Qubits

A qubit is a generalization of a bit (a system with two possible states) capable of occupying a quantum superposition of both states. A quantum gate, on the other hand, is a generalization of a logic gate describing the transformation of one or more qubits once a gate is applied given their initial state. Physical implementation of qubits and gates is challenging for the same reason that quantum phenomena are difficult to observe in everyday life given the minute scale on which they occur. One approach to achieving quantum computers is by implementing superconductors whereby quantum effects are macroscopically observable, though at the price of extremely low operation temperatures.

Superconductors

Unlike typical conductors, superconductors possess a critical temperature at which resistivity plummets to nearly zero and conductivity is drastically increased. In superconductors, the basic charge carriers are pairs of electrons (known as Cooper pairs), rather than single fermions as found in typical conductors.^[13] Cooper pairs are loosely bound and have an energy state lower than that of Fermi Energy. Electrons forming Cooper pairs possess equal and opposite momentum and spin so that the total spin of the Cooper pair is an integer spin. Hence, Cooper pairs are bosons. Two such superconductors which have been used in superconducting qubit models are niobium and tantalum, both d-band superconductors.^[14]

Electrical circuits

At each point of a superconducting electronic circuit (a network of electrical elements), the condensate wave function describing charge flow is well-defined by some complex probability amplitude. In typical conductor electrical circuits, this same description is true for individual charge carriers except that the various wave functions are averaged in macroscopic analysis, making it impossible to observe quantum effects. The condensate wave function becomes useful in allowing design and measurement of macroscopic quantum effects. Similar to the discrete atomic energy levels in the Bohr model, only discrete numbers of magnetic flux quanta can penetrate a superconducting loop. In both cases, quantization results from complex amplitude continuity. Differing from microscopic implementations of quantum computers (such as atoms or photons), parameters of superconducting circuits are designed by setting (classical) values to the electrical elements composing them such as by adjusting capacitance or inductance.

To obtain a quantum mechanical description of an electrical circuit, a few steps are required. Firstly, all electrical elements must be described by the condensate wave function amplitude and phase rather than by closely related macroscopic current and voltage descriptions used for classical circuits. For instance, the square of the wave function amplitude at any arbitrary point in space corresponds to the probability of finding a charge carrier there. Therefore, the squared amplitude corresponds to a classical charge distribution. The second requirement to obtain a quantum mechanical description of an electrical circuit is that generalized Kirchhoff's circuit laws are applied at every node of the circuit network to obtain the system's equations of motion. Finally, these equations of motion must be reformulated to Lagrangian mechanics such that a quantum Hamiltonian is derived describing the total energy of the system.

Manufacturing

Superconducting quantum computing devices are typically designed in the radio-frequency spectrum, cooled in dilution refrigerators below 15 mK (millikelvins) and addressed with conventional electronic instruments, e.g. frequency synthesizers and spectrum analyzers. Typical dimensions fall on the range of micrometers, with sub-micrometer resolution, allowing for the convenient design of a Hamiltonian system with well-established integrated circuit technology. Manufacturing superconducting qubits follows a process involving lithography, depositing of metal, etching, and controlled oxidation as described in.^[16] Manufacturers continue to improve the lifetime of superconducting qubits and have made significant improvements since the early 2000s.^[16]: 4

Josephson junctions

A single Josephson junction where C is a thin layer of insulator and A & B are (superconducting) currents with nonequivalent wave functions

One distinguishable attribute of superconducting quantum circuits is the use of Josephson junctions. Josephson junctions are an electrical element which does not exist in normal conductors. Recall that a junction is a weak connection between two leads of wire (in this case a superconductive wire) on either side of a thin layer of insulator material only a few atoms thick, usually implemented using shadow evaporation technique. The resulting Josephson junction device exhibits the Josephson Effect whereby the junction produces a supercurrent. An image of a single Josephson junction is shown to the right. The condensate wave function on the two sides of the junction are weakly correlated, meaning that they are allowed to have different superconducting phases. This distinction of nonlinearity contrasts continuous superconducting wire for which the wave function across the junction must be continuous. Current flow through the junction occurs by quantum tunneling, seeming to instantaneously "tunnel" from one side of the junction to the other. This tunneling phenomenon is unique to quantum systems. Thus, quantum tunneling is used to create nonlinear inductance, essential for qubit design as it allows a design of anharmonic oscillators for which energy levels are discretized (or quantized) with nonuniform spacing between energy levels, denoted ${\displaystyle \Delta E}$.^[1] In contrast, the quantum harmonic oscillator cannot be used as a qubit as there is no way to address only two of its states.

Phase qubit

The phase qubit possesses a Josephson to charge energy ratio on the order of magnitude ${\displaystyle 10^{6}}$. For phase qubits, energy levels correspond to different quantum charge oscillation amplitudes across a Josephson junction, where charge and phase are analogous to momentum and position respectively as analogous to a quantum harmonic oscillator. Note that in this context phase is the complex argument of the superconducting wave function (also known as the superconducting order parameter), not the phase between the different states of the qubit.

The left-most image shows a Fluxonium superconducting loop consisting of a collection of larger area Josephson junctions and one smaller area Josephson junction, as shown by an electron microscope.^[23] The top right image depicts fluxonium circuit components, and the bottom right image depicts a smaller area Josephson junction.^[23]

Flux qubit

The flux qubit (also known as a persistent-current qubit) possesses a Josephson to charging energy ratio on the order of magnitude ${\displaystyle 10}$. For flux qubits, the energy levels correspond to different integer numbers of magnetic flux quanta trapped in a superconducting ring.

Charge qubit

The charge qubit, also known as the Cooper pair box, possesses a Josephson to charging energy ratio on the order of magnitude ${\displaystyle <1}$. For charge qubits, different energy levels correspond to an integer number of Cooper pairs on a superconducting island (a small superconducting area with a controllable number of charge carriers).^[26] Indeed, the first experimentally realized qubit was the Cooper pair box, achieved in 1999.^[27]

A device consisting of four superconducting transmon qubits, four quantum buses, and four readout resonators fabricated by IBM and published in npj Quantum Information in January 2017^[28]

Gatemon

Another variation of the transmon qubit is the Gatemon. Like the Xmon, the Gatemon is a tunable variation of the transmon. The Gatemon is tunable via gate voltage.

Superconducting circuit consisting of 3 Unimons (blue), each connected to resonators (red), drive lines (green), and joint probe lines (yellow)^[33]

Unimon

In 2022 researchers from IQM Quantum Computers, Aalto University, and VTT Technical Research Centre of Finland discovered a novel superconducting qubit known as the Unimon.^[34] A relatively simple qubit, the Unimon consists of a single Josephson junction shunted by a linear inductor (possessing an inductance not depending on current) inside a (superconducting) resonator.^[35] Unimons have increased anharmocity and display faster operation time resulting in lower susceptibility to noise errors.^[35] In addition to increased anharmocity, other advantages Unimon qubit include decreased susceptibility to flux noise and complete insensitivity to dc charge noise.^[22]

Superconducting Qubit Archetypes ^[36]

Type Aspect	Charge qubit	RF-SQUID qubit (prototype of the Flux Qubit)	Phase qubit
Circuit	Charge qubit circuit. A superconducting island (encircled with a dashed line) is defined between the leads of a capacitor with capacitance ${\displaystyle C}$ and a Josephson junction with energy ${\displaystyle E_{J}}$ biased by voltage ${\displaystyle U}$.	Flux qubit circuit. A superconducting loop with inductance ${\displaystyle L}$ is interrupted by a junction with Josephson energy ${\displaystyle E_{J}}$. Bias flux ${\displaystyle \Phi }$ is induced by a flux line with current ${\displaystyle I_{0}}$.	Phase qubit circuit. A Josephson junction with energy parameter ${\displaystyle E_{J}}$ is biased by current ${\displaystyle I_{0}}$.
Hamiltonian	${\displaystyle H=E_{C}(N-N_{g})^{2}-E_{J}\cos \phi }$ In this case ${\displaystyle N}$ is the number of Cooper pairs to tunnel through the junction, ${\displaystyle N_{g}=CV_{0}/2e}$ is the charge on the capacitor in units of Cooper pairs number, ${\displaystyle E_{C}=(2e)^{2}/2(C_{J}+C)}$ is the charging energy associated with both capacitance ${\displaystyle C}$ and Josephson junction capacitance ${\displaystyle C_{J}}$.	${\displaystyle H={\frac {q^{2}}{2C_{J}}}+\left({\frac {\Phi _{0}}{2\pi }}\right)^{2}{\frac {\phi ^{2}}{2L}}-E_{J}\cos \left[\phi -\Phi {\frac {2\pi }{\Phi _{0}}}\right]}$ Note that ${\displaystyle \phi }$ is only allowed to take values greater than ${\displaystyle 2\pi }$ and is alternatively defined as the time integral of voltage along inductance ${\displaystyle L}$.	${\displaystyle H={\frac {(2e)^{2}}{2C_{J}}}q^{2}-I_{0}{\frac {\Phi _{0}}{2\pi }}\phi -E_{J}\cos \phi }$Here ${\displaystyle \Phi _{0}}$ is magnetic flux quantum.
Potential	${\displaystyle U=-E_{J}\cos \phi }$ . Bias voltage is set such that ${\displaystyle N_{g}={\frac {1}{2}}}$, minimizing the energy gap between ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$, consequently distinguishing the gap from other energy gaps (e.g. gap between ${\displaystyle |1\rangle }$ and ${\displaystyle |2\rangle }$). The difference in gaps allows addressing transitions from ${\displaystyle |0\rangle }$ to ${\displaystyle |1\rangle }$ and vice versa only, without populating other states.	${\displaystyle U=\left({\frac {\Phi _{0}}{2\pi }}\right)^{2}{\frac {\phi ^{2}}{2L}}-E_{J}\cos \left[\phi -\Phi {\frac {2\pi }{\Phi _{0}}}\right]}$ Bias flux is ${\displaystyle \Phi =\Phi _{0}/2}$. Different wells correspond to a distinct number of flux quanta trapped in the superconducting loops. The two lower states correspond to a symmetrical and anti-symmetrical superposition of zero or single trapped flux quanta, sometimes denoted as clockwise and counterclockwise loop current states: ${\displaystyle |0\rangle =\left[|\circlearrowleft \rangle +|\circlearrowright \rangle \right]/{\sqrt {2}}}$ and ${\displaystyle |1\rangle =\left[|\circlearrowleft \rangle -|\circlearrowright \rangle \right]/{\sqrt {2}}}$.	${\displaystyle U=-I_{0}{\frac {\Phi _{0}}{2\pi }}\phi -E_{J}\cos \phi }$, also known as "washboard" potential. Bias current is adjusted to allow wells shallow enough to contain exactly two localized wave functions. A slight increase in bias current causes a selective "spill" of higher energy state (${\displaystyle |1\rangle }$), expressed with a measurable voltage spike (a mechanism commonly used for phase qubit measurement).

In the table above, the three superconducting qubit archetypes are reviewed. In the first row, the qubit's electrical circuit diagram is presented. The second row depicts a quantum Hamiltonian derived from the circuit. Generally, the Hamiltonian is the sum of the system's kinetic and potential energy components (analogous to a particle in a potential well). For the Hamiltonians denoted, ${\displaystyle \phi }$ is the superconducting wave function phase difference across the junction, ${\displaystyle C_{J}}$ is the capacitance associated with the Josephson junction, and ${\displaystyle q}$ is the charge on the junction capacitance. For each potential depicted, only solid wave functions are used for computation. The qubit potential is indicated by a thick red line, and schematic wave function solutions are depicted by thin lines, lifted to their appropriate energy level for clarity.

Note that particle mass corresponds to an inverse function of the circuit capacitance and that the shape of the potential is governed by regular inductors and Josephson junctions. Schematic wave solutions in the third row of the table show the complex amplitude of the phase variable. Specifically, if a qubit's phase is measured while the qubit occupies a particular state, there is a non-zero probability of measuring a specific value only where the depicted wave function oscillates. All three rows are essentially different presentations of the same physical system.

Single qubit gates

A depiction of the Bloch sphere

A single qubit gate is achieved by rotation in the Bloch sphere. Rotations between different energy levels of a single qubit are induced by microwave pulses sent to an antenna or transmission line coupled to the qubit with a frequency resonant with the energy separation between levels. Individual qubits may be addressed by a dedicated transmission line or by a shared one if the other qubits are off resonance. The axis of rotation is set by quadrature amplitude modulation of microwave pulse, while pulse length determines the angle of rotation.^[37]

More formally (following the notation of ^[37]) for a driving signal

${\displaystyle {\mathcal {E}}(t)={\mathcal {E}}^{x}(t)\cos(\omega _{d}t)+{\mathcal {E}}^{y}(t)\sin(\omega _{d}t)}$

of frequency ${\displaystyle \omega _{d}}$, a driven qubit Hamiltonian in a rotating wave approximation is

${\displaystyle H^{R}/\hbar =(\omega -\omega _{d})|1\rangle \langle 1|+{\frac {{\mathcal {E}}^{x}(t)}{2}}\sigma _{x}+{\frac {{\mathcal {E}}^{y}(t)}{2}}\sigma _{y}}$,

where ${\displaystyle \omega }$ is the qubit resonance and ${\displaystyle \sigma _{x},\sigma _{y}}$ are Pauli matrices.

To implement a rotation about the ${\displaystyle X}$ axis, one can set ${\displaystyle {\mathcal {E}}^{y}(t)=0}$ and apply a microwave pulse at frequency ${\displaystyle \omega _{d}=\omega }$ for time ${\displaystyle t_{g}}$. The resulting transformation is

${\displaystyle U_{x}=\exp \left\{-{\frac {i}{\hbar }}\int _{0}^{t_{g}}H^{R}dt\right\}=\exp \left\{-i\int _{0}^{t_{g}}{\mathcal {E}}^{x}(t)dt\cdot \sigma _{x}/2\right\}}$.

This is exactly the rotation operator ${\displaystyle R_{X}(\theta )}$ by angle ${\displaystyle \theta =\int _{0}^{t_{g}}{\mathcal {E}}^{x}(t)dt}$ about the ${\displaystyle X}$ axis in the Bloch sphere. A rotation about the ${\displaystyle Y}$ axis can be implemented in a similar way. Showing the two rotation operators is sufficient for satisfying universality as every single qubit unitary operator ${\displaystyle U}$ may be presented as ${\displaystyle U=R_{X}(\theta _{1})R_{Y}(\theta _{2})R_{X}(\theta _{3})}$ (up to a global phase which is physically inconsequential) by a procedure known as the ${\displaystyle X-Y}$ decomposition.^[38] Setting ${\displaystyle \int _{0}^{t_{g}}{\mathcal {E}}^{x}(t)dt=\pi }$ results in the transformation

${\displaystyle U_{x}=\exp \left\{-i\int _{0}^{t_{g}}{\mathcal {E}}^{x}(t)dt\cdot \sigma _{x}/2\right\}=e^{-i\pi \sigma _{x}/2}=-i\sigma _{x}}$

up to the global phase ${\displaystyle -i}$ and is known as the NOT gate.

Quantum bus

Another method of coupling two or more qubits is by way of a quantum bus, by pairing qubits to this intermediate. A quantum bus is often implemented as a microwave cavity modeled by a quantum harmonic oscillator. Coupled qubits may be brought in and out of resonance with the bus and with each other, eliminating the nearest-neighbor limitation. Formalism describing coupling is cavity quantum electrodynamics. In cavity quantum electrodynamics, qubits are analogous to atoms interacting with an optical photon cavity with a difference of GHz (rather than the THz regime of electromagnetic radiation). Resonant excitation exchange among these artificial atoms is potentially useful for direct implementation of multi-qubit gates.^[40] Following the dark state manifold, the Khazali-Mølmer scheme^[40] performs complex multi-qubit operations in a single step, providing a substantial shortcut to the conventional circuit model.

Cross resonant gate

One popular gating mechanism uses two qubits and a bus, each tuned to different energy level separations. Applying microwave excitation to the first qubit, with a frequency resonant with the second qubit, causes a ${\displaystyle \sigma _{x}}$ rotation of the second qubit. Rotation direction depends on the state of the first qubit, allowing a controlled phase gate construction.^[41]

Following the notation of,^[41] the drive Hamiltonian describing the excited system through the first qubit driving line is formally written

${\displaystyle H_{D}/\hbar =A(t)\cos({\tilde {\omega }}_{2}t)\left(\sigma _{x}\otimes I-{\frac {J}{\Delta _{12}}}\sigma _{z}\otimes \sigma _{x}+m_{12}I\otimes \sigma _{x}\right)}$,

where ${\displaystyle A(t)}$ is the shape of the microwave pulse in time, ${\displaystyle {\tilde {\omega }}_{2}}$ is resonance frequency of the second qubit, ${\displaystyle \{I,\sigma _{x},\sigma _{y},\sigma _{z}\}}$ are the Pauli matrices, ${\displaystyle J}$ is the coupling coefficient between the two qubits via the resonator, ${\displaystyle \Delta _{12}\equiv \omega _{1}-\omega _{2}}$ is qubit detuning, ${\displaystyle m_{12}}$ is stray (unwanted) coupling between qubits, and ${\displaystyle \hbar }$ is Planck's constant divided by ${\displaystyle 2\pi }$. The time integral over ${\displaystyle A(t)}$ determines the angle of rotation. Unwanted rotations from the first and third terms of the Hamiltonian can be compensated for with single qubit operations. The remaining component, combined with single qubit rotations, forms a basis for the su(4) Lie algebra.