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How instantaneous is state preparation in a quantum register, if all possible superpositions are to be initialized equally?

We know that giving a single qubit starting in the state $|0\rangle$, which is a state one can initialize very fast with high fidelity, then we can put it in the superposition state $|\psi \rangle = ...
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5 votes
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Is the CNOT in the standard three-qubit circuit for the GHZ state necessary?

If you initialize three qubits to $|0\rangle$, apply a Hadamard gate to each, then measure each in the computational basis, the result will be an independent coin flip for each bit: that is, any of ...
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Have you ever seen the preparation of the state $a^{*}|0\rangle+b^*|1\rangle$ and $a|0\rangle+b|1\rangle$ from one initial state?

$\langle \psi|$ is not a quantum state, but a linear functional on the set of quantum states. $|\psi\rangle$ is a quantum state, any gates that you can apply to it can only take it to quantum states, ...
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4 votes

How instantaneous is state preparation in a quantum register, if all possible superpositions are to be initialized equally?

You can prepare equal superposition by application of Hadamard gate on each qubit. The result will be state $$ |q\rangle=\frac{1}{\sqrt{2^n}}\sum_{i=0}^{2^n}|i\rangle, $$ i.e. the desired equally ...
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3 votes

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

I think that asking for an exact solution is pointless, because quantum computers don't have infinite precision. You are limited, for example, by accuracy of pulses that control the gates. To ...
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3 votes

How many quantum gates are needed to prepare an arbitrary state?

A canonical reference for gate decompositions is Barenco et al., Elementary gates for quantum computation. In particular, it also contains recipes to decompose an arbitrary $n$-qubit unitary into ...
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3 votes

How to prove that EPR outcomes have equal probability no matter the basis?

The local state (described by density matrix) of each qubit in EPR state is \begin{equation} \rho=\frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \end{equation} It does not depend on ...
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  • 3,004
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How to prove that EPR outcomes have equal probability no matter the basis?

Let $\rho_{AB}$ be a quantum state shared between two parties, Alice and Bob. Suppose Alice performs a POVM measurement $\{M_i\}_i$ on her half of the state. Then the probability that Alice obtains ...
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2 votes
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How to prepare a random 1-qubit superposition for data encoding

Theorem: Suppose $U$ is a unitary operation on a single qubit. Then there exist real numbers $\alpha, \beta, \gamma, \delta$ such that $$ U = e^{i\alpha} R_z(\beta) R_y(\gamma)R_z(\delta) $$ This is ...
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  • 12.3k
2 votes

Have you ever seen the preparation of the state $a^{*}|0\rangle+b^*|1\rangle$ and $a|0\rangle+b|1\rangle$ from one initial state?

$\left<\psi\right|$ is not a state of a quantum system, it is a linear functional that takes a quantum state and returns a scalar. In terms of basic Linear Algebra, it is a row vector rather than a ...
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2 votes
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How to create superposition of states with fixed parity with a quantum circuit?

I actually found the solution to my problem - with one caveat: the amplitudes $a,b,c,d$ can not be completely arbitrary. An $n$-qubit superposition state with even parity will have $2^{n-1}-1$ free ...
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2 votes
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How many quantum gates are needed to prepare an arbitrary state?

I believe this Q&A answers your question about decomposition in detail: Minimum number of 2 qubit gates to build any unitary In short, you are correct that the lower bound for a number of 2-qubit ...
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  • 600
1 vote

How to create superposition of states with fixed parity with a quantum circuit?

To do this in general is equivalent to creating an arbitrary $n-1$-qubit state, and then using the pattern of controlled-nots to determine the bit value of the final qubit, as you did in your solution....
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1 vote
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How to create known quantum state in Qiskit (or any other platform) comprising of two or more bits?

As Martin Vesely mentioned in the comments, you can use the initialize function to perform such a task. For instance, to create the state you desire, you can do the ...
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1 vote

Produce a quantum state with its density matrix an identity matrix up to an constant

In fact, I figured out by myself. When I was reading this website, it posts a density matrix that I want(that is a long website, to find the corresponding part, just search the keyword 'identity'). ...
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1 vote

How can I find the fidelity of the preparation operation $|0\rangle$ of IBMQ?

Ideal measurement Let $\rho_0$ denote the state resulting from the $|0\rangle$ preparation. The fidelity of $\rho_0$ and $|0\rangle$ is $$ F: = F(|0\rangle, \rho_0) = \langle 0|\rho_0|0\rangle. $$ On ...
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