Um ponto de vista simb

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Um ponto de vista simbólico sobre a Simulação de
Algoritmos Quânticos

• António Pereira Rosália Rodrigues
• CEOC-UA CIMA-UE 2006

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Quantum Computation

• Research in Quantum Computation
• building quantum devices
• designing algorithms for quantum devices

How to Simulate it ?
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Simulating Quantum Computation
Vectorial approach state ? vector on a
Hilbert space evolution ? matrix
products advantage easy to implement and trace
drawback exponential growth in space and time
Symbolic approach state ? linear expression
evolution ? algebraic rules advantage
control over complexity drawback convince
Mathematica not to evaluate... yet
Symbolic Quantum Computer Simulator
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qudits qubits
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kets in SQCS
Basis qudit state
Object with head ket
General qudit state
Linear expression of ket objects
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bras in SQCS
Riesz Theorem
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braKets in SQCS
braKet

• conjugate linear
• in the first argument
• linear in the second

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Qudit Systems

1
2
3
n

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The Kronecker product in SQCS
Properties of the Tensor Product (Kronecker
Product)

• Associative
• Noncommutative
• Distributive with respect
• to linear combinations

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Operators in SQCS
The discrete time evolution of a closed quantum
system is described by the action of a unitary
operator
Quantum Algorithm
Initial state Sequence of unitary operators
Measurement
Every linear operator is represented in SQCS by
an object
opname_,n_,f_

• where
• name ? label for the operator
• n ? number of qudits on which the operator acts
• f ? function that defines the action of the
  operatoron the basis qudits states (set of
  rules)

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Operators in SQCS
The Hadamard operator

• Creates a uniform superposition
• Is its own inverse

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Operators in SQCS
The Walsh-Hadamard operator
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Operators in SQCS
The Outer Product operator
Completeness Relation
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Simulating Grovers Algorithm
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Simulating Grovers Algorithm
Classical Database Case
Quantum Computer
Classical Database
f(x)
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Simulating Grovers Algorithm
Quantum Database Case
Quantum Computer
Quantum Database
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Simulating Grovers Algorithm
Step by step
Database of size 2532
Index of the element to be searched for
Number of steps
The Oracle
Grovers operator
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Simulating Grovers Algorithm
Step by step
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Simulating Grovers Algorithm
Step by step
Probability distribution
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Simulating Grovers Algorithm
Step by step
Probability distribution
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Simulating Grovers Algorithm
Step by step
Probability distribution
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Simulating Grovers Algorithm
Step by step
Probability distribution
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Grovers Algorithm Simulation Times
Classical Database Case
Time Number of qubits
Time Database size
Mathematica 5, Pentium IV, 3.0 GHz, 1GB RAM
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Grovers Algorithm Simulation Times
Quantum Database Case
Time Number of qubits
Time Database size
Mathematica 5, Pentium IV, 3.0 GHz, 1GB RAM
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Conclusions Further work
Conclusions

• Symbolic Approach to Quantum Computation
• Provides a suitable environment for testing
  quantum algorithms.
• Allows for larger problem instances.
• Algorithms can be programmed at high-level.
• Useful tool for the teaching of Quantum
  Computation.

Further work

• Measuring Operators.
• A quantum register address manager.
• Simulate other quantum algorithms
  Deutsch-Jozsa, Shor,
• Use SQCS as a tool for the development of new
  quantum algorithms.

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References
1. Nielsen, M.A., Chuang, I.L. Quantum
Computation and Quantum Information. Cambridge
University Press (2000) 2. Kitaev, A.Y., Shen,
A., Vyalyi, M. Classical and quantum
computation. Volume 47 of Graduate Studies in
Mathematics. American Mathematical Society
(2002) 3. Wolfram, S. The Mathematica Book,
Fifth Edition. Wolfram Media, Inc. (2003) 4.
Grover, L.K. A fast quantum mechanical algorithm
for database search. In Proc. 28th Annual ACM
Symposium on the Theory of Computing. (1996)
212-219 5. Biham, E., Biham, O., Biron, D.,
Grassl, M., Lidar, D.A. Grover's quantum search
algorithm for an arbitrary initial amplitude
distribution. Physical Review A 60 (1999)
27-42 6. Pereira, António, Rodrigues, Rosália A
Symbolic Approach to Quantum Computation
Simulation. Lecture Notes in Computer Science
(2006) Vol. 3992. 454 461 7. Pereira, António,
Rodrigues, Rosália Symbolic Quantum Computation
Simulation with Mathematica. Cadernos de
Matemática. Universidade de Aveiro. CM05/I-44
(2005)
Thank You