H. Oracle for f(x) = parity of the number of 1s in x

Codeforces

IDCF1001H

Time1000ms

Memory256MB

Difficulty—

English · Original

Chinese · Translation

Formal · Original

Implement a quantum oracle on _N_ qubits which implements the following function: , i.e., the value of the function is 1 if _x_ has odd number of 1s, and 0 otherwise. ## Input You have to implement an operation which takes the following inputs: * an array of qubits _x_ (input register), * a qubit _y_ (output qubit). The operation doesn't have an output; the "output" of your solution is the state in which it left the qubits. Your code should have the following signature: namespace Solution { open Microsoft.Quantum.Primitive; open Microsoft.Quantum.Canon; operation Solve (x : Qubit\[\], y : Qubit) : () { body { // your code here } } } [samples]

实现一个作用于 #cf_span[N] 个量子比特上的量子预言机，实现以下函数：即，当 #cf_span[x] 中包含奇数个 1 时，函数值为 1，否则为 0。你需要实现一个操作，该操作接收以下输入：该操作没有输出；你的解决方案的“输出”是其最终留下的量子比特状态。你的代码应具有以下签名： ## Input 你需要实现一个操作，该操作接收以下输入：一组量子比特 #cf_span[x]（输入寄存器），一个量子比特 #cf_span[y]（输出量子比特）。该操作没有输出；你的解决方案的“输出”是其最终留下的量子比特状态。你的代码应具有以下签名：namespace Solution { open Microsoft.Quantum.Primitive; open Microsoft.Quantum.Canon; operation Solve (x : Qubit[], y : Qubit) : () { body { // your code here } }} [samples]

**Definitions** Let $ N \in \mathbb{Z}^+ $ be the number of qubits. Let $ \mathcal{H} = (\mathbb{C}^2)^{\otimes N} $ be the Hilbert space of $ N $ qubits. Let $ |x\rangle \in \mathcal{H} $ denote a computational basis state corresponding to an $ N $-bit string $ x \in \{0,1\}^N $. Let $ f: \{0,1\}^N \to \{0,1\} $ be the parity function: $$ f(x) = \bigoplus_{i=1}^N x_i = \text{parity}(x) = \begin{cases} 1 & \text{if } |x|_1 \text{ is odd} \\ 0 & \text{otherwise} \end{cases} $$ **Objective** Implement a unitary operator $ U_f: \mathcal{H} \to \mathcal{H} $ such that for all $ |x\rangle \in \{0,1\}^N $: $$ U_f |x\rangle = (-1)^{f(x)} |x\rangle $$ **Constraints** - The operation acts only on the $ N $ input qubits. - No ancilla qubits may be used unless explicitly permitted (assumed not allowed). - The implementation must be exact and unitary. - The "output" is the final state of the qubits; no measurement is performed.

API Response (JSON)

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