Source: Atharva Vidwans, https://medium.com/predict/a-parallel-approach-race-towards-building-a-perfect-quantum-computer-1f5e26f6fd07

Superconductors

A superconductor is a material conducting electricity without resistance when it becomes colder than a certain temperature. This temperature is called ‘Critical Temperature’. At this temperature, electrons move freely through the material without collision. The simplified plot below shows the relationship between resistivity and temperature for normal material and superconducting materials.

(from: https://saylordotorg.github.io/text_general-chemistry-principles-patterns-and-applications-v1.0/s16-07-superconductors.html)

In general, the resistivity of any material decreases with temperature and then becomes almost constant; see the purple line. But for a superconductor (see the green line), the material's resistivity suddenly drops to zero at the critical temperature.

The resistance drops to zero because the moving electrons in the current bump into atoms of matter. But in superconductors, Cooper pairs are formed in the material, which is absent in normal materials. This presence of Cooper pairs gives rise to zero resistivity. In condensed matter physics, a Cooper pair is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner. At higher temperatures, the Cooper pairs can be easily broken due to the thermal energy in the lattice. But at lower temperatures like the ‘Critical Temperature,’ the electrons are weakly bound to each other. The energy is not enough to break this bond. This increases the electron scattering time.

Josephson Junction

Josephson junction is a critical part of the Transmon qubit.

Schematic-diagram-of-ac-Josephson-junction-Both-electrodes-are-connected-to-an-external.png

(from: https://www.researchgate.net/figure/Schematic-diagram-of-ac-Josephson-junction-Both-electrodes-are-connected-to-an-external_fig1_332245281)

The Josephson effect happens when current flows from one superconductor to another superconductor through a weak link. This link can be an insulator, a non-superconducting metal, or a weak superconductor.

Such a junction of superconducting material and insulator is called a Josephson junction. In the usual case, the insulator tends to block the flow of electrons. But in this junction, the current will actually tunnel through the insulator. This phenomenon is called ‘Quantum Tunnelling.’

Josephson junctions act as an inductor in Transmon qubit, adding non-linearity to the energy-phase diagram.

In its simplest form, a transmon consists of an interconnected island called electrodes connected via Josephson Junction and capacitors. It is similar to an LC oscillator circuit with a parallel combination of one capacitor and one inductor called a Josephson junction(working explained in the previous section). Transmon, in its simplest form, is illustrated below,

Screenshot 2022-11-03 at 16.27.00.png

To understand why the Josephson Junction is used instead of a simple inductor in the LC harmonic oscillator, we check the drawbacks of an inductor for storing Qubits. A regular LC oscillator is as shown below,

(From Wikipedia: https://en.wikipedia.org/wiki/LC_circuit)

Where C is the capacitance, and L is the inductance. The Hamiltonian of the system (H) consists of a capacitive term which is quadratic on the charge of the capacitor, and an inductive term which is quadratic on the flux of the inductor.

The Hamiltonian of the Quantum Harmonic Oscillator (QHO) can be obtained by quantizing the Hamiltonian of a linear LC circuit. Shown below is the energy level of the system,

Screenshot 2022-11-03 at 16.54.28.png

Source: https://www.researchgate.net/figure/a-Circuit-of-an-LC-oscillator-with-inductance-L-and-capacitance-C-We-denote-the-phase_fig2_357039254

We note that the spectrum of the quantized LC oscillator is perfectly harmonic. This is because the energy levels are equally spaced. That is, the energy difference is the same for all states.

The anharmonicity (δ) is defined as the energy difference between |1⟩ →|2⟩ and |0⟩ →|1⟩ transition:

δ = ℏω₁₂−ℏω₀₁ or simply 𝜔₁₂−𝜔₀₁

This value is constant for all the energy levels. The problem with using this system as a qubit is the energy required to reach state |1> from state |0> is the same as the energy required to reach |2> from state |1>. Thus we cannot restrict the system to just two levels.

This can be explained by an example. If the current state is |1> and our intended state is |0>, when we zapped the system with energy equal to the anharmonicity value, then the state transition may occur either to state |2> or to state |0> with equal probability. Therefore, even though the states are well-defined, we cannot use this system as a qubit because it is not possible to restrict the system to two levels. We can accidentally involve some higher energy levels. Hence, the leakage from the qubit subspace is a threat and a drawback of this system.

We need a transition between two uniquely addressable levels that confines the system dynamics within two levels. The energy required for the transition of state |0> to |1> should be unique from other transition energies. This is how the transmon fundamentally differs from an LC oscillator. In transmon, induction is provided by a Josephson Junction instead of a typical coil inductor. Due to this, the inductive energy is not a Quadratic function but a cosine function. This change drastically affects the spectrum, as it disrupts the perfectly harmonic structure. Therefore, the distance between |0> and |1> is different from the |1> and |2> and all the subsequent states.

The changed spectrum due to Josephson Junction is shown below.

Screenshot 2022-11-03 at 16.58.46.png

From the above spectrum, it can be seen that ℏω₁₂ is not the same as ℏω₀₁. Moreover, the distance between the subsequent states keeps on decreasing. Due to this change, if the system is zapped with energy ℏω₀₁ when in-state |1>, the transition will only occur to state |0> with absolute certainty. This system is called transmon.

Operating on Transmon Qubits

The qubit transition frequency depends on the capacitances and inductances in the circuit. Quantum operations are performed by sending electromagnetic impulses at microwave frequencies (around 4–6 kHz) to the resonator coupled to the qubit. This frequency resonates with the energy separation between the energy levels for |0⟩ and |1⟩. The duration of the pulse controls the angle of rotation of the qubit state around a particular axis of the Bloch sphere. Therefore, different pulses form different quantum gates.

Screenshot 2022-11-03 at 17.38.45.png

Maintaining Qubits

As these transmons sit on a Quantum chip, the chip needs constant cooling. Due to this, a dilution refrigerator is used to maintain the low temperature. The inside of a dilution refrigerator is given in the image below,

Inside the Dilution Refrigerator of IBMQ Quantum

Source: IBM Q https://www.flickr.com/photos/shankrad/41285102821/sizes/l/

There are multiple stages in this refrigerator. The temperature decreases in every stage; at the last stage, the temperature is reduced to about ten millikelvins. Dilution refrigerator is necessary basically for two reasons:

The transmons use superconducting material and need a lower temperature.
For quantum-level systems, even the slightest external temperature can excite the system, causing the transition to a higher state. This dilution refrigerator is necessary to avoid accidental excitation due to external temperature.