Quantum
ComputersQuantum
Computers
Despite the ever increasing speed of modern processors, some problems just take too long. Working out how chemicals bind together to make cells function, organizing the best way to distribute a product and minimising traffic delays in a road network are all problems which modern computers struggle with. Another example is finding the prime factors of large numbers. Physicists have discovered that quantum computation could solve these kinds of hard problems. By harnessing the power of the quantum world we can perform all the necessary computational steps simultaneously in a single run of the machine.
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"Efficient linear optics quantum computation", E.Knill, L. Laflamme and G.J.Milburn, Nature 409, 46 (2001).
"Teleportation with the entangled states of a beamsplitter", P.Cochrane and G.J.Milburn, Submitted to Phys. Rev. A (2001).
"Quantum Phase Transitions in a Linear Ion Trap", G.J.Milburn and P.Alsing to appear in Dan Walls Memorial (Springer, 2000).
" Teleportation using coupled oscillator states ", P.T. Cochrane, G.J. Milburn, W.J. Munro, Rev. A 62, 062307 (2000).
"Entangled coherent-state qubits in an ion trap" Munro,W.J. Milburn,G.J. and Sanders,B.C., Phys. Rev. A 62, 052108 (2000).
Until recently it was thought that QC with photons required large third order (Kerr type) optical nonlinearity. Such interactions are typically slow and noisy. W have recently discoverede that single photon sources, passive linear optics and particle detectors are sufficient for implementing reliable quantum algorithms. Feedback from the detectors to the optical elements is required for this implementation. Without feedback, non-deterministic quantum computation is possible. There are classical algorithms for predicting the outcome of an experiment using the above elements whose complexity grows exponentially in the number of modes measured. A non-deterministic single photon source sufficient for quantum computation can be built with an active linear optical element (squeezer) and a photodetector. The same methods also work for fermionic modes.
Ion traps enable as few as one ion to be held in a oscillating electromagnetic field for long periods. Laser cooling enables the ions to be placed in the ground state of their lowest order collective vibrational mode. Two internal electronic states of the ions are used to encode the information. External lasers, directed at individual ions, enable the internal states to entangle the vibrational mode and the electronic state of many ions. Following the suggestion of Cirac and Zoller for an ion trap quantum computer, a number of labs are actively pursuing ion traps for studying large quantum entanglement and information processing.
A variety on interacting many body systems may be 'synthesised' in an ion trap QC architecture, for example the Tavis-Cummings model can be realised in a linear ion trap of N ions with the bosonic degree of freedom appearing as the quantised collective centre-of-mass motion. If each ion is coupled to the vibrational motion using an identical external (classical) laser detuned to the first red-sideband transition, the symmetry is such that the electronic degree of freedom for the ions can be described as a collective spin (N) and the reversible dynamics is well described by the TC model. The TC model is known to exhibit important nonlinear quantum effects including a quantum phase transition\cite{Reiger} in which the (zero temperature) ground state undergoes a morphological change as a parameter is varied and averages of intensive quantities undergo a bifurcation.
Quantum phase transitions occur at zero temperature and are driven by quantum fluctuations. By introducing the new tools of quantum information theory into the study of quantum phase transitions our aim is to determine what is essentially quantum about such phase transitions, illuminating the quantum nature of the correlations in the system, and explaining how the transition to classical behaviour occurs as the temperature of the system is raised, destroying the entanglement in the system which gives it its essentially quantum nature.
Existing work on quantum phase transitions is based on methods adapted from the classical theory of phase transitions . Although these techniques are powerful and allow considerable progress to be made, in many instances they throw out much of what is ``quantum'' about the system in question. For example, quantum phase transitions typically involve the presence of long-range correlations in the quantum system, correlations due to the presence of quantum entanglement. Work on quantum information theory has shown that the presence of entanglement in a quantum system typically gives rise to correlations which cannot be explained using classical statistics. However, the standard technique used to study these correlations in existing theory is based on classical correlation functions which do not show the presence of this essentially quantum element.
In the solid state quantum computer (SSQC) of Kane[Nature 393, 133 (1998)], the qubits are the nuclear spin-half degree of freedom of each phosphorus dopant. We need to be able to control the spin of a single donor nucleus (one qubit A-gate) and also induce a controlled interaction between two neighbouring nuclear spins (two qubit J-gate). We also need to be able to readout the nuclear spin state of a single nuclear spin with high efficiency. By high efficiency we mean that if the nuclear spin is down the measurement must reveal this with a probability approaching unity (and conversely). A mistake in the readout of a nuclear qubit is an important source of error in the operation of the Kane SSQC. One of the main objectives of the theoretical modelling is to calculate this efficiency.
We are thus faced with a very different scenario to that usually encountered in solid state. Typically, the measured signal in a solid state experiment is the average signal made up from measurements on a large number of roughly similar elements. For example in photoluminescence spectroscopy of quantum dots an optical signal is produced that is the incoherent superposition of luminescence from very many dots all with slightly varying physical characteristics ( this leads to a substantial broadening of the PL spectrum). In contrast for the Kane SSQC we need to be able to make a sequence of measurements on one and the same quantum dot at fixed and pre-determined times. Indeed for certain measurements we may need to continuously monitor the state of the nuclear qubit even in the presence of interactions with other nearby qubits .
The nuclear spins are coupled ( via the hyperfine interaction) to a single spin-half electron weakly bound at the donor site. The nuclear spin is controlled by controlling the electronic charge distribution of this weakly bound electron using a surface gate above a single door for single qubit operations. To effect two qubit operations we couple (exchang interaction) the two electrons of neighbouring donors via a surface gate placed the between two electron spins at two donors for two qubit operations. Thus all manipulations on the actual qubits proceed via weakly bound electronic spin states at each donor site.
To readout the nuclear spin of a single qubit we must also work though the electronic states of two donors. One donor is th two be measured nuclear qubit (called the Target) and the other donor forms the first stage of the measurement system (called the Apparatus). The objective of the first stage of the measurement is to transfer an electron from the target donor to the Apparatus donor conditional on the state of the Target nuclear spin. This is done by suitably biasing the A gates (asymmetrically ) on both the Target and Apparatus qubit and sweeping the bias on the J gate between the Target and the apparatus donor. In other words we can induce a static electric {\em polarisation} of the electronic state of the Target and Apparatus donor conditioned on the nuclear spin of the target. This is a static dipole induced over a distance of the donor separation. One of the objectives of the theoretical modelling is to calculate the strength of this electric dipole in a two donor system in Si with the surface gates.
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Recent technological advances enable one to prepare, control and measure the quantum state of a single system thus promoting the new field of quantum information processing. Central to these new developments is a reconsideration of quantum entanglement; the correlations between physical systems that cannot be accounted for using classical physics. It is now widely recognised that control of quantum entanglement enables new classes of measurement, communication and computational systems, which in some cases can dramatically outperform the non-quantum analogues.
The aim of this project is to develop new practical schemes for high precision measurement in optics and electronics, using quantum entanglement and error correction. Entanglement enables a more efficient use of physical resources than classical physics, and increases the precision of measurements. The proposed schemes are all based on current laboratory technologies (cavity QED, ion-traps, and mesoscopic electronics) and reasonable extrapolations of those technologies.