Universal Quantum Machines

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From Flipping Qubits to Programmable Quantum
Processors
Vladimír Buek, Mário Ziman, Mark Hillery,
Reinhard Werner, Francesco DeMartini
VIII-th International Workshop on Squeezed States
and Uncertainty Relations Puebla, 9th June 2003
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Flipping a Bit NOT Gate
0
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Flipping a Bit NOT Gate
0
1
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Universal NOT Gate

• NOT gate in a computer basis

Poincare sphere state space
is antipode of
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Universal NOT Gate Problem
is antipode of

• - Spin flipping is an inversion of the Poincare
  sphere
• - This inversion preserves angels
• The Wigner theorem - spin flip is either unitary
  or anti-unitary operation
• Unitary operations are equal to proper rotations
  of the Poincare sphere
• Anti-unitary operations are orthogonal
  transformations with det-1
• Spin flip operation is anti-unitary and is not
  CP
• In the unitary world the ideal universal NOT
  gate which would flip a
• qubit in an arbitrary (unknown) state does not
  exist

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Measurement-based vs Quantum Scenario
Measurement-based scenario optimally measure and
estimate the state then on a level of classical
information perform flip and prepare the flipped
state of the estimate
Quantum scenario try to find a unitary operation
on the qubit and ancillas that at the output
generates the best possible approximation of the
spin-flipped state. The fidelity of the operation
should be state independent (universality of the
U-NOT)
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Quantum Clickology

• measurement conditional distribution on a
  discrete state space of the aparatus A
  observables with eigenvalues li

Apparatus
System
Measurement
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Quantum Bayesian Inference

• Bayesian inversion from distribution on A to
  distribution on W

• Reconstructed density operator given the result li

• invariant integration measure

K.R.W. Jones, Ann. Phys. (N.Y.) 207, 140
(1991) V.Buek, R.Derka, G.Adam, and P.L.Knight,
Annals of Physics (N.Y.), 266, 454 (1998)
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Optimal Reconstructions of Qubits

• average fidelity of estimation

• Construction of optimal POVMs maximize the
  fidelity F

• POVM via von Neumann projectors Naimark theorem

• Estimated density operator on average

• Optimal decoding of information

• Optimal preparation of quantum systems

S.Massar and S.Popescu, Phys. Rev. Lett. 74, 1259
(1995) R.Derka, V.Buek, and A.K.Ekert, Phys.
Rev. Lett 80, 1571 (1998)
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Measurement-based Flipping of Qubit

• Estimated density operator when just a single
  qubit is available

• Flipping based on this estimation

R.Derka, V.Buek, and A.K.Ekert, Phys. Rev. Lett
80, 1571 (1998)
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Quantum Scenario Universal NOT Gate
C-NOT gate
V.Buek, M.Hillery, and R.F.Werner, J. Mod. Opt.
47, 211 (2000)
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Theorem Optimal Universal NOT Gate
V.Buek, M.Hillery, and R.F.Werner Phys. Rev. A
60, R2626 (1999)
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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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U-NOT via Optical Parametric Amplifier
A.Lamas-Linares, C.Simon, J.C.Howell, and
D.Bouwmeester, Science 296, 712
(2002) F.DeMartini, V.Buek, F.Sciarino, and
C.Sias, Nature 419, 815 (2002)
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Optimal Universal-NOT Gate
See a talk by Franceso DeMartini Contextual
realization of the universal optimal cloning and
u-NOT gates by optical parametric amplification
Wednesday at 1200 Session C1
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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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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

• Buzek-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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Inverse Problem Quantum Simulators
Given a set Fx of quantum operations . Is it
possible to design a processor that performs all
these operations?
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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
• set of maps induced by a given processor (loops)
• quantum processor for a given set of maps
• quantum multi-meters

See a talk by Miloslav Dusek Optical
implementation of a programmable quantum state
discriminator Thursday at 1130 Session D1
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)
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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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POVM Measurement
V.Buek, M.Roko, and M.Hillery, unpublished
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Quantum Loops
Analogy of for-to cycles in classical
programming
data
Quantum processor
program

• Introducing loops control system quantum
  clocks
• Halting problem how (when) to stop the
  computation process

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U-NOT via OPA

• Original qubit is encoded in a polarization state
  of photon

• This photon is injected into an OPA excited by
  mode-locked UV laser

• Under given conditions OPA is SU(2) invariant

• Spatial modes and are described by
  the operators and

• Initial state of a qubit is

• The other mode is in a vacuum

• Evolution stimulated emission

• Evolution spontanous emission