我如何构建一个电路以产生2个量子位的3个结果的相等叠加？

18

给定一个 $2$ 量子位的系统并由此 $4$ 可能的测量结果中的基础 $\{|00\rangle$ ， $|01\rangle$ ， $|10\rangle$ ， $|11\rangle\}$ ，我怎么可以准备状态，其中：

只有 $3$ 的这些 $4$ 的测量结果是可能的（比如， $|00\rangle$ ， $|01\rangle$ ，）？ $|10\rangle$
这些测量是否同样可能？（类似于贝尔州，但有 $3$ 结果）

— 周末
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1

您的意思是写出实际状态或创建电路以准备在给定输入的情况下这种状态？

— 乔苏·埃特塞萨雷塔·马丁内斯（Jusu Etxezarreta Martinez），

@JosuEtxezarretaMartinez，我的意思是电路。

— 周末

@Blue，如何DIT你设法将这些00和11到狄拉克符号？我尝试 $\ket{00}$ 失败了。

— Weekens

1

@weekens如果单击“编辑”，则可以看到MathJax代码。另外，请参阅此。

— Sanchayan Dutta

1

Nirk de Beaudrap在Quirk中的解决方案……

— stestet，

10

分部分解决问题。

说我们已经派出到 $\mid 00 \rangle$ 。我们可以发送给 $\frac{1}{\sqrt{3}} \mid 00 \rangle + \frac{\sqrt{2}}{\sqrt{3}}\mid 01 \rangle$ 由 $\frac{1}{\sqrt{3}} \mid 00 \rangle + (\frac{1}{2} (1+i))\frac{\sqrt{2}}{\sqrt{3}}\mid 01 \rangle + (\frac{1}{2} (1-i))\frac{\sqrt{2}}{\sqrt{3}}\mid 10 \rangle$ 。满足您所有可能性的要求 $\sqrt{SWAP}$ 但具有不同的阶段。如果要在每个相位上使用相移门来获得想要的相位，就像想要使它们全部相等一样。 $\frac{1}{3}$

现在，我们怎样才能从到 $\mid 00 \rangle$ ？如果是 $\frac{1}{\sqrt{3}} \mid 00 \rangle + \frac{\sqrt{2}}{\sqrt{3}}\mid 01 \rangle$ $\frac{1}{\sqrt{2}} \mid 00 \rangle + \frac{1}{\sqrt{2}}\mid 01 \rangle$ , we could do a Hadamard on the second qubit. It is not a easy with this but we can still use a unitary only on the second qubit. That is done by a rotation operator purely on the second qubit by factoring as

I d \otimes U : ∣ 0 ⟩ \otimes (∣ 0 ⟩) \to∣ 0 ⟩ \otimes (\frac{1}{\sqrt{3}} ∣ 0 ⟩ + \frac{\sqrt{2}}{\sqrt{3}} ∣ 1 ⟩)

$Id \otimes U : \; \mid 0 \rangle \otimes (\mid 0 \rangle) \to \mid 0 \rangle \otimes (\frac{1}{\sqrt{3}} \mid 0 \rangle + \frac{\sqrt{2}}{\sqrt{3}} \mid 1 \rangle)$

U = (\begin{matrix} \frac{1}{\sqrt{3}} & \frac{\sqrt{2}}{\sqrt{3}} \\ \frac{\sqrt{2}}{\sqrt{3}} & - \frac{1}{\sqrt{3}} \end{matrix})

$U = \begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{\sqrt{2}}{\sqrt{3}} & \\ \frac{\sqrt{2}}{\sqrt{3}} & -\frac{1}{\sqrt{3}} & \\ \end{pmatrix}$ works. Decompose this into more basic gates if you need to.

总共有：

∣ 00 ⟩ \to \frac{1}{\sqrt{3}} ∣ 00 ⟩ + \frac{\sqrt{2}}{\sqrt{3}} ∣ 01 ⟩ \to \frac{1}{\sqrt{3}} ∣ 00 ⟩ + (\frac{1}{2} (1 + i)) \frac{\sqrt{2}}{\sqrt{3}} ∣ 01 ⟩ + (\frac{1}{2} (1 - i)) \frac{\sqrt{2}}{\sqrt{3}} ∣ 10 ⟩ \to \frac{1}{\sqrt{3}} ∣ 00 ⟩ + \frac{e^{i θ_{1}}}{\sqrt{3}} ∣ 01 ⟩ + \frac{e^{i θ_{2}}}{\sqrt{3}} ∣ 10 ⟩

$\mid 00 \rangle \to \frac{1}{\sqrt{3}} \mid 00 \rangle + \frac{\sqrt{2}}{\sqrt{3}}\mid 01 \rangle\\ \to \frac{1}{\sqrt{3}} \mid 00 \rangle + (\frac{1}{2} (1+i))\frac{\sqrt{2}}{\sqrt{3}}\mid 01 \rangle + (\frac{1}{2} (1-i))\frac{\sqrt{2}}{\sqrt{3}}\mid 10 \rangle\\ \to \frac{1}{\sqrt{3}} \mid 00 \rangle + \frac{e^{i \theta_1}}{\sqrt{3}}\mid 01 \rangle + \frac{e^{i \theta_2}}{\sqrt{3}}\mid 10 \rangle$

— AHusain
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How do I construct U from basic gates? Let's say, from those available on IBM Q Experience.

— weekens

1

@weekens There's an 'advanced' gate called U3 that allows you to implement any single qubit unitary - you input the values for

θ, λ

$\theta, \lambda$ and

ϕ

$\phi$ to implement

U 3 (θ, λ, ϕ) = (\begin{matrix} \cos \frac{θ}{2} & - e^{i λ} \sin \frac{θ}{2} \\ e^{i ϕ} \sin \frac{θ}{2} & e^{i (λ + ϕ)} \cos \frac{θ}{2} \end{matrix}),

$U3 \left( \theta, \lambda, \phi\right) = \begin{pmatrix}\cos\frac{\theta}{2} && -e^{i\lambda}\sin\frac{\theta}{2} \\ e^{i\phi}\sin\frac{\theta}{2} && e^{i\left(\lambda+\phi\right)}\cos\frac{\theta}{2}\end{pmatrix},$ which can be approximated using

θ \approx 1.91, λ = π

$\theta \approx 1.91, \lambda=\pi$ and

ϕ = 0

$\phi = 0$

— Mithrandir24601

To do this in basic gates, it looks like you would need to rotate into the right basis, then do a phase rotation, then rotate back which may require a fair few gates. However, in a sense, the above U3 is basic in that it's a physically implemented gate (i.e. is directly achieved by performing a couple of physical operations on the qubit instead of the many the would be required by stringing lots of 'not-advanced' gates together)

— Mithrandir24601

@Mithrandir24601, thanks for your explanation! I haven't used U3 yet, will experiment with it in nearest time.

— weekens

@AHusain, implemented your approach in Quirks simulator: here

— weekens

8

I'll tell you how to create any two qubit pure state you might ever be interested in. Hopefully you can use it to generate the state you want.

Using a single qubit rotation followed by a cnot, it is possible to create states of the form

α | 0 ⟩ \otimes | 0 ⟩ + β | 1 ⟩ \otimes | 1 ⟩ .

$\alpha \, |0\rangle \otimes |0\rangle + \beta \, |1\rangle \otimes |1\rangle .$

Then you can apply an arbitrary unitary, $U$ , to the first qubit. This rotates the $|0\rangle$ and $|1\rangle$ states to new states that we'll call $|a_0\rangle$ and $|a_1\rangle$ ,

U | 0 ⟩ = | a_{0} ⟩, U | 1 ⟩ = | a_{1} ⟩

$U |0\rangle = |a_0\rangle, \,\,\, U |1\rangle = |a_1\rangle$

Our entangled state is then

α | a_{0} ⟩ \otimes | 0 ⟩ + β | a_{1} ⟩ \otimes | 1 ⟩ .

$\alpha \, |a_0\rangle \otimes |0\rangle + \beta \, |a_1\rangle \otimes |1\rangle .$

We can similarly apply a unitary to the second qubit.

V | 0 ⟩ = | b_{0} ⟩, V | 1 ⟩ = | b_{1} ⟩

$V |0\rangle = |b_0\rangle, \,\,\, V |1\rangle = |b_1\rangle$

which gives us the state

α | a_{0} ⟩ \otimes | b_{0} ⟩ + β | a_{1} ⟩ \otimes | b_{1} ⟩ .

$\alpha \, |a_0\rangle \otimes |b_0\rangle + \beta \, |a_1\rangle \otimes |b_1\rangle .$

Due to the Schmidt decomposition, it is possible to express any pure state of two qubits in the form above. This means that any pure state of two qubits, including the one you want, can be created by this procedure. You just need to find the right rotation around the x axis, and the right unitaries $U$ and $V$ .

To find these, you first need to get the reduced density matrix for each of your two qubits. The eigenstates for the density matrix of your first qubit will be your $|a_0\rangle$ and $|a_1\rangle$ . The eigenstates for the second qubit will be $|b_0\rangle$ and $|b_1\rangle$ . You'll also find that $|a_0\rangle$ and $|b_0\rangle$ will have the same eigenvalue, which is $\alpha^2$ . The coefficient $\beta$ can be similarly derived from the eigenvalues of $|a_1\rangle$ and $|b_1\rangle$ .

— James Wootton
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8

Here is how you might go about designing such a circuit. $\def\ket#1{\lvert#1\rangle}$ Suppose that you would like to produce the state $\ket{\psi} = \tfrac{1}{\sqrt 3} \bigl( \ket{00} + \ket{01} + \ket{10} \bigr)$ . Note the normalisation of ${\small 1}/\small \sqrt 3$ , which is necessary for $\ket{\psi}$ to be a unit vector.

If we want to consider a straightforward way to realise this state, we might want to think in terms of the first qubit being a control, which determines whether the second qubit should be in the state $\ket{+} = \tfrac{1}{\sqrt 2}\bigl(\ket{0}+\ket{1}\bigr)$ , or in the state $\ket{0}$ , by using some conditional operations. This motivates considering the decomposition

| ψ ⟩ = \frac{\sqrt{2}}{\sqrt{3}} | 0 ⟩ | + ⟩ + \frac{1}{\sqrt{3}} | 1 ⟩ | 0 ⟩ .

$\ket{\psi} \;=\; \tfrac{\sqrt 2}{\sqrt 3}\ket{0}\ket{+} \;+\; \tfrac{1}{\sqrt 3}\ket{1}\ket{0}.$ Taking this view it makes sense to consider preparing

| ψ ⟩

$\ket{\psi}$ as follows:

Prepare two qubits in the state $\ket{00}$ .
Rotate the first qubit so that it is in the state $\tfrac{\sqrt 2}{\sqrt 3}\ket{0} + \tfrac{1}{\sqrt 3}\ket{1}$ .
Apply a coherently controlled operation on the two qubits which, when the first qubit is in the state $\ket{0}$ , performs a Hadamard on the second qubit.

Which specific operations you would apply to realise these transformations — i.e. which single-qubit transformation would be most suitable for step 2, and how you might decompose the two-qubit unitary in step 3 into CNOTs and Pauli rotations — is a simple exercise. (Hint: use the fact that both $X$ and the Hadamard are self-inverse to find as simple a decomposition as possible in step 3.)

— Niel de Beaudrap
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0

Here is an implementation of a circuit producing state $|\psi\rangle = \frac{1}{\sqrt{3}}(|00\rangle + |01\rangle + |10\rangle)$ on IBM Q:

Note that $\theta = 1.2310$ for $\mathrm{Ry}$ on $q_0$ . $\theta = \frac{\pi}{4}$ and $\theta = -\frac{\pi}{4}$ for first and second $\mathrm{Ry}$ on $q_1$ .

The $\mathrm{Ry}$ on $q_0$ prepares qubit in superposition $|q_0\rangle = \sqrt{\frac{2}{3}}|0\rangle + \frac{1}{\sqrt{3}}|1\rangle$ . $\mathrm{Ry}$ gates on $q_1$ and $\mathrm{CNOT}$ implements controlled Hadamard gate. When $q_0$ is in state $|0\rangle$ the Hadamard acts on $q_1$ thanks to negation $\mathrm{X}$ . This happens with probability $\frac{2}{3}$ . Since Hadamard turns $|0\rangle$ to $|+\rangle$ , i.e. equally distributed superposition, final states $|00\rangle$ and $|01\rangle$ can be measured with probability $\frac{1}{3}$ . When $q_0$ is in state $|1\rangle$ , controled Hadamard does not act and state $|10\rangle$ is measured. Since $q_0$ is in state $|1\rangle$ with probability $\frac{1}{3}$ , $|10\rangle$ is measured also with probability $\frac{1}{3}$ .

— Martin Vesely
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