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Quantum computing implementation
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 respectively. IBM, IMEC, BBN Technologies, Rigetti, and Intel. Many recently developed QPUs (quantum processing units, or quantum chips) use superconducting architecture. as of 2016, up to 9 fully controllable qubits are demonstrated in the 1D array, and up to 16 in 2D architecture.

Classical computation models rely on physical implementations consistent with the laws of classical mechanics. 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. 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, 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. 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. 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. 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.

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.

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.

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. Manufacturers continue to improve the lifetime of superconducting qubits and have made significant improvements since the early 2000s.

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 transmon, Xmon, and quantronium. For any qubit implementation the logical quantum states 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. Josephson energy can be written as [[File:Energy scales for qubits. png|thumb|A graph of various superconducting qubit archetypes by their Josephson energy to charging energy ratio with a legend on the right. The top left graphic illustrates a unimon electrical circuit. The top right image depicts fluxonium circuit components, and the bottom right image depicts a smaller area Josephson junction. 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.

The charge qubit, also known as the Cooper pair box, possesses a Josephson to charging energy ratio on the order of magnitude 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). Indeed, the first experimentally realized qubit was the Cooper pair box, achieved in 1999. thumb|upright=0.8|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|alt=

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 (illustrated in the table above in row one column one). It was also the first qubit to demonstrate quantum supremacy. The increased ratio of Josephson to charge energy mitigates noise. Two transmons can be coupled using a coupling capacitor. An Xmon is essentially a tunable transmon. The major distinguishing difference between transmon and Xmon qubits is the Xmon qubits is grounded with one of its capacitor pads.

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. thumb|Superconducting circuit consisting of 3 Unimons (blue), each connected to resonators (red), drive lines (green), and joint probe lines (yellow)

The GHz energy gap between energy levels of a superconducting qubit is designed to be compatible with available electronic equipment, due to the terahertz gap (lack of equipment in the higher frequency band). The superconductor energy gap implies a top limit of operation below ~1THz beyond which Cooper pairs break, so energy level separation cannot be too high. On the other hand, energy level separation cannot be too small due to cooling considerations: a temperature of 1 K implies energy fluctuations of 20 GHz. Temperatures of tens of millikelvins are achieved in dilution refrigerators and allow qubit operation at a ~5 GHz energy level separation. Qubit energy level separation is frequently adjusted by controlling a dedicated bias current line, providing a "knob" to fine tune the qubit parameters.

The ability to couple qubits is essential for implementing 2 qubit gates. Coupling two qubits can be achieved by connecting both to an intermediate electrical coupling circuit. The circuit may be either a fixed element (such as a capacitor) or be controllable (like the DC SQUID). In the first case, decoupling qubits during the time the gate is switched off is achieved by tuning qubits out of resonance one from another, making the energy gaps between their computational states different. This approach is inherently limited to nearest neighbor coupling since a physical electrical circuit must be laid out between connected qubits. Notably, D Wave Systems' nearest neighbor coupling achieves a highly connected unit cell of 8 qubits in Chimera graph configuration. Quantum algorithms typically require coupling between arbitrary qubits. Consequently, multiple swap operations are necessary, limiting the length of quantum computation possible before processor decoherence.

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. Following the dark state manifold, the Khazali Mølmer scheme

Following the notation of, and by using selective microwave driving. Off resonant driving can be used to induce differential ac Stark shift, allowing the implementation of all microwave controlled phase gates.

DiVincenzo's criteria is a list describing the requirements for a physical system to be capable of implementing a logical qubit. DiVincenzo's criteria is satisfied by superconducting quantum computing implementation. Much of the current development effort in superconducting quantum computing aim to achieve interconnect, control, and readout in the 3rd dimension with additional lithography layers. The list of DiVincenzo's criteria for a physical system to implement a logical qubit is satisfied by the implementation of superconducting qubits. Although DiVincenzo's criteria as originally proposed consists of five criteria required for physically implementing a quantum computer, the more complete list consists of seven criteria as it takes into account communication over a computer network capable of transmitting quantum information between computers, known as the “quantum internet”. Therefore, the first five criteria ensure successful quantum computing, while the final two criteria allow for quantum communication. A scalable physical system with well characterized qubits. "Well characterized implies that that Hamiltonian function must be well defined i. e the energy eigenstates of the qubit should be able to be quantified A scalable system is self explanatory, it indicates that this ability to regulate a qubit should be augmentable for multiple more qubits. Herein lies the major issue Quantum Computers face, as more qubits are implemented it leads to a exponential increase in cost and other physical implementations which pale in comparison to the enhanced speed it may offer. A fiducial state is one that is easily and consistently replicable and is useful in quantum computing as it may be used to guarantee the initial state of qubits. One simple way to initialize a superconducting qubit is to wait long enough for the qubits to relax to the ground state. Controlling qubit potential with tuning knobs allows faster initialization mechanisms. Long relevant decoherence times Recent strategies to improve device coherence include purifying the circuit materials and designing qubits with decreased sensitivity to noise sources. This criterion may also be satisfied by coupling two transmons with a coupling capacitor.

One of the primary challenges of superconducting quantum computing is the extremely low temperatures at which superconductors like Bose Einstein Condensates exist. Other basic challenges in superconducting qubit design are shaping the potential well and choosing particle mass such that energy separation between two specific energy levels is unique, differing from all other interlevel energy separation in the system, since these two levels are used as logical states of the qubit. Superconducting quantum computing must also mitigate quantum noise (disruptions of the system caused by its interaction with an environment) as well as leakage (information being lost to the surrounding environment). One way to reduce leakage is with parity measurements. At the end of the century in 1999, a paper was published by Yasunobu Nakamura, which exhibited the initial design of a superconducting qubit which is now known as the "charge qubit". This is the primary basis point on which later designs amended upon. These initial qubits had their limitations in respect to maintaining long coherence times and destructive measurements. The further amendment to this initial breakthrough lead to the invention of the phase and flux qubit and subsequently resulting in the transmon qubit which is now widely and primarily used in Superconducting Quantum Computing. The transmon qubit has enhanced original designs and has further cushioned charge noise from the qubit. IBM released the following roadmap publicly that they have set for their quantum computers which also incorporated superconducting qubits and the transmon qubit. 2021: In 2021, IBM came out with their 127 qubit processor. 2022: On November 9, IBM announced its 433 qubit processor called "Osprey". 2023: IBM plan on releasing their Condor quantum processor with 1,121 qubits.