A4. Generate W state

Codeforces

IDCF1002A4

Time1000ms

Memory256MB

Difficulty—

English · Original

Chinese · Translation

Formal · Original

You are given _N_ = 2_k_ qubits (0 ≤ _k_ ≤ 4) in zero state . Your task is to create a generalized [W state](https://en.wikipedia.org/wiki/W_state) on them. Generalized W state is an equal superposition of all basis states on _N_ qubits that have Hamming weight equal to 1: For example, for _N_ = 1, . You have to implement an operation which takes an array of _N_ qubits as an input and has no output. The "output" of the operation is the state in which it leaves the qubits. Your code should have the following signature: namespace Solution { open Microsoft.Quantum.Primitive; open Microsoft.Quantum.Canon; operation Solve (qs : Qubit\[\]) : () { body { // your code here } } } [samples]

给定 #cf_span[N = 2k] 个量子比特（#cf_span[0 ≤ k ≤ 4]），它们均处于零态。你的任务是为它们构造一个广义 W 态。广义 W 态是所有在 #cf_span[N] 个量子比特上汉明重量等于 1 的基态的等权叠加：例如，当 #cf_span[N = 1] 时，。你需要实现一个操作，该操作以一个包含 #cf_span[N] 个量子比特的数组作为输入，且无输出。该操作的“输出”是它作用后量子比特所处的状态。你的代码应具有以下签名： [samples]

**Definitions** Let $ N = 2^k $ for some $ k \in \mathbb{Z} $, $ 0 \leq k \leq 4 $. Let $ \mathcal{H} = (\mathbb{C}^2)^{\otimes N} $ be the Hilbert space of $ N $ qubits. Let $ \{|x\rangle \mid x \in \{0,1\}^N\} $ be the computational basis of $ \mathcal{H} $. The **generalized W state** on $ N $ qubits is defined as: $$ |W_N\rangle = \frac{1}{\sqrt{N}} \sum_{i=1}^{N} |e_i\rangle $$ where $ |e_i\rangle $ is the basis state with Hamming weight 1 and the single $ 1 $ at position $ i $ (i.e., $ |e_i\rangle = |0\rangle^{\otimes (i-1)} \otimes |1\rangle \otimes |0\rangle^{\otimes (N-i)} $). **Constraints** 1. $ N = 2^k $ for $ k \in \{0, 1, 2, 3, 4\} $, so $ N \in \{1, 2, 4, 8, 16\} $. 2. The input state is $ |0\rangle^{\otimes N} $. 3. The operation must be unitary and leave no ancillae uncomputed. **Objective** Implement a unitary operation $ U $ such that: $$ U |0\rangle^{\otimes N} = |W_N\rangle $$

API Response (JSON)

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      "content": "You are given _N_ = 2_k_ qubits (0 ≤ _k_ ≤ 4) in zero state . Your task is to create a generalized [W state](https://en.wikipedia.org/wiki/W_state) on them. Generalized W state is an equal superpositi",
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