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Novel schemes of measurement-based quantum computation
By Jens Eisert
Novel schemes of measurement-based quantum computation
Imagine one has a quantum system with many constituents prepared
in a certain entangled state in a laboratory. This could be a state
of a many-body system like cold atoms in optical lattices, say, a ground
state achieved via cooling, or a different entangled state realized
via cold collisions [2]. This could also be a state of atoms in cavities,
entangled via a light bus. Could this state be used
for quantum computing, based merely on local measurements,
but abandoning the need for any unitary control to realize quantum
gates?
Indeed, it can, if the state is a so-called cluster state:
It was the key insight by Hans Briegel and Robert Raussendorf
that local measurements on a cluster state have the same
computational power as the gate model for quantum computation [3].
This cluster state [3,4] in turn has a number of interesting, but also
quite rare properties in quantum many-body states. So what
if the state is simply not a cluster state, but just some other
state. Can we compute with it? Frankly: Can we find new resources for
measurement-based quantum computing?
Recent work of David Gross and Jens Eisert from Imperial College London
opens up an avenue to follow such a line of thought [1]. Based on ideas
from many-body theory - deriving from of matrix-product, finitely-correlated
and projected entangled pair states [5-7] - this work introduces the
notion of a computational tensor network, and gives rise to a
systematic framework of constructing new such computational models. In fact,
one arrives at new schemes for measurement-based computing: They are
different in the way randomness is compensated, or the way information is
rerouted. One also arrives at resource states that are quite radically
different from the cluster state in their entanglement and correlation properties.
In a detailed followup, written in joint work with Norbert Schuch from
Ignacio Cirac's group at the Max Planck Institute for Quantum Optics
in Munich and David Perez-Garcia from Madrid, many such new resource states
have been explored, including Kitaev's toric code states and variants thereof [8].
This approach might also well shed light on the intriguing question of the
relationship between classical efficient simulatability [9,10] and being a
universal resource for quantum computing.
So, what does one really need to have as a property of a state to
allow for universal, measurement-based quantum computing?
Could one even formulate new "DiVincenzo criteria" of what
one will ultimately need in order to realize a fully fletched
measurement-based quantum computer?
Whereas the answer to this question is largely unknown
(for important steps in this direction, however, see Ref. [11]), the work
in Refs. [1,8] demonstrates clearly what one does not need to have:
The cluster state has no correlations between constituents further away
from each other than being next-to-nearest neighbors. Typical ground
states, say, do not share this property. So is this property really
required? No, one can also have universal resource states
with long-range correlations? Then, one could think that one needs
interactions with maximally entangling power to build up universal
resources? Not quite, as one can have gates with little entangling
power, and still end up with probably universal resources
Or, cluster states are maximally mixed on each site? Does one have to
have this property? Again, the answer is no. Hence, quite intriguingly,
it turns out that one can lessen the requirements of initial states
quite significantly, while retaining a universal resource.
This suggests that to some extent, the theorist does not necessarily
have to approach the experimentalist, asking for the preparation of a
particular state, one that may possibly be too fragile with respect to
decoherence or finite temperature effects. But that the theorist may ask:
What is the state that you can prepare, then trying to construct a computational
model based on this very state that is anyway available.
In this way, it would reverse the aim of engineering the model to the state,
and not the state to the model.
[1] D. Gross, J. Eisert, Phys. Rev. Lett. 98, 220503 (2007).
[2] O. Mandel, M. Greiner, A. Widera, T. Rom, T.W. Haensch, and I. Bloch, Nature 425, 937 (2003).
[3] R. Raussendorf, H.J. Briegel, Phys. Rev. Lett. 86, 5188 (2001).
[4] M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, H.-J. Briegel, quant-ph/0602096.
[5] M. Fannes, B. Nachtergaele, R.F. Werner, Commun. Math. Phys. 144, 443 (1992).
[6] F. Verstraete, J.I. Cirac, cond-mat/0407066.
[7] F. Verstraete, J.I. Cirac, Phys. Rev. A 70, 060302(R) (2004).
[8] D. Gross, J. Eisert, N. Schuch, and D. Perez-Garcia, Phys. Rev. A, in press (2007), arXiv:0706.3401.
[9] M. Van den Nest, W. Dür, G. Vidal, H. J. Briegel, Phys. Rev. A 75, 012337 (2007).
[10] R. Jozsa, quant-ph/0603163.
[11] M. Van den Nest, W. Dür, A. Miyake, H. J. Briegel, New J. Phys. 9 204 (2007);
Phys. Rev. Lett. 97, 150504 (2006).
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