Exercise 3.1.5 (J.Gruska)

[sassa_nf]

Suppose there is a quantum function that can be represented as a unitary matrix U_y whose action is U_y|ψ⟩=(-1)^y|ψ⟩.

Given a conditional U_y, determine y using a network with at most two Hadamard gates.

---

CU_y:

|a⟩ ---*---- |a⟩
       |
|b⟩ -- Uy -- (-1)a*y * |b⟩

As a matrix:

Y=(-1)y

      |1 0 0 0|
CUy = |0 1 0 0|
      |0 0 Y 0|
      |0 0 0 Y|

So, if the first qubit is |0⟩, everything is left unchanged. If the first qubit is |1⟩, the action U_y is applied to the second qubit. (The usual Control gate behaviour).

The solution to the problem then can be expressed as:

|0⟩ -- H --*-- H -- |y⟩
           |
|0⟩ ------ Uy ----- |0⟩

That is, if both qubits on input are 0, then the first qubit of output is always measured to be y.

As a matrix:

Y=(-1)y

|1  0  1  0|   |1 0 0 0|   |1 0  1  0|         |1 0  Y  0|   |1 0  1  0|         |1+Y   0 1-Y   0|
|0  1  0  1| * |0 1 0 0| * |0 1  0  1| * 1/2 = |0 1  0  Y| * |0 1  0  1| * 1/2 = |  0 1+Y   0 1-Y| * 1/2
|1  0 -1  0|   |0 0 Y 0|   |1 0 -1  0|         |1 0 -Y  0|   |1 0 -1  0|         |1-Y   0 1+Y   0|
|0  1  0 -1|   |0 0 0 Y|   |0 1  0 -1|         |0 1  0 -Y|   |0 1  0 -1|         |  0 1-Y   0 1+Y|

So for y=0 we get:

Y=1

|1 0 0 0|
|0 1 0 0|
|0 0 1 0|
|0 0 0 1|

and for y=1 we get:

Y=-1

|0 0 1 0|
|0 0 0 1|
|1 0 0 0|
|0 1 0 0|

Or:

(H ⊗ I) * CUy * (H ⊗ I) |0ψ⟩ = (H ⊗ I) * CUy * (H|0⟩ ⊗ |ψ⟩)
                             = (H ⊗ I) * CUy * 1/sqrt(2) * (|0ψ⟩ + |1ψ⟩)
                             = (H ⊗ I) * 1/sqrt(2) * (|0ψ⟩ + Y*|1ψ⟩)
                             = 1/sqrt(2) * (H|0⟩ + Y*H|1⟩) ⊗ |ψ⟩
                             = 1/2 * (|0ψ⟩ + |1ψ⟩) + Y/2 * (|0ψ⟩ - |1ψ⟩)
                             = (1+Y)/2 * |0ψ⟩ + (1-Y)/2 * |1ψ⟩

Again: if y=0, Y=1, then the computation produces |0ψ⟩; and if y=1, Y=-1, then the computation produces |1ψ⟩.

We can fit whatever on input as ψ, and 0 as the first bit. Then measure the first qubit. It will be equal to y.

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What I find very interesting is this.

We cannot understand the function of a component, and derive the meaning of the circuit: you need to consider the circuit as the whole. For example, we "see" CU_y "does not modify" the first qubit: if the first qubit is |0⟩, it remains |0⟩; if the first qubit is |1⟩, it remains |1⟩. But when used in conjunction with H, which just creates a superposition of |0⟩ and |1⟩, the meaning changes: CU_y no longer preserves the first qubit.

It seems the quantum circuits are not composable. How does one scale the process of designing new solutions to new problems?