um ponto de vista simbólico sobre a simulação de algoritmos quânticos antónio pereira &...

Um ponto de vista simbólico Um ponto de vista simbólico sobre a Simulação de Algoritmos Quânticossobre a Simulação de Algoritmos Quânticos

António Pereira & Rosália Rodrigues

CEOC-UA – CIMA-UE

2006

CEOC – CIMA - 2006 2

Quantum ComputationQuantum Computation

Research in Quantum Computation:Research in Quantum Computation:

• building quantum devices

• designing algorithms for quantum devices

How to Simulate it ?How to Simulate it ?


Simulating Quantum ComputationSimulating Quantum Computation

Vectorial approach:

state vector on a Hilbert space

evolution matrix products

advantageadvantage: easy to implement and trace

drawbackdrawback: exponential growth in space and time

Symbolic approach:

state linear expression

evolution algebraic rules

advantageadvantage: control over complexity

drawbackdrawback: “convince” Mathematica not to evaluate... yet

Symbolic

Quantum

Computer

Simulator


qudits & qubitsqudits & qubits


kets in SQCSkets in SQCS

Basis qudit state

Linear expression of ket objects

Object with head ket

General qudit state


bras in SQCSbras in SQCS

Riesz Theorem:


braKets in SQCSbraKets in SQCS

• conjugate linear

in the first argument

• linear in the second

braKet


Qudit SystemsQudit Systems

………… …………

1 2 3 n


The Kronecker product in SQCSThe 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 SQCSOperators in SQCS

Quantum Algorithm

Initial state +

Sequence of unitary operators +

Measurement

op[name_,n_,f_]

Every linear operator is represented in SQCS by an object

where:

• name ― label for the operator

• n ― number of qudits on which the operator acts

• f ― function that defines the action of the operator

on the basis qudits states (set of rules)

The discrete time evolution of a closed quantum system is described by the action of a unitary operator

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The Hadamard operator

• Creates a uniform superposition

• Is its own inverse

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The Walsh-Hadamard operator

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The Outer Product operator

Completeness Relation:

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Simulating Grover’s AlgorithmSimulating Grover’s Algorithm

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Classical Database Case:

ClassicalDatabase

QuantumComputer

f(x)

n 5

k 2^n;

t Table0,i, 0, k 1;pos RandomInteger,0, k 1;Print"POS", postpos 1 1;

fi_: ti 1;numIt RoundPi4 ArcSin1Sqrtk 12;Print"Iterations ", numIt; oraclef, nGrover n20 0n n n 00n1 n0nestedApplyGrover,1, numItTiming

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Quantum Database Case:

QuantumDatabase

QuantumComputer

n 30;

k 2^n;

pos RandomInteger,0, k 1;Print"POSITION ", posnumIt RoundPi4 ArcSin1Sqrtk 12;Print"Number of Iterations ", numIt;posBin IntegerDigitspos, 2, n; n 2 kron #1GSD0,1,2#1GSD0,1,2& posBin

Grover n20 0n n n 00n1 n0nestedApplyGrover,1, numItTiming

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Step by step:

Database of size25=32

Index of the elementto be searched for

Number of steps

The Oracle

Grover’s operator

In[1]:= SQCS`

In[2]:= numericOn;

makeOuterProductRulesGSD0, 1, 2;makeSpecialOuterProductRules;

makeHadamardRulesGSD0, 1, 2;makeHadamardInverseRules;

In[7]:= n 5;

k 2^n;

pos RandomInteger,0, k 1;Print"POSITION ", pos;POSITION 24

In[10]:= numIt RoundPi4 ArcSin1Sqrtk 12;Print"Number of Iterations ", numIt;Number of Iterations 4

In[11]:= posBin IntegerDigitspos, 2, n; n 2

kron #1GSD0,1,2#1GSD0,1,2& posBin

Out[12]= 21 11 10 00 00 0 5

In[13]:= Grover n20 0n n n

Out[13]= 55 2005 5 21 11 10 00 00 0 5

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In[1]:= SQCS`

In[2]:= numericOn;

makeOuterProductRulesGSD0, 1, 2;makeSpecialOuterProductRules;

makeHadamardRulesGSD0, 1, 2;makeHadamardInverseRules;

In[7]:= n 5;

k 2^n;

pos RandomInteger,0, k 1;Print"POSITION ", pos;POSITION 24

In[10]:= numIt RoundPi4 ArcSin1Sqrtk 12;Print"Number of Iterations ", numIt;Number of Iterations 4

In[11]:= posBin IntegerDigitspos, 2, n; n 2

kron #1GSD0,1,2#1GSD0,1,2& posBin

Out[12]= 21 11 10 00 00 0 5

In[13]:= Grover n20 0n n n

Out[13]= 55 2005 5 21 11 10 00 00 0 5

In[14]:=00nOut[14]=05In[15]:=1 n0

Out[15]= 0.176777015In[16]:=2 nestedApplyGrover,1, 1

Out[16]= 0.1767772.11000

0.1250101010101 1.015In[17]:=3 nestedApplyGrover,2, 1

Out[17]= 0.1767773.7511000


Out[18]= 0.1767775.0312511000

0.6738280101010101 1.015

Step by step:

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In[14]:=00nOut[14]=05In[15]:=1 n0


Out[16]= 0.1767772.11000


Out[17]= 0.1767773.7511000


Out[18]= 0.1767775.0312511000

0.6738280101010101 1.0155 10 15 20 25 30

0.2

0.4

0.6

0.8

1

Probability distribution

Step by step:

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In[14]:=00nOut[14]=05In[15]:=1 n0


Out[16]= 0.1767772.11000


Out[17]= 0.1767773.7511000


Out[18]= 0.1767775.0312511000

0.6738280101010101 1.0155 10 15 20 25 30

0.2

0.4

0.6

0.8

1

Step by step:

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In[14]:=00nOut[14]=05In[15]:=1 n0


Out[16]= 0.1767772.11000


Out[17]= 0.1767773.7511000


Out[18]= 0.1767775.0312511000

0.6738280101010101 1.0155 10 15 20 25 30

0.2

0.4

0.6

0.8

1

Step by step:

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In[14]:=00nOut[14]=05In[15]:=1 n0


Out[16]= 0.1767772.11000


Out[17]= 0.1767773.7511000


Out[18]= 0.1767775.0312511000

0.6738280101010101 1.0155 10 15 20 25 30

0.2

0.4

0.6

0.8

1

Step by step:

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Grover’s Algorithm – Simulation TimesGrover’s Algorithm – Simulation Times

Time × Database sizeTime × Number of qubits

Classical Database Case:

Mathematica 5, Pentium IV, 3.0 GHz, 1GB RAM

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Grover’s Algorithm – Simulation TimesGrover’s Algorithm – Simulation Times

Time × Database sizeTime × Number of qubits

Mathematica 5, Pentium IV, 3.0 GHz, 1GB RAM

Quantum Database Case:

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Conclusions & Further workConclusions & 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.

Symbolic Approach to Quantum Computation: 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.

Conclusions:

Further work:

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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)

ReferencesReferences

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