# Tag Info

1 vote

### Proability of measuing a qubit in a two qubit system without having a perpendicular basis

Lets presume your example is correct then the give homework answer follows from the example. Born's rule in quantum mechanics is a fundamental principle used to calculate the probability of measuring ...
• 564
1 vote

### Prove that there is no polynomial size quantum algorithm for a Simon's problem with no promise on the input

Consider the set of values taken by $f$; that is, consider the range of $f$. For any arbitrary function $f$, this range can be partitioned as follows: There could be a first set of images $A$ that ...
Accepted

### To understand the notation of $| a, a \oplus b \rangle$

As you correctly wrote $|a,a\oplus b\rangle:=|a\rangle\otimes|a\oplus b\rangle$ where $a$ as well as $a\oplus b$ are either $0$ or $1$. The reason for the latter is that the logic symbol $\oplus$ ...

### Prove that there is no polynomial size quantum algorithm for a Simon's problem with no promise on the input

Assume You have a function $f(x) := \begin{cases} 1 & \text{if } x = \widehat{x} \\ 0 & \text{else}\end{cases}$ for an arbitrary value $\widehat{x}$....
• 160

### Creating a uniform superposition of a subset of basis states

Please refer to Algorithm 1 in https://arxiv.org/abs/2306.11747. This is the most efficient deterministic method of preparing the uniform superposition states. Instead of using the probabilistic ...
1 vote

### Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$

A more general perspective on this is offered thinking in terms of operator frames. Suppose we're working in a $\mathbb{C}^d$. Let $(\mathcal O_i)_{i=1}^n$ be a set of Hermitian operators that spans ...
• 24.8k

### Are measurements in $X$ and $Z$ bases being equal enough to prove equality of quantum states?

No, it is not possible, as you can clearly see from @CraigGidney's example. However, I would like to also provide some algebra as to why that is the case. The most general single qubit state has three ...
• 2,391
Accepted

### Are measurements in $X$ and $Z$ bases being equal enough to prove equality of quantum states?

Consider $\frac{1}{\sqrt{2}}\big(|0\rangle + i|1\rangle\big)$ vs $\frac{1}{\sqrt{2}}\big(|0\rangle - i|1\rangle\big)$.
• 36.7k

### What is the IQ plane?

I know it a bit late for this, but in case I can help future people searching for this, this paper explains it quite well at the start: Efficient Z-gates for quantum computing Link: http://dx.doi.org/...
1 vote

### Show that quantum channels act as affine transformations in the Bloch sphere

There is already mistake in the very first line of your computation: When you write $\mathcal{E}(\rho)=[\ldots]=\frac12(I+\sum_l E_l(\vec{r}.\vec{\sigma})E_l^\dagger)$ you forgot to apply the channel ...

• 439
1 vote

### Equivalence of quantum circuits

The circuits are not equivalent in general. As an example, take $A = B = X$. The unitary matrices representing the circuits are different: ...
• 643
Accepted

### Equivalence of quantum circuits

Your problem is that your notation is leading you astray. In your first way of writing the circuit, you have the gate $B$ achieving the change in amplitudes $b'c\rightarrow b'c''$ while in the second ...
• 57.9k
Accepted

### Existence of a two-outcome measurement $M$ such that the induced distributions differs between different density matrices

Yes, you can use $\rho - \sigma$ to construct a measurement where the measurement statistics differ for $\rho$ compared to $\sigma$. The issue is that $\rho - \sigma$ is not necessarily positive, ...
• 6,718
### Is $|A\rangle = \frac{1}{\sqrt2} |00\rangle + \frac{1}{\sqrt2} |01\rangle$ a valid quantum state?
Given that we are talking about the state of 2 qubits, the canonical basis of the state space is $$(|00\rangle,|01\rangle,|10\rangle,|11\rangle)$$ A physically valid quantum state is just a vector in ...