reverses the direction of evolution when the echo condition Ïecho = 2Ï2 is met ..... A useful guide to the components necessary for a quantum computer was provided ...... Naturwissenschaften 23, 807, 823, 844, translated by John D. Trimmer in.
Fundamentals of Flux-based Quantum Computing by Timothy Levi Robertson BA (Princeton University) 1996
A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY
Committee in charge: Professor John Clarke, Chair Professor K. Birgitta Whaley Professor Joel E. Moore
Spring 2005
The dissertation of Timothy Levi Robertson is approved.
Chair
Date
Date
Date
University of California, Berkeley
Spring 2005
Fundamentals of Flux-based Quantum Computing
c 2005 Copyright by Timothy Levi Robertson
Abstract
Fundamentals of Flux-based Quantum Computing by Timothy Levi Robertson Doctor of Philosophy in Physics University of California, Berkeley Professor John Clarke, Chair The study of Quantum Computing necessitates careful examination of the most fundamental questions of Quantum Theory, such as the measurement problem, and may lead to important advances in practical applications such as cryptography, search, and optimization. In order for a Quantum Computer to be practically useful, the design must be scalable to hundreds of quantum bits, or qubits, while maintaining quantum coherence. Qubits constructed from superconducting electronics are promising because of their inherent scalability using established nano-fabrication techniques. Superconducting qubits based on the flux degree of freedom are insensitive to noise from charge fluctuations and can be read-out using a Superconducting QUantum Interference Device (SQUID). When properly designed, a superconducting loop interrupted by three Josephson junctions acts as a quantum two-state system. In this Dissertation, an exact calculation of the energy levels of the three junction flux qubit is used to design samples consisting of one or two qubits to investigate coherence properties. Careful attention is given to the system electronics to minimize external sources of noise acting back on the qubit that result in decoherence. We report measurements on two superconducting flux qubits coupled to a readout SQUID. Two on-chip flux bias lines allow independent flux control of any two of the three elements, as illustrated by a two-dimensional qubit flux map. The application of microwaves yields a frequencyflux dispersion curve for 1- and 2-photon driving of the single-qubit excited state and reveals spurious resonances intrinsic to each qubit. Coherent manipulation of the single-qubit state results in Rabi oscillations, Ramsey fringes, and Hahn spin-echos. This information is used to develop a model of the decoherence caused by the interaction of the qubit with its environment. A detailed model for the interaction of a flux qubit with a readout SQUID predicts the resolution of a measurement and its effect on the qubit. Two adjustable inter-qubit coupling systems that can produce bipolar coupling strength are presented. These systems can be used to produce the quantum Controlled-NOT gate, which when combined with single qubit operations forms a basis for Universal Quantum Computation.
Professor John Clarke Dissertation Committee Chair 1
“Of all the communities available to us, there is not one I would want to devote myself to except for the society of the true searchers, which has very few living members at any one time.” Albert Einstein, quoted by Max Born (1971). “If I have been able to see further, it was only because I stood on the shoulders of giants.” Sir Isaac Newton in a letter to Robert Hooke (February 5, 1675) This work is dedicated to those who have devoted their lives to searching for the truth and then shared what they have found so that others may see a bit further.
i
Contents Contents
ii
List of Figures
v
List of Tables
xiii
Acknowledgments
xiv
1 Introduction
1
1.1
Macroscopic Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Decoherence and Wavefunction Collapse . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Indeterminacy and the Power of Quantum Computing . . . . . . . . . . . . . . . . .
4
1.4
Building a Quantum Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.5
Feasibility of Large Scale Quantum Computation . . . . . . . . . . . . . . . . . . . .
7
2 Flux Qubits 2.1
9
Quantum Two State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1
9
The Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2
Superconducting Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3
The One Junction Flux Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4
The Three Junction Flux Qubit 2.4.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Scalability and Architecture
33
3.1
Standard Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2
Nano-fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3
Infrastructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.1
Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ii
3.4
Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4.1
Flux Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.2
SQUID Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.3
Microwave Excitation and Modulation . . . . . . . . . . . . . . . . . . . . . . 52
4 The 2RQ Chip 4.1
4.2
55
Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1.1
Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.2
SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.3
Flux Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Flux Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 Qubit Manipulation
68
5.1
The Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2
Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3
5.2.1
Measuring δ and d/dΦQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2.2
Spurious Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.3
Measuring T1 and T20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.4
Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Coherent Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3.1
Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.2
Ramsey Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3.3
Spin Echos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Decoherence
92
6.1
The Spin-Boson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2
Relaxation and Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3
Sources of Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.4
6.3.1
Flux Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.3.2
1/f Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3.3
Subgap Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7 Measurement 7.1
102
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
iii
7.2
Calculation of Escape Rates
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.3
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.4
Calculation of Decoherence Due to Shunts . . . . . . . . . . . . . . . . . . . . . . . . 116
7.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8 Qubit Coupling
123
8.1
Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.2
SQUID Coupler
8.3
The INSQUID Coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Bibliography
138
A The ChipObject Design Environment
144
A.1 ChipObject.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 B The 2RQ Rev. J Design
145
B.1 2rq.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 C Calculated Inductances for 2RQ Rev. J
146
C.1 2RQ Rev. J Inductances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
iv
List of Figures 2.1
The Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2
(a) Electron energy level difference. Difference between ground and excited state energy ∆E for an electron in a magnetic field as the magnitude of the Bz field component is varied. Dashed line shows asymptote ∆E = . (b) and (c) Energy eigenvectors. Amplitudes of energy eigenvectors are shown for an electron ground state (b) and excited state (c) in a magnetic field as a function of Bz field strength. Coefficients of the |0i (red) and |1i (blue) parts of the wavefunction are shown. . . . 12
2.3
One-junction flux qubit. (a) Schematic. (b) Symmetric double well potential for flux bias Φx = Φ0 /2, shown diagrammatically as a function of phase difference φ. (c) Flux fluctuation ∆Φ couples to Ω only in second order. (d) Critical current fluctuation ∆I0 produces exponential change in Ω. . . . . . . . . . . . . . . . . . . . 16
2.4
Three quantities for the ground state of the one-junction flux qubit at the degeneracy point calculated using the standard WKB approximation (solid), WKB approximation corrected for the ground state (dashed), and numerical solution for the wavefunctions (points), plotted as a function of the dimensionless screening parameter βL . (a) Splitting frequency between ground state and first excited state, (b) sensitivity parameter Λ, and (c) effect of critical current fluctuations on tunneling rate for three values of δI0 /I0 . Parameters are from Friedman et al.: L = 240 pH and C = 104 fF (Friedman et al., 2000). . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5
Eigenfunctions and energy levels for a one-junction flux qubit. Absolute values of eigenfunctions (thin lines), each offset so that it asymptotes to the corresponding energy level, are shown as a function of the junction phase δ. The thick line is the asymmetric double well potential. Eigenfunctions are labeled according to corresponding single-well harmonic oscillator quantum numbers. Parameters are as in Fig. 2.4, with βL = 1.5; the flux φx ≈ 0.514 × 2π produces a resonance between the 3L (left) and 9R (right) states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6
Numerical solution for the excited states of an asymmetric one-junction flux qubit. (a) Tunneling frequency between the third excited state in the shallow well and the ninth excited state in the deep well as a function of βL for a system on resonance at βL = 1.5. (b) Derived sensitivity to critical current fluctuations. Device parameters are as in Fig. 2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7
Schematic of three junction flux qubit with loop inductance LQ , circulating current JQ , and phase differences γi across Josephson junctions of critical current I0i . . . . 22
v
2.8
Three junction flux qubit potential. Three contours of potential energy [U = 1.4Ej (red), U = 10Ej (green), and U = 30Ej (blue)] are shown. Parameters are αQ = 0.8, βQ = 0.4, Q = 0, and φQ = π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.9
Slice through γt = 0 of three junction flux qubit potential. Parameters are αQ = 0.8, βQ = 0.01, Q = 0, and φQ = π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.10 Unit cell of a larger inductance flux qubit. Slices through both the γt -γs plane and the γa -γs , which coincide along the dotted line, plane are shown. Parameters are αQ = 0.6, βQ = 0.4, Q = 0, and φQ = π. . . . . . . . . . . . . . . . . . . . . . . . 26 2.11 Circulating current at the degeneracy point. Magnitude of qubit circulating current JQ , as indicated by location of potential minimum, is shown as a function of (a) αQ and (b) βQ . Parameters are = 0, φQ = π, βQ = 0.1 (a), and αQ = 0.7 (b). . . . 27 2.12 Flux dependence of three junction qubit potential. The potential energy is shown for γt = γa = 0 for various flux biases. Parameters are αQ = 0.8, βQ = 0.01, and Q = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.13 Calculated energy levels of the three junction flux qubit. Energy as a function of qubit flux bias is shown for αq = 0.63, βQ = 0.15, Q = 0, σQ = 0, EJ /h = 50 GHz, Ec /h = 1 GHz, nk = 5, nl = 10, and ns = 2. Each visible line is actually a doublet split by the intercell tunneling calculated to be ∼ 300 kHz for these parameters. . . 32 3.1
Standard cell. The dashed blue box encloses a uniform collection of circuitry, which is repeated to form the elements of a quantum computer. . . . . . . . . . . . . . . . 34
3.2
Atomic force micrograph of Josephson junction. The junction occurs at the oxide barrier between the “finger” and the “toe.” Two images of the junction toe, formed by evaporations at two different angles, are visible. The current path is indicated by the yellow arrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3
Electron micrograph of Josephson junction. The current path is indicated by the yellow arrows. The dashed box indicates the area of the junction. . . . . . . . . . . . 37
3.4
System filtering. See text in Sec. 3.3.1 for discussion. . . . . . . . . . . . . . . . . . . 40
3.5
Sample box. The Cu box is partitioned into five chambers, with SMA connections for the various signals. The sample chip is visible as the small black square in the center left cavity, which appears darker because of its Pb plating. The sample is connected to the signal leads by wire bonds to Cu pads, which are visible in the sample chamber. The signals then pass out of the sample chamber to the outer chambers through narrow slits. Microwaves couple to the sample through a coax terminated in a ∼ 1 mm loop a few mm above the qubit. A block of In has been placed in the sample chamber to reduce its size, thus increasing the frequency of the corresponding electromagnetic cavity mode. The outer chambers are potted in Cu powder filled epoxy to damp electromagnetic modes. The sense isolation resistors are visible in the top left chamber. The box is closed by a Cu lid, which has been plated with PB over the sample chamber. . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6
Electronics layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
vi
3.7
Digital noise. The power spectral density (PSD) of a random bit sequence approaches the distribution [sin(πf/fclk )fclk /πf]2 , as shown in the figure. The distribution is shown for square edged bit transitions, but bandwidth limits in the system ultimately attenuate the distribution above a cutoff frequency. . . . . . . . . . . . . . . . . . . . 44
3.8
Schematic of one current source; second source is identical. The second potentiometer on the AD5235 chip containing the fine control is unused in our current design. . . . 46
3.9
Current stability over 70 hours (a) without control and (b) controlled by LabVIEW program. The sampling rate was 0.5 Hz, corresponding to a measurement bandwidth of 0.25 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1/2
3.10 Current noise spectrum SI (f) (left-hand ordinate) measured at a current of 275 µA. Right-hand ordinate shows current noise spectrum normalized to the current I. . . . 48 3.11 Transition between the two flux qubit states measured on three consecutive days. Vertical axis shows the bias current IB at which the readout SQUID has a 50% switching probability P versus the flux ΦQ in the qubit. The qubit and SQUID were enclosed in a superconducting cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.12 Pulse waveform. The pulse waveform produced by the arbitrary waveform generator is shown as the black solid line. Depending on if the SQUID switches (red) or remains in the supercurrent state (green), a low-pass filtered version of the pulse waveform is developed on the sense line. An appropriately chosen comparator level (dashed) can distinguish between the two states. . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.13 Fast modulation apparatus. Two high performance mixers connected in series modulate the microwave source. The mixers are driven by a fast pulse generator, whose signal is filtered and distributed to the mixers. . . . . . . . . . . . . . . . . . . . . . 53 3.14 SQUID Suppression. The minimum microwave power, referenced to the input to the dewar, necessary to suppress the SQUID critical current by 1% is plotted as a function of frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.1
(a) Layout of 2RQ Revision J chip. Blue represents Al traces, Gold AuCu traces. Pads near upper edge of chip provide two independent flux lines; wire-bonded Al jumpers couple left and right halves. Pads near lower edge of chip supply current pulses to the readout SQUID and sense any resulting voltage. (b) Photograph of center region of completed device. Segments of flux lines are visible to left and right of SQUID, which surrounds the two qubits. AuCu quasiparticle traps are visible as four light rectangular regions near center of photo. . . . . . . . . . . . . . . . . . . . 56
4.2
P(I) curves across Qubit 2 step. The SQUID switching probability for the qubit in the |0i state (ΦQ2 = 0.48Φ0 , red) and |1i state (ΦQ2 = 0.52Φ0 , blue) is shown as a function of the amplitude of current pulse applied to the SQUID. Both measurements were taken at constant applied flux to the SQUID. Each curve contains 100 points averaged 8, 000 times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
vii
4.3
Raw flux map. The SQUID switching current I50% s , represented as a color, is shown as a function of the two applied flux currents, I1 and I2 . The flux modulation characteristic of the SQUID is clearly visible as an oscillation running from the upper left to lower right. Lines representing contours of constant flux applied to the SQUID are superimposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4
Residual flux map. The difference between the measured and fitted SQUID switching current, represented as a color, is shown as a function of the two applied flux currents, I1 and I2 . The residual is quite small, indicating that the flux model captures all the salient features of the system. Two distinct regions, where the fit is mostly low, seen as blue, or high, seen as red, can be associated with the two datasets which which are composited here. They were collected three days apart, and drifts in the system are evident at this level. Lines representing contours of constant flux applied to the SQUID and constant flux applied to each qubit are superimposed. . . . . . . . . . . 64
4.5
(a) Full SQUID modulation. All the data from the raw flux map shown in Fig. 4.3 is plotted as a function of the SQUID flux ΦS . The fitted SQUID modulation curve Ifit s (ΦS ) is shown in red (b) SQUID periodicity. The data from (a) are shown collapsed into one period, along with Ifit s (ΦS ) in red. In both (a) and (b), much of the deviation of the data from the fit can be attributed to the uncontrolled states of the qubits in the dataset. (c) Qubit modulation. The SQUID switching current is shown as a function of qubit 1 flux ΦQ1 along a line where ΦS is constant. I50% s Transitions in qubit 1 and qubit 2 alternate, and the periodic nature of both qubits is clearly visible. The small deviations from periodicity at ΦQ1 = 0.3Φ0 and 0.75Φ0 can be explained by shifts in ΦS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6
Qubit flux map. The difference between the measured SQUID switching current, and the fitted contribution from the SQUID modulation, represented as a color, is shown as a function of the two applied flux currents, I1 and I2 . The transitions of each qubit are clearly visible as sudden changes in color. Lines representing contours of constant flux applied to each qubit are superimposed. Circle marks a double degeneracy point and square shows a region where qubit 2 is near a degeneracy point and qubit 1 acts as a linear screener. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.1
Spectroscopy pulse sequence. The qubit is stimulated with a burst of microwave radiation and then measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2
Qubit step. Thermal state (light blue), thermal fit (dashed black), and driven state (orange) are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3
Experimental spectroscopy of a three junction flux qubit. Red corresponds to a decrease in qubit screening flux upon application of microwave radiation, and purple to and increase in screening flux. The dashed line shows a fit to Eq. 2.4, with ∆/h = 3.99 GHz and (d/dΦQ )/h = 896 MHz/mΦ0 . Broad horizontal band near 6 GHz is a result of a resonance in the SQUID circuitry and does not directly reflect qubit properties. Narrow two-photon transitions are also visible. Data were taken on qubit 2 of the 2RQ Rev. J sample, described in Chapter 4. . . . . . . . . . . . . . 72
viii
5.4
Multi-photon transitions. Increase in qubit screening flux (white) and decrease (black) are shown for a qubit 2 of the 2RQ Rev. J sample under strong microwave excitation. Red lines are of slope s0 /n for n from 1 to 5, with s0 the slope of the fundamental. Strong dark horizontal features are the result of excess power coupled to the readout SQUID, resulting in corrupted measurement. . . . . . . . . . . . . . . 73
5.5
High resolution spectroscopy. Increase or decrease in qubit screening current shown relative to meaas reflected by change in switching current of readout SQUID I50% s surements in the absence of microwaves. Dashed lines indicate fit to hyperbolic dispersion for 1- and 2-photon qubit excitations. The 2-photon fit is one-half the frequency of the 1-photon fit. Inset containing ∼ 23, 000 points is at higher resolution. Data taken on qubit 2 of 2RQ Rev. J. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.6
Martinis resonances. High resolution spectroscopy of qubit 2 on the 2rqj sample showing 9 spurious resonances over 4 GHz. The dashed circle shows a type-one resonance, while the solid lines indicate type-two resonances. The large circle shows a cluster of 6 resonances. Data taken on qubit 2 of 2RQ Rev. J. . . . . . . . . . . . 75
5.7
Dual qubit spectroscopy. Both qubits of the 2RQ Rev. J sample have been flux biased so that they are near their respective degeneracy points. Induced microwave transitions for each qubit are labeled. A type-one resonance is visible in qubit 1, but not qubit 2, strongly suggesting an intrinsic origin. The lightening of the spectroscopic lines in both qubits below the cross is not observed when the qubits are isolated, and may be evidence on coupling. . . . . . . . . . . . . . . . . . . . . . . . 77
5.8
Qubit relaxation. (a) Spectroscopic peak with fitted Lorentzian (blue) and baseline in the absence of microwaves (green). (b) Observed relaxation of peak as τmeas is incremented and fit to exponential decay (red). Fitted decay time is T1 = 179 ns. Data were acquired on on qubit 2 of 2RQ Rev. J chip at 10 GHz. . . . . . . . . . . 78
5.9
(a) Spectroscopic peaks. Peaks are shown for three different driving powers with fitted Lorentzians. (b) Amplitude dependence. The full width at half maximum (FWHM) of the peak is seen to scale linearly with amplitude, as expected for the strong driving regime. The extrapolated intercept is at 285 MHz, implying T20 = 11 ns. Data taken on qubit 2 of 2RQ Rev. J at 11.1 GHz. . . . . . . . . . . . . . . . . 78
5.10 Excited state spectroscopy. Plot is a composite of four spectroscopic raster scans taken on qubit of UNQ sample. See text for discussion . . . . . . . . . . . . . . . . . 80 5.11 Rabi oscillation. The qubit is stimulated with resonant microwave bursts of varying length τmw and then measured. Data points are shown along with a fit to damped sinusoid (red line). Oscillation decays with a characteristic time of 78 ns and displays a scaled contrast of 63%. Data was taken on 2RQ Rev. J sample at 10 GHz. . . . . 81 5.12 Power dependence of Rabi oscillation. (a)-(c) Rabi oscillations for three different applied microwave powers are shown on the left, with fits to damped sinusoids. (d) Rabi frequency, as determined by the fit, is shown as a function of applied microwave amplitude, in arbitrary units. The blue line indicates a liner fit through the origin. Data was taken on qubit 2 of 2RQ Rev. J sample at 10 GHz. . . . . . . . . . . . . . 82
ix
5.13 Raster scan of Rabi oscillations. The plot shows the switching probability, represented as a color, as a function of both pulse width and applied microwave frequency. The color scale is optical, with switching probability increasing from red to violet. Oscillations that increase in frequency away from the resonance around 10 GHz are clearly visible. However, the fringe visibility and decay time vary irregularly for small changes in the driving frequency. Data was taken on qubit 2 of 2RQ Rev. J sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.14 Ramsey pulse sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.15 Ramsey fringes. (a) Ramsey fringe at 10.0 GHz with decay Ramsey fringe at 10.1 GHz with decay time T2∗ = 5.6 ns. resonance at 10.225 GHz. (a) Ramsey fringe at 10.49375 T2∗ = 8.5 ns. . . . . . . . . . . . . . . . . . . . . . . . . . . .
time T2∗ = 4.8 ns. (b) (c) Ramsey fringe on GHz with decay time . . . . . . . . . . . . . . 85
5.16 Raster scan of Ramsey fringes. (a) oscillations as a function of applied frequency and τRamsey . (b) Fourier transformed fringes. (c) Atari Computer logo, suggestive of Ramsey fringes. See text for discussion. . . . . . . . . . . . . . . . . . . . . . . . . 86 5.17 Raster fit parameters. Parameters extracted by fitting each Ramsey oscillation to (a/2) exp(−τRamsey /T2∗ ) cos(2πψ − fRamsey τRamsey ) + d. . . . . . . . . . . . . . 88 5.18 Spin echo pulse sequence. The first π/2-pulse rotates the spin into the equatorial plane, where it precesses at a rate equal to the detuning. The π-pulse effectively reverses the direction of evolution when the echo condition τecho = 2τ2 is met, canceling low frequency dephasing. The final π/2-pulse rotates the spin out of the equatorial plane for measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.19 Spin echo measurement. The qubit state as reflected by the switching current, encoded as gray scale, is shown as a function of τecho and τ2 . The red line indicates the echo condition Eq. 5.9. Modulation in the τ2 direction is believed to be due to an electromagnetic resonance around 6.7 GHz. Data acquired for 100 MHz detuning at 7.125 GHz on qubit 2 of the 2RQ Rev. J sample. . . . . . . . . . . . . . . . . . . 90 5.20 Spin echo peak decay. See text for discussion. . . . . . . . . . . . . . . . . . . . . . . 91 6.1
Spin-boson model. A spin-1/2 particle, represented by the black arrow, is coupled to a bath of harmonic oscillators, drawn as springs coupled to masses. . . . . . . . . 93
6.2
Sources of decoherence for a flux qubit. . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3
Effects of low frequency flux and critical current fluctuations in a superconducting qubit. (a) Flux modulation from vortices hopping into and out of a loop, and critical current modulation from electrons e− temporarily trapped at defect sites in the junction barrier. (b) A single charge trap blocks tunneling over an area ∆A, reducing the critical current. (c) Fluctuations modify the oscillation frequency, inducing phase noise which leads to decoherence in time-averaged ensembles of sequential measurements of the qubit observable Z. . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.4
Low frequency noise in three-junction flux qubit. (a) Tunneling frequency and (b) Λ vs. Josephson-to-charging energy ratio. Solid lines indicate dependence on large junction ratio γa,b with γc = 28, and dashed lines indicate dependence on small junction ratio γc with γa = γb = 35. EC = 7.4 GHz for all plots. . . . . . . . . . . 98 x
6.5
Temperature dependence of Qubit 2 of the 2RQ Rev. J sample. See text for discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.1
Schematic of flux qubit coupled to readout SQUID.
7.2
Idealized SQUID switching distribution. As Ib is increased, the probability of the SQUID switching increases from 0 to 1 over a characteristic width δIs (∆p) . . . . . . 105
7.3
Transformation from SQUID with shunt admittances Y to equivalent single Josephson junction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.4
SQUID potential energy. Contour plot showing the critical point xc , the minimum x0 and the saddle point x01 . Open circles show the starting point for search taken from analytic approximations. White broken lines show contours where ∂U/∂(γ1 −γ2 ) = 0 (dotted) and ∂U/∂(γ1 + γ2 ) = 0 (dashed). SQUID parameters are: ΦS = 0.05 Φ0 , I0 = 1.96 µA, Ib = 1.49 µA and LS = 500 pH, corresponding to βL = 0.95. . . . . . 109
7.5
SQUID with shunts consisting of a resistor and capacitor in series across each junction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.6
Micrographs of RC-SQUID. (a) Overview showing flux bias line; leads and pads for applying bias current and detecting voltage are labeled at top. The pulse current is split symmetrically between pads “SQUID A” and “SQUID B” so that fluxes generated by this current do not couple to the qubit. (b) Enlarged view showing Josephson junctions, interdigitated capacitors, and AuCu resistors, indicated by the ellipse. (c) Capacitor detail. (d) Junction detail. . . . . . . . . . . . . . . . . . . . . 112
7.7
vs. ΦS for an RC-shunted SQUID. Thick line shows data, thin lines show (a) I50% s calculated modulation curves for βL = 1.05 (top), βL = 0.95 (middle), and βL = 0.86 (∆p) vs. ΦS . (c) Resolution ρ(∆p) vs. ΦS . (bottom). (b) Switching width δIs p/I50% s Measurements made at 24 mK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.8
Measured 90% switching width vs. temperature for several different ΦS . Solid lines show the predicted distribution widths for RC-shunted SQUID with Rs = 40 Ω, Cj = 9.3 fF, I0 = 1.96 µA, Cs = 20 fF and a fit value Rl = 52 Ω. Dashed line shows calculated width for corresponding unshunted SQUID with Rl = ∞ at ΦS = 0.01Φ0 . 114
7.9
Suppression of T ∗ for RC-shunted SQUID 2πkB T ∗ /hω0 vs. ωc /ω0 . Four curves are shown for different values of κ ≡ Cs /Cj . . . . . . . . . . . . . . . . . . . . . . . . . 115
. . . . . . . . . . . . . . . . . . 104
7.10 J(ω) for a flux qubit coupled to an RC-shunted SQUID with parameters given in text. Solid line corresponds to Case A and dashed line corresponds to Case B. . . . 119 7.11 Dephasing (T2 , black) and relaxation (T1 , gray) times for flux qubit coupled to RCshunted SQUID. Solid lines correspond to Case A and dashed lines correspond to Case B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.1
(a) SQUID-based coupling scheme. The admittance Y represents the SQUID bias circuitry. (b) Response of SQUID circulating current JS to applied flux Φs for βL = 0.092 and Ib /Ic (0.45Φ0 ) = 0, 0.4, 0.6, 0.85 (top to bottom). Lower right inset shows JS (Φs ) for same values of Ib near Φs = 0.45Φ0 . Upper left inset shows Ic versus Φs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
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8.2
(a) Variation of K with Ib for Φs = 0.45Φ0 and device parameters described in text. (b) Highest achievable value of Ks versus βL evaluated at Ib = 0.85Ic (0.45Φ0 ); I0 (and hence βL ) is varied for L = 200 pH. . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.3
Pulse sequence for implementing CNOT gate. Energy scales in GHz. Total single˜i (t) + δi (t), where microwave pulses ˜1,2 (t) produce qubit energy bias i (t) = 0i + single-qubit rotations in the decoupled configuration; crosstalk modulation of K(t) is shown (see text). The bias current is pulsed to turn on the interaction in the central region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.4
Chain of flux qubits with intervening dc SQUIDs arranged to provide both variable nearest neighbor coupling and qubit readout. . . . . . . . . . . . . . . . . . . . . . . 129
8.5
Schematic of INSQUID with two flux bias lines coupled to readout dc SQUID with resistive shunts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.6
(a) Circulating current ji in the input loop and (b) circulating current jc in the coupling loop as functions of (Φxi , Φxc ). Range for ji is ±1 and for jc is ±2, with equally spaced contours. The scale is optical, with red representing minimal values and violet representing large values. Parameters are βI = 0.9 and λ = 1/137. . . . . 132
8.7
(c) INSQUID gain G as a function of (Φxi , Φxc ) shown as both a 3D and contour plot; Gain is normalized to the full scale range 1/(1 − β2I λ). The scale is optical, with red representing minimal values and violet representing large values. Parameters are βI = 0.9 and λ = 1/137. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.8
INSQUID figure of merit for flux qubits of amplitudes 10−1 Φ0 (solid), 10−3 Φ0 (short dash), 10−5 Φ0 (long dash), versus Gaussian noise amplitude applied to the coupling loop. Parameters are the same as for Fig. 8.7. . . . . . . . . . . . . . . . . . . . . . . 135
8.9
Two different schemes for switchable qubit arrays: (a) a double buffered scheme with nearest-neighbor couplings and (b) an architecture incorporating a flux transformer which allows arbitrary interactions within a set of qubits. . . . . . . . . . . . . . . . 135
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List of Tables 4.1
Qubit parameters for 2RQ Rev. J. Junction sizes, as determined by measurements of scanning electron micrograph of completed device are shown. Junction areas include a contribution from the edges of the 35 nm thick film. Critical current is estimated by scaling the measured witness junction critical current, as inferred by measurements of the normal state resistance Rn and the relation I0 Rn = (π/2)170 µV, by the area ratios of the junctions. Low, best, and high estimates of junction capacitance are shown for specific capacitances of 65, 87, and 100 fF/µm2 , respectively. Parameters αQ and Q are calculated directly from measured junction areas, including contribution from junction edges. Qubit loop inductance, as calculated by FastHenry, is also reported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2
Calculated qubit 1 energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3
Calculated qubit 2 energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4
SQUID parameters. Loop inductance LS as calculated by FastHenry, SQUID critical current I0 , as measured, screening parameter βL = LS I0 /Φ0 , and the calculated SQUID plasma frequency fp for no current or flux bias, are reported. . . . . . . . . . 59
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Acknowledgements I must begin by thanking those with whom I have worked so closely for so long to produce this Dissertation. The experimental results presented here would have been impossible without the tireless efforts of Paul, Travis, Cheng-en, and Antonio. My good friend and colleague Britton perfectly balanced my sometimes reckless nature with his careful attention to detail so that work proceeded quickly and smoothly. Without his systematic checks I would have probably blown up the lab and never finished anything. So many great teachers I have had the privilege of learning from have helped me develop my interest in science. People like Becking, Waskom, Strange, Schaudt, Miller, Keammerer, Bailey, and Woolever. Jeff Peterson, who first really introduced me to physics, Lyman Page, Jim Peebles, and Dave Wilkinson, who helped me develop at Princeton. At Berkeley it’s been a pleasure to learn quantum mechanics from Gene Cummins who clearly put so much effort into his teaching. Frank Wilhelm has been an invaluable repository of insight. I couldn’t ask for a better advisor than John Clarke. He is a gentleman and a scholar who never spares any effort in educating his students. His openness, commitment to science, and personal and professional integrity have been exemplary to me. It is mainly my family who have made me who I am and without their unceasing support and guidance I would never have gotten as far as I have. My parents have always made sure that I’ve had the challenges and opportunities necessary for my development, no matter what the expense or hardship. My grandparents, particularly Mimi and Papa Bill, have shown me wonderful new things that expanded my mind and horizons. Finally, without my dear wife Jyothi, who brings me so much happiness, I would not have enjoyed this work so much. -Timothy L. Robertson May 15, 2005 Vallejo, CA
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Chapter 1
Introduction The Quantum Theory, developed during the first half of the twentieth century, represents such a radical departure from previous frameworks for comprehending the world that understanding of its consequences has not fully permeated the consciousness of mankind, even a century later. While, as a practical matter, Quantum Theory is responsible for a host of technological advances, ranging from nuclear power to the laser, full acceptance of its foundations has been reluctant, even by those who contributed the most to it. As I write this in 2005, the World Year of Physics, we celebrate the 100th anniversary of Albert Einstein’s annus mirabilis, the year he revolutionized our notions of space and time (Einstein, 1905d), the constitution of matter (Einstein, 1905a), mass and energy (Einstein, 1905b), and the nature of light (Einstein, 1905c). It was in this last work, which built upon Planck’s theory of blackbody radiation (Planck, 1901), where Einstein explained the photoelectric effect by postulating the photon, a discrete packet of electromagnetic energy. This idea, for which he received the Nobel Prize in 1921, planted the seeds for modern Quantum Theory. Einstein himself was quite uncomfortable with Quantum Theory. As the Theory developed, it soon extended beyond the notion of quantization of energy to include such counterintuitive features as superposition of states and entanglement of separated particles. It required abandoning notions of hard causality and determinism for a fuzzy, probabilistic view of events. This so unnerved Einstein that he spent the second half of his career pursuing his intuition that Quantum Theory was in some way wrong or incomplete. The idea of superposition, that a system can be in two or more distinct states simultaneously, is contrary to our everyday experience. When we observe an ordinary object, at a definite time it is in a definite place; we do not see multiple copies of it superposed over different states, and we do not see it jump between different states upon repeated observation, as quantum mechanics allows. Furthermore, we do not observe entanglement in the everyday world. In classical physics systems interact through forces that are consistent with causality; however, Quantum Theory admits another possibility which is strange and unfamiliar. Two systems can become intimately bound to one another in an entangled state, so that an interaction with one causes an effect on the other, not through any force, but simply because of the shared history of the two systems. Einstein
1
found this non-local interaction ridiculous, and held it up as an example of why Quantum Theory must be wrong (Einstein et al., 1935). The absurdity one can reach when combining the principles of superposition and entanglement is dramatically demonstrated in Erwin Schr¨odinger’s “Cat Paradox,” which he proposed contemporaneously to Einstein’s paradox developed with Podolsky and Rosen: One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay and releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The [wavefunction] of the entire system would express this by having it in the living and the dead cat (pardon the expression) mixed or smeared out in equal parts. (Schr¨odinger, 1935) Here, the superposition in the state of the radioactive substance is transfered through an entangling apparatus into superposition in the state of the cat. I know of no cruel physicist who has attempted to carry out this experiment, but there is no dispute about the outcome: one would either have a live or dead cat at the end, and not a superposition of the two.
1.1
Macroscopic Superposition
So where is the line between the microscopic world, where all of the strangeness of Quantum Theory holds, and the macroscopic world, where classical physics holds sway? Surely somewhere between atoms and cats, but what is the nature of this boundary and how far can it be pushed? These questions arose repeatedly over then next half century, to the point where they were sharpened into a more rigorous version of the cat paradox (Leggett and Garg, 1985). One constructs a ring of superconducting metal (Bardeen et al., 1957), where a persistent current can flow without dissipation, generating a magnetic flux. Under appropriate bias conditions, the two states of clockwise and counter-clockwise current flow have equal energy, making them degenerate. A tunneling probability, formed by placing a Josephson junction (Josephson, 1962) in the ring, connects the two states, allowing the system to enter a superposition of states of opposite circulating current. Here the size of the “cat” is not set by the whims of evolution, but rather is controllably determined by the size of the loop. Thus one can systematically explore the boundaries between the quantum and classical worlds. This idea of Macroscopic Quantum Coherence (MQC), soon came to be tested experimentally. The first experiments did not involve a superposition between two stable states as described above, but instead concentrated on the quantum decay of a bound metastable state into an unbound continuum state, in analogy with the radioactive decay of the Schr¨odinger’s Cat Experiment. A current-biased Josephson junction, a few microns in extent, was shown to switch from the supercurrent state to the voltage state exactly as predicted by Quantum Theory (Martinis et al., 1987). 2
Later experiments demonstrated macroscopic superpositions of current states in superconducting loops, as speculated by Leggett and Garg, with the two states involving millions of electrons moving coherently in a superposition of clockwise and counter-clockwise states (Mooij et al., 1999; Friedman et al., 2000).
1.2
Decoherence and Wavefunction Collapse
So why can millions of electrons in a ring a few microns across form a superposition, while the ∼ 1026 particles that make up a cat several centimeters across cannot? The orthodox explanation is given by the Copenhagen Interpretation of quantum mechanics, developed around Niels Bohr in the 1920s (Murdoch, 1989). It draws upon Bohr’s Correspondence Principle, that Quantum Theory should reduce to classical mechanics in the limit of large quantum numbers. This leads to a phenomenological description of the measurement process, where a superposition of states is projected onto a single eigenstate by the act of observation. Thus, in the Copenhagen Interpretation, measurement is a process that happens outside of Quantum Theory. The situation was summarized by John von Neumann in his exposition on the formal mathematical structure of Quantum Theory: We have then answered the question as to what happens in the measurement of a quantity.... To be sure, the “how” remains unexplained for the present. This discontinuous transition... is certainly not of the type described by the time dependent Schr¨ odinger equation. (von Neumann, 1955) For Quantum Theory demands unitary evolution of the wavefunction, and collapse into a single eigenstate is in violation of this postulate. The struggle to understand measurement has led to many different interpretations of Quantum Theory which involve extending or adding new postulates to the Theory. A central part of the problem is the need to define what constitutes a measurement. Is it the act of looking with the eyes, or does the transition of a memory circuit in an automated experiment constitute a measurement? Perhaps, as some have suggested, consciousness plays a special role, and it is our existence as sentient beings which allows us to perform measurements. The difficulty of all these extensions to Quantum Theory is brought sharply into focus in the context of Schr¨odinger’s reductio ad absurdum, the Cat Paradox. Because of the special role our anthropomorphic notions of observation are given in these frameworks, they lead to a situation where the cat is in a superposition of states until the moment when the lid is lifted off of the box and the observation is made, at which time the superposition collapses. However, when one thinks in physical terms, what should be so special about lifting the lid off of the box? Is it the photons which are let in to the previously dark box? Perhaps it is the process of the outside air mixing with the vapors inside the box. Would the result depend on if the box were hermetically sealed? What about the thermal interactions between the box, the cat, and the surrounding room? This line of inquiry leads quickly to the question of where the experimental apparatus ends and the rest of the world begins. We do not find a physical definition of measurement here, but we are lead in the correct direction, which is to consider the interaction between the quantum system and the environment in which it is embedded. 3
The notion of an isolated quantum system is often found in textbooks, but they are extremely rare in nature. Consider the particular radioactive atom that may or may not decay in the Cat Paradox. It is governed by Quantum Theory, but it is in thermal equilibrium with all of the other atoms which make up the experimental apparatus. These atoms interact with those in the room, which through a growing web of interactions connect our “isolated” radioactive atom to all the atoms on Earth. Thus one cannot consider this atom as an isolated system, but must account for all the additional degrees of freedom through which it interacts with its environment. The net effect of these interactions is to destroy quantum superpositions and force the system into a classical state. Generally, the coupling of a particle to additional degrees of freedom increases the effective mass of the particle, as it must drag along the additional degrees of freedom as it changes state. This renormalized mass enters the calculation of the particles action, which becomes much greater than Planck’s constant h, the scale at which quantum effects occur. The result is a particle which acts classically as it becomes too heavy to tunnel quantum mechanically. This argument was made rigorous by Caldeira and Leggett for the specific case of a spin-1/2 particle coupled to a bath of harmonic oscillators (Caldeira and Leggett, 1981, 1983). One finds that the behavior of the particle depends on both the character and strength of the coupling to the bath. In the simplest case, one finds that for weak coupling the particle acts like a quantum system, maintaining a superposition for a characteristic time determined by the particulars including temperature. When the coupling strength is increased beyond a critical threshold, the partial becomes trapped in one state or the other, and the quantum nature of the system is lost. From this treatment we finally find a physical definition of measurement; it occurs when the quantum system is coupled strongly to an environment, forcing the localization of the state because of the increased effective mass of the aggregate system. This process is known as decoherence.
1.3
Indeterminacy and the Power of Quantum Computing
However, at its root, it was not the “spooky” phenomena like superposition and entanglement that were the source of Einstein’s reservations about Quantum Theory. Rather, it was the indeterminate and probabilistic nature of the entire framework which caused Einstein to feel as if something was lost in going from classical to quantum theory. Einstein saw physics as not just a theory, but as a window onto ultimate truth, the Divine. Classical physics shows us God as a supreme clock maker, constructing a machine of exquisite complexity and scope put in motion at the dawn of time. From the perspective of Einstein in the first half of the twentieth century, Quantum Theory reduces God to a mere bookie, setting the odds for various processes and only probabilistically affecting the outcome. This view is summarized in his famous quotation, “God does not play dice with the universe.”(Eisberg and Resnick, 1985) It was not until decades later that a physicist from the next generation, Richard Feynman, came to recognize that this indeterminacy was not a defect or a sign of imperfection, but instead represented a great source of wonder, mystery, and power. Feynman thought about constructing a fully deterministic computer, one based entirely on logic within the realm of classical physics, to simulate a quantum system (Feynman, 1982). Consider two subsystems, one described by m variables and the other by n variables. Classically a full description of the composite system requires m + n variables. However, in quantum mechanics we must in general take the tensor product of the constituent Hilbert spaces, necessitating a total of m × n variables. This leads to exponential 4
growth with size in the resources necessary for a calculation, which will quickly overwhelm any classical computer.1 Feynman conjectured that if a computer harnessed the power of quantum mechanics, such a quantum computer could simulate a quantum physical system so that the required resources would scale linearly with the system complexity. That is, doubling the system size would double the required resources. This conjecture was eventually proved by Seth Lloyd (Lloyd, 1996), and provides the fundamental basis for quantum computation. There have been many applications developed for a quantum computer (Nielsen and Chuang, 2000), and undoubtedly many more will be developed if a quantum computer is ever made. Here we will briefly describe the two most important algorithms that have been invented for a quantum computer. Crudely, both draw upon superposition to test many possibilities at once. Whereas a classical computer gives a single output for a single input, a quantum computer can give the outputs for all the states in a superposed input register simultaneously. By interfering the results with one another, it is possible to select the input which gives the desired output. First we will consider the problem of factoring a large number into its constituent primes. This may seem like a mathematical exercise, but it is actually the basis of most modern cryptography, used for everything from bank transactions to clandestine messages (Rivest et al., 1978). Classically, this factoring problem becomes exponentially more difficult as the number of digits increases. This is because, without prior knowledge, a d digit number has to be tested for divisibility by O(10d ) factors. It is this scaling law which protects the integrity of our secret codes. Peter Shor showed that a quantum computer could factor a large number in a time that scales like a polynomial with the number of bits (Shor, 1994). This means that a quantum computer puts the national security at risk, and hence the great interest in the feasibility of one. The task became all the more urgent in 2001, when the theory behind Shor’s algorithm was tested on a 7-bit quantum computer based on Nuclear Magnetic Resonance (NMR) to factor 15 into 3 × 5 (Vandersypen et al., 2001). This modest accomplishment proves that it is only the difficulty of a practical implementation for a large scale quantum computer, and not some fundamental limitation, that protects our codes. A second algorithm which has potentially more far-reaching applications is known as Grover’s search algorithm (Grover, 1997). On a classical computer, to find a particular element in a list of length k requires, on average, k/2 tests of individual elements. Grover’s algorithm changes the √ scaling law from being linear in the length, to going as k. This produces a tremendous gain in efficiency for large lists. On inspection, Grover’s algorithm might seem like a particular solution to a narrowly defined 1
To illustrate the classical computational resources required to simulate a physical system, consider a state-of-theart desktop machine. First, what size state vector it can store in its memory? A modern processor operates with 32-bit busses, which means it can address 232 memory locations, or about four gigabytes. However, because of the exponential growth of resources necessary for simulating a quantum system, this machine is woefully inadequate. It could, for example, store the state vector of the fermions occupying a 33-site lattice – barely enough to consider a single site and all its nearest-neighbor interactions in three dimensions. Second, how large a Hamiltonian can it store in its memory? To calculate the evolution of a system with n states for a short period of time entails multiplication of an n × n matrix, which means that a system only half again as large can be simulated. One might hope Moore’s Law (Moore, 1965, 2003), that computing resources double every 18 months, would come to the rescue, but because the demands also grow exponentially, a doubling of resources means it can simulate on the order of one more lattice site. From this it should be clear why computer scientists call problems with exponential complexity intractable.
5
problem; however, a broad class of problems can be recast in the form of a search suitable for this quantum speedup.
1.4
Building a Quantum Computer
As the Vandersypen experiment proves, the tenets underlying the theory of quantum computation are sound. The challenge is in constructing a physical implementation on a scale large enough to be useful. A useful guide to the components necessary for a quantum computer was provided by DiVincenzo in the form of five necessary and sufficient criteria (DiVincenzo, 2000). We shall consider them briefly here, and use them as an outline for the remainder of this work.
A scalable physical system with well characterized qubits The qubit or quantum bit is the extension of the classical bit (Shannon, 1948) to allow for the interactions and states allowed by quantum mechanics. Whereas a classical bit is either 0 or 1, a qubit can be in an arbitrary superposition of these two basis states. Also, different qubits can be entangled with one another so their behavior is correlated. Numerous systems have been advanced as candidate qubits, such as nuclear spins in solution (Vandersypen et al., 2001), trapped ions (Cirac and Zoller, 1995), nuclear spins in a solid substrate (Kane, 1998), quantum optics (Turchette et al., 1995), quantum dots (Loss and DiVincenzo, 1998), and many others. Some of these approaches make production of a single qubit easy, while scaling to a system of hundreds or thousands of qubits is unimaginable. Others, like the approach based on the flux produced by superconducting devices which will be the basis of this Dissertation, use straightforward techniques for scaling, but the difficultly lies producing a single qubit of sufficient quality. Flux qubits and scaling of the system will be the subjects of Chapters 2 and 3, respectively. Chapter 4 deals with the design of a particular qubit that will be the focus of the experimental section of this work.
The ability to initialize the state of the qubits to a simple fiducial state A quantum computer must have a reset function of some sort, a way to initialize all the registers to 0, for example. Also, quantum error correction (Shor, 1995) requires a fresh supply of reduced entropy qubits. For most physical systems, including superconducting flux qubits, initialization is a simple thermal process. At low temperature the 0 state is lower in energy than the 1 state, and the system relaxes into its ground state after sufficient time. We shall discuss the special role the superconducting gap plays in the process for superconducting qubits in Chapter 2. In Chapter 5 we will present experimental data on the initialization of a flux qubit, and in Chapter 6 we will give some theoretical background for understanding qubit relaxation.
6
Long relevant decoherence times, much longer than the gate operation time As discussed above, the notion of an isolated quantum system, and hence an isolated qubit, is an artificial one. An environment is always present, and the central challenge in building a quantum computer is isolating the qubits from the environment sufficiently so that they maintain their quantum character long enough for the computation to take place. This must be balanced against the need to interact strongly with the quantum computer in order to control and measure its state. In Chapter 5 we present measurements of the decoherence time of a flux qubit, and in Chapter 6 we give a theoretical basis for understanding decoherence.
A qubit-specific measurement capability One must be able to read out the state of the qubits at the end of a quantum computation. For flux qubits, this is naturally accomplished by the use of the most sensitive known flux detector, the Superconducting QUantum Interference Device, or SQUID. However, the measurement device also links the qubit to the outside environment, and one must pay special care to make sure that the lid is not lifted off of the metaphorical box protecting the qubit at the wrong time, collapsing the wavefunction before the calculation is complete. We develop a theory for quantitatively understanding measurement of a flux qubit by a SQUID in Chapter 7.
A “universal” set of quantum gates For a classical computer, DeMorgan’s theorem tells us that the two-bit operation nor forms a complete set of primitive gates; any logic operation on an arbitrary number of bits can be decomposed in to so many applications of the nor gate (DeMorgan, 1847). For a quantum computer, the situation is similar. Any quantum algorithm can be reduced to a set of one and two qubit gates, with the most common choice for the two qubit gate being the conditional not, or CNOT (DiVincenzo, 1995). In Chapter 8 we describe two structures which could be used as the basis of a two-qubit gate and show how CNOT can be produced from such a controllable interaction.
1.5
Feasibility of Large Scale Quantum Computation
The question of if it is possible to build a large scale quantum computer with any technology is an open one. As well as the technical challenges in combating decoherence and scaling to many qubits, there may still be fundamental reasons why such a device is impossible. The potential gain for mankind that would be realized by such a device justifies the current effort directed at this goal. However, even if we are unable to ever build a useful quantum computer, the issues raised in trying to do so are important from a scientific standpoint. As Freeman Dyson has said, Quantum computation forces us to confront many of the strange and fantastic features of quantum theory, like entanglement, superposition, and the measurement problem. 7
For these reasons, [work on quantum computation] is valuable even if it does not lead to a quantum computer, because it furthers our fundamental understanding.2 How to interpret Quantum Theory has been a problem since its inception, and work on quantum computation requires us to look inside the box at Schr¨odinger’s Cat, first to see when the measurement really takes place, and then to control the measurement problem to our own ends. The situation is summed up in this quotation by Feynman: Turning to quantum mechanics....Might I say immediately, so that you know where I really intend to go, that we always have had (secret, secret, close the doors!) we always have had a great deal of difficulty in understanding the world view that quantum mechanics represents .... It has not yet become obvious to me that there’s no real problem. I cannot define the real problem, therefore I suspect there’s no real problem, but I’m not sure there’s no real problem. So that’s why I like to investigate things... [By studying quantum computing we] squeeze the difficulty of quantum mechanics into a smaller and smaller place.(Feynman, 1982) Building a quantum computer requires rigorous, quantitative understanding of many of the current mysteries of Quantum Theory. Understanding these mysteries may be the ultimate legacy of work in this field.
2
Freeman Dyson, private communication.
8
Chapter 2
Flux Qubits 2.1
Quantum Two State Systems
At the heart of quantum computing is the qubit. This is the quantum analog of the classical bit which is the basis for modern digital computers.(Shannon, 1948) Whereas a classical bit can be either 1 or 0, the quantum bit can be in any superposition of states allowed by quantum mechanics. For a single qubit, this can be written as |ψi = α|0i + β|1i,
(2.1)
where α and β are, in general, complex numbers. Equation 2.1 describes the state of the qubit in the computational basis, as linear combinations of the states |0i and |1i. This representation is special because results of a measurement of the state of the system are usually given in this basis; however, consistent with quantum mechanics, we can change to any other basis which might be more convenient to describe the system. Equation 2.1 appears, prima facie, to describe a four-dimensional configuration space, as each complex coefficient is in one-to-one correspondence with R2 ; however, the normalization condition that hψ|ψi = 1 imposes the constraint |α|2 + |β|2 = 1, reducing the dimensionality of the space by one. Furthermore, when we consider an isolated qubit, the absolute phase of the wavefunction is unimportant; it is only the relative phase factor between α and β which we must consider; thus we can, as is conventionally done, take α to be real. Through these two considerations, the four dimensional configuration space is reduced to a two dimensional one. We note that this process is exactly analogous to the decomposition of the polarization of light into its Stokes’ parameters(Stokes, 1852). Just as the two-dimensional subspace of Stokes’ parameters can be visualized by mapping onto a Poincar´e sphere, so too can the two-dimensional subspace of qubit states. In this case, the sphere is referred to as the Bloch sphere, recognizing its origins in the early history of Nuclear Magnetic Resonance (NMR)(Bloch, 1946; Nielsen and Chuang, 2000). Figure2.1 shows the Bloch sphere and identifies a few selected points in the computational basis. As we shall see, the Bloch
9
|1> ε B |0>+i|1> |0>-|1>
|0>+|1>
δ |0>-i|1>
|0> Figure 2.1. The Bloch sphere
10
sphere provides a useful too for visualizing single qubit states and operations. For example, any two antipodal states on the Bloch sphere are orthogonal to one another.
2.1.1
The Electron
An ideal two state system is a spin-1/2 particle, or, for definiteness, an electron in a magnetic field with the translational degree of freedom suppressed. We shall briefly review the dynamics of an electron in a magnetic field, so that we may draw upon these concepts when we discuss the flux qubit. For further study, a much more eloquent description is given in Vol. III of the Feynman Lectures (Feynman et al., 1965). We will describe the electron in a computational basis formed by the projection of the zcomponent of the spin. To understand the system dynamics, we must introduce the Hamiltonian, which is ~ H = ~µ · B
(2.2)
= µe [Bx (|0ih1| + |1ih0|) + iBy (|0ih1| − |1ih0|) + Bz (|0ih0| − |1ih1|)]
(2.3)
~ Written in matrix for an electron with magnetic moment ~µ of magnitude µe in magnetic field B. form in the computational basis this can be expressed succinctly as 1 −δ Bz Bx − iBy = H = µe . (2.4) Bx + iBy Bz 2 −δ Here, we have introduced which we shall call the qubit energy bias, and δ, which we will refer to as the tunneling matrix element, for reasons which will become clear. The magnetic field vectors are shown diagrammatically in Fig. 2.1. Diagonalization of the Hamiltonian Eq. 2.4 is simple. The resulting two energy levels are separated by an energy difference p ∆E = δ2 + 2 . (2.5) The ground state eigenvector is r |E0 i =
1 + |0i + 2 2∆E
r
1 − |0i − 2 2∆E
r
1 − |1i, 2 2∆E
(2.6)
1 + |1i. 2 2∆E
(2.7)
while the excited state is r |E1 i =
One can see that at = 0, corresponding to a purely transverse magnetic field, the energy eigenstates reduce to symmetric and antisymmetric combinations of the computational basis states. These results are summarized graphically in Fig. 2.2. So how might one implement a qubit, a quantum two state system, in a physical device? A quantum system with only two states is a useful idealization, but there are no true two state systems in nature. Even the most isolated system has some coupling to the rest of the universe, and so these environmental degrees of freedom must be accounted for at some level, with the result that
11
∆E/δ
4 3 2
(a)
1 0
Amplitude — |E0>
1 4⁄5 3⁄5 2⁄5
(b) |0>
1⁄5
|1>
Amplitude — |E1>
0 1 ½ 0
|1>
-½ -1
|0>
(c)
-4
-2
0
ε/δ
2
4
Figure 2.2. (a) Electron energy level difference. Difference between ground and excited state energy ∆E for an electron in a magnetic field as the magnitude of the Bz field component is varied. Dashed line shows asymptote ∆E = . (b) and (c) Energy eigenvectors. Amplitudes of energy eigenvectors are shown for an electron ground state (b) and excited state (c) in a magnetic field as a function of Bz field strength. Coefficients of the |0i (red) and |1i (blue) parts of the wavefunction are shown. 12
they add additional states to a complete description of the system. Many other systems have more than two states, even when the environment is neglected. This brings us to two important ingredients for producing a qubit from a candidate quantum system: isolation and truncation. As we shall see in Chapter 4, interaction of a two-state system with the environment leads to decoherence, which is perhaps the central practical obstacle to building a quantum computer. We will see that if the interaction is not too strong, the two state description can still be valid, with the environment regarded in some sense as a perturbation. Truncation is particularly important for flux qubits. Although a system may have many more states, if the dynamics can be controlled so that the state of the system remains in an appropriate subspace, the two state description may be valid. We refer to these two states as the computational, or simply qubit, states. As an example, if the energy spectrum of the system consists of a low-lying ground and first excited state, separated by an energy gap from other levels, then the system can act as a qubit so long as no excitations cause an appreciable occupation probability of the higher levels.
2.2
Superconducting Circuits
Superconducting circuits are natural candidates for qubits because the superconducting energy gap provides isolation from environmental degrees of freedom. Furthermore, tremendous flexibility in design of the circuit topology makes it practical to engineer circuits which can be truncated. We will analyze different superconducting circuits in this work, so here we present some general techniques and principles we shall draw upon later. When cooled below their critical temperature, the electrons in a superconductor condense into pairs in a highly organized state(Bardeen et al., 1957). Experiments with superconducting qubits are performed, for the most part, well below the critical temperature, where the properties approach those of a pure superconductor. An energy gap develops between the ground state of the superconducting condensate and the excited states of quasiparticles. This gap indicates that many processes, which would otherwise couple the condensate to external environmental degrees of freedom, are forbidden. This gap isolates the condensate from all manner of low energy excitations, most notably the scattering of electrons off photons and other electrons, so that the enormous number of microscopic modes can be completely disregarded; only the collective modes of the condensate must be considered. The circuits we will consider consist of superconductors, characterized by an inductance Li , connecting Josephson junctions, characterized by a critical current I0j . The collective modes of the condensate in these circuits are described by the currents Ji flowing through the inductors and the phase differences γj across the junctions. There will usually be one or more externally applied magnetic fluxes Φx to control the state of the system. We will analyze the classical dynamics of the condensate using Lagrangian mechanics, and analyze the quantum states of the system using a Hamiltonian formalism(Goldstein, 1980; Jos´e and Saletan, 1998). We begin by writing down the potential energy of the system in terms of the Ji and γj (Jackson, 1975; Weiss, 1999) X1 X U= Li J2i + I0j (1 − cos γj ). (2.8) 2 i
j
13
This appears to be an equation in i+j variables, but the geometry of the circuit imposes constraints between the Ji and γ0j . The Ginzburg-Landau theory of superconductivity (Ginzburg and Landau, 1950) states that the phase around a closed loop must be single valued, implying I ∇γ · d~s = 2πn, (2.9) where we have simply performed a line integral of the gradient of the phase γ around a closed loop s. For our circuits of thick superconductors and Josephson junctions, upon introducing the vector ~ this can be written as potential A, I I X 2π ~ · d~s − ∇γ · d~s = A γj Φ0 j on s
=
X 2π(Φx + Li Ji ) − γj Φ0
(2.10)
j on s
= 2πn = 0, where we have defined the phase differences in the opposite sense of the current so that a positive current results in a negative phase difference, and set n to zero for simplicity. We have one such fluxoid equation for each closed loop of the superconducting circuit, which allows us to eliminate a set of variables, such as the Ji , from U. The kinetic energy T of the system comes from charges that condense on the capacitances Cj associated with the Josephson junctions. The kinetic energy then takes the simple canonical form T=
X1 j
2
Cj Vj2
X Φ2 ∂γj 2 0 Cj , = 8π2 ∂t
(2.11)
(2.12)
j
where we have used the ac Josephson relation dγ 2eV = dt h
(2.13)
to replace the voltages across the junctions Vj with quantities proportional to the time rate of change of the junction phases (∂γj /∂t) ≡ γ˙j . The Lagrangian is then formed from the difference of the kinetic and potential energies L = T − U,
(2.14)
and the classical equations of motion can be found by applying the variational principle: ∂2 L ∂L 2 = ∂γ . ∂γ˙j j
(2.15)
The classical steady state solutions can be found by setting all the time derivatives to zero in these Euler-Lagrange equations and solving the resultant system. This concludes the classical solution of the system. 14
To find the quantum solution to the system, we develop the classical Hamiltonian H and then translate the classical variables into quantum operators. To find the classical Hamiltonian we first find the canonical momenta ∂L pj = . (2.16) ∂γ˙j The Hamiltonian is then H=
X
pj γ˙j − L.
(2.17)
j
Finally, to to form the quantum Hamiltonian the canonical momenta are replaced with the corresponding momentum operators pj → −ih(∂/∂γj ) and the classical phase variables γj are regarded as quantum operators. A general procedure for finding the quantum solution involves expanding the Hamiltonian in a complete set of basis functions bk (~γ). In general, an infinite number of such basis functions are required, but by clever selection of the bk an approximation can be found which quickly converges on the solution. By calculating the Hamiltonian matrix elements Z Hmn = bn (~γ)H(~γ)bm (~γ)d~γ (2.18) and the overlap matrix
Z Bmn = bn (~γ)bm (~γ)d~γ,
(2.19)
one finds the energy levels are simply the eigenvalues of the matrix ← → ← → ← → K = B −1 H .
(2.20)
Furthermore, the wavefunctions are given by a linear combination of the basis functions, weighted ← → according to the eigenvectors of K .
2.3
The One Junction Flux Qubit
We now turn to the analysis of two specific superconducting circuits to see how thy act as qubits. In the next section we will analyze the three-junction flux qubit, which will be the subject of the remainder of this work; however in this section we consider the one-junction flux qubit [Fig. 2.3(a)], consisting of a single Josephson junction of critical current I0 and capacitance C in a loop of inductance L biased with an applied flux Φx . At the degeneracy point Φx = Φ0 /2, the energy vs. flux curve is a degenerate double-well potential given by V(φ) =
Φ20 2βL cos φ + (φ + π + 2πΦx /Φ0 )2 , 2 8π L
(2.21)
in terms of the junction phase φ and the dimensionless screening parameter βL ≡ 2πLI0 /Φ0 . The two states of lowest energy are approximately symmetric and antisymmetric combinations of localized states in the left and right wells characterized by clockwise and counterclockwise circulating currents, between which the “phase particle” tunnels at frequency Ω/2π [Fig. 2.3(b)]. Fluctuations in the flux tilt the potential wells, weakly changing the tunneling frequency in second order [Fig.
15
(a)
(b)
L C
I0
Ω0
Ω
Φ0/2 (c)
∆U
(d) δΩ~exp(∆I0)
δΩ~(∆Φ)2 ∆Φ
∆I0
Figure 2.3. One-junction flux qubit. (a) Schematic. (b) Symmetric double well potential for flux bias Φx = Φ0 /2, shown diagrammatically as a function of phase difference φ. (c) Flux fluctuation ∆Φ couples to Ω only in second order. (d) Critical current fluctuation ∆I0 produces exponential change in Ω. 2.3(c)]; however, critical current fluctuations directly modulate the barrier height, producing an exponential change in the qubit tunneling frequency [Fig. 2.3(d)]. We now calculate the tunnel splitting, or more precisely the energy difference between the ground and first excited state, for the one-junction flux qubit using three different methods. The purpose of this pedagogical exercise is to understand in which regimes certain approximations are valid. We will build on this insight to analyze other qubits later. Our first approach is to approximate the potential with a quartic polynomial and quote an analytic result for the tunneling frequency in the semi-classical WKB approximation, (Leggett, 1987) h i Ω = ω0 exp −η(βL − 1)3/2 . (2.22) Here ω0 ≡ 2[(βL − 1)/LC]1/2 is the classical frequency of small oscillations in the bottom of the wells and η ≡ (8I0 CΦ30 /π3 h2 )1/2 is a parameter which describes the “degree of classicality,” and hence determines when quantum tunneling is important (Leggett, 1987). Figure 2.4(a) plots Ω/2π vs. βL for stated values of L and C. However, the semi-classical approximation is valid only in the regime where the bound states in 16
each well nearly form a continuum, which is far from the case we consider here with only one bound state in each well. To obtain the correct splittings for the ground state in the WKB approximation one must modify Eq. 2.22. A more accurate result is (Garg, 2000) r mω0 φ2m κ −S0 /h Ω = 2ω0 e e , (2.23) πh where S0 is the action along the tunneling direction Z φm p S0 = 2mV(φ)dφ, (2.24) −φm
and κ is a correction factor
Z φm " κ = 0
# mω0 1 p dφ. − 2mV(φ) φm − φ
(2.25)
Here m = C (Φ0 /2π)2 is the effective mass of the tunneling particle, and ±φm are the positions of the minima of the symmetric double well potential. The great advantage of this formulation of the WKB approximation, beyond its validity for ground state splittings, is that the limits of the integrals are at the true extrema of the potential rather than the classical turning points, making the calculation more tractable. By evaluating Eqs. (2.23)-(2.25) numerically, we obtain a second result for Ω, shown in Fig. 2.4(a) as a function of βL . We see that the two forms of the WKB approximation are similar in overall shape, with Ω vanishing at βL = 1 where ω0 becomes zero, and decreasing exponentially at larger values of βL . However, the two forms disagree quantitatively at small values of βL and diverge from one another at large values of βL . Thus, we turn to a full quantum mechanical solution of the degenerate double-well potential to resolve this discrepancy. To solve for the ground state wavefunction we choose as our basis set 12 simple harmonic oscillator wavefunctions centered in the left well and 12 more centered in the right well. We use the Hamiltonian " 2 # Φ0 2 ∂ H(φ) = 2βL cos(φ) + (π + φ + φx )2 + LC , (2.26) 8π2 L ∂φ where φx ≡ 2πΦx /Φ0 . The results for φx = 0 are shown as points in Fig. 2.4(a). For large values of βL the full solution approaches the modified WKB expression, Eq. (2.23), asymptotically, but the difference diverges at small values of βL . The standard WKB approximation gives a tunneling frequency which is inconsistent with the full solution almost everywhere. To characterize the sensitivity of a particular qubit to fluctuations in the junction critical currents, it is useful to introduce the quantity I0 dΩ . (2.27) Λ= Ω dI0 We shall explore this in more detail in Sec. 6.3.2, but here we simply present Fig. 2.4(b), which shows Λ vs. βL for the three calculations. The two semi-classical approximations predict that Λ vanishes at certain values of βL , but this is an artifact of the apparent maxima in Fig. 2.4(a); the full quantum treatment shows no zero. Figure 2.4(c) plots the fractional change in tunneling frequency, δΩ/Ω, vs. βL for the three calculations for three fractional changes in critical current, δI0 /I0 . We note that for βL & 1.1 the three approaches differ by no more than a factor of about two. 17
Ω/2π (GHz)
10
(a)
1
0.1
0.01
80
Λ
60 40 20
(b)
0
δI0/I0=10-3
dΩ/Ω
10-1
δI0/I0=10-4
10-2
δI0/I0=10-5
10-3 10-4
(c) 1
1.05
1.1
βL
1.15
1.2
1.25
Figure 2.4. Three quantities for the ground state of the one-junction flux qubit at the degeneracy point calculated using the standard WKB approximation (solid), WKB approximation corrected for the ground state (dashed), and numerical solution for the wavefunctions (points), plotted as a function of the dimensionless screening parameter βL . (a) Splitting frequency between ground state and first excited state, (b) sensitivity parameter Λ, and (c) effect of critical current fluctuations on tunneling rate for three values of δI0 /I0 . Parameters are from Friedman et al.: L = 240 pH and C = 104 fF (Friedman et al., 2000).
18
10
9R 8R 7R 6R 5R 4R 3R 2R 1R 0R
Energy (K)
3L 7.5 2L 1L 5 0L 2.5 0
-2.5 -3
-2
-1
0 δ
1
2
3
Figure 2.5. Eigenfunctions and energy levels for a one-junction flux qubit. Absolute values of eigenfunctions (thin lines), each offset so that it asymptotes to the corresponding energy level, are shown as a function of the junction phase δ. The thick line is the asymmetric double well potential. Eigenfunctions are labeled according to corresponding single-well harmonic oscillator quantum numbers. Parameters are as in Fig. 2.4, with βL = 1.5; the flux φx ≈ 0.514 × 2π produces a resonance between the 3L (left) and 9R (right) states. Excited States The first demonstration of a one-junction flux qubit did not employ ground states, however, but excited states in deep, tilted potential wells (Friedman et al., 2000). The WKB approximation is again unsuitable, for two main reasons. First, treating asymmetric potentials is more difficult, because of different prefactors for the two wells, but this can be overcome (Averin et al., 2000). More importantly, resonant tunneling, which causes a dramatic increase in the tunneling rate when two energy levels are aligned, is entirely absent from the WKB approximation. Thus, the only way to calculate the sensitivity to critical current fluctuations is to solve the Schr¨ odinger equation for the energy levels numerically. We adopt the approach of Sec. IV.A with a different basis set. We use 60 harmonic oscillator wavefunctions centered between the minima of the two wells, so that B becomes the identity matrix. To reproduce the experimental conditions, (Friedman et al., 2000) we set βL = 1.5 and find the energy levels for successive values of applied flux φx . We find that the energy difference between the third and ninth excited states has a local minimum at φx ≈ 0.514 × 2π, corresponding to the condition for resonant tunneling. The potential, wavefunctions, and energy levels for this situation are shown in Fig. 2.5. Fixing φx at this value and sweeping βL , we calculate the relevant quantities for low frequency critical current fluctuations. The results are shown in Fig. 2.6.
19
In Fig. 2.6(a) we see that near the resonant point βL = 1.5, Ω decreases with increasing barrier height, as one would expect from a semi-classical analysis, but reaches a local minimum at a slightly higher value. As βL is increased further, Ω increases because the energy levels are no longer resonant. At the minimum, the derivative quantity Λ vanishes, as the changing barrier height balances the loss of resonance, indicating that the system is immune to small critical current fluctuations at this point. We note that on resonance, where Λ is almost optimally bad, the system is immune to flux noise, because the energy is a minimum as a function of flux. Thus, one can exchange sensitivity to critical current fluctuations for sensitivity to flux noise as appropriate.
2.4
The Three Junction Flux Qubit
Since the publication of its invention in 1999 (Orlando et al., 1999), the three junction flux qubit has become one of the most important devices for flux-based quantum computing. It consists of three Josephson junctions, one characterized by a somewhat smaller critical current than the other two, embedded in a superconducting loop. A schematic of the three junction flux qubit is shown in Fig. 2.7. As in the one junction flux qubit, the two qubit states have currents flowing in opposite directions around the loop. However, we saw that in the ground state of the one junction flux qubit a double well potential was formed by a delicate balancing of the loop inductance against the qubit critical current, so βL ≈ 1. The dynamics of the three junction flux qubit are dominated by the small junction, while the two large junctions can be though of as adding to the loop inductance through their Josephson inductance. Thus, the three junction flux qubit can be designed to have a much smaller geometric inductance than the one junction flux qubit. Furthermore, the stringent requirement on βL , which places tight tolerances on LQ , I0 , and their product, is replaced with a requirement on αQ , the ratio of the critical current of the small junction to that of the larger junctions. Because the critical current of a junction scales with its area, a requirement on αQ translates into a requirement of the relative sizes of the junctions, which are defined in a common nano-lithographic process. We can begin to understand the three junction flux qubit by considering the potential energy of the system. Following the procedure described in Sec. 2.2, the potential energy has an inductive and a Josephson contribution,U = Ul + UJ . The inductive contribution is simply 1 Ul = LQ J2Q , 2
(2.28)
where JQ is the current circulating in the qubit. The Josephson contribution is UJ =
Φ0 [I01 (1 − cos γ1 ) + I02 (1 − cos γ2 ) + I03 (1 − cos γ3 )] , 2π
(2.29)
where the I0i are the critical currents of the three junctions and the γi are the phase differences across the three junctions. The phase differences and the circulating current are connected through the fluxoid condition LQ JQ γ1 + γ2 + γ3 = φQ + 2π (2.30) Φ0 where φQ = 2πΦQ /Φ0 is the flux bias applied to the qubit in units of phase.
20
Ω/2π (GHz)
10
(a)
8 6 4 2 70
(b)
60
Λ
50 40 30 20 10 1.35
1.4
1.45
1.5 βL
1.55
1.6
1.65
Figure 2.6. Numerical solution for the excited states of an asymmetric one-junction flux qubit. (a) Tunneling frequency between the third excited state in the shallow well and the ninth excited state in the deep well as a function of βL for a system on resonance at βL = 1.5. (b) Derived sensitivity to critical current fluctuations. Device parameters are as in Fig. 2.4. 21
JQ =
Φ0 2πLQ
(γ1 + γ2 +γ3 − ϕQ) ϕQ=2πΦQ/Φ0
γ2
I02
I01
LQ γ1
I03 γ3 Figure 2.7. Schematic of three junction flux qubit with loop inductance LQ , circulating current JQ , and phase differences γi across Josephson junctions of critical current I0i .
22
To facilitate analysis of the potential, we introduce the dimensionless parameters 2πLQ 1 1 1 −1 + + , βQ = Φ0 I01 I02 I02
(2.31)
which characterizes the relative importance of Josephson and geometric inductances, αQ =
2I03 , I01 + I02
(2.32)
which describes the relative size of the small junction, and Q =
I01 − I02 , I01 + I02
(2.33)
which characterizes asymmetry between the two large junctions. Equations 2.28 and 2.29 show that while UJ is bounded, Ul can grow without limit as the magnitude of JQ increases. Through the fluxoid condition (Eq. 2.30), J2Q , and hence Ul , are minima when γ1 + γ2 + γ3 = φQ . (2.34) This can be seen in Fig. 2.8, where the constant potential energy surfaces become flat as the circulating current dominates the energy (blue and green surfaces). When Ul is small, the periodic structure of UJ becomes important, and we have an array of local minima near the plane defined by Eq. 2.34. The structure of the potential becomes clearer if we rotate into a coordinate system defined by a set of variables aligned with the plane of zero inductive energy. We define the total phase variable γt =
αQ (φQ − γ1 − γ2 − γ3 ) , 1 + 2αQ
(2.35)
which runs normal to the plane of zero inductive energy. We decompose the in-plane coordinates into a symmetric 1 γs = [2αQ (γ3 − φQ ) − γ1 − γ2 ] (2.36) 2(1 + 2αQ ) and an antisymmetric mode 1 γa = (γ1 − γ2 ). 2
(2.37)
The Josephson energy is then UJ = EJ [(1 + Q ) cos(γa − γs − γt ) + (1 − Q ) cos(γa + γs + γt ) γt +αQ cos(2γs + φQ − ) , (2.38) αQ and the inductive energy depends only on γt Ul = EJ
(1 + 2αQ )2 (2Q − 1) 2αQ βQ (1 + 2αQ − 2Q )
γ2t .
(2.39)
If we first consider the small βQ regime, we see that Ul dominates, so solutions of minimum potential energy will have γt ≈ 0. We thus reduce the three dimensional problem to a two dimensional one. When the qubit is biased with a half quantum of flux (φQ = π), we see the array 23
δ2 -2π
-3π 3π
-π
0
π
2π
3π
2π π δ3 0 -π -2π -3π -3π
-2π
-π δ1
0
π
2π
3π
Figure 2.8. Three junction flux qubit potential. Three contours of potential energy [U = 1.4Ej (red), U = 10Ej (green), and U = 30Ej (blue)] are shown. Parameters are αQ = 0.8, βQ = 0.4, Q = 0, and φQ = π.
24
2π
π
P2 P0
δs
0 2.8
Energy (EJ)
P1
-π
-2π -2π
2 1 0
-1 -1.425
-π
0 δa
π
2π
Figure 2.9. Slice through γt = 0 of three junction flux qubit potential. Parameters are αQ = 0.8, βQ = 0.01, Q = 0, and φQ = π.
25
π
π/2
P0
δs
0 5.2 Energy (EJ)
-π/2
4 3 2 1 0 1 -1.68
P1
-π 1 ½ 0 -½ δt
-π/2
0 δa
π/2
π
Figure 2.10. Unit cell of a larger inductance flux qubit. Slices through both the γt -γs plane and the γa -γs , which coincide along the dotted line, plane are shown. Parameters are αQ = 0.6, βQ = 0.4, Q = 0, and φQ = π. of degenerate local minima shown in Fig. 2.9. Closer inspection reveals that the minima come in pairs separated by a barrier, such as those marked by the points P0 and P1 . Thinking of the system semi-classically, we can imagine a “particle” at P0 representing the state of the system, which can tunnel through the barrier to P1 . Quantum mechanically, the ground state energy of of the system is split by this tunneling amplitude, so the the ground state is a symmetric superposition of states localized at P0 and P1 and the first excited state is an anti-symmetric combination. If we define a state localized at P0 to be our computational basis |0i state, and a√state localized at P1 to be the qubit |1i state, √ then the ground state at ΦQ = Φ0 /2 is (|0i + |1i)/ 2 and the first excited state is (|0i − |1i)/ 2. However, as well as the intracell tunneling path leading from P0 to P1 , another intercell path exists, typified by the arrow pointing to P2 . As shown, the barrier to intercell tunneling is much higher than the intracell one, so it splits the energy levels much less. As we shall see below, because the tunnel splitting is exponential in the barrier height, one can readily choose qubit parameters so that the intercell tunnel splitting is hundreds of times less than the intracell splitting and can be disregarded in many instances. We can better understand the nature of the two computational basis states if we look perpendicularly to the γa –γs plane, in the γt direction. Figure 2.10 shows two slices through 26
1.
(a)
|JQ|/I03
0.8
(b)
0.6 0.4 0.2 0. 0.
0.2
0.4
αQ
0.6
0.8
1. 0.
1.
2.
βQ
3.
4.
5.
Energy (EJ)
Figure 2.11. Circulating current at the degeneracy point. Magnitude of qubit circulating current JQ , as indicated by location of potential minimum, is shown as a function of (a) αQ and (b) βQ . Parameters are = 0, φQ = π, βQ = 0.1 (a), and αQ = 0.7 (b).
ϕQ=0.9π
ϕQ=0.95π
-π 0 π δt
-π 0 π δt
ϕQ=π
ϕQ=1.05π
ϕQ=1.1π
-π 0 π δt
-π 0 π δt
2 0
-2
-π 0 δt
π
Figure 2.12. Flux dependence of three junction qubit potential. The potential energy is shown for γt = γa = 0 for various flux biases. Parameters are αQ = 0.8, βQ = 0.01, and Q = 0. the three-dimensional potential space for a large-inductance flux qubit (βQ = 0.4). It is clear that the minima do not lie exactly in the γa –γs plane, but actually above and below it in γt . Recalling that γt is proportional to the circulating current JQ , we see that P0 and P1 correspond to circulating currents in opposite directions. For the parameters listed in the caption to Fig. 2.10, the minima lie at γt = ±0.2205; for comparison, a circulating current of JQ = ±I03 , that is one the size of the critical current of the small junction, would correspond to γt = ±αQ βQ (1 + 2αQ − 2Q )/(1 + 2αQ )(2Q − 1) = ±0.2400. Furthermore, as βQ tends toward zero, the magnitude of JQ in the two basis states tends toward I03 . Thus, we can describe the |0i state as one of (say) clockwise circulating current, and the |1i state as one of counter-clockwise circulating current. For αQ less than some critical value, the potential is a single well so no persistent current can be supported, rendering the device unsuitable as a qubit. For a qubit with vanishing inductance (βQ → 0), this critical value is α = 1/2; however, as the inductance is increased, a smaller value of αQ can support a persistent current. For example, for βQ = 0.1, the potential is bifurcated at αQ ≈ 0.41 (Fig. 2.11). For αQ > 1 a double well potential still exists, but the barrier to intercell tunneling is lower than the barrier to intracell tunneling. This leads to quantum states of the system which are not suitable for a conventional qubit. The circulating current JQ interacts with the flux bias ΦQ to form a screening energy. When
27
ΦQ = (n + 1/2)Φ0 this screening energy is equal and opposite for the two basis states |0i and |1i, and so cancels out. However, as the flux bias is moved away from the half flux quantum point, it raises the energy of one basis state and lowers the energy of the other. Classically we would expect this energy difference to be Φ0 Φ0 = 2JQ ΦQ − ≈ 2I03 ΦQ − , (2.40) 2 2 where we have associated this energy with the qubit bias energy . This energy can be seen in Fig. 2.12, which shows the effective one-dimensional potential for a small-βQ flux qubit, obtained by setting γa = γt = 0. The qubit bias energy manifests itself by raising the bottom of one well relative to the other.
2.4.1
Hamiltonian
To fully understand the dynamics and quantum states of the three junction flux qubit, we must introduce the kinetic energy and form the Hamiltonian. Kinetic energy in the system comes from charges on the junction capacitance, and can be written as 1 T= 2
Φ0 2π
2
(C1 + Cs )γ˙1 2 + (C2 + Cs )γ˙2 2 + (C3 + Cs )γ˙3 2 .
(2.41)
Here, we have associated a capacitance Ci with each junction, which we will take to be proportional to the critical current and hence the area of the corresponding junction, and a stray capacitance Cs , which is taken to be the same for all junctions. Equation 2.41 can be recognized to be of the form Ci Vi2 /2, where the voltage developed across each junction is Vi = (Φ0 /2π)dγi /dt. Applying the coordinate transformations (Eqs. 2.35-2.37) and parameter transformations (Eqs. 2.31-2.33) defined above, the kinetic energy takes the form 1 + σQ −Q −Q γ˙ a γ˙ a 2 h · γ˙ s · γ˙ s , (2.42) −Q 1 + 2αQ + 3σQ σQ (αQ − 1)/αQ T= 4Ec −Q σQ (αQ − 1)/αQ (αQ + 2α2Q + σQ + 2α2Q σ)/2α2Q γ˙ t γ˙ t where we have introduced the charging energy Ec = e2 /(C1 + C2 ), the capacitance ratio σQ = 2Cs /(C1 + C2 ), and used a matrix notation to represent all the terms. Note that the mass tensor in Eq. 2.42 becomes diagonal in a qubit with no stray capacitance and no asymmetry. We now form the Hamiltonian H using the standard techniques of classical mechanics (Goldstein, 1980; Jos´e and Saletan, 1998). First we form the Lagrangian, L = T − U, and then calculate ˙ ~ = (pa , ps , pt ). We can write this as the vector equation p ~ = ∂L/∂~γ, the canonical momenta p ~ where γ˙ = (γ˙ a , γ˙ s , γ˙ t ). The Hamiltonian is then
which can be expressed as
~ · ~γ˙ − L, H=p
(2.43)
→ 1 ← ~ · M −1 · p ~ + U, H= p 2
(2.44)
28
where the inverse mass matrix is ← → M −1 =
4Ec h i× 2 2 2 h (1 + σQ ) − Q
1 + σQ 2Q 1+2αQ 4αQ Q 1+2αQ
2Q 1+2αQ (σQ +αQ )(1+σQ )+2αQ 2 (1−2Q +2σQ +σ2Q ) 2
(1+2αQ ) (αQ +σQ ) 4αQ [σQ +σQ 2 +αQ (Q 2 −σQ −σ2Q )] 2 (1+2αQ ) (αQ +σQ )
4αQ Q 1+2αQ 4αQ [σQ +σQ 2 +αQ (Q 2 −σQ −σ2Q )]
. (2.45) 2 (1+2αQ ) (αQ +σQ ) 2 2 2 2αQ [2αQ (1+σQ )+1+4σQ +3σQ −Q ] 2 (1+2αQ ) (αQ +σQ )
Solving the Hamiltonian To find the quantum mechanical solution to the system, we will expand the Hamiltonian in a complete set of basis functions. The potential energy is 2π-periodic in both γa and γs , while the kinetic energy is quadratic for a qubit without asymmetry or stray capacitance. This suggests that expansion in plane waves is appropriate for these two components. However, while γt also has a quadratic kinetic energy and terms in the potential energy which are periodic, the inductive potential is quadratic in γt . This suggests that harmonic oscillator wavefunctions are more appropriate for the expansion in γt , but the periodic terms may present some difficulty. Thus we choose for our basis functions the product state s t |ψklm i = |ψa k i|ψl i|ψm i,
(2.46)
where the first two factors are plane waves {a,s}
|ψk
1 i = √ e−ikγ{a,s} . 2π
(2.47)
The third factor is a simple harmonic oscillator wavefunction in γt |ψtm i
mt ωt = 2n 2 πh(n!)2
1/4
r Hm
2 mt ωt γt e−mt ωt γT /2h , h
(2.48)
where Hm [γ] is the mth degree Hermite polynomial. The mass for the harmonic oscillator is mt =
h2 (αQ + σQ )[(1 + σQ )2 − 2Q ](1 + 2αQ )2 8Ec α2Q [(1 + σQ )(1 + 2αQ + 3σQ ) − 2Q ]
and the natural frequency is v u 2EJ αQ (1 − 2Q )[(1 + σQ )(1 + 2αQ + 3σQ ) − 2Q ] 2Ec u t . ωt = h Ec βQ (1 + 2αQ − 2Q )(αQ + σQ )[(1 + σQ )2 − 2Q ]
(2.49)
(2.50)
As we saw in Sec. 2.2, we need to calculate the matrix elements ← → H klmk 0 l 0 m 0 = hψklm |H|ψk 0 l 0 m 0 | i. 29
(2.51)
To do this we first Fourier transform the potential energy with respect to γa and γs and write it as " 2 # 2 X X ← → −i(kγa +lγs ) 1 U = Ej (2.52) U kl e + mt ω2T γ2t , 2 k=−2 l=−1
← → where U is a function of eiγt : 0 (1 − Q )eiγt ← → 1 i(φQ − γt ) α U kl = αe 0 2 0 (Q − 1)e−iγt
0 (Q − 1)e−iγt 0 0 0 (1 − Q )eiγt
0
γt
αe−i(φQ − α ) . 0
(2.53)
← → Note that the U 00 component is in the middle of the matrix. We can now write down the Hamiltonian expansion and address the terms individually. X ← → ← →−1 s s a H klmk 0 l 0 m 0 = hψa k |hψl |pi M i,j pj |ψl 0 i|ψk 0 i i={a,s} j={a,s}
← → ← →−1 s a a s hψ + 2 M −1 |p |ψ i + M hψ |p |ψ i hψtm |pt |ψtm 0 i 0 0 a s as k l k l at 2 2 X X
+
← → s −i(pγa +qγs ) s hψtm | U pq |ψtm 0 ihψa |ψl 0 i|ψa k |hψl |e k0 i
p=−2 q=−1
+ hψtm |
1 p2t + mt ω2t |ψtm 0 i. 2mt 2 (2.54)
The first term is easily evaluated by replacing the momentum operators pi with −ih(∂/∂γi ) and performing the integrals: X
Zπ Zπ
i={a,s} −π −π j={a,s}
← → −h2 M −1 ij 4π2
ei(kγa +lγs )
∂ ∂ −i(k 0 γa +l 0 γs ) e dγa dγs ∂γi ∂γj ← → ← →−1 →−1 2← = h2 (k2 M −1 aa + 2kl M as + l M ss )δkk 0 δll 0 , (2.55)
where we have introduced the Kronecker delta function δab . For the next term, the first factor involving pa and ps is easily evaluated using the same technique as the first term. The second factor, involving pt , is evaluated by representing the momentum p operator with harmonic oscillator annihilation and creation operators, pt = i mt ωt h/2(a†t − at ): r ← →−1 a ← →−1 s mt ωt h t † t a s M at hψk |pa |ψk 0 i + M as hψl |ps |ψl 0 i × i hψm |at |ψm 0 i − hψtm |at |ψtm 0 i 2 r 3 √ √ mt ωt h (kδkk 0 + lδll 0 )( n + 1δn 0 ,n+1 − nδn 0 +1,n ). (2.56) = −i 2 Evaluating the next set of terms involves calculating two factors. The second factor is simply another plane wave overlap integral, giving s −i(pγa +qγs ) s hψa |ψl 0 i|ψa k |hψl |e k 0 i = δk,p+k 0 δl,q+l 0 .
30
(2.57)
The first factor involves overlap integrals between harmonic oscillator wavefunctions and exponentials. The formula required to evaluate this factor is (Witschel, 1996) hψtm |ekγt |ψtm 0 i
0 1/2
= (m!m !)
e
h2 k2 2mt ωt
min{m,m 0 }
X j=0
(m+m 0 −2j)/2 0 m m hk2 . j! 2mt ωt j j
The final term is trivial. It is simply the overlap of a harmonic oscillator basis functions, giving 1 p2t t 2 t hψm | + mt ωt |ψm 0 i = m + 2mt 2
(2.58)
harmonic oscillator Hamiltonian with 1 2
hωt δmm 0 .
(2.59)
← → Thus, all the matrix elements in H klmk 0 l 0 m 0 can be written down in closed form. The matrix is infinite dimensional in this basis, but we can truncate it for numerical diagonalization by letting k and k 0 run from −nk to nk , l and l 0 run from −nl to nl , and m and m 0 run from 0 to nm . ← → The eigenvalues of H then give the energy levels and the eigenvectors give the coefficients of the wavefunctions. Experience has shown that (nk , nl , nm ) = (5, 10, 2) gives good results for small inductance flux qubits, say βQ < 0.1. For larger values of βQ , nm must be increased to capture the effect of the inductance. The results of such a numerical calculation are shown in Fig. 2.13. The states of opposite circulating current can be identified by the way they disperse hyperbolically in opposite directions. The difference in energy between these states fit quite well to Eq. 2.4, with a root-mean-square (rms) deviation of less than 20 MHz, indicating that a three junction flux qubit truncated to these two states is well approximated as an ideal two state system. To conclude this section on calculation of the three junction flux qubit energy levels, we apply the WKB formalism described in Eq. 2.23 to intracell tunneling. When |Q |
