Feedback control of superconducting quantum circuits

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(1)Feedback control of superconducting quantum circuits.

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(3) Feedback control of superconducting quantum circuits. Proefschrift. ter verkrijging van de graad van doctor aan de Technische Universtiteit Delft, op gezag van de Rector Magnicus Prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties, in het openbaar te verdedigen op 17 oktober 2014 om 10:00 uur. door. Diego RISTÈ nanoscience ingenieur geboren te Jesi, Italië.

(4) Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. L.P. Kouwenhoven. Copromotor: Dr. L. DiCarlo. Samenstelling promotiecommissie: Rector Magnicus,. voorzitter. Prof. dr. ir. L.P. Kouwenhoven,. Technische Universiteit Delft, promotor. Dr. L. DiCarlo,. Technische Universiteit Delft, copromotor. Prof. dr. M.H. Devoret,. Yale University, New Haven. Prof. dr. D.P. DiVincenzo,. Rheinisch-Westfälische. Technische. Hochschule. Aachen en Forschungszentrum Jülich Prof. dr. ir. R. Hanson,. Technische Universiteit Delft. Prof. dr. ir. J.E. Mooij,. Technische Universiteit Delft. Prof. dr. A. Wallra,. Eidgenössische Technische Hochschule Zürich. Prof. dr. Y.M. Blanter,. Technische Universiteit Delft, reservelid. Copyright. c. 2014 by Diego Ristè. All rights reserved.. No part of this book may be reproduced, stored in a retrieval. system, or transmitted, in any form or by any means, without prior permission from the copyright owner.. ISBN: 978-90-8593-196-6 Casimir PhD Series Delft-Leiden 2014-23 Cover: Tremani www.tremani.nl Printed by Gildeprint Drukkerijen www.gildeprint.nl An electronic version of this thesis is available at www.library.tudelft.nl/dissertations.

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(7) Contents. Contents 1. 2. INTRODUCTION. 1. 1.1. Basic criteria of quantum computing. 1.2. The seven stages of development. 1.3. Superconducting quantum circuits. 1.4. . . . . . . . . . . . . . . . . . .. 3. . . . . . . . . . . . . . . . . . . .. 3. Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. QUANTUM FEEDBACK WITH SUPERCONDUCTING CIRCUITS 2.1. 2.2. 3. 1. . . . . . . . . . . . . . . . . . . . .. 7. Digital feedback control in quantum computing. . . . . . . . . . . . .. 8. 2.1.1. Classication of quantum feedback. . . . . . . . . . . . . . . .. 8. 2.1.2. Protocols using digital feedback . . . . . . . . . . . . . . . . .. 8. 2.1.3. Experimental realizations of digital feedback. . . . . . . . . .. 10. Implementation of digital feedback in 3D cQED . . . . . . . . . . . .. 11. 2.2.1. Concepts in digital feedback . . . . . . . . . . . . . . . . . . .. 11. 2.2.2. Closing the loop in cQED. . . . . . . . . . . . . . . . . . . . .. 12. 2.2.3. Feedback controllers. . . . . . . . . . . . . . . . . . . . . . . .. 13. INITIALIZATION BY MEASUREMENT OF A TWO-QUBIT SUPERCONDUCTING CIRCUIT. 17. 3.1. Qubit state initialization . . . . . . . . . . . . . . . . . . . . . . . . .. 18. 3.2. High-delity readout using a Josephson parametric amplier . . . . .. 18. 3.2.1. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .. 18. 3.3. Characterization of JPA-backed qubit readout and initialization . . .. 19. 3.3.1. . . . . . . .. 22. 3.4. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 3.4.1 3.5. Repeated quantum nondemolition measurements. Further developments. . . . . . . . . . . . . . . . . . . . . . .. 24. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25. 3.5.1. Device fabrication. 25. 3.5.2. Conguration of the JPA for qubit readout. . . . . . . . . . .. 25. 3.5.3. Readout error model . . . . . . . . . . . . . . . . . . . . . . .. 27. . . . . . . . . . . . . . . . . . . . . . . . .. vii.

(8) Contents 4. FAST RESET OF A SUPERCONDUCTING QUBIT USING DIGITAL FEEDBACK 4.1. 4.2. 5. 33. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 4.1.1. Experimental setup for qubit reset. . . . . . . . . . . . . . . .. 34. 4.1.2. Passive qubit initialization to steady state . . . . . . . . . . .. 34. Qubit reset based on digital feedback . . . . . . . . . . . . . . . . . .. 37. 4.2.1. Characterization of the reset protocol. . . . . . . . . . . . . .. 4.2.2. Speed-up enabled by fast reset. 37. . . . . . . . . . . . . . . . . .. 38. 4.3. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40. 4.4. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40. 4.4.1. Readout and feedback congurations . . . . . . . . . . . . . .. 40. 4.4.2. Measurement of the transmon populations . . . . . . . . . . .. 40. 4.4.3. Model for measurement and feedback. . . . . . . . . . . . . .. 43. 4.4.4. Comparison of feedback protocols . . . . . . . . . . . . . . . .. 44. DETERMINISTIC ENTANGLEMENT BY PARITY MEASUREMENT AND DIGITAL FEEDBACK Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. 5.2. Two-qubit parity meter. 48. 5.3. 5.4. 5.5. 5.6. . . . . . . . . . . . . . . . . . . . . . . . . .. 5.2.1. Engineering the cavity as a parity meter . . . . . . . . . . . .. 48. 5.2.2. Two-qubit evolution during parity measurement. 51. Entanglement generation by parity measurement. . . . . . . .. . . . . . . . . . . .. 53. 5.3.1. Probabilistic entanglement by measurement and postselection. 53. 5.3.2. Deterministic entanglement by measurement and feedback . .. 53. Conclusion 5.4.1. 6. 47. 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Further developments. 55. Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . .. 55. 5.5.1. Device parameters. 55. 5.5.2. Readout signal processing. 5.5.3. Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Supplementary gures. . . . . . . . . . . . . . . . . . . . . . . . . . .. 56 56 57. REVERSING QUANTUM TRAJECTORIES WITH ANALOG FEEDBACK. 67. 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 68. 6.2. Measurement-induced qubit dephasing. . . . . . . . . . . . . . . . . .. 68. 6.2.1. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .. 68. 6.2.2. Observing measurement-induced qubit dephasing . . . . . . .. 6.3. 6.4. viii. 55. . . . . . . . . . . . . . . . . . . . . . .. Tracking and undoing the measurement-induced qubit evolution. 70. . . . . . .. 70. 6.3.1. Correlating measurement record to qubit kickback. 6.3.2. Suppression of measurement kickback by analog feedback. 6.3.3. Optimum weighing of the measurement record. Conclusion. 70. . .. . .. 72. . . . . . . . .. 74. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75.

(9) Contents 6.5. 7. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75. 6.5.1. Device parameters. . . . . . . . . . . . . . . . . . . . . . . . .. 75. 6.5.2. Analog feedback for phase cancellation . . . . . . . . . . . . .. 76. 6.5.3. Pump leakage suppression . . . . . . . . . . . . . . . . . . . .. 78. 6.5.4. Weight function optimization. . . . . . . . . . . . . . . . . . .. 79. 6.5.5. Mode-matching theory for the optimal weight function . . . .. 80. MEASUREMENT OF QUASIPARTICLE TUNNELING IN A TRANSMON BY PROJECTIVE READOUT AND RESET Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86. 7.2. Measurement of quasiparticle tunneling and induced decoherence. 87. . .. 7.2.1. Evidence of charge-parity uctuations. . . . . . . . . . . . . .. 87. 7.2.2. Real-time detection of quasiparticle tunneling . . . . . . . . .. 87. 7.2.3. Measurement of QP-tunneling-induced qubit decoherence. . .. 88. 7.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91. 7.4. Conclusion. 93. 7.4.1 7.5. 8. 85. 7.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Further developments. . . . . . . . . . . . . . . . . . . . . . .. 93. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93. 7.5.1. Device parameters. 93. 7.5.2. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .. 93. 7.5.3. Extraction of QP tunneling rates . . . . . . . . . . . . . . . .. 94. 7.5.4. Validation of the charge-parity detector. . . . . . . . . . . . .. 95. . . . . . . . . . . . . . . . . . . . . . . . .. CONCLUSIONS. 99. 8.1. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100. 8.2. Quantum network with 3D cQED . . . . . . . . . . . . . . . . . . . .. 100. 8.2.1 8.3. 8.4. Remote entanglement stabilization. . . . . . . . . . . . . . . .. 101. Parity measurements and feedback in 2D cQED . . . . . . . . . . . .. 103. 8.3.1. Tripartite entanglement by simultaneous parity measurements. 104. 8.3.2. Bit-ip error correction code. 105. 8.3.3. Towards logical qubits. Conclusions. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. 108. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 108. Summary. 111. Samenvatting. 113. Acknowledgements. 115. List of Publications. 119. Curriculum Vitae. 121. Bibliography. 123. ix.

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(11) Symbols and abbreviations AWG. a (a† ) b (b† ) cQED. C(Vint ) cfb C e e, o EC EJ EN (ρ) FPGA. F Fb0 (Fb1 ) Fp fm fr G g H, L JPA. MA , MB , ... MI ( MQ ) MP ng n ¯ ph nqp n ¯ ss P Perr P|0i (P|1i ) Pt ( Pb ) P|iji,ss P|iji,τ pM ij M pij,kl psuccess QIP QND. arbitrary waveform generator annihilation (creation) operator for the intra-cavity eld annihilation (creation) operator for the outgoing eld after JPA circuit quantum electrodynamics fraction of counts for a bin centered at. Vint. analog feedback gain concurrence elementary charge even or odd charge-parity, respectively charging energy Josephson energy logarithmic negativity of state. ρ. eld-programmable gate array charge-parity detection delity feedback protocol targeting. |0i (|1i). two-qubit parity readout delity measurement frequency cavity resonance frequency JPA amplitude gain qubit-cavity coupling strength. M. possible measurement outcomes for a single-qubit measurement Josephson parametric amplier subsequent measurements measurement with. φ = 0 (φ = π/2). two-qubit parity measurement charge oset average intra-cavity photon number quasiparticle density steady-state intra-cavity photon number with no drive charge-parity measurement result, equal to 1 (-1) for even (odd) initialization error population of state. |0i (|1i). parity measurement result for top (bottom) qubit pair, equal to steady-state population in level population in level. |iji. at time. probability of nal state probability of nal state. |iji τ following. e or o. initialization. |ji for initial state |ii and measurement M |kli for initial state |iji and measurement M. success probability for an entangled pair quantum information processing quantum nondemolition. xi.

(12) QP. Q. quasiparticle quality factor. RST. reset. RTS. random telegraph signal. θ Rα R~n,ϕ R~n,ϕ Rij. qubit rotation of amplitude. θ. around axis. tomographic pre-rotation around axis as. R~n,ϕ ,. but with. ϕ = ϕ¯ + δϕ. n ¯,. α (x, y. or. rotated by. z) ϕ¯ around z. set by feedback. hP (0)P (τ )iij , |ii (|ji) qubit absolute coherence |ρ01 | P conditioned qubit coherence, C(Vint )r(Vint ) closed-loop qubit coherence r qubit coherence r with no measurement drive applied open-loop qubit coherence r qubit coherence postselected on a bin centered at Vint characteristic qubit relaxation time, 1/Γ01 charge parity autocorrelation function, for initial (nal) qubit state. r rcon rcl roff rol r(Vint ) T1 T2ϕ T2∗ T2,echo t+1 (t−1 ) Teq Tr τinit τFb τP VD. hVH i V|ki VI (VQ ) Vint VP Vth Vth+ (Vth− ) w w∞ wmm wopt xqp α αk Γrts Γij. characteristic pure dephasing time characteristic Ramsey dephasing time,. [(T2ϕ )−1 + (2T1 )−1 ]−1. characteristic echo decay time dwell time for even (odd) charge parity characteristic equilibration time refrigerator temperature initialization time time between end of measurement and end of conditional operation parity measurement duration digitization threshold average homodyne voltage average homodyne voltage for qubit in. |ki. homodyne voltage for measurement phase. φ = 0 (π/2). running integral of homodyne voltage instantaneous homodyne voltage for a two-qubit parity measurement postselection threshold voltage postselection threshold for even (odd) parity with. e (o ) = 0.01. weight function for the homodyne record theoretical optimum. w = hVI i. for innite-bandwidth detection. optimum weight function from mode-matching theory experimental optimum weight function quasiparticle density. nqp. w. normalized to the Cooper-pair density. quasiparticle-tunneling contribution to qubit relaxation, intra-cavity coherent state for qubit in. Γeo 10 /Γ10. |ki. quasiparticle-tunneling rate obtained from real-time measurement transition rate from qubit state. xii. |ii. to. |ji.

(13) 0. Γll kk0 ∆f ∆ ∆r ∆t ∆topt ∆texp E e ( o ) m p M ij (˜ η) η θ κin (κout ) κ κJPA ν ρ(0) ρi ρss ρθ σ φ ϕ ϕe |Φ+ i 2χ |Ψ+ i

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(15) ψ ωr ωRF ω01 ω12. transition rate from. |ki. to. |k 0 i. and from charge parity. l. to. l0. qubit frequency dierence between the two charge parities superconducting gap of aluminum detuning between measurement drive and cavity resonance,. 2π(fr −fm ). Ramsey interval optimal Ramsey interval. 1/4∆f. for charge-parity detection. experiment repetition time eciency of entanglement generation,. EN (ρ). parity readout error for prepared even (odd) state measurement amplitude (in. s−1 ). measurement input amplitude at the AWG (in probability of obtaining the measurement result. V) M for initial state |iji. (experimental) quantum eciency transverse rotation angle for qubit state preparation photon decay rate through the input (output) cavity port total intra-cavity photon decay rate JPA bandwidth (full-width half maximum of gain) single-spin density of states at the Fermi energy density matrix for the two-qubit maximal superposition state experimental approximation of the ideal state.

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(17) ψ. |ii. steady-state single-qubit density matrix experimental approximation of the ideal state. cos θ |0i + sin θ |1i. standard deviation of a Gaussian microwave pulse relative phase between measurement pulse and JPA pump phase of the conditional qubit pulse, dening the rotation axis. |00i + |10i). deterministic AC Stark phase between symmetric odd Bell state,. √1 (|01i 2. and. |11i. transmon-cavity dispersive shift symmetric even Bell state,. √1 (|00i 2. two-qubit maximal superposition. + |11i) state (|00i + |01i + |10i + |11i)/2. cavity resonant angular frequency measurement pulse angular frequency qubit transition angular frequency transition angular frequency between transmon states. xiii. |1i. and. |2i.

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(19) Chapter 1. INTRODUCTION. 1.1 Basic criteria of quantum computing After the birth of quantum mechanics in the early 20. th. century, the discussion of. whether it is a valid description of the physical world thrived for decades. It was only in the eighties, with the visionary ideas of R. P. Feynman. 2 tests. 1. and new experimental. that quantum eects started to be considered not just as an oddity, but as a. potential resource. In particular, Feynman realized that only by harnessing quantum mechanics we would be able to accurately simulate the physical world.. Later it. was realized that quantum mechanics would excel in solving specic problems, such as factoring large numbers. 3. 4. or searching ordered databases .. More recently, new. quantum algorithms have been devised to eciently solve systems of linear equa-. 5. 6. tions , perform linear ts , and looking further ahead, calculate the ground state of. 7,8 . molecules. Since the design of quantum algorithms escapes our classical intu-. ition, it is likely that more applications await discovery, while prototypical quantum computers are already testing the simplest ones. The elementary object which can be described by quantum eects is a system consisting of only two energy levels.. Because of its simplicity, the vast ma jority of. experimental eorts revolve around the identication and manipulation of two-level systems, or quantum bits (qubits).. The basic characteristic of a qubit is its ability. to be in a superposition state, i.e., in a linear combination of the classical states 0 and 1.. Identifying such quantum objects in a physical system is not trivial, and. constitutes the rst of the famous criteria for quantum computing formulated by D. P. DiVincenzo. 9. in 2000. Over the years, a variety of physical systems have been. proposed as candidates for quantum information processing (QIP), but only a few of them have satised the remaining criteria. A second prerequisite is the capability. 1.

(20) 1. INTRODUCTION to initialize the qubit register into a known state, before the start of computation. As we will see later, initialization at start is not sucient, as ancillary qubits in the ground state are continuously needed for repeated quantum error correction. A third requirement, the implementation of a universal set of quantum gates, involves the manipulation of multiple qubits.. Not only do these qubits need to be individually. controllable, but they also need to interact in order to perform multi-qubit gates. Controlled interaction between qubits is a pathway to another quantum phenomenon, the entanglement of multiple ob jects. In an entangled state, qubits exhibit stronger correlations than allowed by the laws of classical physics.. Whereas the individual. qubit states are a priori not dened, measuring one of them will force the other qubits to collapse into a state which is one-to-one related with the measurement result.. Superposition and entangled states are at the heart of the computational. power of QIP. The next requirement, considered a major bottleneck to the development of QIP in some systems, are qubit coherence times long enough for a reliable computation. Unlike classical bits, quantum states are highly susceptible to any unwanted interaction with the environment and the measurement system, inevitably causing loss of information and errors over the course of a computation. But what is a long enough coherence time? The answer heavily depends on the chosen path to robust quantum computing.. However small, inevitable gate and measurement errors, in addition to. decoherence, will corrupt any computation relying on distinct qubits in most physical implementations. Schemes have been devised for the so-called fault-tolerant quantum computing. 10,11 ,. where a certain number of physical qubits is used to encode a state in. a logical qubit. Such encoding allows the detection of errors in the protected state and their conditional correction. Dierent encoding schemes provide dierent thresholds on the tolerated errors. In addition to preparing, manipulating, and preserving singleand multi-qubit states, a quantum computer needs the capability of measuring the qubit register to obtain the result of a computation. The need for denite measurement results, obtained by coupling the quantum system to a classical measurement apparatus, often conicts with the goal of isolating the qubits to protect their coherence. Since DiVincenzo enumerated the criteria, several phsyical systems have developed to the point were all ve criteria are satised. Some examples are photons, trapped ions and atoms, spin or charge states in quantum dots, impurities in diamond, and Josephson-junction qubits. 12 .. So what is holding back the realization of a quantum. computer and favors one architecture over the other?. While it is true that all. basic constituents are there, the road to robust quantum computing is intricate. M. H. Devoret and R. J. Schoelkopf explicated this path in Ref. 13 for superconducting circuits, but most considerations are applicable to any physical platform. In particular, making error correction possible requires a paradigm shift in quantum control.. Qubit measurement, so far conned to the passive roles of providing the. nal results of algorithms and characterization of protocols and gates, needs to take the active function of real-time error verication.. 2.

(21) 1.2. The seven stages of development. 1.2 The seven stages of development In the roadmap laid out in Ref. 13, the DiVincenzo criteria comprise the rst two stages, those of operations and algorithms using physical qubits.. The third stage. includes the development of quantum nondemolition measurement.. While in the. criteria above measurement only serves the purpose of determining the qubit state at the end of computation, making measurement nondemolition requires leaving the qubits in a state consistent with the measurement result.. This is crucial in the. context of error correction, where in many protocols ancilla qubits are repeatedly used and measured during computation.. Furthermore, the realization of quantum. gates conditioned in real time on the measurement result completes the third QIP stage, the state of the art in all physical implementations. Measurement and conditional qubit control prepare the stage for the next steps, aimed at building self-correcting quantum circuits.. Already implementing a single. logical qubit, requisite of stage four, poses a challenge in terms of physical qubits required, whose number steeply increases with the error threshold one can aord. This encoding will be successful when the lifetime of the logical qubit will surpass that of its physical constituents, a milestone which has not been achieved in any system to date. The fth and sixth stage will replicate the features of the rst two, quantum gates and algorithms, but now at the level of logical qubits. Clearly, the scale of the circuit will increase proportionally as each physical qubit is replaced by its own section of the circuit. Finally, the seventh and last stage combines all the underlying steps to realize fault-tolerant quantum computing, which not only preserves qubit states, but is also robust to gate and measurement errors.. 1.3 Superconducting quantum circuits Solid-state architectures for quantum computing have seen a fast and steady advancement in the last decade and are now approaching the state of the art in trapped ions. 14 ,. particularly for spins in diamond. 15. and superconducting qubits. 13 .. Among the. leading technologies in QIP, superconducting circuits are considered one of the most exible platforms in terms of design and potential for scaling, stemming from decades of research and developments in nanofabrication with semi- and superconducting materials.. Nonetheless, the size of a prototypical quantum processor is similar for all. these architectures a few qubits. extension to a larger processor:. There are several challenges to face during the. the need to address each qubit individually, both. in manipulation and readout; maintaining qubit coherence times in spite of invasive hardware for control and readout; selective control of qubit-qubit interactions; and calibrating and characterizing the operation of the circuit, just to name a few. One of the most successful and widespread uses of superconducting qubits is in the circuit quantum electrodynamics (cQED) architecture. 1618 .. Here, qubits are coupled. 3.

(22) 1. INTRODUCTION to one or more microwave resonators, serving several purposes. Resonators are used to read the qubits to which they are coupled, to protect them from the continuum of the electromagnetic environment, or to mediate the interaction between them. The increased understanding of decoherence sources and improvements in circuit design over the years culminated with the demonstration of few-qubit algorithms. 1926 .. Mastering single- and two-qubit gates was demonstrated with verication protocols such as process tomography. 27,28. and randomized benchmarking. 27,29 .. In order to. assess the performance of any of these protocols, one typically needed to repeat it thousands of times, and average all the results together in order to overcome the system noise. As a result, the experimentalist could only measure the average eect of a process and from it infer the errors, correct the process, then repeat. Not only is this a slow and often manual procedure, it does not allow the correction of stochastic events such as qubit relaxation, uctuations in the qubit transition frequency, or random errors.. What is needed is the capability of the system to react quickly to. such events, rst detecting and then applying the appropriate correction in real time, conditionally on the detected error. Two ingredients are required for the transition from open-loop control, relying on average measurements and ensemble-state analysis, to closed-loop control of qubits, where the operations are conditioned in real time. First, a readout with high singleshot delity is necessary to determine the qubit state with a single measurement. In addition to providing high-delity state assignment, an ideal measurement is pro jective. 30 ,. meaning that the post-measurement state is pure and is in one-to-one relation. with the measurement result. This ensures that the input of the feedback loop reects the state of the qubit at the time of the measurement. Even a perfect match between measurement result and qubit state does not guarantee the success of a feedback loop.. The second requisite is a fast loop compared to qubit relaxation time.. Not. only does the measurement result need to be consistent with the state immediately after measurement, but this state must be preserved until the time the conditional operation is applied. The loop time is set by the speed of electronics (chapter 2). In parallel to the improvements in gate delities, qubit coherences, and increase in circuit complexity, a steady increase in single-shot readout delity. 3135. was preparing. the stage for a prominent role of measurement in quantum computing. A second event paved way for the realization of quantum feedback with the existing electronics: the extension of coherence times by two orders of magnitude achieved in the 3D cQED architecture. 36. (Fig. 1.1).. In relation to the seven stages above, cQED is currently. addressing the third stage of QND measurement and feedback, while experimental eorts are directed to the realization of a highly coherent logical qubit.. Two key. ingredients of error correction based on measurement are demonstrated for the rst time with superconducting qubits in this thesis: a parity measurement to detect error syndromes, and the feedback control required to correct such errors.. 4.

(23) 1.4. Thesis overview a. b. c. 250 µm. 200 µm. Figure 1.1 | 3D cQED architecture. a, 3D cavity of identical design as those used in the experiments of chapters 3, 4 (aluminum) and chapters 5-7 (copper). b, Typical transmon geometry, with zoom-in (c) on a Josephson junction.. 1.4 Thesis overview This thesis focuses on the development of feedback control of superconducting qubits and its rst applications to one- and two-qubit protocols. In chapter 2, we describe the implementation of a digital feedback loop, i.e., based on pro jective measurement, in a 3D cQED architecture, providing the basis for the following chapters. Chapter 3 presents the realization of the required high-delity qubit projective measurement, enhanced by Josephson parametric amplication. This readout allows us to initialize a two-qubit device by measurement, purifying it from a steady-state excitation.. In. chapter 4 we present our rst application of digital feedback for the fast reset of a transmon qubit. Here, feedback allows us to purify the initial state deterministically and to repeat experiments faster and with lower error than by passive initialization. In chapter 5, we extend the use of feedback to the two-qubit setting.. By engineer-. ing the qubit-state dependent cavity dispersive shifts, we transform the cavity into a parity meter.. Starting from an initial superposition state, we postselect on the. result of the parity measurement to generate an entangled state. Feedback on one of the qubits turns the entanglement generation from probabilistic to deterministic, obtaining the target entangled state every time. Chapter 6 introduces analog feedback as a tool to counteract measurement-induced dephasing, the limiting factor for the quality of measurement-based entanglement.. Here, we demonstrate its eectiveness. in a simpler one-qubit setting, reproducing the conditions of each parity subspace. A qubit superposition is weakly measured through the cavity, whose output signal is fed into the feedback controller to partially cancel the measurement-induced phase kickback.. Beyond the above applications in QIP, in chapter 7 we apply feedback. to measure the fundamental limits of qubit coherence imposed by quasiparticle tun-. 5.

(24) 1. INTRODUCTION neling.. By converting a single-junction transmon into a charge-parity detector, we. nd that quasiparticle tunneling only limits qubit coherence in the millisecond range. This indicates that transmon qubit coherence times can be increased by at least an order of magnitude before quasiparticle tunneling becomes the bottleneck. Finally, in chapter 8 we give an overview of the ongoing experiments enabled by the developments presented here, including the stabilization of entanglement between remote qubits, the generation of three-qubit entanglement by measurement, and the realization of the three-qubit bit-ip error correction code.. 6.

(25) Chapter 2. QUANTUM FEEDBACK WITH SUPERCONDUCTING CIRCUITS. This chapter explains the concept of digital feedback and illustrates some of its applications in QIP. We then describe our rst implementations in a solid-state system, using superconducting qubits in a 3D cQED architecture and commercial electronics.. Part of this chapter has been published in Physical. Review Letters 109, 240502 (2012). 7.

(26) 2. QUANTUM FEEDBACK WITH SUPERCONDUCTING CIRCUITS. 2.1 Digital feedback control in quantum computing Moving from proof-of-principle demonstrations of quantum gates and algorithms to fully edged quantum hardware requires closing the loop between qubit measurement and control.. There are dierent categories of quantum feedback control, depending. on the type of measurement and feedback law used.. For clarity, we rst oer a. classication of quantum feedback, similarly to that used in classical feedback. Then, we focus on the particular class of discrete-time, digital feedback, which is the most used in this thesis work.. 2.1.1 Classication of quantum feedback A rst distinction is between continuous-time and discrete-time feedback. In the rst case, measurement and control are continuous in time and concurrent. An example is the stabilization of a qubit state using continuous partial measurement. 3742 .. In. discrete-time feedback, instead, the conditional control is applied only after a measurement has been performed and processed. The implementations presented in this thesis are all discrete-time. analog and digital.. This class can be further divided into two categories,. We speak of. analog. feedback when the measurement result as-. sumes a continuum of values and the feedback law is a continuous function of the result. An example is the experiment in chapter 6, where the feedback controller rst integrates the signal produced by a weak measurement and then applies the resulting coherent operation on the qubit. If the measurement has a nite set of possible results, instead, the possible feedback actions are also nite. We refer to this as. digital. feedback. The simplest example is feedback-based qubit reset (chapter 4), in which a strong pro jective measurement collapses the qubit into either the ground or excited state.. Here, a. π. rotation brings the qubit to ground.. Another interesting example. is digital feedback using ancilla-based partial measurement. 43,44 .. In this case, the. measurement output is discrete, showing that partial measurement is not necessarily associated with analog feedback. In many applications, digital feedback is a way to force determinism into one of the most controversial aspects of quantum mechanics, namely the measurement, whose result is intrinsically probabilistic.. When we look. at the action of digital feedback as a black box, we expect to see a denite output qubit state for a given input. In an ideal feedback scheme, measurement results and the conditioned operations vary at every run of the protocol, but the overall process is deterministic and the output state is always the same. For example, one can pro ject a two-qubit superposition in a well-dened Bell state by combining a parity measurement with digital feedback (chapter 5).. 2.1.2 Protocols using digital feedback Several QIP protocols call for the use of digital feedback.. One of the requirements. for a quantum computer is ecient qubit initialization .. Often, the steady state. 9. 8.

(27) 2.1. Digital feedback control in quantum computing of a physical qubit does not correspond to a pure computational state.. Therefore,. active initialization methods have been used across dierent QIP architectures. Examples are laser or microwave initialization. 23,49 . control. 4548. and initialization by relaxation rate. An alternative method, recently used with defect centers in diamond. 50. and superconducting qubits (chapter 3), relies on pro jective measurement to initialize the qubits into a pure state.. However, measurement alone cannot produce the de-. sired state with certainty, since the measurement result is probabilistic and generally dierent every time.. Closing a feedback loop based on this measurement turns the. unwanted outcomes into the desired state. A qubit register needs to be initialized in a pure state not only at the beginning of computation, but often during the computation as well. For example, performing multiple rounds of error correction requires the ancilla qubits to be reset to the ground state after each parity check using a qubit as a detector (e.g. of. 52 charge. or photon. 53 ), parity. 51 .. When. a fast reset can be. used to increase the sampling rate without keeping track of all the past measurement records. Similarly, in the multi-qubit setting, digital feedback is the key element turning measurement-based protocols from probabilistic to deterministic. An example is the generation of entanglement by parity measurement. 54 .. A parity measurement projects. an initial superposition state into an entangled state with a well-dened parity, i.e., with either even or odd number of qubit excitations (chapter 5). However, once again, the result of the parity measurement is random. After running the protocol open-loop multiple times, the average nal state has no specic parity and is unentangled. Only by forcing a denite parity with feedback can one generate a target entangled state deterministically. A variation of closed-loop control, named feedforward, decouples the measured qubits from those acted on by the feedback.. Feedforward schemes have already. found application in quantum communication, with the main ob jective of transmitting quantum information securely at a distance.. In quantum teleportation, a measure-. ment on the Bell basis of two qubits projects a third qubit, at any distance, into the state of the rst, apart from a single-qubit rotation. 10 .. The measurement result de-. termines which qubit rotation, if any, needs to be applied by the feedback controller in order to teleport the original state. An extension of teleportation is entanglement swapping. 10 .. This protocol transfers entanglement to two qubits which never interac-. ted with each other. This idea is at the basis of quantum repeaters. 55 ,. which aim to. distribute entanglement to larger distances than allowed by a lossy communication channel. Here, measurement and feedback are used in every step to rst purify. 56. and. then deterministically transfer entangled pairs to progressively farther nodes. In quantum computing, feedforward operations are at the basis of an outstanding goal in the eld, which is preserving a logical qubit for a time longer than the coherence time of the constituent physical qubits. The simplest protocol is the bit-ip code. 57 ,. which encodes a quantum state into three qubits, and uses measurement of two-qubit. 9.

(28) 2. QUANTUM FEEDBACK WITH SUPERCONDUCTING CIRCUITS operators (syndromes) in combination with feedback to correct for a similar structure is the phase-ip code, which protects only from. σz. σx. error.. Of. rotations.. To. obtain protection from rotations along any axis, the minimum size of the encoding is ve qubits.. In this application, pro jective measurement is more than a tool to. detect which of these errors has already occurred.. In fact, an arbitrary error is not. just a single rotation along any of these axes. Nevertheless, any process on a single qubit can be decomposed into a combination of Pauli rotations. Measuring the error syndromes forces one and only one of these errors to happen. This greatly simplies the feedback step, which is now restricted to a nite set of correcting rotations. While few-qubit error correction schemes are in principle capable of correcting for any error, they impose experimentally inaccessible measurement and gate delities. A more realistic approach is oered by topologically protected circuits such as surface codes. 11 ,. where errors as high as. physical qubits required. 58 .. 1% are tolerated at the expense of a larger number of. One cycle in a surface code, aimed at maintaining a logical. state encoded in a square lattice of qubits, includes the projective measurements of 4-qubit operators as error syndromes. When an error is a detected on a data qubit, the corrective, coherent feedback operation is replaced by a change of sign in the operators for the following measurements involving that qubit. Beyond protecting a state from external perturbations, performing fault-tolerant quantum computing will require robustness to gate errors.. In surface codes, single- and two-qubit gates on. logical qubits are also based on pro jective measurements and in some cases require feedback to apply conditional rotations.. 9. In addition to the gate model , digital feedback is central to the paradigm of measurement-based quantum computing. 59 .. In this approach, also called one-way. computation, the initial state is an entangled state of a large number of qubits. All logical operations are performed by means of projective measurements. To make computation deterministic, at each computational step feedback selects the measurement bases, conditional on the results of the previous measurements.. 2.1.3 Experimental realizations of digital feedback Experimentally, digital feedback has been employed for entanglement swapping with trapped ions atomic. 64,65. 60. and for the unconditional teleportation of photonic. 61 ,. ionic. 62,63 ,. and. qubits. In linear optics, feedforward has been used to implement segments. of one-way quantum computing. 6671. and for photon multiplexing. 72 .. In the solid. state, the rst approach to feedback, of the analog type, was used to stabilize Rabi oscillations of a superconducting qubit indenitely. 42 .. Soon after, digital feedback. with high-delity projective measurement was introduced in the solid state, also using superconducting circuits. 73,74 .. Recently, digital feedback has been extended to multi-. qubit protocols with superconducting qubits. 10. 75,76. and NV centers in diamond. 77 ..

(29) 2.2. Implementation of digital feedback in 3D cQED |0Ú. (a). (b). H. H UH UL. |yÚ L. |0Ú. |0Ú. |yÚ. H. T1. |1Ú. |0Ú. |0Ú. |yÚ. X. L. |1Ú. (c). |1Ú. |0Ú. X. L. |0Ú. |1Ú. Figure 2.1 | Concept of a single-qubit digital feedback loop and possible errors. a, The measurement is digitized into either H or L for qubit declared in |0i or |1i. A. dierent unitary rotation is applied for each result. Errors occurring in case of qubit relaxation between measurement and action (b) or wrong measurement assignment (c).. 2.2 Implementation of digital feedback in 3D cQED Here we illustrate the basic components of a feedback loop and its working principle. We then detail the rst implementation of digital feedback with a solid-state qubit, using a 3D cQED architecture. 36 ,. and the following improvements in speed and delity.. 2.2.1 Concepts in digital feedback The basic ingredients for a digital feedback loop are: 1) pro jective qubit readout and 2) control conditional on the measurement result (see Fig. 2.1a for the simplest singlequbit loop). The main challenge for (1) is to obtain a high-delity readout which is also nondemolition, i.e., leaving the qubits in a state consistent with the measurement result. A mismatch between measurement result and post-measurement qubit state will trigger the wrong feedback action (Fig. 2.1b).. The requirement for (2) is to. minimize the time between measurement and feedback action. This is often referred to as the feedback loop latency, which is a combination of dierent sources: the time for the signal to travel from the sample to the feedback controller, the time for the feedback controller to process the signal and digitize it into one of discrete results, and the delay to the execution of the conditional qubit gates. If a transition between levels occurs in one of the measured qubits during this interval, for instance because of spontaneous relaxation, its state becomes inconsistent with the chosen feedback action, resulting in the wrong nal state (Fig. 2.1c). In feedforward protocols, such as error-correction or teleportation, the feedback action is applied to data qubits, which are dierent from the measured ancilla qubits. In this case, the loop also needs to be fast compared to the data qubit coherence times. The simplest example of digital feedback consists of a single qubit and two possible measurement results, ideally corresponding to the qubit pro jection in (Fig. 2.1 and chapter 3).. |0i. or. |1i. Here we consider the eect of these errors in this case,. modeling the qubit as a classical three-level system, where the third level includes the possibility of transitions out of the qubit subspace.. This is relevant in the case. 11.

(30) 2. QUANTUM FEEDBACK WITH SUPERCONDUCTING CIRCUITS of transmon qubits with a sizeable steady-state excitation. 73,74 .. pM ij. with initial state. the probability of obtaining the measurement result. post-measurement state. |ji,. to. and with. τFb. With. Γij τFb. We indicate with. |ii. and. we indicate the transition rates from. |ii. the time between the end of measurement and the end of the. conditional operation.. cos(θ) |0i + sin(θ) |1i. |ji.. M. For perfect pulses, the combined errors. θ Perr. for initial state. are, to rst order:. θ=0 H Perr = pL 00 + p01 + Γ01 τFb ,. (2.1). θ=π L Perr = pH 11 + p10 + p12 + (Γ10 + Γ12 )τFb , and weighted combinations thereof for other. θ.. A simple way to improve feedback. θ = 0 remains θ=0 + p12 + Γ12 τFb . The second cycle comPerr pensates errors arising from relaxation to |0i between measurement and pulse in the rst cycle. However, it does not correct for excitation from |1i to |2i. For this reason, delity is to concatenate two cycles.. unchanged, for. θ =π. While the dominant error for. it decreases to. adding more cycles does not signicantly reduce the error, unless the population in. |2i. is brought back to the qubit subspace.. deterministic. This can be done. 73,74. for example by a. π pulse returning the population from |2i to |1i, or with a more complex. feedback loop capable of resolving and manipulating all three states.. 2.2.2 Closing the loop in cQED Until recently, the available qubit coherence times of superconducting qubits bottlenecked both achievable readout delity and required feedback speed. control, the development of circuit quantum electrodynamics (3D cQED) ence times. 36. 16,78. For feedback. with 3D cavities. constitutes a watershed. The new order of magnitude in qubit coher-. (> 10 µs),. combined with Josephson parametric amplication. pro jective-readout delities up to. 99% and. 79,80 ,. allows. feedback realizable with o-the-shelf elec-. tronics. In chapter 3 we detail our implementation of high-delity projective readout of a transmon qubit in 3D cQED. Here, we focus on the processing of the signal in real time by the feedback controller, and on the resulting feedback action. The input to a feedback loop in cQED is the homodyne signal obtained by amplication and demodulation of the qubit-dependent cavity transmission or reection (chapter 3). The response of the feedback controller is one or more qubit microwave pulses, which are generated and sent to the device (Fig. 2.2). This loop has a signicant spatial extension, as the qubits sit in the coldest part of a dilution refrigerator, while the feedback controller is at room temperature.. A full round trip, from the. measurement signal exiting the cavity to the conditional microwave pulses entering, takes. 5 − 10 m. of cable, which translates into a delay of. 25 − 50 ns,. without account-. ing for delays due to lters or other microwave components. This physical limitation, which would require fast cryogenic electronics to be overcome, is only a small fraction of the total latency. A ma jor source of delay is the processing time in the controller,. 12.

(31) 2.2. Implementation of digital feedback in 3D cQED Vout. Controller. wc. Vthr. Cavity. ). q' q. wq. Qubit. (. Amplification. Figure 2.2 | Simplied schematics of a single-qubit feedback loop in cQED. Upon application of a measurement tone at ωc , the signal Vout obtained from processing of the cavity output, carrying information on the qubit state, is input to the feedback controller and compared to a preset threshold Vth . If Vout > Vth (or Vth < Vth ), the conditional rotation θ (θ0 ) is applied to the qubit. combined with the generation or triggering of the microwave pulses for the conditional qubit rotations.. The details of this process depend on the type of controller.. We describe the rst implementations below.. 2.2.3 Feedback controllers The rst realization of a digital feedback controller used commercial components for data sampling, processing, and conditional operations. 76 .. The core of the controller is. an ADwin-Gold, a processor with a set of analog inputs and congurable analog and digital outputs. The ADwin samples the readout signal once, at a set delay following a trigger from an arbitrary waveform generator (Tektronix AWG5014). This delay is optimized to maximize readout delity. A routine determines the optimum threshold for digitizing the readout signal. the measurement.. This voltage is then used to assign. arbitrary waveform generator (Tektronix AWG520) to produce a outcome is. L.. H. or. L. to. For the reset function in chapter 4, the ADwin triggers another. π. pulse when the. Pulse timings and signal delays in the feedback cycle are illustrated in. Fig. 2.3. The total time between start of the measurement and end of the feedback pulse is. ≈ 2.6 µs,. mainly limited by the processing time of the ADwin.. To cut the loop time, our second generation of digital feedback used a complex programmable logic device (CPLD, Altera MAX V), acquiring the signal following a. 8-bit. ADC, in place of the ADwin.. This home-assembled feedback controller has. the major advantages of a programmable integration window and a response time of. 0.11 µs. (Fig. 2.4), an order of magnitude faster than the ADwin.. As the feedback. response time is now comparable or faster than the typical cavity decay time, active depletion of the cavity. 81. will be required to take full advantage of the CPLD speed. and further shorten the feedback loop. Further developments in the feedback controller replaced the CPLD with a eld-. 13.

(32) 2. QUANTUM FEEDBACK WITH SUPERCONDUCTING CIRCUITS. a. ADwin-Gold. Tektronix AWG520. trigger low-pass filter. I MW sources. fridge out. Q. fridge in. b cavity population. AWG5014 ADwin. readout homodyne signal acquisition. trigger. p. AWG520. Figure 2.3 | Digital feedback loop with an ADwin controller. a, Schematics of the feedback loop, consisting of an ADwin, sampling the signal, and a Textronix AWG520, conditionally generating a qubit π pulse. b, Timings of the feedback loop. The measurement pulse, here 400 ns long, reaches the cavity at t = 0. The ADwin, triggered by an AWG5014, measures one channel of the output homodyne signal (red: qubit in |0i, blue: |1i), delayed by ∼ 200 ns due to a low-pass lter at its input side (Fig. 4.5). After comparison of the measured voltage at t = 0.6 µs to the reference threshold, the AWG520 is conditionally triggered at t = 2.54 µs, resulting in a π pulse reaching the cavity at 2.62 µs. Figure adapted from Ref. 73. 14.

(33) 2.2. Implementation of digital feedback in 3D cQED. a. Tektronix AWG520. ADC. trigger. CPLD. Q. MW sources. fridge out. b. I. fridge in. cavity population. AWG5014. integration window. homodyne signal. CPLD. earliest trigger. readout trigger. p. integration. 0. 0.1. running integral 0.2. Relative time (ȝs). 0.3. AWG520. Figure 2.4 | Digital feedback loop with a CPLD-based controller. a, Schematics of the feedback loop, with an ADC and a CPLD (or FPGA) board replacing the ADwin in Fig. 2.3. b, Timings of the feedback loop. The CPLD samples the signal at every clock cycle (10 ns) and then integrates it over a window set by a marker of an AWG5014. The internal delay of the CPLD breaks down into the analog-to-digital conversion (60 ns) and the processing to compare the integrated signal to a calibrated threshold, determining the binary output (50 ns). These timings are multiples of the clock (reduced to 4 ns in a recent FPGA-based implementation ). The total delay in chapter 6 is increased to 2 µs to let the cavity return to the ground state before the conditional π pulse. 82. 15.

(34) 2. QUANTUM FEEDBACK WITH SUPERCONDUCTING CIRCUITS a. b. c. Figure 2.5 | Hardware comparison for feedback control in the bit-ip code. As summarized in section 8.3.2, the bit-ip code requires a two-bit digital feedback, acting on three qubits. Scaling the system in Fig. 2.3 would take an AWG520 for each qubit (a). A recent implementation performs signal processing and pulse generation on FPGA boards, resulting in the compact controller shown in b, c. 82. programmable-gate-array (FPGA, chapter 5) to increase the on-board memory and enable more complex signal processing.. For example,. it allows dierent weights. for the measurement record and maximal correlation with the qubit evolution (see chapter 6).. A FPGA-based controller has also been employed for digital feedback. at ETH Zurich. 75 .. Current developments in our group and at Yale. 83. include the. pulse generation on a FPGA board, eliminating the need of an external AWG and its latency. For comparison, Fig. 2.5 shows the setup that would be required for the 3-qubit repetition code (section 8.3.2) using our rst generation of feedback (a) and the most recent one based on FPGAs (b,. 16. c)..

(35) Chapter 3. INITIALIZATION BY MEASUREMENT OF A TWO-QUBIT SUPERCONDUCTING CIRCUIT. We demonstrate initialization by joint measurement of two transmon qubits in 3D cQED. Homodyne detection of cavity transmission is enhanced by Josephson parametric amplication to discriminate the two-qubit ground state from single-qubit. 98.1% delity. Measurement and postselec4.7% residual excitation per qubit achieve 98.8%. excitations non-destructively and with tion of a steady-state mixture with. delity to the ground state, thus outperforming passive initialization.. This chapter has been published with minor dierences in Physical Review Letters 109, 050507 (2012).. 17.

(36) 3. INITIALIZATION BY MEASUREMENT OF A TWO-QUBIT SUPERCONDUCTING CIRCUIT. 3.1 Qubit state initialization The abilities to initialize, coherently control and measure a multi-qubit register set the overall eciency of a quantum algorithm. 10 .. In systems where qubit transition. energies signicantly exceed the thermal energy, initialization into the ground state can be achieved by waiting several multiples of the qubit relaxation time. T1 12 .. While. this passive method has been standard in superconducting qubit systems, the recent breakthrough to light. longer.. T1. improvements. 36. in cQED. 16,78. have brought its many shortcomings. First, the wait time between computations have become proportionately. Second, commonly observed. 84,85. residual qubit excitations can produce ini-. tialization errors exceeding the lowest single- and two-qubit gate errors now achieved. < 0.1%. (. and. < 1% 86 ,. respectively).. quantum error correction pared to. 51. Third, moving forward, multiple rounds of. will require re-initialization of ancilla qubits fast com-. T1 .. An ecient method for active initialization is to use a high-delity, quantum nondemolition (QND) readout. 87. to collapse qubits into known states.. In cQED, sig-. nicant progress in QND readout has been achieved using bifurcation in nonlinear resonators. 33. and parametric amplication. this chapter and Ref. 88,. T1. 34,35 .. However, until the experiments in. has limited the best qubit readout delity to. 86%.. 3.2 High-delity readout using a Josephson parametric amplier In this chapter, we demonstrate ground-state initialization of two superconducting qubits by joint measurement and postselection. qubits in a 3D cQED architecture. 36. to realize a high-delity, nondemolition readout. transmission at. ∼ 10. We combine long-lived transmon. with phase-sensitive parametric amplication. 79,80. Homodyne measurement of cavity. intra-cavity photons discriminates the two-qubit ground state. from single-qubit excitations with. 98.1±0.3% delity (limited by T1 ) and up to 99.6%. correlation between the measurement result and the post-measurement state. We use this readout to purify the two-qubit system against a residual excitation of per qubit, achieving probabilistic ground-state preparation with. 98.8%. ∼ 4.7%. delity.. 3.2.1 Experimental setup Our system consists of an Al 3D cavity enclosing two superconducting transmon qubits, labeled. QA and QB , with transition frequencies ωA(B) /2π = 5.606 (5.327) GHz, T1A(B) = 23 (27) µs, and Ramsey dephasing times. relaxation times. 18.

(37) 3.3. Characterization of JPA-backed qubit readout and initialization ∗ T2A(B) = 0.45 (4.2) µsi . The fundamental mode of the cavity (TE101) resonates at ωr /2π = 6.548 GHz (for qubits in ground state) with κ/2π = 430 kHz linewidth, 16 and couples with g/2π ∼ 75 MHz to both qubits. The measured dispersive shifts 2χA(B) /2π = −3.7 (−2.6) MHz place the system in the strong dispersive regime of cQED. 91 .. Qubit readout in cQED typically exploits dispersive interaction with the cavity. A readout pulse is applied at or near resonance with the cavity, and a coherent state builds up in the cavity with amplitude and phase encoding the multi-qubit state. 16,92 .. We optimize readout of QA by injecting a microwave pulse through the cavity at ωRF = ωr − χA , the average of the resonance frequencies corresponding to qubits in |00i and |01i, with left (right) index denoting the state of QB (QA ) (Figs. 3.1a,d). This choice maximizes the phase dierence between the pointer coherent states. Homodyne detection of the output signal, itself proportional to the intra-cavity state, is overwhelmed by the noise added by the semiconductor amplier (HEMT), precluding high-delity single-shot readout (Fig. 3.1c). We introduce a Josephson parametric amplier (JPA). 79. at the front end of the amplication chain to boost the. readout signal by exploiting the power-dependent phase of reection at the JPA (see Figs. 3.1a,b). Depending on the qubit state, the weak signal transmitted through the cavity is either added to or subtracted from a much stronger pump tone incident on the JPA, allowing single-shot discrimination between the two cases (Fig. 3.1c).. 3.3 Characterization of JPA-backed qubit readout and initialization The ability to better discern the qubit states with the JPA-backed readout is quantied by collecting statistics of single-shot measurements. benchmark the readout includes two measurement pulses, long, with a central integration window of. MB ,. a. π. pulse is applied to. QA. 300 ns. The sequence used to. MA. and. (Fig. 3.2a).. MB ,. each. 700 ns. Immediately before. in half of the cases, inverting the population of. ground and excited state (Fig. 3.2b). We observe a dominant peak for each prepared state, accompanied by a smaller one overlapping with the main peak of the other case.. We hypothesize that the main peak centered at positive voltage corresponds. to state. |00i,. and that the smaller peaks are due to residual qubit excitations, mix-. ing the two distributions. To test this hypothesis, we rst digitize the result of with a threshold voltage. Vth ,. MA. chosen to maximize the contrast between the cumulat-. ive histograms for the two prepared states (Fig. 3.2c), and assign the value. H(L) to MB. the shots falling above (below) the threshold. Then we only keep the results of corresponding to. MA = H .. Indeed, we observe that postselecting. 91%. of the shots. i Q is a A 1 GHz below. double-junction qubit with a random, non-tunable magnetic ux oset placing it ∼ from its ux sweet spot, limiting its T2∗ 89 . The relatively short T2∗ of QB compared to Ref. 36 most likely results from qubit-frequency sensitivity to charge noise 89 and to photon number uctuations arising from residual cavity excitation 90 .. 19.

(38) 3. INITIALIZATION BY MEASUREMENT OF A TWO-QUBIT SUPERCONDUCTING CIRCUIT a. RF. (. ). transmons. c. Null. |00Ú Pump+RF. |01Ú Pump+RF HEMT Pump. b. d. 50 W 20 dB. RF, |01Ú. RF,|00Ú. |01ÚRF|00Ú. 163°. average. eij JPA. Figure 3.1 | JPA-backed dispersive transmon readout. a, Simplied diagram of the experimental setup, showing the input path for the readout signal carrying the information on the qubit state (RF, green) and the stronger, degenerate tone (Pump, grey) biasing the JPA. Both microwave tones are combined at the JPA and their sum is reected with a phase dependent on the total power (b), amplifying the small signal. An additional tone (Null) is used to cancel any pump leakage into the cavity. The JPA is operated at the low-signal gain of ∼ 25 dB and 2 MHz bandwidth. c, Scatter plot in the I − Q plane for sets of 500 single-shot measurements. Light red and blue: readout signal obtained with an RF tone probing the cavity for qubits in |00i and |01i, respectively. Dark red and blue: the Pump tone is added to the RF. d, Spectroscopy of the cavity fundamental mode for qubits in |00i and |01i. The RF frequency is chosen halfway between the two resonance peaks, giving the maximum phase contrast (163◦ , see inset on the right). reduces the overlaps from. ∼6. to. 2%. and from. ∼9. to. 1%. in the. H. and. L. regions, re-. spectively (Fig. 3.2d). This substantiates the hypothesis of partial qubit excitation in the steady state, lifted by restricting to a subset of measurements where the register to be in. |00i.. MA. declares. Further evidence is obtained by observing that moving the. threshold substantially decreases the fraction of postselected measurements without. ∼ +0.1 (0.2)%. signicantly improving the contrast [. keeping. 85 (13)%. of the shots]. (Fig. 3.3b). Postselection is eective in suppressing the residual excitation of any of the two qubits, since the. |00i,. |01i. and. |10i. distributions are both highly separated from. and the probability that both qubits are excited is only. ∼ 0.2%. .. The performance of the JPA-backed readout and the eect of initialization by measurement are quantied by the optimum readout contrast. This contrast is dened as the maximum dierence between the cumulative probabilities for the two prepared. 20.

(39) 3.3. Characterization of JPA-backed qubit readout and initialization b. a M0. RA(q) 0,p. M1,unconditioned. q=0 q=p. M1. M0=H. M0=L. c. d. M1,conditioned on M0=H. q=0 q=p. Vth. Figure 3.2 | Ground-state initialization by measurement. a, Pulse sequence used to distinguish between the qubit states (MB ), upon conditioning on the result of an initialization measurement MA . The sequence is repeated every 250 µs. b, Histograms of 500 000 shots of MB , without (red) and with (blue) inverting the population of QA with a π pulse. c, Histograms of MA , with Vth indicating the threshold voltage used to digitize the result. d, MB conditioned on MA = H to initialize the system in the ground state, suppressing the residual steady-state excitation. The conditioning threshold, selecting 91% of the shots, matches the value for optimum discrimination of the state of QA . states (Fig. 3.3a). contrast of. Without initialization, the use of the JPA gives an optimum. 84.9%, a signicant improvement over the 26% obtained without the pump. tone. Comparing the deviations from unity contrast without and with initialization, we can extract the parameters for the error model shown in Fig. 3.3c.. The model. (section 3.5.3) takes into account the residual steady-state excitation of both qubits,. ∼ 4.7% each, and the error probabilities for the qubits prepared in the four |00i occurs with 99.8 ± 0.1% delity, this probability is reduced to 98.8% in the time τ = 2.4 µs between MA and MB , chosen to fully deplete the cavity of photons before the π pulse preceding MB . We note that τ could be reduced by increasing κ by at least a factor of two without compromising T1A 93 . By correcting for partial equlibration during τ , we calculate by the Purcell eect an actual readout delity of 98.1±0.3%. The remaining indelity is mainly attributed. found to be. basis states. Although the pro jection into. 21.

(40) 3. INITIALIZATION BY MEASUREMENT OF A TWO-QUBIT SUPERCONDUCTING CIRCUIT to qubit relaxation during the integration window. As a test for readout delity, we performed single-shot measurements of a Rabi oscillation sequence applied to sian pulse preceding. MB ,. QA , with variable amplitude of a resonant 32 ns Gaus-. and using ground-state initialization as described above. (Fig. 3.3d). The density of discrete dots reects the probability of measuring depending on the prepared state.. By averaging over. ∼ 10 000. H. or. L. shots, we recover the. sinusoidal Rabi oscillations without (white) and with (black) ground-state initialization. As expected, the peak-to-peak amplitudes (. 85.2. and. 96.7%,. respectively) equal. the optimum readout contrasts in Fig. 3.3a, within statistical error.. 3.3.1 Repeated quantum nondemolition measurements In an ideal pro jective measurement, there is a one-to-one relation between the outcome and the post-measurement state. We perform repeated measurements to assess the QND nature of the readout, following Refs. two consecutive measurements,. MB. and. MC ,. 31,32.. The correlation between. is found to be independent of the ini-. tial state over a large range of Rabi rotation angles. θ. (see Fig. 3.4a).. A decrease. in the probabilities occurs when the chance to obtain a certain outcome on low (for instance to measure. MB = H. for a state close to. readout errors or to the partial recovery arising between. MB. |01i) and. MB. is. and comparable to. MC .. We extend the. readout model of Fig. 3.3c to include the correlations between each outcome on. MB. and the post-measurement state. The deviation of the asymptotic levels from unity,. PH|H = 0.99 and PL|L = 0.89, is largely due to recovery during τ , as demonstrated in Fig. 3.4b. From the model, we extrapolate the correlations for two adjacent measurements,. PH|H (τ = 0) = 0.996 ± 0.001. and. PL|L (τ = 0) = 0.985 ± 0.002,. corresponding. to the probabilities that pre- and post-measurement state coincide. In the latter case, mismatches between the two outcomes are mainly due to qubit relaxation during. MC .. Multiple measurement pulses, as well as a long pulse, do not have a signicant eect on the qubit state , supporting the QND character of the readout at the chosen power.. 3.4 Conclusion We have demonstrated the simultaneous projection by measurement of two qubits into the ground state.. This technique allows us to correct for residual single-qubit. excitations, preparing the register in. |00i. with. 98.8%. probability.. Initialization will. be imperfect when the population of the doubly-excited state is relevant, a problem that can be addressed by choosing a dierent conguration of the joint readout, fully discriminating one of the computational states from the other three. The logical extension of this work is to use the knowledge gained by projection to condition further coherent operations on or more qubits In the example presented in the following. 22.

(41) 3.4. Conclusion. a. b 96.6%. c. P|00Ú,ss P|01Ú,ss P|10Ú,ss P|11Ú,ss. d. M0 t. 0.9p p. 1.1p. P|00Ú,t P|01Ú,t P|10Ú,t P|11Ú,t eH 01 eH 01 H e 11 e0 L 0. H. L -0.1p 0. -p. 0.1p. p. Figure 3.3 |Analysis of readout delity. a, Cumulative histograms for MB without and with conditioning on MA = H , obtained from data in Figs. 3.2c,d . The optimum. threshold maximizing the contrast between the two prepared states is the same in both cases. Deviations of the outcome from the intended prepared state are: 8.9% (1.3%) for the ground state, 6.2% (2.1%) for the excited state without (with) conditioning. Therefore, initialization by measurement and postselection increases the readout contrast from 84.9% to 96.6%. b, Readout contrast (purple) and postselected fraction (black) as a function of Vth . c, Schematics of the readout error model, including the qubit populations in the steady state and at τ = 2.4 µs after MA . Only the arrows corresponding to readout errors are shown. d, Rabi oscillations of QA without (empty) and with (full dots) initialization by measurement and postselection. In each case, data are taken by rst digitizing 10 000 single shots of MB into H or L, then averaging the results. Error bars on the average values are estimated from a subset of 175 measurements per point. For each angle, 7 randomly-chosen single-shot outcomes are also plotted (black dots at 0 or 1). The visibility of the averaged signal increases upon conditioning MB on MA = H .. 23.

(42) 3. INITIALIZATION BY MEASUREMENT OF A TWO-QUBIT SUPERCONDUCTING CIRCUIT. M2 M0. q. M1. H t L. a. b. |01Ú. Figure 3.4 | Projectiveness of the measurement. a, Conditional probabilities for two consecutive measurements MB and MC , separated by τ = 2.4 µs. Following an initial measurement pulse MA used for initialization into |00i by the method described, a Rabi pulse with variable amplitude rotates QA by an angle θ along the x-axis of the Bloch sphere, preparing a state with P|01i = sin2 (θ/2). Red (blue): probability to measure MC = H(L) conditioned on having obtained the same result in MB , as a function of the initial excitation of QA . Error bars are the standard error obtained from 40 repetitions of the experiment, each one having a minimum of 250 postselected shots per point. Deviations from an ideal projective measurement are due to the nite readout delity, and to partial recovery after MB . The latter eect is shown in b, where the conditional probabilities converge to the unconditioned values, PH = 0.91 and PL = 0.09 for τ T1 , in agreement with Fig. 3.2, taking into account relaxation between the π pulse and MC . Error bars are smaller than the dot size. chapter, measuring of a qubit and applying a. π. pulse conditioned on having pro jected. onto the excited state deterministically prepares the ground state.. 3.4.1 Further developments Josephson parametric amplication has become a standard technique for the highdelity readout of qubits in cQED. Since this experiment, pro jective readout of transmon. 75,94,95. and ux. 96. qubits has been obtained using dierent varieties of Josephson. junction-based ampliers. The technology for these ampliers is now evolving to accommodate the needs of growing quantum circuits. One approach is to increase the amplier bandwidth to read out multiple qubits, each coupled to a dedicated resonator. 44 .. Recent implementations in this direction included Josephson junctions in a. transmission line. 97 ,. in low-Q resonators. 98,99 ,. or in a circuit realizing a superconduct-. ing low-inductance undulatory galvanometer (SLUG). 100 .. Another line of research. aims at increasing quantum eciency by minimizing the losses between cavity and amplier, now dominated by the microwave components required to suppress the. 24.

(43) 3.5. Methods pump leakage. lators. 101 ,. This can been done, for instance, by improving the design of circu-. or by removing the need for circulators entirely operating the amplier in. transmission. 97,102 .. The method of postselection on measurement has also been used. 44,103. in cQED to detect weak values. and for the study of quantum trajectories. 104106 .. 3.5 Methods 3.5.1 Device fabrication The qubits are patterned on a sapphire substrate (C-plane,. 430 µm. thickness, single-. side polished) by electron-beam lithography and Al double-angle evaporation with. 0.4 mBar). The qubit design is very similar to ∼ 300 × 300 nm2 and 250 × 500 µm2 antennas. Qubit QA is a double-junction transmon, and qubit QB is of the singlejunction type. The 3D cavity is machined from Al alloy 6082 (AlSi1MgMn) in two 2 halves with internal cross section 35.4 × 10.4 mm and total height 24.3 mm. intermediate oxidation (. O2 ,. 10 min,. that pioneered in Ref. 36, with junction sizes. 3.5.2 Conguration of the JPA for qubit readout The resonance frequency of the JPA, tunable from. 25 µm2. biasing its 20 SQUID loops (. IC = 5 µA) 79. 4.8. to. 6.8 GHz,. is set by ux. ux-threading area in each, critical currents. with a home-built superconducting coil driven by a voltage-controlled. current source. The bias point is actively stabilized with an ADwin-GOLD processor running a proportional-integral feedback loop.. This active control of the bias point. is eective because the duty cycle of measurement, which shifts the output level (Figs. 3.6 and 3.1c), is only. 0.3%. < 0.1 quanta), with ωRF /2π = 6.5446 GHz. The JPA is operated in the phase-sensitive mode (added noise a continuous wave (CW) pump tone at the readout frequency,. −94 dBm) bends the JPA resonance lineshape 6.564 GHz down to ωRF , making the phase of the reected signal highly sensitive. (Figs. 3.5 and 3.1c). The pump power ( from. to variations in the incident power (Fig. 3.1b). The frequency shift due to the pump tone determines the tradeo between JPA gain and bandwidth, their product being. 25 dB) is chosen to fully resolve the coherent states corresponding |01i in 300 ns. In this conguration, the JPA bandwidth is 2 MHz, and does not limit the measurement rate, which is set by κ. A phase shifter in constant. The gain (. to qubits in. |00i. and. the output line is used to maximize the sensitivity of the in-phase (I) quadrature after demodulation. The relative phase between the measurement pulse and the pump is. |00i ∼ 5 min to cancel any phase drift between. adjusted to maximize the contrast between the histograms for qubits prepared in and. |01i.. This calibration is repeated every. the two generators.. An additional CW tone (null), split from the pump generator,. is injected to the cavity via the output port. This tone is used to cancel the leakage from the pump into the cavity arising from the limited isolation (. ∼ 40 dB. in total). 25.

(44) 3. INITIALIZATION BY MEASUREMENT OF A TWO-QUBIT SUPERCONDUCTING CIRCUIT. Agilent E8257D. R&S Agilent Agilent SMB100 E8257D E8257D. Tektronix AWG5014. PM. trigger. S 1 2. S. 1. 2.  I. Q. I. Q. 1 2 S. drive QB. 1 S 2. 20 dB. 20 dB. 30 dB. 30 dB. transmons. (. ). null. AlazarTech ATS9870. I. . RF drive QA. Q. SRS 445A. Miteq AFS3 (x2). 30 dB. Caltech. 30 dB. HEMT. pump. Bias DAC ADwin. 30 dB. 300 K 3K 20 dB. 15 mK 10 dB 50 . 20 dB. coil. JPA JPA. Figure 3.5 | Detailed schematic of the experimental setup. Complete wiring of electronic components outside and inside the 3 He/4 He dilution refrigerator (Leiden Cryogenics CF-650). The measurement tone (RF) and qubit drives enter the cavity through the same transmission line. The RF is combined with a stronger pump tone (∼ +26 dB) at the JPA, which is ux-biased with an external superconducting coil. The pump signal is split into two arms, one being directed to the cavity through a circulator, in order to suppress the photon leakage from the pump port. The estimated signal loss from the cavity to the JPA is < 1dB. After being added to the pump, the readout signal is amplied at 3 K (Caltech Cryo1-12, 0.06 dB noise gure) and at room temperature (two Miteq ampliers, 4 − 8 GHz, 0.8 dB and 2 dB noise gure at 6.5 GHz). It is then demodulated (0 Hz intermediate frequency) and re-amplied (SRS445A, 25 V/V gain) before being digitized by an AlazarTech ATS9870 (1 GS/s, 8 bits). The ouptut signal is also fed to an ADwin-GOLD processor, running a proportional-integral control loop which actively stabilizes the JPA bias point.. 26.

(45) 3.5. Methods. Figure 3.6 | Small-signal gain of the JPA at the bias point. The peak gain is 25 dB and the full width at half maximum is κJPA /2π = 2 MHz.. of the two circulators between JPA and cavity.. Cancellation is achieved by tuning. the amplitude and phase of the null tone to suppress the residual one-photon peak visible in qubit spectroscopy. 91 .. The nulling is crucial to realize high-delity single-. qubit pulses by avoiding Stark shifts. continuously measuring the qubits.. ∼ 10. to. Nulling also prevents leakage photons from. The chosen power of the RF pulse corresponds. intra-cavity photons in the steady state, calibrated by qubit spectroscopy. (not shown), and is the minimum power that achieves a complete separation of the histograms at the chosen JPA bias point.. 3.5.3 Readout error model We rst characterize the readout errors by considering the probabilities of measuring. H. or. L. for the four input states.. As shown in Figs. 3.2 and 3.3, the measured. deviations from unity contrast are due to the combination of readout errors and steady-state excitations.. From Fig. 3.3. a. we extract four values at the optimum. readout threshold, one per prepared state, with and without initialization. We denote. M ij. M (H or L) for the pre|iji, and by P|iji,ss and P|iji,τ the populations of the qubits in |iji in the steady state and at τ = 2.4 µs following initialization, respectively. By using M the measured T1A,B and taking P|iji,ss and ij as free parameters, we calculate the two-qubit populations immediately before MB , with or without initialization, and by. the probability of obtaining the measurement result. measurement state. the probabilities for each measurement outcome.. Good agreement of the simulated. sequence with the data is obtained with the values in the top half of Table 3.1, also supported by the experiment in Fig. 3.4a (see below). The readout process is further analyzed by considering the probabilities of measuring. H. L for the four input states, and for all the possible post-measurement states H For example, p01,00 indicates the probability that measuring |01i gives result H and the post-measurement state is |00i. Again, we assume symmetry or. (Fig. 3.9). the. 27.