chap4 - Chapter 4 Fourier Sampling Simons Algorithm 4.1...

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Chapter 4 Fourier Sampling & Simon’s Algorithm 4.1 Reversible Computation A quantum circuit acting on n qubits is described by an 2 n × 2 n unitary operator U .S i n c e U is unitary, UU = U U = I . This implies that each quantum circuit has an inverse circuit which is the mirror image of the original circuit and which carries out the inverse operator U . The circuits for U and U are the same size and have mirror image gates. Examples: H = H CNOT = CNOT R θ = R θ 37
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38 CHAPTER 4. FOURIER SAMPLING & SIMON’S ALGORITHM 4.2 Simulating Classical Circuits Let us frst consider whether given any classical circuit there is an equivalent quantum circuit. More concretely, suppose there is a classical circuit that computes a Function f ( x ) { 0 , 1 } m on input x { 0 , 1 } n , is there a quantum circuit that does the same? Obviously such a quantum circuit must map computational basis states to computational basis states (i.e. it must map each state oF the Form | x ° to the state | f ( x ) ° ). A unitary transFormation taking basis states to basis states must be a permutation. (Indeed, iF U | x ° = | u ° and U | y ° = | u ° ,th en | x ° = U 1 | u ° = | y ° .) ThereFore we need the input, or domain, to be the exact same number oF bits as the range: m = n . What is more, the Function f ( x ) must be a permutation on the n -bit strings. Since this must hold aFter every application oF a quantum gate, it Follows that iF a quantum circuit computes a classical Function, then it must be reversible :i t must have an inverse. How can a classical circuit C which takes an n bit input x and computes f ( x ) be made into a reversible quantum circuit that computes the same Func- tion? The circuit must never lose any inFormation, so how could it compute a Function mapping n bits to m<n bits (e.g. a boolean Function, where m = 1)? The solution to this problem is to have the circuit take the n input qubits in the state | x ° and send them to | x ° , while in the process taking some m qubits in the | 0 ° state to | f ( x ) ° . Then the inverse map is simple: the n -bit string | x ° goes back to x , and | f ( x ) ° goes to an m bit string oF 0’s: | 0 ° . However, this is not always perFectly easy, some times to make the circuit work it needs scratch qubits in the input. A scratch qubit is a qubit that starts out in the | 0 ° state, and ends in the | 0 ° state. Its purpose is to be used in computations inside oF the circuit. OF course, since the quantum circuit does not alter these qubits, the inverse circuit also leaves them alone. While these bits are a necessary ingredient to a reversible quantum circuit, they are not the main character and are oFten let out oF circuit diagrams. Take a look at ±igure 4.2 For the Full picture. How is this done? It is a Fact that any classical AND and OR gates can be simulated with a C-SWAP gate and some scratch | 0 ° qubits (±igure 4.2).
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chap4 - Chapter 4 Fourier Sampling Simons Algorithm 4.1...

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