Superconducting quantum computing is a branch of solid state quantum computing that implements superconducting electronic circuits using superconducting qubits as artificial atoms, or quantum dots. For superconducting qubits, the two logic states are the ground state and the excited state, denoted ${\displaystyle |g\rangle {\text{ and }}|e\rangle }$respectively.^[1] Research in superconducting quantum computing is conducted by companies such as Google,^[2] IBM,^[3] IMEC,^[4] BBN Technologies,^[5] Rigetti,^[6] and Intel.^[7] Many recently developed QPUs (quantum processing units, or quantum chips) use superconducting architecture.

As of May 2016^[update], up to 9 fully controllable qubits are demonstrated in the 1D array,^[8] and up to 16 in 2D architecture.^[3] In October 2019, the Martinis group, partnered with Google, published an article demonstrating novel quantum supremacy, using a chip composed of 53 superconducting qubits.^[9]

Background

Classical computation models rely on physical implementations consistent with the laws of classical mechanics.^[10] Classical descriptions are accurate only for specific systems consisting of a relatively large number of atoms. A more general description of nature is given by quantum mechanics. Quantum computation studies quantum phenomena applications beyond the scope of classical approximation, with the purpose of performing quantum information processing and communication. Various models of quantum computation exist, but the most popular models incorporate concepts of qubits and quantum gates (or gate-based superconducting quantum computing).

Superconductors are implemented due to the fact that at low temperatures they have infinite conductivity and zero resistance. Each qubit is built using semiconductor circuits with an LC circuit: a capacitor and an inductor.^{[citation needed]}

Superconducting capacitors and inductors are used to produce a resonant circuit that dissipates almost no energy, as heat can disrupt quantum information. The superconducting resonant circuits are a class of artificial atoms that can be used as qubits. Theoretical and physical implementations of quantum circuits are widely different. Implementing a quantum circuit had its own set of challenges and must abide by DiVincenzo's criteria, conditions proposed by theoretical physicist David P DiVincenzo,^[11] which is set of criteria for the physical implementation of superconducting quantum computing, where the initial five criteria ensure that the quantum computer is in line with the postulates of quantum mechanics and the remaining two pertaining to the relaying of this information over a network.^{[citation needed]}

We map the ground and excited states of these atoms to the 0 and 1 state as these are discrete and distinct energy values and therefore it is in line with the postulates of quantum mechanics. In such a construction however an electron can jump to multiple other energy states and not be confined to our excited state; therefore, it is imperative that the system be limited to be affected only by photons with energy difference required to jump from the ground state to the excited state.^[12] However, this leaves one major issue, we require uneven spacing between our energy levels to prevent photons with the same energy from causing transitions between neighboring pairs of states. Josephson junctions are superconducting elements with a nonlinear inductance, which is critically important for qubit implementation.^[12] The use of this nonlinear element in the resonant superconducting circuit produces uneven spacings between the energy levels.^{[citation needed]}

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]

Bose-Einstein condensates

Once cooled to nearly absolute zero, a collection of bosons collapse into their lowest energy quantum state (the ground state) to form a state of matter known as Bose–Einstein condensate. Unlike fermions, bosons may occupy the same quantum energy level (or quantum state) and do not obey the Pauli exclusion principle. Classically, Bose-Einstein Condensate can be conceptualized as multiple particles occupying the same position in space and having equal momentum. Because interactive forces between bosons are minimized, Bose-Einstein Condensates effectively act as a superconductor. Thus, superconductors are implemented in quantum computing because they possess both near infinite conductivity and near zero resistance. The advantages of a superconductor over a typical conductor, then, are twofold in that superconductors can, in theory, transmit signals nearly instantaneously and run infinitely with no energy loss. The prospect of actualizing superconducting quantum computers becomes all the more promising considering NASA's recent development of the Cold Atom Lab in outer space where Bose-Einstein Condensates are more readily achieved and sustained (without rapid dissipation) for longer periods of time without the constraints of gravity.^[15]

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.

Technology

Manufacturing

Superconducting quantum computing devices are typically designed in the radio-frequency spectrum, cooled in dilution refrigerators below 15 mK 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

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, given that the spacing between every energy level and the next is exactly the same.

Qubit archetypes

The three primary superconducting qubit archetypes are the phase, charge and flux qubit. Many hybridizations of these archetypes exist including the fluxonium,^[17] transmon,^[18] Xmon,^[19] and quantronium.^[20] For any qubit implementation the logical quantum states ${\displaystyle \{|0\rangle ,|1\rangle \}}$ are mapped to different states of the physical system (typically to discrete energy levels or their quantum superpositions). Each of the three archetypes possess a distinct range of Josephson energy to charging energy ratio. Josephson energy refers to the energy stored in Josephson junctions when current passes through, and charging energy is the energy required for one Cooper pair to charge the junction's total capacitance.^[21] Josephson energy can be written as

${\displaystyle U_{j}=-{\frac {I_{0}\Phi _{0}}{2\pi }}\cos \delta }$,

where ${\displaystyle I_{0}}$ is the critical current parameter of the Josephson junction, ${\displaystyle \Phi _{0}={\frac {h}{2e}}}$ is (superconducting) flux quantum, and ${\displaystyle \delta }$ is the phase difference across the junction.^[21] Notice that the term ${\displaystyle cos\delta }$ indicates nonlinearity of the Josephson junction.^[21] Charge energy is written as

${\displaystyle E_{C}={\frac {e^{2}}{2C}}}$,

where ${\displaystyle C}$ is the junction's capacitance and ${\displaystyle e}$ is electron charge.^[21] Of the three archetypes, phase qubits allow the most of Cooper pairs to tunnel through the junction, followed by flux qubits, and charge qubits allow the fewest.

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.

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.

Fluxonium

Fluxonium qubits are a specific type of flux qubit whose Josephson junction is shunted by a linear inductor of ${\displaystyle E_{J}\gg E_{L}}$ where ${\displaystyle E_{L}=(\hbar /2e)^{2}/L}$.^[24] In practice, the linear inductor is usually implemented by a Josephson junction array that is composed of a large number (can be often ${\displaystyle N>100}$) of large-sized Josephson junctions connected in a series. Under this condition, the Hamiltonian of a fluxonium can be written as:

${\displaystyle {\hat {H}}=4E_{C}{\hat {n}}^{2}+{\frac {1}{2}}E_{L}({\hat {\phi }}-\phi _{\mathrm {ext} })^{2}-E_{J}\cos {\hat {\phi }}}$.

One important property of the fluxonium qubit is the longer qubit lifetime at the half flux sweet spot, which can exceed 1 millisecond.^[24][25] Another crucial advantage of the fluxonium qubit biased at the sweet spot is the large anharmonicity, which allows fast local microwave control and mitigates spectral crowding problems, leading to better scalability.^[26][27]

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).^[28] Indeed, the first experimentally realized qubit was the Cooper pair box, achieved in 1999.^[29]

Transmon

Transmons are a special type of qubit with a shunted capacitor specifically designed to mitigate noise. The transmon qubit model was based on the Cooper pair box^[31] (illustrated in the table above in row one column one). It was also the first qubit to demonstrate quantum supremacy.^[32] The increased ratio of Josephson to charge energy mitigates noise. Two transmons can be coupled using a coupling capacitor.^[1] For this 2-qubit system the Hamiltonian is written

${\displaystyle \stackrel{^<!--\; ^\; -->}{H}=\frac{\mathrm{\hslash <!--\; \hslash \; -->}J}{2}({\mathrm{\sigma <!--\; \sigma \; -->}}_1^x{\mathrm{\sigma <!--\; \sigma \; -->}}_2^x+{\mathrm{\sigma <!--\; \sigma \; -->}}_1^y}$Zdroj:https://en.wikipedia.org?pojem=Superconducting_qubits
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