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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.


Schematics of the N-qubit model described in Example A.Straight edges represent the Heisenberg interactions, while the triple edge represents the three-body interaction among qubits 1–3. The red part in the upper figure corresponds to H(1) acting on qubits 1 and 2, while the remainder including a local term Z3 on qubit 3 corresponds to H(2) acting on all the N qubits. The Zeno projection P1 on qubit 1 transforms the upper Hamiltonians to the lower model, where the state of qubit 1 is frozen, while we are left with a Heisenberg chain with the local term Z3 and a control  on qubit 2. The Lie algebra of the upper system is only two dimensional, while the lower allows us to perform full control over the system apart from the frozen qubit 1.
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f3: Schematics of the N-qubit model described in Example A.Straight edges represent the Heisenberg interactions, while the triple edge represents the three-body interaction among qubits 1–3. The red part in the upper figure corresponds to H(1) acting on qubits 1 and 2, while the remainder including a local term Z3 on qubit 3 corresponds to H(2) acting on all the N qubits. The Zeno projection P1 on qubit 1 transforms the upper Hamiltonians to the lower model, where the state of qubit 1 is frozen, while we are left with a Heisenberg chain with the local term Z3 and a control on qubit 2. The Lie algebra of the upper system is only two dimensional, while the lower allows us to perform full control over the system apart from the frozen qubit 1.

Mentions: In Example A, we consider N qubits (Fig. 3, upper), the first two of which are manipulated via the control Hamiltonians H(1)=X1X2, and complement it with H(2) consisting of the nearest-neighbor Heisenberg interactions involving all the qubits but the first two, together with a coupling term acting on the first three qubits and a local term on the third, that is,


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)

Schematics of the N-qubit model described in Example A.Straight edges represent the Heisenberg interactions, while the triple edge represents the three-body interaction among qubits 1–3. The red part in the upper figure corresponds to H(1) acting on qubits 1 and 2, while the remainder including a local term Z3 on qubit 3 corresponds to H(2) acting on all the N qubits. The Zeno projection P1 on qubit 1 transforms the upper Hamiltonians to the lower model, where the state of qubit 1 is frozen, while we are left with a Heisenberg chain with the local term Z3 and a control  on qubit 2. The Lie algebra of the upper system is only two dimensional, while the lower allows us to perform full control over the system apart from the frozen qubit 1.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f3: Schematics of the N-qubit model described in Example A.Straight edges represent the Heisenberg interactions, while the triple edge represents the three-body interaction among qubits 1–3. The red part in the upper figure corresponds to H(1) acting on qubits 1 and 2, while the remainder including a local term Z3 on qubit 3 corresponds to H(2) acting on all the N qubits. The Zeno projection P1 on qubit 1 transforms the upper Hamiltonians to the lower model, where the state of qubit 1 is frozen, while we are left with a Heisenberg chain with the local term Z3 and a control on qubit 2. The Lie algebra of the upper system is only two dimensional, while the lower allows us to perform full control over the system apart from the frozen qubit 1.
Mentions: In Example A, we consider N qubits (Fig. 3, upper), the first two of which are manipulated via the control Hamiltonians H(1)=X1X2, and complement it with H(2) consisting of the nearest-neighbor Heisenberg interactions involving all the qubits but the first two, together with a coupling term acting on the first three qubits and a local term on the third, that is,

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.