Superconducting quantum computing is an implementation of a quantum computer in superconducting electronic circuits. Research in superconducting quantum computing is conducted by Google, as of May 2016 [update] , up to nine fully controllable qubits are demonstrated in a 1D array, up to sixteen in a 2D architecture.Microsoft, IBM, Rigetti, and Intel.
Quantum computing is the use of quantum-mechanical phenomena such as superposition and entanglement to perform computation. A quantum computer is used to perform such computation, which can be implemented theoretically or physically.
Superconductivity is a phenomenon of exactly zero electrical resistance and expulsion of magnetic flux fields occurring in certain materials, called superconductors, when cooled below a characteristic critical temperature. It was discovered by Dutch physicist Heike Kamerlingh Onnes on April 8, 1911, in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum mechanical phenomenon. It is characterized by the Meissner effect, the complete ejection of magnetic field lines from the interior of the superconductor during its transitions into the superconducting state. The occurrence of the Meissner effect indicates that superconductivity cannot be understood simply as the idealization of perfect conductivity in classical physics.
An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. To be referred to as electronic, rather than electrical, generally at least one active component must be present. The combination of components and wires allows various simple and complex operations to be performed: signals can be amplified, computations can be performed, and data can be moved from one place to another.
More than two thousand superconducting qubits are in a commercial product by D-Wave Systems, however these qubits implement quantum annealing instead of a universal model of quantum computation.
D-Wave Systems, Inc. is a quantum computing company, based in Burnaby, British Columbia, Canada. D-Wave claims to be the world's first company to sell computers which exploit quantum effects in their operation, although some researchers have expressed skepticism in these claims.
Quantum annealing (QA) is a metaheuristic for finding the global minimum of a given objective function over a given set of candidate solutions, by a process using quantum fluctuations. Quantum annealing is used mainly for problems where the search space is discrete with many local minima; such as finding the ground state of a spin glass. It was formulated in its present form by T. Kadowaki and H. Nishimori (ja) in "Quantum annealing in the transverse Ising model" though a proposal in a different form had been made by A. B. Finilla, M. A. Gomez, C. Sebenik and J. D. Doll, in "Quantum annealing: A new method for minimizing multidimensional functions".
Classical computation models rely on physical implementations consistent with the laws of classical mechanics.It is known, however, that the classical description is only accurate for specific cases, while the more general description of nature is given by quantum mechanics. Quantum computation studies the application of quantum phenomena, that are beyond the scope of classical approximation, for information processing and communication. Various models of quantum computation exist, however the most popular models incorporate the concepts of qubits and quantum gates. A qubit is a generalization of a bit - a system with two possible states, that can be in a quantum superposition of both. A quantum gate is a generalization of a logic gate: it describes the transformation that one or more qubits will experience after the gate is applied on them, given their initial state. The physical implementation of qubits and gates is difficult, for the same reasons that quantum phenomena are hard to observe in everyday life. One approach is to implement the quantum computers in superconductors, where the quantum effects become macroscopic, though at a price of extremely low operation temperatures.
Computation is any type of calculation that includes both arithmetical and non-arithmetical steps and follows a well-defined model, for example an algorithm.
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.
Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.
In a superconductor, the basic charge carriers are pairs of electrons (known as Cooper pairs), rather than the single electrons in a normal conductor. The total spin of a Cooper pair is an integer number, thus the Cooper pairs are bosons (while the single electrons in the normal conductor are fermions). Cooled bosons, contrary to cooled fermions, are allowed to occupy a single quantum energy level, in an effect known as the Bose-Einstein condensate. In a classical interpretation it would correspond to multiple particles occupying the same position in space and having an equal momentum, effectively behaving as a single particle.
In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.
In quantum mechanics, a boson is a particle that follows Bose–Einstein statistics. Bosons make up one of the two classes of particles, the other being fermions. The name boson was coined by Paul Dirac to commemorate the contribution of Indian physicist and professor of physics at University of Calcutta and at University of Dhaka, Satyendra Nath Bose in developing, with Albert Einstein, Bose–Einstein statistics—which theorizes the characteristics of elementary particles.
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. These particles obey the Pauli exclusion principle. Fermions include all quarks and leptons, as well as all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.
At every point of a superconducting electronic circuit (that is a network of electrical elements), the condensate wave function describing the charge flow is well-defined by a specific complex probability amplitude. In a normal conductor electrical circuit, the same quantum description is true for individual charge carriers, however the various wave functions are averaged in the macroscopic analysis, making it impossible to observe quantum effects. The condensate wave function allows designing and measuring macroscopic quantum effects. For example, only a discrete number of magnetic flux quanta penetrates a superconducting loop, similarly to the discrete atomic energy levels in the Bohr model. In both cases, the quantization is a result of the complex amplitude continuity. Differing from the microscopic quantum systems (such as atoms or photons) used for implementations of quantum computers, the parameters of the superconducting circuits may be designed by setting the (classical) values of the electrical elements that compose them, e.g. adjusting the capacitance or inductance.
Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electrical elements interconnected by wires. Where the elements roughly correspond to real components the representation can be in the form of a schematic diagram or circuit diagram. This is called a lumped element circuit model. In other cases infinitesimal elements are used to model the network in a distributed element model.
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ or Ψ.
In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.
In order to obtain a quantum mechanical description of an electrical circuit a few steps are required. First, all the electrical elements are described with the condensate wave function amplitude and phase, rather than with the closely related macroscopic current and voltage description used for classical circuits. For example, a square of the wave function amplitude at some point in space is the probability of finding a charge carrier there, hence the square of the amplitude corresponds to the classical charge distribution. Second, generalized Kirchhoff's circuit laws are applied at every node of the circuit network to obtain the equations of motion. Finally, the equations of motion are reformulated to Lagrangian mechanics and a quantum Hamiltonian is derived.
An electric current is a flow of electric charge. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).
Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named volt. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for volt uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by ∆V, but more often simply as V, for instance in the context of Ohm's or Kirchhoff's circuit laws.
Kirchhoff's laws are two equalities that deal with the current and potential difference in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhoff. This generalized the work of Georg Ohm and preceded the work of James Clerk Maxwell. Widely used in electrical engineering, they are also called Kirchhoff's rules or simply Kirchhoff's laws. These laws can be applied in time and frequency domains and form the basis for network analysis.
The devices are typically designed in the radio-frequency spectrum, cooled down in dilution refrigerators below 100mK and addressed with conventional electronic instruments, e.g. frequency synthesizers and spectrum analyzers. Typical dimensions on the scale of micrometers, with sub-micrometer resolution, allow a convenient design of a quantum Hamiltonian with the well-established integrated circuit technology.
A distinguishing feature of superconducting quantum circuits is the usage of a Josephson junction - an electrical element non existent in normal conductors. A junction is a weak connection between two leads of a superconducting wire, usually implemented as a thin layer of insulator with a shadow evaporation technique. The condensate wave functions on the two sides of the junction are weakly correlated - they are allowed to have different superconducting phases, contrary to the case of a continuous superconducting wire, where the superconducting wave function must be continuous. The current through the junction occurs by quantum tunneling. This is used to create a non-linear inductance which is essential for qubit design, as it allows a design of anharmonic oscillators. A quantum harmonic oscillator cannot be used as a qubit, as there is no way to address only two of its states.
The three superconducting qubit archetypes are the phase, charge and flux qubits, though many hybridizations exist (Fluxonium, are to be mapped to the different states of the physical system, typically to the discrete (quantized) energy levels or to their quantum superpositions. In the charge qubit, different energy levels correspond to an integer number of Cooper pairs on a superconducting island. In the flux qubit, the energy levels correspond to different integer numbers of magnetic flux quanta trapped in a superconducting ring. In the phase qubit, the energy levels correspond to different quantum charge oscillation amplitudes across a Josephson junction, where the charge and the phase are analogous to momentum and position correspondingly of a quantum harmonic oscillator. Note that the phase here is the complex argument of the superconducting wavefunction, also known as the superconducting order parameter, not the phase between the different states of the qubit.Transmon, Xmon, Quantronium ). For any qubit implementation, the logical quantum states
In the table below, the three archetypes are reviewed. In the first row, the qubit electrical circuit diagram is presented. In the second, the quantum Hamiltonian derived from the circuit is shown. Generally, the Hamiltonian can be divided to a "kinetic" and "potential" parts, in analogy to a particle in a potential well. The particle mass corresponds to some inverse function of the circuit capacitance, while the shape of the potential is governed by the regular inductors and Josephson junctions. One of the first challenges in qubit design is to shape the potential well and to choose the particle mass in a way that the energy separation between specific two of the energy levels will differ from all other inter-level energy separations in the system. These two levels will be used as the logical states of the qubit. The schematic wave solutions in the third row of the table depict the complex amplitude of the phase variable. In other words, if a phase of the qubit is measured while the qubit is in a specific state, there is a non-zero probability to measure a specific value only where the depicted wave function oscillates. All three rows are essentially three different presentations of the same physical system.
|Charge Qubit||RF-SQUID Qubit (prototype of the Flux Qubit)||Phase Qubit|
, where is the number of Cooper pairs to tunnel the junction, is the charge on the capacitor in units of Cooper pairs number, is the charging energy associated with both the capacitance and the Josephson junction capacitance , and is the superconducting wave function phase difference across the junction.
, where is the charge on the junction capacitance and is the superconducting wave function phase difference across the Josephson junction. is allowed to take values greater than , and thus is alternatively defined as the time integral of voltage along the inductance .
The potential part of the Hamiltonian, , is depicted with the thick red line. Schematic wave function solutions are depicted with thin lines, lifted to their appropriate energy level for clarity. Only the solid wave functions are used for computation. The bias voltage is set so that , minimizing the energy gap between and , thus making the gap different from other energy gaps (e.g. the gap between and ). The difference in gaps allows addressing transitions from to and vice versa only, without populating other states, thus effectively treating the circuit as a two-level system (qubit).
The potential part of the Hamiltonian, , plotted for the bias flux , is depicted with the thick red line. Schematic wave function solutions are depicted with thin lines, lifted to their appropriate energy level for clarity. Only the solid wave functions are used for computation. Different wells correspond to a different number of flux quanta trapped in the superconducting loops. The two lower states correspond to a symmetrical and an antisymmetrical superposition of zero or single trapped flux quanta, sometimes denoted as clockwise and counterclockwise loop current states: and .
The so-called "washboard" potential part of the Hamiltonian, , is depicted with the thick red line. Schematic wave function solutions are depicted with thin lines, lifted to their appropriate energy level for clarity. Only the solid wave functions are used for computation. The bias current is adjusted to make the wells shallow enough to contain exactly two localized wave functions. A slight increase in the bias current causes a selective "spill" of the higher energy state (), expressed with a measurable voltage spike - a mechanism commonly used for phase qubit measurement.
The GHz energy gap between the 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. In addition, the superconductor energy gap implies a top limit of operation below ~1THz (beyond it, the Cooper pairs break). On the other hand, the energy level separation cannot be too small due to cooling considerations: a temperature of 1K implies energy fluctuations of 20 GHz. Temperatures of tens of mili-Kelvin achieved in dilution refrigerators allow qubit operation at a ~5 GHz energy level separation. The qubit energy level separation may often be adjusted by means of controlling a dedicated bias current line, providing a "knob" to fine tune the qubit parameters.
An arbitrary single qubit gate is achieved by rotation in the Bloch sphere. The rotations between the 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 the pulse length determines the angle of rotation.
More formally, following the notation of,for a driving signal
of frequency , a driven qubit Hamiltonian in a rotating wave approximation is
where is the qubit resonance and are Pauli matrices.
In order to implement a rotation about the axis, one can set and apply the microwave pulse at frequency for time . The resulting transformation is
that is exactly the rotation operator by angle about the axis in the Bloch sphere. An 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 is essential for implementing 2-qubit gates. Coupling two qubits may be achieved by connecting them to an intermediate electrical coupling circuit. The circuit might be a fixed element, such as a capacitor, or controllable, 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 is to be lay out in between the connected qubits. Notably, D-Wave Systems' nearest-neighbor coupling achieves a highly connected unit cell of 8 qubits in the Chimera graph configuration. Generally, quantum algorithms require coupling between arbitrary qubits, therefore the connectivity limitation is likely to require multiple swap operations, limiting the length of the possible quantum computation before the processor decoherence.
Another method of coupling two or more qubits is by coupling them to 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, hence eliminating the nearest-neighbor limitation. The formalism used to describe this coupling is cavity quantum electrodynamics, where qubits are analogous to atoms interacting with optical photon cavity, with the difference of GHz rather than THz regime of the electromagnetic radiation.
One popular gating mechanism includes two qubits and a bus, all 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.
More formally, following the notation of,the drive Hamiltonian describing the system excited through the first qubit driving line is
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 due to the first and third terms of the Hamiltonian can be compensated with single qubit operations. The remaining part is exactly the controlled-X gate.
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 by an adjust DC-SQUID magnetometer. A more general readout scheme includes a coupling to a microwave resonator, where the resonance frequency of the resonator is shifted by the qubit state.
The list of DiVincenzo's criteria for a physical system to implement a logical qubit is satisfied by the superconducting implementation. The challenges currently faced by the superconducting approach are mostly in the field of microwave engineering.
In engineering, a transfer function of an electronic or control system component is a mathematical function which theoretically models the device's output for each possible input. In its simplest form, this function is a two-dimensional graph of an independent scalar input versus the dependent scalar output, called a transfer curve or characteristic curve. Transfer functions for components are used to design and analyze systems assembled from components, particularly using the block diagram technique, in electronics and control theory.
Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm for integer factorization, formulated in 1994. Informally, it solves the following problem: Given an integer , find its prime factors.
A polaron is a quasiparticle used in condensed matter physics to understand the interactions between electrons and atoms in a solid material. The polaron concept was first proposed by Lev Landau in 1933 to describe an electron moving in a dielectric crystal where the atoms move from their equilibrium positions to effectively screen the charge of an electron, known as a phonon cloud. This lowers the electron mobility and increases the electron's effective mass.
Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements.
In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.
A trapped ion quantum computer is one proposed approach to a large-scale quantum computer. Ions, or charged atomic particles, can be confined and suspended in free space using electromagnetic fields. Qubits are stored in stable electronic states of each ion, and quantum information can be transferred through the collective quantized motion of the ions in a shared trap. Lasers are applied to induce coupling between the qubit states or coupling between the internal qubit states and the external motional states.
In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems. The canonical ensemble gives the probability of the system X being in state x as
In quantum computing, and more specifically in superconducting quantum computing, flux qubits are micrometer sized loops of superconducting metal interrupted by a number of Josephson junctions, functioning as quantum bits. The junction parameters are engineered during fabrication so that a persistent current will flow continuously when an external magnetic flux is applied. As only an integer number of flux quanta are allowed to penetrate the superconducting ring, clockwise or counter-clockwise currents are developed in the loop to compensate a non-integer external flux bias. When the applied flux through the loop area is close to a half integer number of flux quanta, the two lowest energy eigenstates of the loop will be a quantum superposition of the clockwise and counter-clockwise currents. The two lowest energy eigenstates differ only by the relative quantum phase between the composing current-direction states. Higher energy eigenstates correspond to much larger persistent currents, that induce an additional flux quantum to the qubit loop, thus are well separated energetically from the lowest two eigenstates. This separation, known as the "qubit non linearity" criteria, allows operations with the two lowest eigenstates only, effectively creating a two level system. Usually, the two lowest eigenstates will serve as the computational basis for the logical qubit.
The Jaynes–Cummings model is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity, with or without the presence of light. It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity.
A quantum bus is a device which can be used to store or transfer information between independent qubits in a quantum computer, or combine two qubits into a superposition. It is the quantum analog of a classical bus.
Quantum dissipation is the branch of physics that studies the quantum analogues of the process of irreversible loss of energy observed at the classical level. Its main purpose is to derive the laws of classical dissipation from the framework of quantum mechanics. It shares many features with the subjects of quantum decoherence and quantum theory of measurement.
In quantum computing, and more specifically in superconducting quantum computing, the phase qubit is a superconducting device based on the superconductor-insulator-superconductor (SIS) Josephson junction, designed to operate as a quantum bit, or qubit. The phase qubit is closely related, yet distinct from, the flux qubit and the charge qubit, which are also quantum bits implemented by superconducting devices.
In physics, Berry connection and Berry curvature are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase. These concepts were introduced by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics. Such phases have come to be known as Berry phases.
In quantum computing, the quantum Fourier transform is a linear transformation on quantum bits, and is the quantum analogue of the inverse discrete Fourier transform. The quantum Fourier transform is a part of many quantum algorithms, notably Shor's algorithm for factoring and computing the discrete logarithm, the quantum phase estimation algorithm for estimating the eigenvalues of a unitary operator, and algorithms for the hidden subgroup problem.
Circuit quantum electrodynamics provides a means of studying the fundamental interaction between light and matter. As in the field of cavity quantum electrodynamics, a single photon within a single mode cavity coherently couples to a quantum object (atom). In contrast to cavity QED, the photon is stored in a one-dimensional on-chip resonator and the quantum object is no natural atom but an artificial one. These artificial atoms usually are mesoscopic devices which exhibit an atom-like energy spectrum. The field of circuit QED is a prominent example for quantum information processing and a promising candidate for future quantum computation.
In rotational-vibrational and electronic spectroscopy of diatomic molecules, Hund's coupling cases are idealized cases where specific terms appearing in the molecular Hamiltonian and involving couplings between angular momenta are assumed to dominate over all other terms. There are five cases, traditionally notated with the letters (a) through (e). Most diatomic molecules are somewhere between the idealized cases (a) and (b).
In quantum computing, and more specifically in superconducting quantum computing, a transmon is a type of superconducting charge qubit that was designed to have reduced sensitivity to charge noise. The transmon was developed by Robert J. Schoelkopf, Michel Devoret, Steven M. Girvin and their colleagues at Yale University in 2007. Its name is an abbreviation of the term transmission line shunted plasma oscillation qubit; one which consists of a Cooper-pair box "where the two superconductors are also capacitatively shunted in order to decrease the sensitivity to charge noise, while maintaining a sufficient anharmonicity for selective qubit control".
In quantum optics, a superradiant phase transition is a phase transition that occurs in a collection of fluorescent emitters, between a state containing few electromagnetic excitations and a superradiant state with many electromagnetic excitations trapped inside the emitters. The superradiant state is made thermodynamically favorable by having strong, coherent interactions between the emitters.
The pressuron is a hypothetical scalar particle which couples to both gravity and matter theorised in 2013. Although originally postulated without self-interaction potential, the pressuron is also a dark energy candidate when it has such a potential. The pressuron takes its name from the fact that it decouples from matter in pressure-less regimes, allowing the scalar-tensor theory of gravity involving it to pass solar system tests, as well as tests on the equivalence principle, even though it is fundamentally coupled to matter. Such a decoupling mechanism could explain why gravitation seems to be well described by general relativity at present epoch, while it could actually be more complex than that. Because of the way it couples to matter, the pressuron is a special case of the hypothetical string dilaton. Therefore, it is one of the possible solutions to the present non-observation of various signals coming from massless or light scalar fields that are generically predicted in string theory.
Electric dipole spin resonance (EDSR) is a method to control the magnetic moments inside a material using quantum mechanical effects like the spin–orbit interaction. Mainly, EDSR allows to flip the orientation of the magnetic moments through the use of electromagnetic radiation at resonant frequencies. EDSR was first proposed by Emmanuel Rashba.