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Quantum Computing MAS 725 Hartmut Klauck NTU 20.2.2012

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Quantum Computing Suppose we have a supply of qubits in some basis state We want to apply a unitary transformation Corresponding to a function we wish to compute So that measuring the resulting state gives us a desired output This unitary transformation must be the product of “few” simple, local, unitary transformations I.e., unitary transformation on 2 or 3 qubits This means to build a quantum circuit

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What is the difference to probabilistic computations Randomized/probabilistic computations can be described similarly Keeping track of the probability distribution as a vector of probabilities Important difference: amplitudes can be negative, allowing for negative interference effects Undesirable computations can “cancel each other out”

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“Parallel” Computing?

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What can we do with the resulting state? Measuring now just yields the uniform distribution on x,f(x) We do (at least) get “really random” values of x To determine interesting properties of f we usually have to work harder and employ interference effects.

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Time for the first algorithm Deutsch’s Algorithm Setup: Input is in a Black Box A function f:{0,1} {0, 1} unknown to us (is the input) Access: We can read f(0) or read f(1) Either f(0)=f(1) or f(0) f(1) Problem: decide which is the case

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Deutsch’s Problem How many times do we have to query the Black Box? Any deterministic algorithm that must not err has to read f(0) and f(1). Any randomized algorithm with only 1 query has error probability ½ (i.e. is as good as tossing a coin for an answer....)

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Quantum queries In a deterministic query algorithm the result of previous queries determines the next position to query Randomized queries: In our case there is a probability distribution on 0 and 1, that determines whether we query f(0) or f(1) Quantum queries can be in superposition instead We have to model a quantum query as a unitary operator

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Quantum queries Definition of query operation: U f |i i |a i =|i i |a © f(i) i for all i,a 2 {0,1}; © is XOR operation: 0 © 0=0; 0 © 1=1; 1 © 1=0 Then U f is defined for all basis states and ) U f is defined

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First Idea: “Parallel” computation

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Deutsch’s Algorithm

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Idea for the analysis: Effect of U f on |x i 1/2 1/2 (|0 i - |1 i ): results in (-1) f(x) |x i 1/2 1/2 (|0 i -|1 i ) Because: U f |0 i 1/2 1/2 ( |0 i -|1 i ) = |0 i 1/2 1/2 ( |f(0) i -|f(0) © 1 i ) = (-1) f(0) |0 i 1/2 1/2 ( |0 i -|1 i )

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Deutsch’s Algorithm U f |1 i 1/2 1/2 ( |0 i -|1 i ) = |1 i 1/2 1/2 ( |f(1) i -|f(1) © 1 i ) = (-1) f(1) |1 i 1/2 1/2 ( |0 i -|1 i ) We can disregard the second qubit now

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Deutsch’s Algorithm Now apply Hadamard transform (on the remaining qubit) State before the transformation: f(0) = f(1): § 1/2 1/2 (|0 i +|1 i ) f(0) f(1): § 1/2 1/2 (|0 i -|1 i ) Case 1: f(0)=f(1): H § 1/2 1/2 (|0 i +|1 i ) = § |0 i Case 2: f(0) F(1): H § 1/2 1/2 (|0 i -|1 i ) = § |1 i Measurement decides between both cases perfectly

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Deutsch’s Algorithm H UfUf H |0 i |1 i Measurement H

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Quantum Circuits We write algorithms as circuits on n qubits, unitary transformations are written as boxes Initial states of qubits are denoted on the left side

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Deutsch-Josza Algorithm f:{0,1} n {0,1} Is f balanced (50% 0, 50% 1) or constant? Promise: f is either balanced or constant, if not the output of the algorithm can be anything We show an algorithm with 1 query and no error Any deterministic algorithm needs : 2 n /2+1 queries! Why? We give an adversary argument: Fix f depending on the queries of the algorithm, f(x 1 )=0,...,f(x l )=0 Before l > 2 n /2 the algorithm cannot make a “safe” decision Algorithm can be forced to make an error (f might still be balanced or constant)

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Deutsch Josza Algorithm H UfUf H n |0 i n |1 i Measure H n nnn

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Reminder: Hadamard Transformation

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U f Gate/Black Box U f |x i |a i =|x i |a © f(x) i for all x 2 {0,1} n,a 2 {0,1} f:{0,1} n ! {0,1} © ist XOR Operation: 0 © 0=0; 0 © 1=1; 1 © 1=0

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Deutsch Josza Algorithm

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Deutsch Josza Then Hadamard Amplitude of |0 n i : f constant ) § 1 f balanced ) 0

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A remark The Deutsch Josza problem can be solved easily by a randomized algorithm that is allowed to err with some small probability We will soon see quantum algorithms that “do better”

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Deutsch-Josza Algorithm Deterministic: 2 n /2+1 queries Quantum: 1 query, O(n) gates (local transformations), no error Randomized algorithms are also efficient, but they need to make errors

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