z (3.31) affects a specific transition between levels n and m, using: i ∂ t a n (t) = a n (t) e i E n−En t ∆I dc (t) T nn + a m (t) e i E n−Em t ∆I dc (t) T nm (3.32) i ∂ t a m (t) = a n (t) e i E m−En t ∆I dc (t) T mn + a m (t) e i E m−Em t ∆I dc (t) T mm (3.33) By defining a vector A(t) = (a n (t), a m (t)) this can be written in matrix form as: [ i ∂ t A(t) = ∆I dc (t) ] T nn e −i(E m−E n )t/ T nm e i(E m−E n )t/ T mn T mm A(t) = ∆I dc (t) M A(t) (3.34) Since T mn = T nm , M can be expressed in terms of the Pauli Matrices: M = T mn cos ω mn t σ x + T mn sin ω mn t σ y + T nn − T mm 2 σ z + T mm + T nn 2 I (3.35) 53
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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
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 66 and 67: at time t. r is not restricted to b
Page 68 and 69: 3.1.2 Effects of a Time Dependent P
Page 70 and 71: In some cases, it is possible to so
Page 72 and 73: Figure 3.1: Examples of Numerical S
Page 74 and 75: • The energy difference between t
Page 76 and 77: like this: V = ( V (−1, −1), V
Page 78 and 79: Figure 3.2: Simulation of LC Oscill
Page 82 and 83: with ω mn = Em−En . Multiplying
Page 84 and 85: α, it can be ignored. Thus, the in
Page 86 and 87: e solved exactly: A(t + ∆t) = e
Page 88 and 89: qubits would be simulated using: A(
Page 90 and 91: This calculation assumes that the s
Page 92 and 93: Decoherence consists of two parts:
Page 94 and 95: Note the difference in signs of the
Page 97 and 98: Chapter 4 Designing the Phase Qubit
Page 99 and 100: mutual inductance between the qubit
Page 101 and 102: During the measurement, the | 1 〉
Page 103 and 104: the right impedance transformation
Page 105 and 106: excitations. Since these are a pote
Page 107 and 108: Figure 4.3: Squid I/V Traces - a) L
Page 109 and 110: Figure 4.5: Qubit Integrated Circui
Page 111 and 112: The geometry of the qubit junction
Page 113 and 114: squid loop. Thus, this tool can be
Page 115: now, amorphous silicon seems to pro
Page 118 and 119: Figure 5.1: L-Edit Mask Layout Tool
Page 120 and 121: Figure 5.2: Fabrication Building Bl
Page 122 and 123: Figure 5.3: Photolithography and Et
Page 124 and 125: times the removal can be a bit tric
Page 126 and 127: Figure 5.4: Clearing Vias from Nati
Page 128 and 129: 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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(independent of which axis it is),
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of E xy is measured by expressing i
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For a measurement of two particles
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10.2.1 Photons The first and most n
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10.2.3 Ion and Photon In the attemp
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238
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11.1 State Preparation Since the ab
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Figure 11.1: Bell State Preparation
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Table 11.1: Entangled State Density
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Figure 11.2: Bell Measurements - a)
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11.2.3 Statistical Analysis For the
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11.3 Calibration The most difficult
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Table 11.4: Sequence Parameters - R
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ally determine an optimal value for
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ages are based on this method, incl
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equires to converge. Since the algo
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Table 11.7: Bell Violation Results
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Table 11.9: Error Budget - Capaciti
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Bell inequality. To make this claim
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11.5.2 Dependence of S on Sequence
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point where the measurement happens
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The second qubit is not driven and
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Figure 11.6: Resonator Coupled Samp
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From the resulting states, the 16 p
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Table 11.14: Error Budget - Resonat
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sponding to a violation by 244.0σ.
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educe energy decay and dephasing an
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John F. Clauser, Michael A. Horne,
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J. Majer, J. M. Chow, J. M. Gambett
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Matthias Steffen, M. Ansmann, Rados
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