Linked Questions

7 votes
3 answers

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 ...
Sam Jaques's user avatar
  • 2,066
5 votes
2 answers

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 ...
As10's user avatar
  • 109
4 votes
2 answers

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 ...
Sam Palmer's user avatar
10 votes
1 answer

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{...
orlp's user avatar
  • 211
4 votes
1 answer

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 \begin{equation} |\psi\rangle = \frac{1}{\sqrt{2^{n}}...
BlackHat18's user avatar
  • 1,335
4 votes
1 answer

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 ...
Amir Naveh's user avatar
1 vote
1 answer

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

In, 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 ...
glS's user avatar
  • 25.4k
4 votes
0 answers

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

See article here: There are two steps in this procedure that I am curious about. First off, they suppose one can construct a circuit which efficiently performs ...
QCQCQC's user avatar
  • 449
2 votes
0 answers

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 ...
Philips's user avatar
  • 31
0 votes
0 answers

procedure for making a probability distribution into a quantum circuit

I found this article on loading a probability distribution into a quantum circuit, the Grover-Rudolph scheme: I understand the idea of progressively dividing ...
kerrwe's user avatar
  • 31