Third in a series: Even big superconductors are quantized.

While diamond nitrogen-vacancy centers offer one attractive implementation of quantum qubits, many other systems have been proposed. In theory, at least, any system with clearly identifiable quantum states can serve the purpose. The challenge lies in finding a system in which those states can be manipulated and measured by external forces and can be fabricated in large enough numbers for practical computations.

The techniques of integrated circuit manufacturing are a natural fit for high volume production of thin film structures, and indeed researchers have investigated qubits based on semiconductor quantum dots. As seen in the discussion of nitrogen-vacancy centers, however, scaling to large numbers of very small circuit elements that depend on short range interactions is problematic. An alternative approach, discussed in this article, seeks to exploit the macroscopic quantum behavior of superconductors.

In superconductors, all electrical resistance vanishes below some critical temperature. In the superconducting state, conduction electrons form ordered pairs, known as Cooper pairs. While lattice vibrations are still present, they effectively nudge the electrons along rather than impeding their progress. The physics of superconductivity is beyond the scope of this article, but it has a number of implications that are relevant to potential qubit devices. For example, at the transition to the superconducting state, magnetic fields are expelled from the material. Moreover, any magnetic flux through a loop of superconducting material is quantized: the combination of external flux and flux induced by current flowing in the loop will be a multiple of the magnetic flux quantum (*h*/2*e*).

Because the superconductor has no resistance, a current induced in the loop will persistent indefinitely, as long as the critical temperature is maintained. Moreover, this behavior is independent of the physical dimensions of the loop: it could be a few microns in diameter, or a few meters. Superconducting coils are used in the fabrication of the world’s most powerful magnets.

These behaviors derive from the nature of superconductivity: the formation of Cooper pairs, a quantum phenomenon, takes place on a macroscopic scale. Implementing qubits, however, requires a non-linear element of some kind. Multiple energy levels need to exist, and non-linear spacing between them is necessary to differentiate between the chosen basis states (|0> and |1>) and other energy states of the system. In superconductors, this non-linear element is the Josephson junction.

**Josephson junctions add non-linearity**

A Josephson junction is simply a weak link between two superconducting wires, such as a layer of AlO_{2} between two aluminum segments. It must be thin enough to allow quantum tunneling, with the barrier height defined by the dielectric properties and thickness of the insulator. Superconducting qubits are generally made from niobium, aluminum, or related alloys and have critical temperatures of 10 Kelvin (-263 °C) or less. Superconductors with critical temperatures as high as 138 K (-135 °C) do exist. High critical temperatures are desirable for many applications because they reduce cooling costs. For qubits, however, it’s necessary to eliminate as many potential noise sources as possible. Most proposed qubits, superconducting or not, operate at milliKelvin temperatures.

The simplest possible superconducting qubit design is a loop connecting a Josephson junction to an inductor. Applying a sinusoidal magnetic pulse to the inductor induces a persistent current in the loop, which flows in a superposition of clockwise and counterclockwise directions. This device, an RF-SQUID, is well characterized and is used commercially in applications requiring extremely sensitive magnetic field measurements. It is a macroscopic device in that parameters like the inductance and the thickness of the insulating barrier are not quantized: they can be adjusted to any convenient values during the circuit design and fabrication process.

In some ways, this flexibility is a boon. RF-SQUIDS do not require atomic-level deposition, state-of-the-art lithography, or other nanometer-scale process steps. Typical dimensions are on the order of microns. On the other hand, the devices face all the same sources of process variation as any other analog thin film circuit element. Dielectric thickness and purity, metal dimensions, and so on may vary from one device to the next. These in turn will cause variations in the inductance and tunneling barrier height, the two key parameters that define the behavior of the qubits.

**Different points of view: energy and flux**

More precisely, as explained in Part 1 of this series, any qubit must have two clearly defined states, |0> and |1>. The physical interpretation of these states depends on the system being used. In qubits based on nitrogen-vacancy centers in diamond, the states are derived from energy levels of the electrons associated with the center. In qubits based on Josephson junctions, there are two possible sets of basis states, discussed in more detail by R. Harris and colleagues at D-Wave Systems.

The Hamiltonian for the basic RF-SQUID can be written:

Where Q_{q} is the charge accumulated at the junction, C_{q} is the junction capacitance, and V(φ_{q}) is the potential energy as a function of the phase change across the junction. For appropriately chosen device parameters, the potential energy will be bistable, with a potential energy barrier between two local minima. These two states, |g> and |e> form the “energy” basis for the qubit.

Alternatively, the “flux” basis uses the symmetric and antisymmetric combinations of the energy eigenstates:

The choice of basis does not change the underlying physics of the qubit: the two basis states describe the same device, behaving in the same way. The choice of basis does, however, have implications for the design of the surrounding computational system. Using the energy basis facilitates long coherence times, as the energy states involved are relatively stable in the face of noise. The flux basis offers two important design advantages, however. First, because it does not depend on manipulations of energy states, all computations can take place with the system in its thermodynamically favorable ground state. Second, use of the flux basis facilitates manipulation of the qubit states through induction, a particularly convenient control and readout mechanism.

Unfortunately, the inductance present in the system is subject to process variation, making computations using the flux basis more sensitive as well. Harris and colleagues discussed modifications to the basic RF-SQUID design that can compensate for process variation while still maintaining the quantum characteristics of the system.

**Figure: **Unit cell of D-Wave Systems processor, showing array of qubits. Image courtesy D-Wave Systems, Inc.

There are other sources of noise in the system, too. One of these, somewhat surprisingly for a system that operates at milliKelvin temperatures, is heat. While the superconducting components of the system are lossless, a working circuit will also need resistors, capacitors, and other conventional elements. These dissipate energy, leading to localized heating. Thermal conductivity decreases with temperature; systems operating near absolute zero do not dissipate heat very well. Local heating adds noise, degrading the quantum superposition. Worse, it can be hazardous to the health and reliability of the system. The transition from normal to superconducting behavior is discontinuous; the normal state resistance just above the critical temperature can be substantial. If the critical temperature is exceeded, the sudden transition to the normal state can cause potentially damaging thermal and mechanical stress.

The flux basis offers an important algorithmic advantage, too. The qubits can be treated as a collection of interacting magnetic spins, a system known to physicists as an Ising spin glass. Most discussions of practical quantum computing algorithms have assumed gate-based computation, similar to conventional digital logic. However, as Eric Ladizinsky, chief scientist at D-Wave Systems, explained, the Ising model is fundamentally a thermodynamic optimization problem. As such, it can be used to describe a wide variety of systems with interacting components, ranging from gas-liquid phase transitions to social phenomena like urban segregation. A system of superconducting qubits as described here doesn’t just model an Ising spin glass, it *is* one, potentially offering a computationally elegant solution to problems of this type. Researchers at D-Wave Systems describe this approach as an “adiabatic quantum optimization (AQO) processor.” The AQO has, D-Wave claims, actually been implemented in commercially available hardware, while gate-based computational systems have not. The next and final article in this series considers how the two approaches differ at the algorithmic level, and some of the strengths and weaknesses of both.

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