E1. Bernstein-Vazirani algorithm

Codeforces

IDCF1002E1

Time2000ms

Memory256MB

Difficulty—

English · Original

Chinese · Translation

Formal · Original

You are given a quantum oracle - an operation on _N_ + 1 qubits which implements a function . You are guaranteed that the function _f_ implemented by the oracle is scalar product function (oracle from problem D1): Here (an array of _N_ integers, each of which can be 0 or 1). Your task is to reconstruct the array . Your code is allowed to call the given oracle only once. You have to implement an operation which takes the following inputs: * an integer _N_ - the number of qubits in the oracle input (1 ≤ _N_ ≤ 8), * an oracle _Uf_, implemented as an operation with signature _((Qubit\[\], Qubit) => ())_, i.e., an operation which takes as input an array of qubits and an output qubit and has no output. The return of your operation is an array of integers of length _N_, each of them 0 or 1. Your code should have the following signature: namespace Solution { open Microsoft.Quantum.Primitive; open Microsoft.Quantum.Canon; operation Solve (N : Int, Uf : ((Qubit\[\], Qubit) => ())) : Int\[\] { body { // your code here } } } [samples]

你被给定一个量子预言机——一个作用于 #cf_span[N + 1] 个量子比特上的操作，它实现了一个函数。你保证该预言机实现的函数 #cf_span[f] 是点积函数（来自问题 D1 的预言机）：这里（一个包含 #cf_span[N] 个整数的数组，每个整数为 0 或 1）。你的任务是重构数组。你的代码仅允许调用给定的预言机一次。你需要实现一个操作，该操作接收以下输入：你的操作的返回值是一个长度为 #cf_span[N] 的整数数组，每个元素为 0 或 1。你的代码应具有以下签名： [samples]

**Definitions** Let $ N \in \mathbb{Z}^+ $. Let $ f: \{0,1\}^N \to \{0,1\} $ be a function implemented by a quantum oracle, defined as: $$ f(x) = x \cdot a \mod 2 = \bigoplus_{i=1}^N x_i a_i $$ where $ a = (a_1, a_2, \dots, a_N) \in \{0,1\}^N $ is an unknown target string. **Constraints** - The oracle acts on $ N+1 $ qubits and implements the unitary: $$ U_f: |x\rangle |b\rangle \mapsto |x\rangle |b \oplus f(x)\rangle $$ for $ x \in \{0,1\}^N $, $ b \in \{0,1\} $. - The algorithm may invoke the oracle **exactly once**. - The goal is to determine $ a \in \{0,1\}^N $. **Objective** Reconstruct $ a = (a_1, \dots, a_N) $ using a single query to $ U_f $, via a quantum circuit that outputs $ a $ as a classical bit string.

API Response (JSON)

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