Peter Shor’s demonstration [1] in the mid 1990s of an efficient algorithm for factorizing prime numbers has triggered an immense interest in various aspects of quantum computation. Researchers have proposed several ways to implement quantum computers, ranging from systems that store information in trapped atoms [2] or ions [3] to computers based on condensed matter systems such as Josephson junctions [4] and quantum dots [5]. Such computers would rely on the phenomena of quantum coherence and quantum entanglement among a set of such “qubits” (Fig. 1). Despite these efforts, quantum computers of any useful size still seem far beyond the scope of present day technology, mainly because of the difficulties in maintaining the necessary coherence of all the qubits. Achievable error probabilities for qubit manipulations are still far above the value of ~10-4 required for efficient fault-tolerant quantum computation [6]. A key challenge for quantum computation research is to achieve this precision.
One approach towards this goal is to use quantum geometric phases (that is, the effects of moving a set of quantum parameters around a curved parameter space) [7, 8, 9] to implement quantum gates that manipulate states of physical qubits. Such gates would be the quantum computing equivalent of the logic gates found on today’s microchips. The idea of using geometric phase is known as holonomic or geometric quantum computation, and has become one of the key approaches to achieving quantum computation that is resilient against errors. In 1999, Zanardi and Rasetti [10] laid the theoretical foundations of holonomic quantum computation by showing that any quantum circuit can be generated by using suitable Hamiltonians that depend on experimentally controllable parameters, such as those related to the manipulation of a bosonic mode in a quantum optical system [11]. At the same time, Jones et al. [12] demonstrated experimentally a quantum gate based on geometric phase that was able to entangle a pair of nuclear spins in a nuclear magnetic resonance (NMR) setup. This experiment provided the first explicit example of geometric quantum computation and helped to boost the interest in this field.
Holonomic quantum computation
