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### Forming states of the form $\sqrt{p}\vert 0\rangle+\sqrt{1-p}\vert 1\rangle$

I'm curious about how to form arbitrary-sized uniform superpositions, i.e., $$\frac{1}{\sqrt{N}}\sum_{x=0}^{N-1}\vert x\rangle$$ for $N$ that is not a power of 2. If this is possible, then one can ...
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### Convert a quantum Phase Oracle into a Probability Oracle

Suppose we have an oracle $O_f$ that given an initial state $|x\rangle$ maps it into the following state: $$O_f : |x\rangle \mapsto e^{if(x)} |x\rangle$$ Now, assuming that $f(x) \in [0,1]$, is it ...
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### Rotations to encode $f(x)$ into ancilla qubit for quantum Monte Carlo

I'm trying to understand the quantum monte-carlo algorithm starting at the most basic version. A key step is rotating (Algorithm 1 p.g 8), an ancilla bit by rotation $R$ with respect to the value of a ...
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361 views

### Is there an efficient circuit implementing the unitary $U|x\rangle|0\rangle=|x\rangle\Big(\sqrt{1 - x/2^n}\,|0\rangle+\sqrt{x/2^n}|1\rangle\Big)?$

Given an $n$-qubit register $|x\rangle$, does there exist an efficient circuit implementing unitary operation $U$ such that U |x\rangle|0\rangle = |x\rangle\Big(\sqrt{1 - x/2^n}\, |0\rangle + \sqrt{...
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279 views

### How to prepare a quantum state of the form $\frac1{2^{n/2}}\sum_{x \in \{0, 1\}^{n}} |x\rangle |y_x\rangle$ with $y_x$ random variables?

Let's say I am given an efficiently samplable probability distribution $D$, over $n$ bit strings. I want to efficiently prepare the following state |\psi\rangle = \frac{1}{\sqrt{2^{n}}...
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331 views

### Preparation of states that correspond to efficiently integrable probability distributions

I have been trying to implement methods from paper Creating superpositions that correspond to efficiently integrable probability distributions by Grover and Rudolph. It is stated that there exists an ...
1 vote
174 views

### How does the induction step in the Grover-Rudolph scheme to prepare superpositions from probabilities work?

In https://arxiv.org/abs/quant-ph/0208112, the authors discuss a scheme to, given a discrete probability distribution $\mathbf p\equiv (p_i)_i$, under some assumptions on $\mathbf p$, prepare the ...
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139 views

### What's the circuit to create superpositions corresponding to efficiently integrable probability distributions?

See article here: https://arxiv.org/abs/quant-ph/0208112 There are two steps in this procedure that I am curious about. First off, they suppose one can construct a circuit which efficiently performs ...
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109 views

### Converting from amplitude encoding to basis encoding

The question is inspired from Preparing a quantum state from a classical probability distribution which shows how basis encoding $\frac{1}{\sqrt n}\sum_{x=0}^{n-1}|x\rangle|p(x)\rangle$ may be ...
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