ADIABATIC PROTOCOLS FOR OPERATOR MEASUREMENT BASED ENTANGLEMENT AND QUANTUM COMPUTING

Abstract

The generation and exploitation of entangled states has become one of the most important technological challenges of the twenty-first century. In order to reliably generate multi-particle entangled states, one needs controllable two (or more) qubit interactions, and there have been many proposals for suitable systems. We have recently begun investigating adiabatic transfer protocols (Greentree et al. 2004), which are distinguished from conventional, nonadiabatic approaches by their robustness and relative insensitivity to noise. Focussing on the phosphorus in silicon paradigm we have shown how such transport leads to new quasi-2D architectures (Hollenberg et al. 2006), and surprising particle coupling schemes based on operator measurements (Devitt et al. 2005; Greentree et al. 2006), which in turn may lead to efficient mechanisms for syndrome identification in quantum error correction. Here we describe the realisation of these operator measurements more fully based on adiabatic passage of a control particle, highlighting the applicability of our protocols by showing how they can improve the scalability of other implementations. Introduction The pursuit of devices which can exploit quantum coherence is widely seen as one of the most important and challenging tasks facing twenty-first Century Technology. Defined by the use of coherent quantum systems (at their simplest: qubits, two-state systems) such devices enable a range of tasks, from quantum key distribution via single qubits (Bennett and Brassard 1984) or entangled states (Ekert 1991), to protocols for quantum teleportation (Bennett et al. 1993), efficient quantum simulation (Feynmann 1982; Abrams and Lloyd 1997) and factorisation (Shor 1994). At the heart of most such protocols is the exploitation of entanglement, where qubits that have been entangled, in some sense feel an interaction even if space-like separated. In fact, entanglement is a ubiquitous feature of quantum systems: the task for quantum information devices is to generate controlled and isolated entanglement, so that only the desired qubits are entangled with each other, and deleterious entanglement to the environment avoided. Broadly speaking, entanglement can be generated in one of two ways, either by direct interactions between qubits, or by collective measurement of two (or more) particles in particular bases that are, by their very nature, entangled. Such projective measurements are called operator measurements, and a flexible mechanism to realise them forms the basis of this paper. Operator measurements are intimately linked to the concept of quantum error correction (QEC) (Nielsen and Chuang 2000), and as QEC is an extremely important task for quantum computing, it is possible that our schemes may prove extremely useful in future quantum computers (QC). Operator measurements are also compatible with the concept of ‘repeat-until-success’ quantum computing (Lim et al. 2005; Barrett and Kok 2005; Lim et al. 2006) which has potential advantages for QC in realistic settings. Our method for generating operator measurements is based on protocols to transfer particles adiabatically around a quantum network. There are now several different protocols for such spatial adiabatic passage, which are all extensions of the original STIRAP (STImulated Raman Adiabatic Passage) (Vitanov et al. 2001) protocol. Spatial adiabatic passage protocols can be mediated via electromagnetic pulses (Brandes et al. 2001; Siewert et al. 2006), or direct control of wavefunction overlap (Greentree et al. 2004; Eckert et al. 2004; Petrosyan 2006). All such schemes share a common theme, namely that a particular eigenstate, which is in the null space of the Hamiltonian, can be smoothly evolved as a function of time. Such protocols are therefore extremely robust, requiring little regard for timing provided that the adiabaticity criterion (Vitanov et al. 2001) is respected. For concreteness, we will review the MRAP protocol (Multiple Recipient Adiabatic Passage) introduced by us earlier (Devitt et al. 2005; Greentree et al. 2006), and show explicitly how it can be used to mediate operator measurements. We will then comment on some of the implications of this protocol for QC designs, especially when concepts such as defect-tolerance are introduced. Multiple-Recipient Adiabatic Passage Consider a network, defined by four quantum dots that share a single electron, and two attached qubits, as depicted in Fig. 1(a). Employing quantum communication terminology, we suppose that Alice wishes to send a particle (with information encoded in the electronic spin states) to either Bob 1 or Bob 2, or to some arbitrary superposition of the Bobs, Australian Institute of Physics 17th National Congress 2006 – Brisbane, 3-8 December 2006 RiverPhys Paper No. 474 using the network. For clarity we refer to the particle transferred through the network as the bus particle. We label the quantum dots, A (Alice), 1 B (Bob 1), 2 B (Bob 2), C (Central). Also, the Bobs have qubits 1 Q and 2 Q which will be used later. We further assume that each participant has control of the coupling of their quantum dot to the central dot, and we label this coupling Ωα for α = A, B1, B2. With all dots maintained at constant energy, the Hamiltonian for this four dot system (ignoring the spin degree of freedom) is ( ) ( ) ( ) . . 2 2 1 1 c h C B t C B t A C t H B B A + Ω + Ω + Ω = where the couplings are allowed to vary as a function of time (for example by local control of surface gates to modify the barrier potential). The null space of the Hamiltonian above is a two-dimensional space, spanned by the vectors