LECTURE III FROM FLIPPING QUBITS TO PROGRAMMABLE QUANTUM PROCESSORS

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LECTURE III FROM FLIPPING QUBITS TO
PROGRAMMABLE QUANTUM PROCESSORS
Puebla, September 08 2004
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Content

• Yesterday I was talking about quantum cloners and
  U-NOT gate, today I will utilize the network we
  have found for
• programmable quantum computer programs via
  quantum states
• some CP maps via unconditional quantum
  processors
• arbitrary CP maps via probabilistic programming
• estimation of quantum channels (maps)

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No-Cloning Theorem U-QCM
W.Wootters and W.H.Zurek, Nature 299, 802
(1982) V.Buek and M.Hillery, Phys. Rev. A 54,
1844 (1996) S.L.Braunstein, V.Buek, M.Hillery,
and D.Bruss, Phys. Rev. A 56, 2153 (1997)
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There is Something in This Network
S.L.Braunstein, V.Buek, and M.Hillery, Phys.
Rev. A 63, 052313 (2001)
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Quantum Information Distributor
- Covariant device with respect to SU(2)
operations - POVM measurements - eavesdropping
S.L.Braunstein, V.Buek, and M.Hillery, Phys.
Rev. A 63, 052313 (2001)
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Model of Classical Processor
data register
output register
Classical processor
0010110111
1101110110
program register
Heat
1110010110
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Quantum Processor
data register
output data register
Quantum processor
Quantum processor
program register
Quantum processor fixed unitary transformation
Udp Hd data system, S(Hd) data states Hp
program system, S(Hp) program
states
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Two Scenarios

• Measurement-based strategy - estimate the state
  of program
• Quantum strategy use the quantum program
  register
• conditional
  (probabilistic) processors
• 
  unconditional processors

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C-NOT as Unconditional Quantum Processor

• program state
• program state
• general pure state
• unital operation, since
• program state is 2-d and we can apply 2 unitary
  operations

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Question
Is it possible to build a universal programmable
quantum gate array which take as input a quantum
state specifying a quantum program and a data
register to which the unitary operation is
applied ?
on a qubit an A number of operations can be
performed
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No-go Theorem

• no universal deterministic quantum array of
  finite extent can be realized
• on the other hand a program register with d
  dimensions can be used to implement d unitary
  operations by performing an appropriate sequence
  of controlled unitary operations

M.A.Nielsen I.L.Chuang, Phys. Rev. Lett 79, 321
(1997)
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C-NOT as Probabilistic Quantum Processor

• Vidal Cirac probabilistic implementation of

G.Vidal and J.I.Cirac, Los Alamos arXiv
quant-ph/0012067 (2000) G.Vidal, L.Mesanes, and
J.I.Cirac, Los Alamos arXiv quant-ph/0102037
(2001).
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C-NOT as Probabilistic Quantum Processor
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Description of Quantum Processors

• definition of Udp via Kraus operators
• normalization condition
• induced quantum operation
• general pure program state
• can be generalized for mixed program states

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Universal Probabilistic Processor

• Quantum processor Udp
• Data register rd, dim Hd D
• Quantum programs Uk program register rp, dim
  Hp

• Nielsen Chuang
• N programs Þ N orthogonal states
• Universal quantum processors do not

• Buek-Hillery-Ziman
• Probabilistic implementation
• Uk operator basis,
• program state

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Implementation of Maps via Unconditional Quantum
Processors
U
r
Set of operations
Mark Hillery, Mário Ziman, and Vladimír Buzek,
Phys. Rev. A 66, 042302 (2002)
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Black box Problem

• Having a black box (with no memory) processing
  one qubit in a time, how can we determine its
  parameters?

• How many different states do we need for a
  complete guess?

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Complete Estimation

• For a complete estimation one needs four
  different states, which are linearly independent.

?
?
?
?

• What to do, if we do not have them?

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Entangled States
?
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Entangled States

• Using one completely entangled state in the form
• one gets
• with
• By estimation of the output state we are able to
  completely determine the operation itself

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Incomplete Estimation

• One needs to create a general and reasonable rule
  that would stand for the missing information
• The basic question is, what guess should one make
  if NO information is available
• For symmetry reasons only two candidates are
  relevant
• Identity
• Contraction to the complete mixture
• Identity is not suitable, as it can not be an
  average operation (it is an extremal operation,
  as any unitary operation)

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Data Consistency

• How to check that the (incomplete) data are
  consistent?
• Contractivity of the distance
• Fidelity monotone
• Due to technical reasons, we use the distance
  contractivity to test our data consistency

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General Operation

• The density operator
• The general operation
• With and

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Complete Positivity

• Every guess must be completely positive in
  general it is hard to achieve analytically
• Check is done by applying an operation in the
  form
• on to the maximally entangled state

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Strategy

• The problem can be divided in to two relatively
  separate parts
• How is the identity transformed?
• How are the remaining three pure basis vectors
  transformed?
• Priorities
• Identity and pure states MUST be transformed
  according the existing data
• Transformation MUST be completely positive
• The Identity is transformed to the identity or as
  close as possible (measuring in distance)
• The remaining pure states are transformed to the
  same state as identity or as close as possible
  (measuring in distance)

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Example Specific Channel

• Let us assume a specific transformation map,
  channel

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No Known State

• A universal guess complete mixture

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One Known State

• Two possibilities
• Output state is closer to the complete mixture
  then the input state -gt complete mixture is not
  affected by the transformation
• Output state is farther from the complete mixture
  -gt complete mixture is transformed, namely is
  moved towards the output state vector

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Two Known States

• If the states sum to identity, strategy is the
  same as in the previous case
• If not, a rather complicated situation arises. In
  some cases
• the Identity is not transformed and the third,
  perpendicular pure state is transformed to the
  same state as identity

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Three Known States

• Here o lot of data is already available
• If the three states sum to identity, we turn back
  to the previous case
• In other case, only numerical solutions are
  possible. The only open question is the
  transformation of identity, then all the
  perpendicular pure states are given

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Conclusions Open Questions

• programmable quantum computer programs via
  quantum states programs can be outputs of
  another QC
• some CP maps via unconditional quantum
  processors
• arbitrary CP maps via probabilistic programming
• controlled information distribution
  (eavesdropping)
• simulation of quantum dynamics of open systems
• quantum processor for a given set of maps
• quantum multi-meters
• estimation of quantum channels (maps)

M.Hillery, V.Buzek, and M.Ziman Phys. Rev. A 65,
022301 (2002). M.Dusek and V.Buzek Phys. Rev. A
66, 022112 (2002). M.Hillery, M.Ziman, and
V.Buzek Phys. Rev. A 66, 042302 (2002)