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Cambridge University Press eBooks

Superconducting multiqubit devices

Alexandre M. Zagoskin

Published Jul 14, 2011• 0 citations
Publisher

Abstract

Conquer with skill, not with numbers. A. V. Suvorov, from his military maxims, ca. 1795, translated from Russian by the author Physical implementations of qubit coupling Coupling by linear passive elements. Capacitive coupling To have good basic elements is not enough – it is necessary to be able to connect them in a controllable way, without losing quantum coherence. Any simple effective coupling Hamiltonian (like in Eqs. (3.81, 3.82)) must be somehow implemented “in metal”. Here superconducting circuits provide a wide variety of coupling schemes to choose from (see, e.g., Wendin and Shumeiko, 2005). We will begin with the simplest case, when the interaction between the qubits is realized using linear elements (conventional capacitances and inductances), the coupling circuit stays in its ground state and adiabatically follows the evolution of the qubits (Averin and Bruder, 2003) – that is, it remains “passive”. For this to happen, the excitation energy of the coupler, ħω res , must be much higher than the interlevel spacing in the qubits (where ω res is the resonance frequency of the coupler). In other words, the evolution of the coupler is much faster than that of the qubits, and the coupler can indeed adjust to changes in the state of the latter. In the case of a purely capacitive or purely inductive coupling this condition is automatically satisfied, as then ω res → ∞.

DOI
10.1017/cbo9780511844157.005

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Good order is the foundation of all good things. E. Burke, Reflections on the Revolution in France , 1790 Josephson effect Superconductivity: A crash course The transition from the theoretical description of hypothetical building blocks of a quantum coherent device to something which can be actually fabricated and controlled is made much easier by the existence of superconductivity . This phenomenon, roughly speaking, allows a macroscopic quantum coherent flow of electrons in a sufficiently cold piece of an appropriate material by establishing a specific long-range order among them. Due to this one can, for example, use macroscopically different states of a superconductor as quantum states of a qubit, with the obvious advantage over “microscopically quantum” systems (like actual atoms) from the point of view of control, measurement and, last but not least, fabrication of structures with the desired parameters and on the desired scale. The phenomenon of superconductivity – the history of its discovery, experimental manifestations, theoretical explanation, open questions, relevance to other branches of physics, and technological applications – requires a thorough treatment, which can be found in any number of books. For our purposes Tinkham (2004) will provide more than sufficient background. We will not wander into the field of exotic/high-temperature superconductors for two simple reasons: the quantum coherent behaviour of the kind we need to realize qubits or other quantum coherent devices has not yet been properly and/or routinely achieved in these systems; and the fabrication of such devices is not yet reliable enough or even feasible.

Cambridge University Press eBooks•Jul 14, 2011•1 citation
Conquer with skill, not with numbers. A. V. Suvorov, from his military maxims, ca. 1795, translated from Russian by the author Physical implementations of qubit coupling Coupling by linear passive elements. Capacitive coupling To have good basic elements is not enough – it is necessary to be able to connect them in a controllable way, without losing quantum coherence. Any simple effective coupling Hamiltonian (like in Eqs. (3.81, 3.82)) must be somehow implemented “in metal”. Here superconducting circuits provide a wide variety of coupling schemes to choose from (see, e.g., Wendin and Shumeiko, 2005). We will begin with the simplest case, when the interaction between the qubits is realized using linear elements (conventional capacitances and inductances), the coupling circuit stays in its ground state and adiabatically follows the evolution of the qubits (Averin and Bruder, 2003) – that is, it remains “passive”. For this to happen, the excitation energy of the coupler, ħω res , must be much higher than the interlevel spacing in the qubits (where ω res is the resonance frequency of the coupler). In other words, the evolution of the coupler is much faster than that of the qubits, and the coupler can indeed adjust to changes in the state of the latter. In the case of a purely capacitive or purely inductive coupling this condition is automatically satisfied, as then ω res → ∞.
Abstract This chapter presents the principles of superconducting qubits. We start our discussion with a presentation of the Josephson effect. We then introduce the Cooper pair box and the transmon, the most widespread superconducting qubit. We study how to execute single-qubit gates by driving a superconducting qubit with electromagnetic pulses, and how to read out the state of a qubit by a transmission measurement. The last part of the chapter is devoted to the presentation of some two-qubit gates. Specifically, we illustrate the implementation of the iSWAP and square root of iSWAP gates and of the cross-resonance gate. The last section discusses tunable couplers
We have proposed and experimentally verified a tunable interqubit coupling scheme for large-scale integration of superconducting qubits. The key feature of the scheme is the insertion of connecting pads between the qubit and tunable coupling element. In such a way, the distance between two qubits can be increased considerably to a few millimeters, leaving enough space for arranging control lines, readout resonators, and other necessary structures. The increased interqubit distance provides more wiring space for the flip-chip process and reduces crosstalk between qubits and from control lines to qubits. We use the term tunable coupler with capacitively connecting pad (TCCP) to name the tunable coupling part that consists of a transmon coupler and capacitively connecting pads. With the different placement of connecting pads, different TCCP architectures can be realized. We have designed and fabricated a few multiqubit devices in which TCCP is used for coupling. The measured results show that the performance of the qubits coupled by TCCP, such as T1{T}_{1}T1​ and T2{T}_{2}T2​, was similar to that of traditional transmon qubits without TCCP. Meanwhile, our TCCP also exhibited a wide tunable range of the effective coupling strength and a low residual ZZZZZZ interaction between the qubits by properly tuning parameters on the design. Finally, we successfully implemented an adiabatic controlled-ZZZ gate with TCCP. Furthermore, by introducing TCCP, we also discuss the realization of the flip-chip process and tunable coupling qubits between different chips.
Superconducting quantum circuits typically use capacitive charge-based linear coupling schemes to control interactions between elements such as qubits. While simple and effective, this coupling scheme makes it difficult to satisfy competing circuit design requirements such as maintaining large qubit anharmonicity and coherence along with a high degree of qubit connectivity and packing density. Moreover, tunable interactions using linear coupling elements produce dynamical variations in mode hybridization, which can induce non-adiabatic transitions, resulting in leakage errors and limiting gate speeds. In this work we attempt to address these challenges by proposing a junction-based coupling architecture based on SQUID (superconducting quantum interference device) couplers with relatively small Josephson energies. SQUID couplers provide intrinsic cross-Kerr interactions that can be controlled by external fluxes and that do not rely on mode hybridization. The small Josephson energies of the coupler maintain the interaction at a perturbative scale, which limits undesired higher-order mixing between coupled elements while achieving a sufficiently strong cross-Kerr interaction originating from diagonal coupling elements. Based on these properties, we show that a SQUID coupler can be used to implement a fast, adiabatic, and high-fidelity controlled-Z gate without introducing extra modes, and the operation is robust against junction asymmetry for high-frequency qubits. Although unconventional crosstalk may arise due to junction asymmetries and parasitic hybridization with spectator qubits, we show that these effects are sufficiently small for realistic circuit parameters. As an example of the utility of such junction-based coupling schemes, we present a scalable tiling strategy for a miniaturized superconducting quantum processor based on merged-element transmon qubits.
The aim of this review is to provide quantum engineers with an introductory guide to the central concepts and challenges in the rapidly accelerating field of superconducting quantum circuits. Over the past twenty years, the field has matured from a predominantly basic research endeavor to a one that increasingly explores the engineering of larger-scale superconducting quantum systems. Here, we review several foundational elements—qubit design, noise properties, qubit control, and readout techniques—developed during this period, bridging fundamental concepts in circuit quantum electrodynamics and contemporary, state-of-the-art applications in gate-model quantum computation.
Recent advances in design and fabrication have increased the energy-relaxation times of isolated superconducting transmon qubits to the hundreds-of-microseconds regime, with reported values exceeding 500 μμμs. However, the same progress has not automatically translated to multiqubit processors, where qubits are embedded in connected qubit-coupler lattices and often exhibit much shorter lifetimes than isolated qubits. To identify possible sources of this discrepancy, here we use finite-element simulation to investigate how surface participation ratios and the resulting surface dielectric loss change when a qubit is embedded in a flip-chip qubit-coupler lattice. Controlled comparisons show that higher connectivity can indeed lead to larger surface loss: in the simulated lattice, connecting a qubit to two and four couplers increases the surface loss by factors of 1.3 and 1.8, respectively. We attribute this change to the combined effects of added edge fields from coupling claws, field redistribution over the larger connected metal network, and hybridization with coupler modes. We further examine how this connectivity-induced surface-loss penalty depends on the geometric design parameters of both the qubit electrodes and the coupling claws, and derive guidelines for designing low-loss multiqubit processors.
Good order is the foundation of all good things. E. Burke, Reflections on the Revolution in France , 1790 Josephson effect Superconductivity: A crash course The transition from the theoretical description of hypothetical building blocks of a quantum coherent device to something which can be actually fabricated and controlled is made much easier by the existence of superconductivity . This phenomenon, roughly speaking, allows a macroscopic quantum coherent flow of electrons in a sufficiently cold piece of an appropriate material by establishing a specific long-range order among them. Due to this one can, for example, use macroscopically different states of a superconductor as quantum states of a qubit, with the obvious advantage over “microscopically quantum” systems (like actual atoms) from the point of view of control, measurement and, last but not least, fabrication of structures with the desired parameters and on the desired scale. The phenomenon of superconductivity – the history of its discovery, experimental manifestations, theoretical explanation, open questions, relevance to other branches of physics, and technological applications – requires a thorough treatment, which can be found in any number of books. For our purposes Tinkham (2004) will provide more than sufficient background. We will not wander into the field of exotic/high-temperature superconductors for two simple reasons: the quantum coherent behaviour of the kind we need to realize qubits or other quantum coherent devices has not yet been properly and/or routinely achieved in these systems; and the fabrication of such devices is not yet reliable enough or even feasible.