# Flux qubit

In quantum computing, and more specifically in superconducting quantum computing, **flux qubits** (also known as **persistent current qubits**) are micrometer sized loops of superconducting metal interrupted by a number of Josephson junctions, functioning as quantum bits. The flux qubit was first proposed by Terry P. Orlando et al. at MIT in 1999.[1] 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 (screen or enhance) 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.

Computational operations are performed by pulsing the qubit with microwave frequency radiation which has an energy comparable to that of the gap between the energy of the two basis states. Properly selected pulse duration can put the qubit into a quantum superposition of the two basis states while subsequent pulses can manipulate the probability weighting that the qubit will be measured in either of the two basis states, thus performing a computational operation.

## Fabrication

Flux qubits are fabricated using techniques similar to those used for microelectronics. The devices are usually made on silicon or sapphire wafers using electron beam lithography and metallic thin film evaporation processes. To create Josephson junctions, a technique known as shadow evaporation is normally used; this involves evaporating the source metal alternately at two angles through the lithography defined mask in the electron beam resist. This results in two overlapping layers of the superconducting metal, in between which a thin layer of insulator (normally aluminum oxide) is deposited.

## Flux qubit parameters

The flux qubit is distinguished from other types of superconducting qubit such as the charge qubit or phase qubit by the coupling energy and charging energy of its junctions. In the charge qubit regime the charging energy of the junctions dominates the coupling energy, while in a flux qubit the situation is reversed and the coupling energy dominates. Typically in a flux qubit the coupling energy is 10-100 times greater than the charging energy. It is this ratio that allows the Cooper pairs to flow continuously around the loop, rather than tunnel discretely across the junctions as in a charge qubit.

## Coupling

Coupling between two or more qubits is essential to implement many-qubit gates. The two basic coupling mechanisms are the direct inductive coupling and coupling via a microwave resonator. In the direct coupling, the circulating currents of the qubits inductively affect one another - clockwise current in one qubit induces counter-clockwise current in the other. In the Pauli Matrices formalism, a *σ*_{z}*σ*_{z} term appears in the Hamiltonian, essential for the controlled NOT gate implementation.[2] The direct coupling might be further enhanced by kinetic inductance, if the qubit loops are made to share an edge, so that the currents will flow through the same superconducting line. Inserting a Josephson junction on that joint line will add a Josephson inductance term, and increase the coupling even more. To implement a switchable coupling in the direct coupling mechanism, as required to implement a gate of finite duration, an intermediate coupling loop may be used. The control magnetic flux applied to the coupler loop switches the coupling on and off, as implemented, for example, in the D-Wave Systems machines. The second method of coupling utilizes an intermediate microwave cavity resonator, commonly implemented in a coplanar waveguide geometry. By tuning the energy separation of the qubits to match the one of the resonator, the phases of the loop currents are synchronized, and a *σ*_{x}*σ*_{x} coupling is implemented. Tuning the qubits in and out of resonance (for example, by modifying their bias magnetic flux) controls the duration of the gate operation.

## Readout

Like all quantum bits, flux qubits require a suitably sensitive probe coupled to it in order to measure its state after a computation has been carried out. Such quantum probes should introduce as little back-action as possible onto the qubit during measurement. Ideally they should be decoupled during computation and then turned "on" for a short time during read-out. Read-out probes for flux qubits work by interacting with one of the qubit's macroscopic variables, such as the circulating current, the flux within the loop or the macroscopic phase of the superconductor. This interaction then changes some variable of the read-out probe which can be measured using conventional low-noise electronics. The read-out probe is typically the technology aspect that separates the research of different University groups working on flux qubits.

Prof. Mooij's group at Delft in the Netherlands,[3] along with collaborators, has pioneered flux qubit technology, and were the first to conceive, propose and implement flux qubits as they are known today. The Delft read-out scheme is based on a SQUID loop that is inductively coupled to the qubit, the qubit's state influences the critical current of the SQUID. The critical current can then be read-out using ramped measurement currents through the SQUID. Recently the group has used the plasma frequency of the SQUID as the read-out variable.

Dr. Il'ichev's group at IPHT Jena in Germany[4] are using impedance measurement techniques based on the flux qubit influencing the resonant properties of a high quality tank circuit, which, like the Delft group is also inductively coupled to the qubit. In this scheme the qubit's magnetic susceptibility, which is defined by its state, changes the phase angle between the current and voltage when a small A.C. signal is passed into the tank circuit.

Recently Prof. Petrashov's group at Royal Holloway [5] are using an Andreev interferometer probe to read out flux qubits. This read-out uses the phase influence of a superconductor on the conductance properties of a normal metal. A length of normal metal is connected at either end to either side of the qubit using superconducting leads, the phase across the qubit, which is defined by its state, is translated into the normal metal, the resistance of which is then read-out using low noise resistance measurements.

## References

- Orlando, T. P.; Mooij, J. E.; Tian, Lin; Van Der Wal, Caspar H.; Levitov, L. S.; Lloyd, Seth; Mazo, J. J. (1999). "Superconducting persistent-current qubit".
*Physical Review B*.**60**(22): 15398–15413. arXiv:cond-mat/9908283. doi:10.1103/PhysRevB.60.15398. - Nielsen, Michael A.; Chuang, Isaac L. (2000).
*Quantum Computation and Quantum Information*. Cambridge University Press. ISBN 0-521-63235-8. - University of Delft - Flux Qubit Website Archived 2008-03-01 at Archive.today
- University of Jena - Flux Qubit Website Archived February 14, 2007, at the Wayback Machine
- Royal Holloway University of London - Flux Qubit Website

- Devoret, Michel H.; Martinis, John M. (2005). "Implementing qubits with superconducting integrated circuits".
*Experimental Aspects of Quantum Computing*: 163–203. doi:10.1007/0-387-27732-3_12. ISBN 978-0-387-23045-0.