User:WISE Physics/Superconducting quantum computing

Superconducting Quantum Computing
Superconducting quantum computing implements quantum computing with superconducting electronic circuits. Research in superconducting quantum computing is conducted by companies such as Google, IBM, IMEC, BBN Technologies, Rigetti, and Intel. Currently, up to 9 fully controllable qubits are demonstrated in 1D array and up to 16 in 2D architecture. In October 2019, the Martinis group, partnered with Google, published an article demonstrating novel quantum supremacy using a chip comprised of 53 superconducting qubits. More than 2,000 superconducting qubits are in a commercial product by D-Wave Systems. However, these qubits implement quantum annealing rather than a universal model of quantum computation.

Background[edit]
Classical computation models rely on physical implementations consistent with the laws of classical mechanics. Classical descriptions are accurate only for specific systems with large amounts of atoms, while the more general description of nature is given by quantum mechanics. Quantum computation is the study of quantum phenomena applications that are beyond the scope of classical approximation, which informs artificial intelligence processing and communication. Various models of quantum computation exist, the most popular of which incorporate concepts of qubits and quantum gates.

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 superconducting quantum computing is to implement superconductors, whereby quantum effects are macroscopically observable, though at the price of extremely low operation temperatures.

In superconductors, the basic charge carriers are pairs of electrons (known as Cooper pairs) rather than single electrons found in normal conductors. These cooper pairs have a lower energy than the Fermi energy that opens up a gap in the energy state that allows for the ability of superconductivity. The total spin of a Cooper pair is an integer. Therefore, Cooper pairs are bosons, where single electrons in normal conductors are fermions. Once cooled to nearly 0 Kelvin, a collection of bosons collapse into their lowest energy quantum state, known as the ground state, to form Bose-Einstein Condensate. Unlike fermions, bosons may occupy the same state, making Bose-Einstein Condensate unique to bosons. Rather than behaving discreetly, bosons in Bose-Einstein Condensate act as a single particle, exhibiting macroscopically observable quantum behavior. Because interactive forces between bosons are minimized, Bose-Einstein Condensate acts as a superconductor. Classically, Bose-Einstein Condensate may be conceptualized as multiple particles occupying the same position in space and having equal momentum.

Superconductors are implemented due to the fact that at low temperatures they have almost infinite conductivity and almost zero resistance. Each qubit is built using semiconductor circuits with an LC circuit, a capacitor and an inductor.

Capacitor and inductors are specifically used as they do not produce heat during superconduction which can blemish quantum information. For superconducting quantum circuits we construct artificial atoms to resemble qubits. We map the ground and excited states of these atoms to the 0 and 1 state as these are discrete and distinct energy values 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 particles of light 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 causing transitions between neighboring pairs of states. This is where implementing the Josephson Junction becomes imperative. The use of this junction allows us to create the uneven space required in the energy levels of our superconducting circuit.

At each point of a superconducting electronic circuit (a network of electrical elements), the wavefunction describing charge flow is well-defined by some complex probability amplitude. In typical conductor electrical circuits, this same quantum description is true for individual charge carriers except that the various wave functions are averaged in the macroscopic analysis, therefore making it impossible to observe quantum effects. The condensate wavefunction allows design and measurement of macroscopic quantum effects. For example, only discrete numbers of magnetic flux quanta penetrate a superconducting loop, similar to the discrete atomic energy levels in the Bohr model. In both cases, quantization resulting from complex amplitude continuity. Differing from microscopic quantum systems (such as atoms or photons) used for implementing quantum computing, parameters of superconducting circuits may be 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. First, all electrical elements must be described by the condensate wavefunction amplitude and phase, rather than by the closely related macroscopic current and voltage description used for classical circuits. For instance, the square of the wavefunction amplitude at an arbitrary point in space corresponds to the probability of finding a charge carrier there. Hence, the squared amplitude corresponds to a classical charge distribution. The second requirement for obtaining 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.

Technology[edit]
Superconducting quantum computing devices are typically designed in the radio-frequency spectrum, cooled down in dilution refrigerators below 15mK (milli-Kelvin) and addressed with conventional electronic instruments, e.g. frequency synthesizers and spectrum analyzers. Typical dimensions fall on the scale of micrometers with sub-micrometer resolution, allowing for the convenient design of a Hamiltonian system with well-established integrated circuit technology.

One distinguishable feature of superconducting quantum circuits is the usage of a Josephson junction - an electrical element which is nonexistent in typical conductors. Recall that a junction is a weak connection between two leads of a wire (in this case a superconductive wire), usually implemented as a thin layer of insulator by shadow evaporation technique. The condensate wavefunctions on the two sides of the junction are weakly correlated, meaning that they are allowed to have different superconducting phases. This distinction is contrary to that of a continuous superconducting wire, wherein the superconducting wavefunction must be continuous. Current through the junction occurs by quantum tunneling, seemingly instantaneously “tunneling” from one side of the junction to the other. This phenomenon is used to create non-linear inductance which is essential for qubit design as it allows a design of anharmonic oscillators. On the contrary, quantum harmonic oscillator cannot be used as a qubit, as there is no way to address only two of its states.

Transmons:

Qubit archetypes[edit]
The three superconducting qubit archetypes are phase, charge and flux qubits, with the primary two archetypes being the charge and flux qubits which depend on amplitude and phase respectively. Many hybridizations of these archetypes exist, including Fluxonium, transmon, Xmon, Quantronium. For any qubit implementation, the logical quantum states map to different states of the physical system, which are typically either discrete (quantized) energy levels or quantum superpositions. In charge qubits, energy levels correspond to an integer number of Cooper pairs on a superconducting island; in flux qubits, energy levels correspond to integer numbers of magnetic flux quanta trapped in a superconducting ring; in phase qubits, energy levels correspond to quantum charge oscillation amplitudes across a Josephson junction, where charge and phase are analogous to momentum and position of a quantum harmonic oscillator. Note that in this particular context phase is the complex argument of the superconducting wavefunction (also known as the superconducting order parameter), not the phase between different states of the qubit.

In the table below, the three archetypes are reviewed. In the first row, a qubit 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. Particle mass corresponds to some inverse function of the circuit capacitance, while the shape of the potential is governed by regular inductors and Josephson junctions. The basic challenges of qubit design are shaping the potential and selecting particle mass such that the energy difference between two specific energy levels differs from all other inter-level energy separations in the system. These two levels are used as logical states of the qubit. Schematic wave solutions in the third row of the table depict the complex amplitude of the phase variable. Specifically, if a qubit’s phase is measured while the qubit occupies a specific state, there is a non-zero probability of measuring a specific value only where the depicted wavefunction oscillates. All three rows are essentially three different presentations of the same physical system.

Single qubits[edit]
The GHz energy gap between energy levels of a superconducting qubit is intentionally designed to be compatible with available electronic equipment, due to the terahertz gap - lack of equipment in the higher frequency band. Additionally, the superconductor energy gap implies a top limit of operation below ~1THz, beyond which Cooper pairs break. Similarly, energy separation cannot be too small due to cooling considerations: a temperature of 1K implies energy fluctuations of 20 GHz. Temperatures of tens of milli-Kelvin (achieved in dilution refrigerators) allow qubit operation at ~5 GHz energy level separation. Qubit energy level separation may often be adjusted by controlling a dedicated bias current line, providing a "knob" to fine-tune the qubit parameters.

Single qubit gates[edit]
An arbitrary 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 the 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 the microwave pulse, while pulse length determines the angle of rotation.

Following the notation of for a driving signal

of frequency, a driven qubit Hamiltonian in a rotating wave approximation is formally written

where is qubit resonance and  are Pauli matrices.

To implement a rotation about the axis, one can set  and apply a microwave pulse at frequency  for time. The resulting transformation is, which is exactly the

rotation operator by angle  about the  axis in Bloch sphere representation. Any arbitrary rotation about the axis can be implemented in a similar way. Showing the two rotation operators is sufficient for universality, as every single qubit unitary operator may be presented as  (up to a global phase, that is physically unimportant) by a procedure known as the  decomposition.

For example, setting results with a transformation, that is known as the NOT gate (up to the global phase ).

Coupling qubits[edit]
Coupling qubits is essential for implementing 2-qubit gates. Coupling two qubits may be achieved by connecting them to an intermediate electrical coupling circuit. This circuit may be either a fixed element (such as a capacitor) or a controllable element (such as a DC-SQUID). In the first case, decoupling the qubits (during the time the gate is off) is achieved by tuning the qubits out of resonance one from another, i.e. making the energy gaps between their computational states different. This approach is inherently limited to allow nearest-neighbor coupling only, as a physical electrical circuit must be laid out between the connected qubits. Notably, D-Wave Systems' nearest-neighbor coupling achieves a highly connected unit cell of 8 qubits in Chimera graph configuration. Generally, quantum algorithms require coupling between arbitrary qubits. The connectivity limitation, therefore, is likely to require multiple swap operations, limiting the length of the possible quantum computation before processor decoherence.

Another method of coupling two or more qubits is by way of an intermediate quantum bus. The 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 one with the other, eliminating the nearest-neighbor limitation. The formalism used to describe this coupling is cavity quantum electrodynamics, where 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 may be used for direct implementation of multi-qubit gates. Following the dark state manifold, the Khazali-Mølmer scheme performs complex multi-qubit operations in a single step, providing a substantial shortcut to the conventional circuit model.

Cross resonant gate[edit]
One popular gating mechanism is by using 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 rotation of the second qubit. The rotation direction depends on the state of the first qubit, allowing a controlled phase gate construction. Following the notation of, the drive Hamiltonian describing the system excited through the first qubit driving line is formally written, where is the shape of the microwave pulse in time,  is the resonance frequency of the second qubit,  are the Pauli matrices,  is the coupling coefficient between the two qubits via the resonator,  is the qubit detuning,  is the stray (unwanted) coupling between qubits, and  is Planck constant divided by. The time integral over 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.

Qubit readout[edit]
Architecture-specific readout (measurement) mechanisms exist. The readout of a phase qubit is explained in the qubit archetypes table above. A state of the flux qubit is often read using an adjust DC-SQUID magnetometer. A more general readout scheme includes coupling to a microwave resonator, where the resonance frequency of the resonator is depressively shifted by the qubit state.

DiVincenzo's criteria[edit]
The list of DiVincenzo's criteria for a physical system to implement a logical qubit is satisfied by the superconducting implementation. Although DiVincenzo's criteria as originally proposed consists of five criterium required for physically implementing a quantum computer, the more complete list consists of seven criterium 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 criterium ensure successful quantum computing, while the final two criterium allow for quantum communication.


 * 1) A scalable physical system with well characterized qubits. As the superconducting qubits are fabricated on a chip, the many-qubit system is readily scalable, with qubits allocated on the 2D surface of the chip. Much of the current development effort is to achieve an interconnect, control and readout in the third dimension, with additional lithography layers. The demand of well characterized qubits is fulfilled with (a) qubit non-linearity, accessing only two of the available energy levels and (b) accessing a single qubit at a time, rather than the entire many-qubit system, by per-qubit dedicated control lines and/or frequency separation (tuning out) of the different qubits.
 * 2) The ability to initialize the state of qubits to a simple fiducial state. One simple way to initialize a qubit is to wait long enough for the qubit to relax to its energy ground state. In addition, controlling the qubit potential by the tuning knobs allows faster initialization mechanisms.
 * 3) Long relevant decoherence times. Decoherence of superconducting qubits is affected by multiple factors. Most of it is attributed to the quality of the Josephson junction and imperfections in the chip substrate. Due to their mesoscopic scale, the superconducting qubits are relatively short lived. Nevertheless, thousands of gate operations have been demonstrated in many-qubit systems.
 * 4) A “universal” set of quantum gates. Superconducting qubits allow arbitrary rotations in the Bloch sphere with pulsed microwave signals, thus implementing arbitrary single qubit gates.  and  couplings are shown for most of the implementations, thus complementing the universal gate set.
 * 5) Qubit-specific measurement ability. In general, single superconducting qubit may be addressed for control or measurement.
 * 6) Interconvertibility of stationary and flying qubits.
 * 7) Reliable transmission of flying qubits between specified locations.

Challenges
Challenges faced by the superconducting approach are mostly lie the field of microwave engineering.

References[edit]

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External links[edit]

 * IBM Quantum offers access to over 20 quantum computer systems.
 * The IBM Quantum Experience offers free access to writing quantum algorithms and executing them on 5 qubit quantum computers.
 * IBM's roadmap for quantum computing shows 65 qubit systems available in 2020 and 127 qubits to be available sometime in 2021.

Categories:


 * Quantum information science
 * Quantum electronics
 * Superconductivity " (citations missing)