Geometry Enters Quantum Computing on a Microchip

The development of a quantum information processor has become a major goal of many research labs around the world. It is however a fairly recent and promising idea to implement the necessary logic operations purely by taking advantage of geometry. A research team at ETH has now managed for the first time to realize a geometric operation on a quantum bit ('qubit') created on a microchip, and report their work in this week's Science Express.

Geometry is one of the oldest and most fundamental sciences. It was Euclid who formulated geometry in flat space in an axiomatic form, and his treatment remained the standard for about two millennia before the concept of curvature entered the field. This was the point when the significance of geometry expanded from simple applications in engineering and astronomy to become an important ingredient of modern physical theories. For example in general relativity, which generalized Newtonian gravity, the gravitational force is very beautifully explained by a curvature of space and time. In the last century, geometry has become of even broader importance in mathematics and science, through the emergence of the field of 'topology' which investigates the local and global structure of space. While Plato thought of geometry as the study of static and timeless systems, in this more contemporary view, geometry is described in terms of transformation.

An illustrative example of how geometry induces transformation is the following. Raise your right arm above your head, with your thumb pointing left. Now bring your hand down so that it is in front of you, and then move your arm out to the right, without twisting it. Now raise your arm back above your head, and you will find that your thumb now points in a different direction, namely to the front. During the whole process, your arm did not twist, and yet it ends up rotated! Formally speaking this little experiment deals with a vector living on the surface of a sphere (see right inset of Figure 1) and changing its position without locally changing its direction. When the vector reaches its initial position again it ends up rotated by a certain angle due to a global property of the surface that is its curvature. For the case of a sphere, this angle turns out to be just proportional to the area enclosed by the path on the surface. Such intrinsic transformations induced by following paths through a geometric landscape are called holonomies. Corresponding effects are widespread in science with particularly important cases occurring in quantum physics, something that was first clearly pointed out by Berry only in the 1980's. In particular, such geometrical transformations are regarded as promising candidates to implement logic in a future quantum information processor.

A quantum computer is a device that processes information based on the laws of quantum mechanics, the theory that describes the properties of the microscopic world. Such computers operate by manipulating information (encoded as bits, that is 0's and 1's) stored as quanta of energy in individual atoms, photons or even in electrical circuits (see Figure 1). Since such quantum systems are perfectly fine with occupying several places at once, a qubit can therefore register '0' and '1' at the same time! A quantum processor harnesses this quantum mechanical strangeness to perform many computations at the same time. Such massive parallelism can be used to speed up a variety of computations, including factoring large numbers, which is a problem of interest in cryptography, searching databases and simulating the behavior of other quantum systems, which is desirable for the design of any devices on the nanometer scale.

The concept of using geometry to perform quantum computation was proposed several years ago and first geometric quantum logic operations have been demonstrated in nuclear magnetic resonance as well as ion trap setups only shortly after. It has also been shown theoretically that any arbitrary computation can be built from one and two-qubit holonomic operations. Geometrical quantum computation may have particular advantages. It is very flexibly customizable since many paths are possible, and even more importantly it may show an enhanced robustness to noise and errors, which constitutes the limiting factor of all experimental attempts to create a quantum computer so far.

The Quantum Device Lab at ETH Zurich (see Figure 2) works on the realization of a quantum information processor with superconducting circuits fabricated on a microchip in a scheme called circuit QED (circuit quantum electrodynamics). The qubit is realized as a small electronic circuit made from Aluminum (see Figure 1). In this circuit the number of electrons in a small rectangular box, called the island, can be controlled one by one with electric and magnetic fields to realize the qubit states '0' and '1' or any superposition of the two possibilities. As the number of charges on the island can be made stable for long times it is possible to manipulate the information stored this way using high frequency microwave signals applied to the circuit. In fact using these techniques, one can induce a subtle change in the qubit that is equivalent to the geometric angle occurring when taking a vector along a closed path on a curved surface, as shown in Figure 1. This subtle change is known as geometric phase, or in this special case Berry's phase. Since in quantum information theory the phase of a state is an important ingredient of its actual information content, it is possible to use the demonstrated controlled phase changes to realize logic gates in a superconducting quantum computer.

While some microscopic objects such as atoms and photons inherently behave quantum mechanical, in an electrical circuit one has to make some effort in order to be able to observe a subtle quantum effect such as the Berry phase. First the lifetime of a generated quantum state, has to be long enough to controllably manipulate it in a geometric fashion before it has been irreversibly corrupted. In the experiments performed at ETH Zurich, this is achieved by protecting the quantum electrical circuit from environmental noise using high quality resonator acting as a very effective filter. This trick together with careful chip design and fabrication of the circuits enabled experiments with coherence times among the largest ever observed in solid state based qubits. Many technical challenges were met in the reported experiment. Due to the small energy difference of the states representing '0' and '1' in the qubit circuit, ultra low operating temperatures of 0.02 degree above the absolute zero of temperature are necessary. This task is accomplished using a dilution refrigerator, equipped with ultra low noise and GHz frequency electronics which enables the experimenters to read-out the state of their qubits using individual microwave photons requiring them to detect extremely low powers of about 10^-17 Watt. In comparison, a cell phone emits about 10²³ (100 000 000 000 000 000 000 000) photons per seconds.

The big advantage of solid state approaches to quantum computing is the fact that as soon as the underlying physics is well understood and robust fabrication processes are developed, it should be straightforward to scale the circuits up to more and more qubits; a benefit most competing technologies do not share currently. While a collaborating group at Yale University who works with similar devices have recently demonstrated the coherent transfer of quantum information between two distant qubits mediated just by a single microwave photon, (recently featured on the cover of Nature) the ETH team has now contributed another milestone - the first observation of a geometric operation in any solid state based qubit.

J.M. Fink, P.J. Leek, A. Blais, A. Wallraff