User:Srjmas/sandbox/Superconducting quantum computing

Superconducting quantum computing is an implementation of a quantum computer in superconducting electronic circuits. Research in superconducting quantum computing is conducted by Google, IBM, Intel and the Canadian startup D-Wave.

Background and Technology

In a superconductor, the charge carriers condense to a single quantum wave function, thus an Avogadro number of particles behave as a single one. At every point of a superconducting electronic circuit, i.e. 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. 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 used for quantum computers implementations, such as atoms or photons, 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.

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 motions are reformulated to Lagrangian Mechanics and a quantum Hamiltonian is derived.

The devices are typically designed in the radio-frequency spectrum, cooled down in dilution refrigerators bellow 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.

Qubits Archetypes

The three superconducting qubit archetypes are the phase, charge and flux qubits, though many hybridizations exist (fluxonium, transmon, x-mon, quantronium). For any qubit implementation, the logical states $\{|0\rangle ,|1\rangle \}$ are to be mapped to the different states of the physical system, typically the discrete (quantized) energy levels or their quantum superposition. 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 number of magnetic flux quantum trapped in a superconducting ring. In the phase qubit, the energy levels correspond to different quantum charge oscillation amplitude 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.

	Charge Qubit	RF-SQUID Qubit (prototype of the Flux Qubit)	Phase Qubit
Circuit	Charge qubit circuit A superconducting island defined between the leads of a capacitor with capacitance $C$ and a josephson junction with energy $E_{J}$ is biased by voltage $V_{0}$	Flux qubit circuit A superconducting loop with inductance $L$ is interrupted by a junction with josephson energy $E_{J}$. Bias flux $\Phi$ is induced by a flux line with a current $I_{0}$	Phase qubit circuit Josephson junction with energy parameter $E_{J}$ is biased by a current $I_{0}$
Hamiltonian	$H=E_{C}(N-N_{g})^{2}-E_{J}\cos \phi$, where $N$ is the number of Cooper pairs to tunnel the junction, $N_{g}=CV_{0}/2e$ is the charge on the capacitor in units of Cooper pairs number, $E_{C}=(2e)^{2}/2(C_{J}+C)$ is the charging energy associated with both the capacitance $C$ and the Josephson junction capacitance $C_{J}$, and $\phi$ is the superconducting wave function phase difference across the junction.	$H={\frac {q^{2}}{2C_{J}}}+{\frac {\phi ^{2}}{2L}}-E_{J}\cos \left[{\frac {2e}{\hbar }}(\phi -\Phi )\right]$, where $q$ is the charge on the junction capacitance $C_{J}$ and $\phi$ is the superconducting wave function phase difference across the Josephson junction. $\phi$ is allowed to take values greater than $2\pi$, and thus is alternatively defined as the time integral of voltage along the inductance $L$.	$H={\frac {(2e)^{2}}{2C_{J}}}q^{2}-I_{0}{\frac {\Phi _{0}}{2\pi }}\phi -E_{J}\cos \phi$, where $C_{J}$ is the capacitance associated with the Josephson junction, $\Phi _{0}$ is the magnetic flux quantum, $q$ and $\phi$ are the charge and the phase across the junction, correspondingly.
Potential	Charge qubit potential The potential part of the Hamiltonian, $E_{J}\cos \phi$ 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.	Flux qubit potential. The potential part of the Hamiltonian, ${\frac {\phi ^{2}}{2L}}-E_{J}\cos \left[{\frac {2e}{\hbar }}(\phi -\Phi )\right]$, plotted for the bias flux $\Phi =\Phi _{0}/2$, 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.	phase qubit potential The so called 'washboard' potential part of the Hamiltonian, $-I_{0}{\frac {\Phi _{0}}{2\pi }}\phi -E_{J}\cos \phi$ 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 ($|1\rangle ,$), expressed with a measurable voltage spike - a mechanism commonly used for phase qubit measurement.

Single Qubits

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. On the other hand, the energy level separation cannot be too small due to cooling considerations, as a temperature of 1K implies energy fluctuations of 20GHz. Temperatures of tens of mili-Kelvin achieved in dilution refrigerators allow qubit operation at a ~5GHz energy level separation.

The ability to address two specific energy levels of the superconducting qubit, rather than the infinite number of levels in quantum harmonic oscillator is closely related to the sinusoidal non-linearity of the Josephson junction, hence making the former a crucial part of a qubit design. The popular fabrication technique is the shadow evaporation, with nanometric resolution achieved with Electron-beam lithography. The non-linearity of the Josephson junction, together with the fabrication difficulty, making it the analogue of the transistor in classical computing.

The rotations between the different energy levels of a single qubit are induced by microwave pulses sent to an antennae or transmission line coupled to the qubit, with a frequency resonant with the energy separation between the levels. 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.

Coupling Qubits

It also provides a way to couple different adjust qubits by tuning them in resonance with each other. This sort of coupling is described by a $\sigma _{z}\sigma _{z}$ compounds in the effective two-level Hamiltonians, essentially allowing a construction of a controlled NOT gate.

The qubits are often coupled to microwave cavities, modeled by quantum harmonic oscillators, for various applications: qubit interconnect, readout, shared quantum bus, quantum simulations. The formalism used to describe this coupling is cavity quantum electrodynamics, where qubits are analogue to atoms interacting with optical photon cavity, with the difference of GHz rather than THz regime of the electromagnetic radiation. Coupling of two qubits via an intermediate resonator coordinates their phases, resulting with a $\sigma _{x}\sigma _{x}$ coupling in the two-level Hamiltonian, allowing a construction of controlled phase gates.

DiVincenzo's Criteria

The DiVincenzo's criteria for a physical system to implement a logical qubit is satisfied by the superconducting implementation.

1. A scalable physical system with well characterised qubits. As the superconducting qubits are fabricated on a chip, the many qubits system is readily scallable, 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 charecterised 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 whole qubits system as a whole by per-qubit dedicated control lines and/or frequency separation (tuning out) of the different qubits
2. The ability to initialise the state of the 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 supercnducting qubits are relatively short lived. Nevertheless, thousandths of gate operations were demonstrated in multi-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. $\sigma _{z}\sigma _{z}$ and $\sigma _{x}\sigma _{x}$ couplings are shown for most of the implementations, thus complementing the universal gate set.
5. A qubit-specific measurement capability. In general, single superconducting qubit may be addressed for control or measurement.