Basic Ideas of Quantum Computation

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Basic Ideas of Quantum Computation

• Pochung Chen
• Department of Physics, NTHU
• QIS Winter School, 01/06/2006

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Outline

• What is classical computation
• What is quantum mechanics
• What is quantum computation
• Quantum bit (qubit)
• Quantum operation
• Quantum circuit diagram
• Quantum algorithm
• Implementation of quantum computation
• Minimal requirement

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Classical Computation

• Classical bit 0 or 1
• Logic gates AND, OR, XOR
• 0 OR 11, 1 AND 00
• Circuits
• Error correction
• 0L(000),1L(111)
• (000)-gt(001)-gt(000)
• Algorithms
• Implementation
• Computation power
• Church-Turing thesis

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Classical ? Quantum Computation

• Classical bit ? Quantum bit (qubit)
• Logic gates ? Quantum gates
• Circuits ? Quantum circuits
• Error correction ? Quantum error correction
• Algorithms ? Quantum algorithms
• Implementation ? Implementation
• Computation power ? Computation power

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Quantum Mechanics

• States and Hilbert space
• ?gt, ltx?gt ?(x)
• ?1gt? ?2gt, ?1gt ? ?2gt ? ?3gt,
• dimH12,dimH13
• Superposition
• ?gta?1gt b?2gt
• Hamiltonians and quantum evolution
• H ?gt i?t?gt, ?(t)gt e-iHt?(0)gt
• Observables and Quantum measurement
• ?gt--gt ?igt with probability wi

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Quantum Bit ? Qubit

• Qubit
• 0 ?0gt, 1 ?1gt
• Superposition
• a0gtb1gt is allowed
• a, b are not directly available
• dim(1 qubit)2
• dim(n qubit)2n
• Large Hilbert space, phase coherence
• Why we need quantum algorithms

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Quantum Gates Basic Components I

• Qubit
• Single qubit
• Multiple qubit
• Quantum operation
• Single qubit operation
• Multiple qubit operation
• Quantum measurement

n
U
y
M
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Quantum Gates Basic Components II

• Doing nothing
• Single qubit gate
• Two qubits gate

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Quantum Gates Basic Components III
U
V
x
Mn
x
M
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Frequently Used Gates

• Hadamard
• Pauli-X
• Pauli-Y
• Pauli-Z
• Phase
• p/8

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Frequently Used Gates

• SWAP
• Controlled-NOT
• Controlled-Z
• Controlled-Phase
• Controlled-U

Z
S
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Controlled Operation

Control Qubit
Symmetry
Target Qubit
X
X

Apply NOT when control0
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Connection to Classical Computation

• Classical computation is (in principle)
  irreversible
• Quantum computation is (in principle) reversible
• Connection? Reversibility? Dissipation?

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Energy and Computations

• Laudauers principle
• Suppose a computer erases a single bit of
  information. The entropy of theenvironment
  increases by at least kBln2, here kB is
  Boltzmanns constant.
• Reversibly computation
• If all computer could be done reversibly, then
  Landauers principle imply no lower bound on the
  amount of energy dissipated by the computer!

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Quantum Gates are Reversible

• For any unitary matrix U, we have UUI
• Is it possible to simulate classical gate by
  quantum gate?
• The answer is, of course, yes.

U
U
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Fredkin Gate
AND
NOT
CROSSOVER
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Toffoli gate
FANOUT
NAND
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Quantum Algorithms
Quantum Fourier Transform
Hidden Subgroup
Quantum Search
Discrete log
Order-finding
Factoring
Break cryptosystems (RSA)
Speedup for some NP problems
Search for crypto keys
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Universal Quantum Computation

• Universal classical gates
• AND, OR, NOT
• Able to compute arbitrary classical function
• Universal quantum gates?
• Universal ? any unitary operation may be
  approximated to arbitrary accuracy by a quantum
  circuit involving only those gates
• Three universality construction

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Two-level Unitary Gates are Universal

• For any U
• Unitary matrix acting on a d-dimensional Hilbert
  space
• There exist U1,U2,,Ud
• Unitary matrix acting on a 2-dimensional Hilbert
  space
• Construct U exactly
• U U1U2Ud
• Two-level unitary gates

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Single qubit and CNOT Gates are Universal
Implement two-level gate by single qubit gates
and CNOT gates
U
U

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A Discrete Set of Universal Gates

• CNOTSingle qubit gates are universal
• But difficult to implement them in
  error-resisting way
• Use a discrete set of gates?
• Can not exactly implement arbitrary gate
  (continuous v.s. discrete)
• Can approximate any unitary gate
• Solovay-Kitaev theorem
• Any U on n-qubits may be approximated within a
  distance ? using O(n44nlog(n44n/ ?))gates.
• No constructive algorithm

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Discrete Universal Gate Sets

• Four member standard gates set
• HadamardCNOTphasep/8
• Alternative gates set
• CNOT, Hadamard, Phase, Toffoli

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Implementation of QComputation

• Guiding principles
• Di Vincenzo Criteria
• Quantum computer contenders
• What is on market now?
• Detailed review on quantum dot based QC
• Brief review on selected systems
• Modeling and fighting the decoherence
• How to model the decoherence?
• How to fight the decoherence?

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Guiding principles Di Vincenzo Criteria

• Be a scalable physical system with well-defined
  qubits
• Be initializable to a simple fiducial state such
  as 000...gt
• Have much longer decoherence times
• Have a universal set of quantum gates
• Permit high quantum efficiency, qubit-specific
  measurements

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Quantum Circuit and Implementation

• Quantum algorithm
• quantum circuit diagram, quantum gates
• Physical implementation
• limited set of accessible quantum gates
• Quantum compiler
• Implement desired quantum gates using accessible
  gates (using single qubitCNOT)

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Implement Controlled-U

• Uexp(ia)AXBXC, ABCI
• Multiple qubit control and target?

C-NOT operation

Single qubit operation
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Implement C2(U)

• UV2
• Toffoli gate
• UX, V(1-i)/(IiX)/2

V
V
V
U

• Quantum subroutine

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Implement Toffoli Gate

• General Cn(Uk)?

T
T
S
T

H
T
T
T
T
H
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Life is Not so Easy

• Quantum computer is so great, why cant I buy one
  yet ?
• Decoherence, decoherence, and decoherence !!!
• Ideal qubit
• Pure state gt Pure density matrix
• Non-ideal qubit
• Mix state gt Mixed density matrix
• Why de-coherence kills the quantum computer?
• Consider measurement operators M1gtlt,
  M2-gtlt-
• Ideal qubit p(1)1, p(2)0
• Non-ideal qubit p(1)1/2, p(2)1/2

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Modeling and fighting the Decoherence

• What is decoherence?
• Why decoherence kills a quantum computer
• How to model decoherence?
• The idea of quantum open system
• How to fight the decoherence? (from
  implementation point of view)
• Dynamical decoupling
• Decoherence free subspace
• Quantum feedback control
• Quantum Zeno effect
• Quantum error correction

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Modeling the decoherenceQuantum open system

• Closed system

• Open system

U
Utot
? (t)U?U
?
?
?(t)e(?(0))
?env
Utotexp(-iHtott) HtotHsysHenvHint e(?)Trenv
Utot(???env)Utot
Uexp(-iHt) Closed system ? Unitary Open system
? Non-Unitary
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Modeling the decoherenceQuantum open system
Born-Markovian approximation
Master equation in Lindblad form
Operator sum representation
Bad guy!

• The dilemma of quantum computation
• Controllability ? prefer strong interaction
• Fight decoherence ? prefer weak interaction

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Decoherence Optimatization

• The major obstacle in quantum computation is the
  de-coherence
• The quantum gate operation time is determined by
  the physical system used to implement quantum
  computation
• It is desirable to minimize the number of gates
  used for a quantum algorithm

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Fighting the decoherence

• Dynamical decoupling
• Eliminate the coupling between system and bath
• Decoherence free subspace
• Hide in somewhere the symmetry can protect you
• Quantum feedback control
• Instantaneous error correction via feedback
• Quantum Zeno effect
• The strange effect of quantum measurement
• Quantum error correction
• The last resort, expensive in resource

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Quantum Error-Correction

• Encoded qubit
• Physical qubit ? logical qubit
• Error-detection or syndrome diagnosis
• Knowing the syndrome without knowing the logical
  qubit
• Recovery
• Recover from the syndrome

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Error Detectiong

• Three qubit bit flit code
• Fgta 0Lgtb 1Lgt a000gtb111gt
• Bit-flip error
• Fgt?a100gtb011gt
• Syndrome measurement
• P1100gtlt100011gtlt011
• ltFP1Fgt1
• Still dont know a, b
• Recovery action

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Dynamical decoupling Motivation

• The total Hamiltonian of the open system
• (Periodic) Control Hamiltonian acting on system
• The stroboscopic dynamics at TNNTc
• The effective Hamiltonian
• Can we average out the sys-bath interaction?

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Dynamical decoupling Decoupling group

• Consider a discrete decoupling group
• Assigning thea time evolution operator to thes
  unitary representation
• Average Hamiltonian
• Decoupling by group symmetrization
• Strong and fast control needed (unphysical)

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Decoherence free subspaceprotected by symmetry

• If no special assumptions are made on the
  coefficient matrix aaß and on the initial
  conditions then a necessary and sufficient con
  for a subpace to be decoherence-free Spankgt is
  that all basis states are dgenerate eigenstates
  of all the Lindblad operators Fa

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Quantum feedback control

• Starting from the master equation
• View the evolution as a continuous measurement
• Derived the stochastic master equation
• Use measurement result (current) to perform
  feedback feedback

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Quantum Zeno effect

• Measurement and collapse
• Evolution superoperator
• Measurement superoperator
• Evolutionmeasurements
• Zeno Hamiltonian

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Summary

• What is quantum computation
• The power and the limit of quantum computer
• Toward the implementation of QComputer
• Decoherence and open quantum system