Introduction to Quantum Computing

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A (light) introduction to  Quantum Computing Eric Marcos barcelonaqbit.com

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What QC is?

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Quantum cryptography Quantum simulation Quantum computing

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Quantum cryptography Quantum simulation Quantum computing

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What QC is NOT?

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“QC can solve classically unsolvable problems”

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“QC is exponentially faster than classical computation”

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A little bit of history

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1900 Max Plank E = ħv 1905 Albert Einstein Annus mirabilis 1924 Louis de Broglie λ = ħ / p 1927 Werner Heisenberg σxσp ≥ ħ / 2 1928 Paul Dirac (i - ∂ - m) ψ = 0 Quantum Mechanics

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1931 Kurt Gödel Incompleteness Theorems 1936 Alan Turing Turing Machines 1952 Stephen Kleene Church-Turing Thesis 1956 Noam Chomsky Chomsky Hierarchy 1971 Stephen Cook Leonid Levin NP-Completeness Theory of Computation

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1982 Yuri Manin Richard Feynmann Quantum Computing ideas… 1985 David Deutsch Quantum Turing Machines 1992 Deutsch-Josza Algorithm 1994 Peter Shor Shor’s Algorithm 1996 Lov Grover Grover’s Algorithm Quantum Computing 2001 IBM Stanford First physical implementation of Shor’s Algorithm

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So… What QC is?

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First… some thoughts on classical computing

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What is a BIT?

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BIT x ∈ { false , true } OPERATIONS NOT(x) = ¬x AND(x, y) = x ∧ y TRUTH VALUES

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BIT x ∈ { 0 , 1 } OPERATIONS NOT(x) = 1 - x AND(x, y) = x · y INTEGER VALUES

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VECTOR VALUES (0,1) (1,0)

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( ) VECTOR VALUES (0,1) = True = |1> (1,0) = False = |0> x ∈ { , } 1 0 ( ) 0 1

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( ) VECTOR VALUES x ∈ { , } 1 0 ( ) 0 1 OPERATIONS ?

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( ) NOT( ) = 1 0 ( ) 0 1 NOT(x ̅) = x ̅ ( ) 0 1 1 0 VECTOR VALUES

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In CQ bits are just vectors and operations are just matrices

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In CQ bits are just unit vectors and operations are just unitary matrices

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“I am a Quantum Engineer, but on Sundays I Have Principles.” John Stewart Bell

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“Oh oh, mamma mia, here I go again, my my, how can I resist you” Benny Andersson (ABBA)

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AND(x ̅, y ̅) = ?? VECTOR VALUES OPERATIONS

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AND(x ̅, y ̅) = ?? We need to combine x ̅ and y ̅ into a joint state VECTOR VALUES OPERATIONS

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COMBINING VECTOR BITS x ̅ ⊗ y ̅ = z ̅ In CQ the joint state of 2 vector bits is its tensor product

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COMBINING VECTOR BITS w ̅ ⊗ x ̅ ⊗ y ̅ = z ̅ In CQ the joint state of N vector bits is its tensor product

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COMBINING VECTOR BITS x ̅ ⊗ y ̅ = ( ) x0 · y0 x0 · y1 x1 · y0 x1 · y1

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COMBINING VECTOR BITS |0> ⊗ |0>

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COMBINING VECTOR BITS |0>|0>

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COMBINING VECTOR BITS |00>

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COMBINING VECTOR BITS ( ) 1 0 ( ) 1 0 ⊗

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COMBINING VECTOR BITS ( ) 1 0 0 0 |00> =

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COMBINING VECTOR BITS ( ) 0 1 0 0 |01> =

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COMBINING VECTOR BITS ( ) 0 0 1 0 |10> =

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COMBINING VECTOR BITS ( ) 0 0 0 1 |11> =

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In CQ a register of N bits is represented by a vector space of 2N dimensions

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We can store 2N bits of information with just N qbits!

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AND(x ̅, y ̅) VECTOR VALUES OPERATIONS

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AND(x ̅ ⊗ y ̅) VECTOR VALUES OPERATIONS

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VECTOR VALUES OPERATIONS quantum gates AND x ̅ y ̅ x ̅ (output) y ̅ (trash)

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VECTOR VALUES OPERATIONS quantum circuits AND x ̅ y ̅ w ̅ OR

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VECTOR VALUES OPERATIONS quantum circuits AND x ̅ y ̅ w ̅ OR I I

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VECTOR VALUES OPERATIONS quantum circuits AND x ̅ y ̅ w ̅ OR I I I ⊗ AND = Uand OR ⊗ I = Uor

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(OR ⊗ I) × (I ⊗ AND) × (w ̅ ⊗ x ̅ ⊗ y ̅)

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Uor × Uand × z ̅0 = z ̅1

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Reversible computation

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(reminder) In CQ bits are just unit vectors and operations are just unitary matrices

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UNIT VECTOR ||v|| = = √( ∑ vi 2 ) = 1

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( ) 1 0 0 0 |00> =

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( ) 0.5 0.5 0.5 0.5 |??> =

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( ) 1 0 0 0 0.5 × ( ) 0 1 0 0 + 0.5 × ( ) 0 0 1 0 + 0.5 × ( ) 0 0 0 1 + 0.5 ×

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0.5 × |00> + 0.5 × |01> + 0.5 × |10> + 0.5 × |11>

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|Ψ> = |0> + |1> = THE QUBIT , ∈ ℂ ( )

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|Ψ> = |0> + |1> = THE QUBIT , ∈ ℂ ( ) Quantum superposition

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Operating on a qubit in a superposition state means operating on all the values at once THE GOOD NEWS

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THE GOOD NEWS ( ) 0.5 0.5 0.5 0.5 Uf × This is like applying Uf to |00>, |01>, |10> and |11> in just one computation step!

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Quantum parallelism THE GOOD NEWS

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Superposition only works if the quantum system is isolated THE BAD NEWS

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THE BAD NEWS ( ) 0.5 0.5 0.5 0.5 Uf × = ( ) ?? ?? ?? ??

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Reading the qubit destroys superposition and makes it collapse to one of the base states randomly THE BAD NEWS

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THE BAD NEWS ( ) 0.5 0.5 0.5 0.5 Uf × = |Ψ> read(|Ψ>) p(|00>) = ?? p(|01>) = ?? p(|10>) = ?? p(|11>) = ?? p = ||||2

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How can we build a superposition state?

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HADAMARD GATE ( ) 1 1 1 -1 H = 1/√2

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HADAMARD GATE H |0> = ( ) 1/√2 1/√2 |0> H |0> + |1> √2 H |0> H |1> = ( ) 1/√2 -1/√2

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CNOT GATE ( ) 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 CNOT =

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CNOT GATE |0> |0> H |0> + |1> √2 CNOT |00> + |11> √2

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Quantum entanglement

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COMPUTING FUNCTIONS :|> → |()> Generally not possible because of reversible computation : {0,1} → {0,1}

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COMPUTING FUNCTIONS :|>|0> → |x>|()> : {0,1} → {0,1}

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COMPUTING FUNCTIONS |0> |0> H |0> + |1> √2 U |(0)> + |(1)> √2

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COMPUTING FUNCTIONS |0> H |0> + |1> √2 U |0> - |1> √2 |Ψ> = 1/2[(-1)f(0)+(-1)f(1)]|0> + 1/2[(-1)f(0)-(-1)f(1)]|1> H |Ψ>

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The Deutsch algorithm : {0,1} → {0,1} Given , tell me whether is constant (all outputs are the same for whatever input) or balanced (same number of 1s and 0s)

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Uses the phase kick-back technique and quantum interference to determine whether a function is constant or balanced in just one call The Deutsch algorithm

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The Deutsch-Josza algorithm : {0,1}n → {0,1} Given , tell me whether is constant (all outputs are the same for whatever input) or balanced (same number of 1s and 0s)

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“I think I can safely say that nobody understands quantum mechanics.” Richard Feynman

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@ericmarcosp [email protected] github.com/ericmarcos/pyqu