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Questions tagged [quantum-computation]

Quantum Computation deals with considering computation as fundamentally physical, as well as replacing the classical binary digit (bit) with the quantum binary digit (qubit). While the classical bit is either 0 or 1, the qubit can be in a superposition of these states. Computation systems that use quantum phenomena, such as superposition and entanglement, can solve certain complex problems very quickly.

Related Links

Quantum Computing on Wikipedia

(Textbook) Quantum Computation and Quantum Information: The de-facto standard textbook for learning about quantum computing.

(Video Series) Quantum computing for the determined: Khan-academy-style videos explaining quantum computation, by Michael Nielson (co-author of Quantum Computation and Quantum Information).

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Finding probabilities of outcomes of a qubit

Given a qubit in state $\mid 0\rangle$, first find the probabilities of the outcomes from measuring it in the basis of $\mid a\rangle =\dfrac{1}{\sqrt{2}}(1,i)$ and $\mid b\rangle =\dfrac{1}{\sqrt{2}}(1,-i)$. Following the first measurement, the…
Random Person
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Lower bounds for classical counting algorithm

This is Exercise 6.14 from Nielsen & Chuang. Prove that any classical counting algorithm with a probability at least $3/4$ for estimating $M$ correctly to within an accuracy $c\sqrt{M}$ for some constant $c$ and for all values of $M$ must make…
Rahul Sarkar
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a quantum algorithm with high probability on a 4 to 1 function

Let $f : $ {0,1}$^n \rightarrow $ {0,1}$^n$ be a 4-to-1 function, such that there exist distinct and non-zero $a,b\in $ {0,1}$^n$ such that for all $x\in$ {0,1}$^n$: $f(x) = f(x ⊕ a) = f(x ⊕ b) = f(x ⊕ a ⊕ b)$ . Note that ⊕ is a bit-wise xor, and…
Rachel Bernoulli
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Doubts regarding Local Hamiltonian Problem

I have background in CS but not in Quantum Computing/Quantum Complexity Theory. I am trying to understand the Local Hamiltonian Problem (the formal definition as below): Local Hamiltonians or Q5SAT: Let $H_j\ (for\ j=1,...r)$ be 5-local Hamiltonians…
TheoryQuest1
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Show that surrounding a $CNOT$ with Hadamard gates switches the role of the control-bit and target-bit of $CNOT$

So I wish to show that surrounding a CNOT with Hadamard gates switches the role of the control-bit and target-bit of CNOT. Explicitly I want to show ($H\otimes H$)CNOT($H\otimes H$) is the 2-qubit gate where the second bit controls whether the…
user16319
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Let H be the Hadamard operator, prove $H^{\otimes n} \left| 0 \right>^{\otimes n} = \sum_{i=0}^{2^n -1} \left| i \right>$

Let H be the Hadamard operator. $$ H = (\left| 0 \right> \left< 0 \right| + \left| 0 \right> \left< 1 \right| + \left| 1 \right> \left< 0 \right| -\left| 1 \right> \left< 1 \right| )$$ prove that. $$H^{\otimes n} \left| 0 \right>^{\otimes n} =…
Mario Vega
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Pauli-Z gate clarification

In this following document: https://people.cs.umass.edu/~strubell/doc/quantum_tutorial.pdf Page 16 says the following about Pauli Z gate However the right-hand most (ket-bra) side of the equation doesn't seem to evaluate to the 2x2 matrix. Is this…
Prikshet Sharma
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Understanding the Pauli-Y gate in the Bloch sphere

I'm having some trouble understanding the Bloch representation of qubits in some cases. The canonical representation $\cos(\psi/2) |0\rangle + \sin(\psi/2)e^{i\theta}|1\rangle$ has the first coefficient, $\cos(\psi/2)$, always a nonnegative real…
Miguel
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Function that grows faster then BB(n) using quantum computation?

Is it possible to define a function in terms of quantum computers that grows faster then any that is defined in terms of turing machines?
user347955
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Matrix for Control-Z gate

Can someone help me to prove that this Matrix: I - 2|11...1><11...1| represent (n-1) Control-Z gate (Z operator is applied to n-th qubit only if all remaining qubits are state |1>)
Hai Duc
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Quantum algorithm writing

I'm interested in learning to write quantum algorithms, but I don't know where to start. Could someone recommend some resources for me?
Sheepman
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Special Quantum Gate

Given $\alpha|1\rangle+\beta|0\rangle$, to transform to $|\phi\rangle=\alpha|0\rangle+\beta|1\rangle$ , we use Quantum NOT gate: NOT = $\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$. For $\alpha|0\rangle-\beta|1\rangle$, what kind of gate do we need…
ask
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Can you make a Gaussian-weighted superposition state with polynomially many gates?

Suppose I have N register qubits, so they can represent the range [0, 2^N-1]. They are initialized in the all-zeroes state. I want my final state to approximate $$|\phi \rangle = \frac{1}{2^{2^N-1}} \sum_{m=0}^{2^N-1} {2^N-1 \choose m} |m \rangle…
Craig
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2 exercises on reversible quantum process and quantum measurements

For question 7, Suppose that there exist such a reversible quantum process such that $$\lvert\varphi^{\perp}\rangle=U\lvert\varphi\rangle$$ where $U$ is a unitary matrix. Then,$$ 0 = \langle\varphi\lvert\varphi^{\perp}\rangle=\langle\varphi\lvert U…
Tony
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An $N$-qubit system keeps $2^N$ pieces of information or should we say $2^{N+1}$?

Say my quantum register is of size $3$ qubits. This means I'll need $2^3$ complex numbers to describe all of its possible arrangements. But each complex number requires $2$ real numbers, so maybe I could say I'll need $2^{3}\times2 = 2^4$ real…
R. Chopin
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