ECS 189A多量子位电路标识

通过测量前两位的奇偶性和最后两位的校验性来确定原始状态。绘制一个包括两个测量值的量子电路，以便可以从测量结果。（提示：为此，您需要两个辅助量子位。）（c） （1分）假设你只得到奇偶校验信息（不知道状态）。对于每个奇偶性测量结果，解释您将应用什么酉来恢复原始状态以执行多数票。（d） （1点）假设我们现在处于叠加态α|000+β|111？。假设我们应用X1和然后应用（c）中的协议。我们得到了什么？（可选问题：X2、X3的情况如何？）3.（3分）在讲座中，我们学习了隐形传送。（参见课本第258-260页协议的解释。）在那次讨论中，我们使用了EPR态|EPR？ECS 189A: Assignment 62/24/2023Due: 3/3/20231. Multi-qubit circuit identities.(a) (1 Point) Reduce the presented circuit to Pauli, rotation, or their products.X =?(b) (1 Point) What is U? (Here |Y+? is the +1 eigenstate of Y .)|Y+? |Y+?= U|Y+? |Y+?.(c) (1 Point) Reduce the presented circuit to Pauli, rotation, or their products.Y2. (4 Points) In Project 1, you were asked to find a quantum circuit that computes the parity of bits.Checking a parity can be useful, because we can use those operations to check if an error has occurred. Wewill go through an example that exemplifies this point.(a) (1 Point) Consider two classical bit-strings 00
0 and 111, representing a binary information. Considerthe following three-step process.1. You are given a bit-string 000 (or equivalently, with 111).2. For each bit, flip the bit withprobabilityp.3. Perform a majority vote. If the majority is 1, return 111. If the majority is 0, return 000.If the bit-string obtained is different from the original bit-string. We say that a logical error has occurred.What is the probability of logical error occurring?(b) (1 Point) Now consider two quantum states |000? and |111?. Suppose one of the qubits is flipped(by applying X). No matter which qubit is flipped, it turns out that we can always revert the state back1to the original state by measuring the parity of the first two bits and the parity of the last two bits.Draw a quantum circuit that includes two measurements, such that the parities can be obtained from themeasurement outcomes. (Hint: You will need two ancillary qubits for this purpose.)(c) (1 Point) Suppose you are only given the parity information (without knowing the state). For eachparity measurement outcome, explain what unitary you will apply to get back the original state to performthe majority vote.(d) (1 Points) Suppose now we are in a superposition state α|000?+β|111?. Suppose we applied X1 andthen applied the protocol in (c). What do we get? (Optional Problem: What about the case of X2, X3?)3. (3 Points) In the lecture, we learned about teleportation. (See Page 258-260 of the textbook for theexplanation of the protocol.) In that discussion, we used the EPR state |EPR? = 1√2(|00?+ |11?). (In thetextbook, this is denoted as |Φ+?.) Here we use a modified EPR state:|EPR′? = 1√2(|00? ? |11?).(Everything else about the protocol will be exactly the same.) In this problem, we will learn how thischange modifies the state that Alice gets. (Note: Please take a look at the teleportation circuit in the slidesfor Lecture 12.)(a) (1 Point) There is a single-qubit unitary operator U acting on the first qubit of |EPR?, such that(U ? I)|EPR? = |EPR′?. What is this unitary?(b) (1 Point) Using (a), we can see that our new teleportation protocol is exactly the same as our oldprotocol except for the fact that we now have additional gates between CNOT and measurements. Whatare these gates? (Hint 1. You may find it helpful to draw a circuit diagram. Hint 2. You may find thecheat sheet for circuit identities helpful.)(c) (1 Point) Suppose the sender in the teleportation protocol prepares her/his state as |ψ?. Afterrunning through our new teleportation protocol, what state does the sendee get
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