Limits...
Exponential rise of dynamical complexity in quantum computing through projections.

Burgarth DK, Facchi P, Giovannetti V, Nakazato H, Pascazio S, Yuasa K - Nat Commun (2014)

Bottom Line: After discussing examples, we go on to show that this effect is generally to be expected: almost any quantum dynamics becomes universal once 'observed' as outlined above.Conversely, we show that any complex quantum dynamics can be 'purified' into a simpler one in larger dimensions.We conclude by demonstrating that even local noise can lead to an exponentially complex dynamics.

View Article: PubMed Central - PubMed

Affiliation: Institute of Mathematics, Physics and Computer Science, Aberystwyth University, Aberystwyth SY23 3BZ, UK.

ABSTRACT
The ability of quantum systems to host exponentially complex dynamics has the potential to revolutionize science and technology. Therefore, much effort has been devoted to developing of protocols for computation, communication and metrology, which exploit this scaling, despite formidable technical difficulties. Here we show that the mere frequent observation of a small part of a quantum system can turn its dynamics from a very simple one into an exponentially complex one, capable of universal quantum computation. After discussing examples, we go on to show that this effect is generally to be expected: almost any quantum dynamics becomes universal once 'observed' as outlined above. Conversely, we show that any complex quantum dynamics can be 'purified' into a simpler one in larger dimensions. We conclude by demonstrating that even local noise can lead to an exponentially complex dynamics.

No MeSH data available.


Zeno effect in quantum control.(a) We control a quantum system by switching on and off a set of given Hamiltonians {H(1),…, H(n)}. (b) We perform projective measurements P at regular time intervals during the control to check whether or not the state of the system belongs to a given subspace  of the global Hilbert space. (c) In the limit of infinitely frequent measurements (Zeno limit), the system is confined in the subspace , where it evolves unitarily with the Zeno Hamiltonians {(1),..., (n)} (Zeno dynamics). The Zeno dynamics can explore the subspace  more thoroughly than the purely unitary control without measurement.
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f1: Zeno effect in quantum control.(a) We control a quantum system by switching on and off a set of given Hamiltonians {H(1),…, H(n)}. (b) We perform projective measurements P at regular time intervals during the control to check whether or not the state of the system belongs to a given subspace of the global Hilbert space. (c) In the limit of infinitely frequent measurements (Zeno limit), the system is confined in the subspace , where it evolves unitarily with the Zeno Hamiltonians {(1),..., (n)} (Zeno dynamics). The Zeno dynamics can explore the subspace more thoroughly than the purely unitary control without measurement.

Mentions: In a typical quantum control scenario, it is assumed that the system of interest (say the quantum register of a quantum computer, or the spins in an NMR experiment) can be externally driven by means of sequences of unitary pulses , activated by turning on and off a set of given Hamiltonians {H(1),…, H(n)} (Fig. 1a). If no limitations are imposed on the temporal durations τ of the pulses, it is known27 that by properly arranging sequences composed of {U(1),…, U(n)} one can in fact force the system to evolve under the action of arbitrary transformations of the form U=eΘ with the anti-Hermitian operators Θ being elements of the real Lie algebra formed by the linear combinations of iH(j) and their iterated commutators, and so on. Full controllability is hence achieved if the dimension of is large enough to permit the implementation of all possible unitary transformations on the system, that is, , with d being the dimension of the system (without loss of generality the Hamiltonians can be assumed to be traceless, since the global phase does not play any role).


Exponential rise of dynamical complexity in quantum computing through projections.

Burgarth DK, Facchi P, Giovannetti V, Nakazato H, Pascazio S, Yuasa K - Nat Commun (2014)

Zeno effect in quantum control.(a) We control a quantum system by switching on and off a set of given Hamiltonians {H(1),…, H(n)}. (b) We perform projective measurements P at regular time intervals during the control to check whether or not the state of the system belongs to a given subspace  of the global Hilbert space. (c) In the limit of infinitely frequent measurements (Zeno limit), the system is confined in the subspace , where it evolves unitarily with the Zeno Hamiltonians {(1),..., (n)} (Zeno dynamics). The Zeno dynamics can explore the subspace  more thoroughly than the purely unitary control without measurement.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4214416&req=5

f1: Zeno effect in quantum control.(a) We control a quantum system by switching on and off a set of given Hamiltonians {H(1),…, H(n)}. (b) We perform projective measurements P at regular time intervals during the control to check whether or not the state of the system belongs to a given subspace of the global Hilbert space. (c) In the limit of infinitely frequent measurements (Zeno limit), the system is confined in the subspace , where it evolves unitarily with the Zeno Hamiltonians {(1),..., (n)} (Zeno dynamics). The Zeno dynamics can explore the subspace more thoroughly than the purely unitary control without measurement.
Mentions: In a typical quantum control scenario, it is assumed that the system of interest (say the quantum register of a quantum computer, or the spins in an NMR experiment) can be externally driven by means of sequences of unitary pulses , activated by turning on and off a set of given Hamiltonians {H(1),…, H(n)} (Fig. 1a). If no limitations are imposed on the temporal durations τ of the pulses, it is known27 that by properly arranging sequences composed of {U(1),…, U(n)} one can in fact force the system to evolve under the action of arbitrary transformations of the form U=eΘ with the anti-Hermitian operators Θ being elements of the real Lie algebra formed by the linear combinations of iH(j) and their iterated commutators, and so on. Full controllability is hence achieved if the dimension of is large enough to permit the implementation of all possible unitary transformations on the system, that is, , with d being the dimension of the system (without loss of generality the Hamiltonians can be assumed to be traceless, since the global phase does not play any role).

Bottom Line: After discussing examples, we go on to show that this effect is generally to be expected: almost any quantum dynamics becomes universal once 'observed' as outlined above.Conversely, we show that any complex quantum dynamics can be 'purified' into a simpler one in larger dimensions.We conclude by demonstrating that even local noise can lead to an exponentially complex dynamics.

View Article: PubMed Central - PubMed

Affiliation: Institute of Mathematics, Physics and Computer Science, Aberystwyth University, Aberystwyth SY23 3BZ, UK.

ABSTRACT
The ability of quantum systems to host exponentially complex dynamics has the potential to revolutionize science and technology. Therefore, much effort has been devoted to developing of protocols for computation, communication and metrology, which exploit this scaling, despite formidable technical difficulties. Here we show that the mere frequent observation of a small part of a quantum system can turn its dynamics from a very simple one into an exponentially complex one, capable of universal quantum computation. After discussing examples, we go on to show that this effect is generally to be expected: almost any quantum dynamics becomes universal once 'observed' as outlined above. Conversely, we show that any complex quantum dynamics can be 'purified' into a simpler one in larger dimensions. We conclude by demonstrating that even local noise can lead to an exponentially complex dynamics.

No MeSH data available.