Superconducting Quantum Computing

WOMANIUM Global Quantum Media Project Initiative — Winner of Global Quantum Media Project

Introduction

Superconducting quantum computing has emerged as a groundbreaking field that holds immense promise for revolutionizing computation. This article provides a comprehensive introduction to the key components and techniques employed in superconducting quantum computing. We explore the fundamental concepts and techniques, including Quantum Harmonic Oscillators, Superconducting Qubits (specifically the transmon qubit), Josephson junctions, finite element simulation, superconducting qubit readout, single superconducting qubit gates, and two-qubit gates. By understanding these building blocks, we gain insights into the underlying principles and engineering considerations that drive the development of superconducting quantum computers, paving the way for unprecedented computational capabilities and scientific advancements.

DiVincenzo Criteria for Superconducting Qubits

To evaluate the suitability of superconducting qubits as a physical architecture for quantum computation, the DiVincenzo criteria provide a comprehensive framework. These criteria include: 1) the presence of a scalable physical quantum system with well-characterized qubits, 2) the ability to initialize qubit states, 3) the availability of fast logic gates in comparison to the lifetime of quantum information, 4) a universal set of quantum gates, and 5) individual qubit readout. Notably, preserving the fragile quantum information for extended periods is crucial for superconducting qubits. Prolonging the lifetime of quantum information ensures the preservation of complex quantum states during computations. Additionally, at the conclusion of an algorithm, the ability to accurately read out individual qubits is essential.

Harmonic Oscillators

Harmonic oscillators play a significant role in the description of various physical systems, often encountered either as harmonic oscillators or spins. Interestingly, superconducting qubits can also be interpreted as harmonic oscillators, depending on the context and perspective. In general, a harmonic oscillator is characterized by a system where energy oscillates between kinetic and potential forms. Classical examples include a mass attached to a spring or a pendulum. However, in the realm of superconducting qubits, we are particularly interested in the behavior of a specific type of harmonic oscillator known as an LC circuit. This circuit consists of an inductor and a capacitor connected in parallel, serving as an electrical circuit analogue for other types of harmonic oscillators. In the LC circuit, the potential energy is stored as flux within the inductor, while the kinetic energy is associated with the oscillations of voltage across the capacitor. This conceptual framework allows us to analyze and understand the behavior of superconducting qubits within the context of harmonic oscillators.

Examples of Harmonic Oscillators: Mechanical and Electrical. Image Credits: [1]

Quantum Harmonic LC Oscillator

Quantum harmonic LC oscillators provide a captivating insight into the quantum mechanical behavior of electrical circuits, particularly LC circuits. It may initially seem puzzling to consider the quantum nature of an electrical circuit, given their ubiquitous presence in everyday devices such as phones and filtering schemes. However, it is crucial to recognize that the potential for quantum behavior exists in any physical system. There is no fundamental limitation preventing a broad range of physical systems from exhibiting quantum mechanical phenomena.

Device Description and Spectroscopy. Image Credits:[2]

(a) The transmon, analogous to a charged quantum rotor in a constant magnetic field (~ng), is strongly influenced by a “gravitational” force that keeps it near φ = 0 when EJ/EC is large. Tunneling between adjacent cosine wells, representing a complete 2π rotor movement, acquires an Aharonov-Bohm-like phase due to ng. The tunneling probability decreases exponentially with EJ/EC, leading to a reduced charge dispersion. (b) The cosine potential, represented by a solid black line, corresponds to specific eigenenergies and the squared moduli of the eigenfunctions. Image Credits: [3]

The Hamiltonian operator of a quantum harmonic oscillator characterizes the energy of the system in terms of the squared voltage (V) and squared current (I) operators.

Charge and flux are canonically conjugate:

Solving a harmonic oscillator can be achieved in different basis, such as the position and momentum basis or, equivalently, in terms of voltages and currents within the circuit. By employing dimensionless quantities, in the context of a superconducting circuit, voltage can be mapped to charge, which in turn can be mapped to the number of cooper pairs, a dimensionless quantity.

Cooper Pair Number:

Superconducting Phase (SC-Phase):

Writing coefficients as energies:

Cooper pairs, as the charge carriers in a superconducting circuit, describe the capacitive component of the circuit. Furthermore, rather than focusing on currents, variable transformations are employed to introduce the concept of the superconducting phase. Superconductivity is characterized by a macroscopic wavefunction in which all the Cooper pairs behave coherently, reflecting the behavior of the superconducting phase. This enables the formulation of a Hamiltonian or a quantum description of the LC oscillator that aligns with principles found in quantum textbooks. The connection between this quantum behavior and Maxwell’s equations provides a reference point for understanding the emergence of quantum mechanical phenomena in LC circuits.

Superconducting Quantum Circuits

Superconducting quantum circuits offer a fascinating perspective on the quantum behavior of electrical circuits. The question arises: can an ordinary LC oscillator, typically associated with classical behavior, exhibit quantum mechanical characteristics? Empirical evidence provides a resounding affirmation. Experimental measurements directly demonstrate the quantized nature of excitations in superconducting LC oscillators. The observed peaks in the measurement plot correspond to the discrete energy quanta stored within the circuit. Those familiar with driven oscillators may recognize these peaks as coherent states.

Schrödinger equation

This raises a fundamental query: why would a seemingly ordinary circuit behave quantum mechanically? The answer lies in the quantum-classical correspondence, which states that quantum effects manifest when interactions with the environment are sufficiently weak.

In the case of superconducting circuits, dissipation and energy loss are virtually nonexistent during the transport of DC currents. Consequently, a DC current introduced into a superconducting loop can persist indefinitely. While this approximation is not entirely accurate at higher frequencies, it is a valid approximation for demonstrating quantum mechanical behavior. The superconducting condensate, representing a degree of freedom within the microscopic state of the system, can indeed exhibit quantum mechanical phenomena. In the provided plot (a), the state of the resonator is represented by the Poisson distribution of photons.

Qubit Spectroscopy and Geometric Phase Gate. (a) Qubit spectrum with ¯n ≈ 2, demonstrating the dependence of qubit transition frequency on cavity photon count. (b) Phasor representation of cavity state: cn arrow represents complex amplitude of |ni; circle area proportional to |cn| 2 = p(n). © Example of SNAP gate: two π pulses on qubit along different axes create trajectory enclosing geometric phase, enabling selective rotation on each cn. (d) Final state after © operation, resulting in controllable phase evolution on each cn. (e) Quantum circuit representation of gates used: Q for qubit, C for cavity state. Ry|n(φ) denotes a rotation by angle φ around y conditioned on n cavity photons; “%” represents modulo operator. Image Credits: [4]

Nonlinear Superconducting Oscillators

Nonlinear superconducting oscillators play a pivotal role in the realization of superconducting qubits and their controllability in quantum computation. While the harmonic oscillator itself has its utility, the incorporation of a Josephson junction, a crucial circuit element, introduces nonlinearity. A Josephson junction consists of two superconductors separated by a tunnel barrier, enabling the passage of current through quantum mechanical tunneling effects. This arrangement effectively creates a nonlinear inductor, providing the necessary control mechanisms for superconducting qubits.

a) The circuit diagram features a transmon qubit composed of two Josephson junctions (CJ and EJ) connected in parallel to a large capacitance (CB) and a gate capacitance (Cg). b) The simplified schematic depicts a transmon device design with a split Cooper pair box coupled to a short section of twin-lead transmission line. The transmission line, formed by extending the superconducting islands, is approximately λ/20 in length. The short line section can be represented as a lumped-element capacitor, causing increased capacitances (Cg1, Cg2, and CB’) that impact the effective capacitances (CB and Cg) in the circuit. Image Credits: [3]

A scanning electron micrograph image showcases the Josephson junction, depicting the overlap region where it is formed and the connecting leads to the rest of the circuit. With dimensions on the order of 100 nm, the fabrication of Josephson junctions requires sophisticated techniques like Electron Beam Lithography.

Electron Beam Lithography. Image Credits: [1]

By replacing the linear inductor in the LC oscillator with a Josephson junction, the spacing between energy levels in the harmonic oscillator is altered, giving rise to inharmonicity in the qubit. This inharmonicity defines the speed limit of the qubit’s operation and is of utmost importance. This modification, known as a transmon qubit, has gained tremendous popularity due to its tunable properties primarily determined by the shunting capacitance and specific characteristics of the Josephson junction. In practice, the size of the Josephson junction plays a critical role in setting key parameters. Despite its relatively simple electrical circuit configuration, the transmon qubit represents one of the most widely used types of superconducting qubits in current research and applications.

(a) Schematic of a parallel LC-oscillator (quantum harmonic oscillator, QHO) circuit, comprising an inductance L in parallel with a capacitance C. The phase φ on the superconducting island is referenced to ground. (b) Energy potential diagram for the QHO, showing equally spaced energy levels separated by ~ωr. © Josephson qubit circuit, featuring a nonlinear inductance LJ (represented by the Josephson subcircuit in the dashed orange box) shunted by a capacitance Cs. (d) The Josephson inductance modifies the quadratic energy potential (dashed red) into a sinusoidal shape (solid blue), resulting in no n-equally spaced energy levels. This enables the isolation of the two lowest energy levels |0⟩ and |1⟩, creating a computational subspace with an energy separation ~ω01 different from ~ω12. Image Credits: [5]

The figures illustrate different types of Josephson junctions: (a) a conventional tunnel junction, (b) a coplanar junction with high-Tc superconductors, and © a coplanar junction utilizing 2D materials. Additionally, (d) and (e) show top views of two distinct geometries of Josephson junctions based on 2D materials. In both geometries, the 2D material is positioned parallel to the page between thin-film superconducting electrodes. Geometry A assumes infinitely wide superconducting electrodes, while Geometry B features superconducting electrodes of width W extending infinitely in the z-direction. The red lines represent the distribution of supercurrent in the superconducting electrodes and the 2D material. Within the superconducting electrodes, the supercurrent primarily concentrates within a width equivalent to the London penetration depth 𝜆𝐿 from the edges of the superconductors. The blue arrows indicate the direction of the supercurrent. Notably, the main text considers individual regions ①, ②, and ③ when calculating the magnetic field resulting from the supercurrent. Image Credits: [6]

Josephson Relations:

Ambegaokar-Baratoff Relation:

Other types of Superconducting Qubits

Fluxonium qubit:

(a) Circuit model of fluxonium comprising three elements. (b) Utilization of a linear chain of Josephson junctions to implement a large-value inductance L. © Profile, spectrum, and eigenstates of the particle-in-a-box potential at an integer flux through the loop, displaying a 0–1 qubit transition reminiscent of a transmon. (d) Similar to © but at a half-integer flux bias, where the tunnel-split qubit states are separated from non-computational states by a gap resulting from the excitation of plasma-like oscillations of the phase φ. Image Credits: [7]

Flux Qubit

(a) Optical microscope image illustrates the intersection and galvanic coupling of eleven flux qubits (B1 to B11) with a λ/2 coplanar waveguide (CPW) resonator (resonator B). The CPW resonator, measuring 5.73 mm in length, resonates at the frequency frB ≈ 9.8 GHz. (b) A detailed view displays the coupling capacitor that terminates both ends of the CPW resonator, with a calculated capacitance of approximately 5.0 fF. © Colored atomic force microscopy (AFM) micrograph of qubit B4, revealing a unitary junction with a surface area of 0.0526 ± 0.0008 µm² and an α value of 0.5. (d) Colored AFM micrograph of qubit B6, possessing the same unitary junction surface area and α ratio as B4, along with a narrow constriction in its loop. (e) Close-up view of the 30-nm width constriction present in qubit B6. Image Credits: [8]

Designing a Transmon Qubit

When designing a transmon qubit, understanding the quantum states it encompasses is crucial. However, one challenge with engineered quantum systems is that interpreting the underlying physical dynamics and quantum states can be complex. In the case of superconducting qubits, the key variables are the superconducting phase and the Cooper pairs residing on the superconducting islands. These variables serve as canonically conjugate quantities employed in the description of superconducting qubits.

Solving the Hamiltonian

Solving the Schrödinger equation with the Hamiltonian of the transmon qubit in different bases, similar to solving a regular harmonic oscillator in position or momentum bases, allows us to represent the wavefunctions in those respective bases. The states of the transmon qubit can be described as distributions of superconducting phase, where each energy level represents an eigenstate of the transmon. Alternatively, due to the canonical conjugacy, they can also be represented as distributions of charge.

Phase Basis

Charge Basis

The accompanying plots illustrate flips, with the left plot depicting the first excited state and the right plot representing the ground state. Numerical techniques implemented in Python or Mathematica can easily solve for these states.

(a) The plotted probabilities (|⟨n|ψm⟩|²) depict the presence of n Cooper pairs in the transmon eigenstates m = 0 and 1, considering three different EJ/EC ratios (ng = 1/2) within the charge basis. As the EJ/EC ratio becomes significantly larger (EJ/EC ≫ 1), the solutions approach discrete versions of harmonic oscillator wave functions with increasing width. (b) The fluctuations in the number of Cooper pairs (n) are represented as a function of the EJ/EC ratio for the first three transmon levels. The solid lines represent numerically computed results, while the dashed lines correspond to the asymptotic solution (2.14). In the ground state, there are fluctuations of approximately 1 Cooper pair, while the first excited state exhibits fluctuations of approximately 2 Cooper pairs when the EJ/EC ratio is 100. Image Credits: [3]

To facilitate the design process of superconducting qubits, an open-source Python package called SC Qubits can be highly beneficial. This toolkit provides a range of functionalities and features a user-friendly graphical user interface (GUI) that allows for parameter tuning of popular superconducting qubits, including transmons and fluxoniums. Utilizing SC Qubits can serve as an excellent starting point for designing and exploring various aspects of superconducting qubits.

Tailoring Properties of a Transmon Qubit

The properties of the transmon qubit can be tailored by manipulating the parameters of the circuit, allowing for enhanced control over its behavior and mitigation of environmental effects. One significant challenge in transmon qubits is the presence of noisy charges oscillating in the surrounding environment. However, by adjusting the ratio of parameters that describe the Josephson junction and the capacitor, it is possible to alleviate the sensitivity to such errors.

Increasing this ratio effectively suppresses the susceptibility to this type of noise, offering a means to enhance the stability and reliability of the transmon qubit. This engineering approach showcases the ability to optimize the performance of the transmon qubit by customizing the circuit parameters to counteract specific environmental disturbances.

(a) and (b) The relative and absolute anharmonicity at the degeneracy point are plotted as functions of the EJ/EC ratio. The minimum pulse duration (τp) and the dephasing time due to charge fluctuations (T2) are plotted as functions of the EJ/EC ratio. Initially, an increase in the EJ/EC ratio from the charge regime leads to a significant increase in the minimum pulse duration, which diverges at the “anharmonicity barrier” when the anharmonicity (α) crosses zero. Above the barrier, the operation time scales with a weak power law (∼ (EJ/EC)^(1/2)), while the dephasing time due to charge noise increases exponentially with (EJ/EC)^(1/2). Image Credits: [3]

(a) and (b) The relative and absolute anharmonicity at the degeneracy point are plotted against the EJ/EC ratio. The dashed curves depict the perturbative result. The minimum pulse duration (τp) and the dephasing time due to charge fluctuations (T2) are plotted against the EJ/EC ratio. Initially, increasing the EJ/EC ratio in the charge regime results in a significant rise in the minimum pulse duration, which diverges at the “anharmonicity barrier” when the anharmonicity (α) crosses zero. Above this barrier, the operation time scales with a weak power law (∼ (EJ/EC)^(1/2)). Additionally, the dephasing time due to charge noise increases exponentially with (EJ/EC)^(1/2). Image Credits: [3]

Superconducting Qubit Readout

In order to interact with superconducting qubits and extract information about their states, we employ a technique called superconducting qubit readout. This involves coupling the superconducting qubit to a harmonic oscillator, commonly known as a superconducting resonator, which is typically constructed using the same materials on the same chip as the qubit itself. The interaction between the qubit and the resonator can be described by an interaction Hamiltonian, often in the dispersive limit of the James-Cummings Hamiltonian.

When engineering this circuit, an intriguing phenomenon arises known as the frequency shift of the qubit or resonator based on the state of the other system. For example, if an excitation is introduced to the qubit, causing it to transition from the ground state to the first excited state, the resonator frequency experiences a slight shift. This shift can be measured by analyzing the transmission or reflection of a microwave signal through the resonator at different frequencies. As a function of the signal frequency, two distinct responses are observed depending on the state of the qubit.

FIG. 2 illustrates the proposed implementation of cavity Quantum Electrodynamics (QED) using superconducting circuits. The schematic layout and equivalent lumped circuit representation are presented. The 1D transmission line resonator is constructed from a full-wave section of superconducting coplanar waveguide, fabricated through conventional optical lithography techniques. A Cooper-pair box qubit is positioned between the superconducting lines and capacitively coupled to the center trace, achieving a strong electric dipole interaction between the qubit and a single photon within the cavity. The box comprises two small Josephson junctions (approximately 100 nm × 100 nm) arranged in a 1 µm loop, allowing for the adjustment of the effective Josephson energy through an external flux Φext. Input and output signals are coupled to the resonator via capacitive gaps in the center line, enabling measurement of cavity transmission amplitude and phase, as well as the manipulation of qubit states using dc and rf pulses. Multiple qubits can be placed at different antinodes of the standing wave to generate entanglement and perform two-bit quantum gates over distances of several millimeters. Image Credits: [9]

To extract information about the qubit state, we analyze the amplitude and phase of the AC signal passing through the resonator. By examining the signal at a specific frequency on the x-axis and evaluating the amplitude or phase, we can differentiate between different qubit states. Linear superconducting resonators play a critical role in this readout process, as they provide essential resources for accurately determining the states of superconducting qubits.

FIG. 19. (a) Experimental setup for dispersive qubit readout: an arbitrary waveform generator shapes and generates the resonator probe tone, which is sent to the cryostat. The reflected signal (S11) is amplified using a parametric amplifier and a low-noise HEMT amplifier, downconverted through heterodyne mixing, and digitized. (b) Reflection magnitude (|S11|) and phase (θ) response of the resonator for the qubit ground state (|0⟩) and excited state (|1⟩), separated by a frequency 2χ/2π. c)Complex plane representation of the reflection signal, with Re[S11] and Im[S11] components. The maximum state discrimination occurs when probing the resonator between the two resonances, maximizing the distance between the states. Image Credits: [5]

Implementing and Controlling Superconducting Qubits

In practice, the implementation of superconducting qubits involves a series of steps. It begins with designing a circuit layout and patterning it onto a chip. Once the chip is fabricated, it is placed into a package that serves as an interface between the chip and the necessary wiring for controlling and reading out the qubit’s state. This packaging ensures the secure connection of wires and facilitates the transmission of signals. Subsequently, the packaged qubit system is placed within a dilution refrigerator, which is a specialized cooling apparatus. This cooling stage is essential because superconducting qubits require extremely low temperatures to maintain their superconducting behavior. Many superconducting materials, including aluminum, exhibit superconductivity only below temperatures of 10 Kelvin or lower. To observe quantum mechanical effects, the qubit system is typically cooled down to temperatures around 10 millikelvin, significantly colder than outer space. Dilution refrigerators, such as those manufactured by companies like Bluefors, are commercially available for this purpose.

IBM Superconducting Quantum Computer, Quantum processor is in the middle while all structure is the dilution refrigerator. Image Credits: [10]

To control the qubit system, commercially available control electronics are utilized, such as microwave generators or DC signal generators. These control electronics provide the necessary signals to manipulate and measure the qubit state. Importantly, the control of the system can be accomplished using conventional off-the-shelf computers, which serve as the control interface for the qubit experiments. This integration of control electronics and standard computer interfaces allows for efficient and precise control of superconducting qubits, making them accessible for experimental manipulation and investigation.

Logical and Physical Layer of Superconducting Quantum Processor: Image Credits: [11]

Fabrication

The fabrication process of superconducting qubits involves several steps utilizing standard semiconductor fabrication techniques. Initially, a wafer is prepared, on which multiple chips will be patterned to form superconducting quantum processors. Upon zooming in, an electron micrograph reveals a chip containing multiple qubits arranged in a line and interconnected to enable qubit-qubit interactions. Further zooming in reveals a distinct X-shaped structure representing one capacitor paddle of the circuit, which corresponds to a transmon qubit. Upon closer examination, a loop containing two Josephson junctions becomes apparent, which confers tunability to the transmon qubit. Zooming in even further to a scale of 100nm, the Josephson junction itself is observed, characterized by an overlap region.

Image Credits: [1]

To achieve these intricate structures, standard semiconductor fabrication processes are employed, typically on high-resistivity silicon or sapphire substrates. Deposition techniques such as electron-beam evaporation, sputtering, or molecular beam epitaxy are utilized to deposit high-quality superconducting materials. Additionally, shadow evaporation techniques are employed to fabricate the Josephson junctions. This comprehensive fabrication process allows for the realization of superconducting qubits with precise control over their design and properties.

Representing Qubits

The representation of qubits and their control is a fundamental aspect of quantum computing. One commonly used representation is the Bloch sphere, which serves as a geometric depiction of the qubit states. Each point on the surface of the sphere corresponds to a unique state of the qubit.

Bloch Sphere: Image Credits [12]

By utilizing this representation, it becomes possible to describe any arbitrary single qubit wavefunction parametrically using polar coordinates. This parameterization enables the execution of arbitrary single qubit rotations. In practice, these rotations are accomplished by setting appropriate values for the angles theta and phi on the Bloch sphere.

Microwave pulses are commonly employed as control signals to manipulate qubits and apply desired gates to single qubits. This approach grants the ability to implement a broad range of operations on individual qubits by precisely adjusting the values of theta and phi within the Bloch sphere representation. The utilization of microwave pulses as control mechanisms has proven to be an effective means of manipulating qubits in various quantum computing platforms.

Tunable Transmons

In practical quantum computing systems, tunability of qubits is a crucial requirement for enabling the manipulation of qubit-qubit interactions. To achieve this tunability, Josephson junctions are commonly engineered or employed as proxies. By incorporating a loop structure with two Josephson junctions, magnetic flux can be threaded through, leading to adjustable properties of this qubit element. While this element still behaves similarly to a Josephson junction, the flux tunability allows for turning interactions on and off between qubits. The tunability is achieved through the use of SQUID (Superconducting Quantum Interference Device) technology and on-chip current bias. Additionally, different materials can be utilized to fabricate Josephson junctions that can be effectively tuned using voltage control, offering an additional layer of control for superconducting qubits. This enhanced tunability through the manipulation of external parameters, such as flux and voltage, significantly expands the capabilities and control over superconducting qubits in quantum computing systems.

(a) The side-gate tuning of a nanowire (NW) circuit and mesoscopic Josephson coupling fluctuations are observed. The optical image of device 2 reveals the NW junction and the proximal side gate for voltage control. The fluctuation in the fundamental transition frequency (f01) is induced by sweeping the voltage, showing an inhomogeneously broadened linewidth (γ/2π = 13.2 ± 0.3 MHz). A downward trend in f01 is observed as Vg decreases, with fluctuations around the resonator fundamental at Vg < -15 V. (b) Multiple avoided crossings are observed around Vg = -22 V, as shown in the enlarged plot. c) At Vg = -22.3 V, the NW circuit fully hybridizes with the resonator. The minimum splitting allows extraction of the NW circuit-resonator coupling strength (g/2π = 34 ± 1 MHz). Image Credits: [13]

Finite Element Simulations

Accurately predicting parameter values in a “distributed” circuit, where elements are not discrete but distributed across space, poses a unique challenge in superconducting qubit design. Unlike traditional lumped element circuits that feature discrete capacitors and inductors, superconducting circuits involve capacitance and inductance values that are distributed across the device, with even metal wires exhibiting associated inductance. However, with the aid of commercially available simulation packages, finite element simulations can be employed to accurately determine the desired properties of these distributed circuits.

Finite element simulations are powerful tools that enable detailed analysis of the electromagnetic behavior within complex geometries. By applying these simulations to superconducting qubit designs, it becomes possible to capture the distributed nature of capacitance and inductance within the device. These simulations allow for the extraction of crucial parameter values needed for understanding and optimizing the performance of superconducting qubits.

By leveraging commercially available simulation packages specifically tailored for superconducting circuits, researchers and engineers can obtain valuable insights into the electromagnetic characteristics and properties of these distributed circuits. This enables the precise prediction of parameter values necessary for the design, analysis, and optimization of superconducting qubits in the pursuit of achieving high-fidelity quantum operations.

Single Qubit Gates

Performing single-qubit gates in superconducting qubits involves the utilization of microwave drives that are precisely tuned to the resonant frequency of the qubit transition. By controlling the phase of the microwave signal or pulse, we can achieve arbitrary rotations of the qubit around different axes. The phase of the signal is commonly represented in quadrature coordinates. It is important to mention that the microwave signal is typically modulated with a pulse envelope before being transmitted to the qubit, enabling the execution of these rotations. These microwave signals for qubit manipulation are generated using commercially available hardware, further emphasizing the practical accessibility and utilization of off-the-shelf components in superconducting qubit experiments.

(a) The integrated Josephson quantum processor consists of Al on sapphire, with a linear array of Xmon variant transmon qubits (Q0-Q4) and coplanar waveguide resonators for individual state readout. Control wiring is connected from the chip’s edge.(b) The circuit diagram shows direct qubit coupling facilitated by the nodal connectivity of the Xmon qubit. Frequency-domain multiplexing enables qubit measurement using a single readout line. Microwave control lines provide individual qubit driving through capacitive coupling, while frequency control is achieved via inductively coupled dc lines. c) A schematic representation demonstrates an entangling operation using a controlled-Z gate (UCZ). Qubits at rest with distinct frequencies undergo a state-dependent rotation near resonance, and then return to their initial frequencies. Image Credits: [14]

Microwave pulses are commonly employed for both the control and readout of superconducting qubits. However, when it comes to setting the qubit bias parameter, which governs the tunability of the qubit, a lower frequency signal is often utilized. While single-qubit gates typically rely on microwave frequencies, it is important to note that the bias parameter can be adjusted using signals at a different frequency range, although it is not exclusively limited to lower frequencies. This distinction highlights the versatility of the control techniques employed in superconducting qubits, allowing for the manipulation of various parameters using signals of different frequencies depending on the specific requirements of the qubit architecture and experimental setup.

Two Qubit Gates

Two-qubit gates play a crucial role in generating entanglement between adjacent qubits in superconducting systems. However, one limitation of superconducting qubits is that their locations are fixed once patterned on a chip, preventing their physical movement during an experiment. In contrast, other quantum systems like trapped ions and neutral atoms offer the advantage of being able to manipulate and move qubits, providing additional opportunities for performing entangling operations.

In superconducting qubits, entanglement can be achieved through various techniques. Dynamic tuning involves changing the qubit frequencies to bring about entanglement between the qubits. By adjusting the coupling strengths through parametric gates, entanglement can also be generated. These gates involve modulating the coupling strengths to achieve the desired entanglement. Additionally, multi-photon gates drive two or more photon transitions, leading to the creation of entanglement between the qubits.

The fluxonium-based two-qubit quantum processor. Image Credits: [15]

It is noteworthy that the fidelities with which these entangling operations can be performed in superconducting qubits have significantly improved, surpassing 99% in many cases. However, it is important to acknowledge that the progress in this field is rapid, and even recent tables and measurements may now be outdated due to ongoing advancements and improvements in superconducting qubit technologies.

Control Hardware for Superconducting Quantum Computing

Control Software for Superconducting Quantum Computing

Conclusion

In conclusion, the field of superconducting quantum computing is rapidly evolving through advancements in device design, fabrication techniques, experimental setups, device testing and control, theory and analysis. Materials development, including the use of 3D transmons, 2D tantalum, and innovative processing techniques, contributes to improving qubit performance and exceeding the threshold of 10⁴ operations per qubit lifetime. Efforts towards fault-tolerant quantum computation involve scaling surface code logical qubits and real-time quantum error correction. Near-term systems enable early exploration of crucial aspects such as system calibration, control crosstalk, and connectivity trade-offs, while also allowing the implementation of prototype algorithms and proof-of-principle error correction. These advancements pave the way for the realization of quantum advantage and highlight the tremendous potential of superconducting quantum computing across various applications.

References

Photo by Sigmund on Unsplash

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