All Questions
Tagged with quantum-computer linear-algebra
22 questions
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For any unitary $U = e^{iA}$, question about the matrix element $A_{pq}$ of matrix $A$ and its functional derivative [closed]
Any unitary operation $U \in U(N)$ can be represented as $U = e^{iA}$, where $A$ is an arbitrary $N \times N$ Hermitian matrix. In the case of a time-independent Hamiltonian $H$, we have $A = -itH$ (...
- 177
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118
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Measuring quantum states without violating no-cloning
In Nielsen and Chuang exercise 2.64, the following problem is given:
Suppose Bob is given a quantum state chosen from a set $\{ \lvert \psi_1 \rangle, \ldots , \lvert \psi_m \rangle \}$ of linearly ...
- 131
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1
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67
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Existence of unitary operator that transforms phase gates
Let me first introduce the entire problem:
Let $H$ be an Hermitian operator, $W$ be an Unitary operator and let $S$ be the standard phase gate: $\begin{pmatrix}1 & 0 \\ 0 & i\end{pmatrix}$. ...
- 51
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3
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Unitary equivalence between Hermitian operators
Take two (non-zero) Hermitian operators $A$ and $B$. I want to proof that there exists no unitary operator $W$ such that:
$$W^{\dagger}AW = A + B$$
For my research I proved this for some specific case ...
- 51
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5
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Mathematical explanation of bra-ket notation in quantum mechanics
$\newcommand{\hp}[1]{\hphantom{#1}}$
We have the entangled state of two pairs of qubits:
$$
|\psi \rangle =\frac{1}{2}|0011\rangle-\frac{1}{2}|0110\rangle-\frac{1}{2}|1001\rangle+\frac{1}{2}|1100\...
- 253
1
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1
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137
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Outer Product Other form [closed]
The outer product of a ket $|\psi\rangle$ with a bra $\langle\phi|$ according to the textbook Quantum Computing Explained by D. McMahon, behaves likes an operator. He illustrates this by applying an ...
- 31
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307
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CNOT quantum gate output when the control qubit is already in a superposition of states
The CNOT gate output states are clearly defined when the control qubit is in either of the the "pure" state of 0 or 1, as in the following diagram:
However, when the control bit is already ...
user315366
3
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2
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120
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Why does entanglement negativity not satisfy the triangle inequality in the usual sense?
I am a bit puzzled, I’ve read in some places, like the original paper by Vidal, that
$$
\mathcal{N}(\sum_n a_n \rho_n)\leq \sum_n a_n \mathcal{N}(\rho_n)
$$
whenever $a_n \geq 0$ and $\sum_n a_n =1$. ...
- 1,471
1
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1
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85
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Quantum Computing: Preparation of the Bell state Notation [closed]
I was watching some lectures on qubits. They were talking about how to generate a Bell state. They described it as follows:
Prepare state 00:
$$\left |0 \right> \otimes \left |0 \right>$$
Apply ...
- 348
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3
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2k
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Representing Quantum Gates in Tensor Product Space
I want to write the matrix form of a single or two qubit gate in the tensor product vector space of a many qubit system. Ill outline a simple example:
Both qubits, $q_0$ and $q_1$ start in the ground ...
- 11
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2
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466
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Rank of a density matix
I was just trying to understand the meaning of rank of a density matrix. I came across the following post, which says that the rank of density matrix is the number of non-zero eigenvalues. And for a ...
user214336
2
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2
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1k
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Understanding projective measurements as a special case of POVM measurements ("third postulate" in Nielsen and Chuang)
I am working through Nielsen and Chuang's book and am confused about a detail from sections 2.2.3 and 2.2.5.
On page 88 of my copy (section 2.2.5), they write
Projective measurements can be ...
- 199
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3
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562
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Derive unitary matrix for Tofolli gate
I know that for two qubits, the CNOT matrix is given by
$CNOT=P_0 \otimes I + P_1 \otimes X.$
But I cannot figure out how to get the matrix if I have two control qubits, acting on a third, like in ...
- 81
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1
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596
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Why is the projection operator corresponding to $\tilde M$ given by $P_m\otimes I_B$?
Nielsen and Chuang, Chapter 2 (Box 2.6):
Suppose $M$ is any observable on a system $A$, and we have some
measuring device which is capable of realizing measurements of $M$.
Let $\tilde M$ ...
0
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1
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283
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Which set of basis states can a quantum system of qubits actually collapse to?
I was watching a video on "How Does a Quantum Computer Work?".
I'm confused about what they mean by: "Although the qubits can exist in any combination of states, when they are measured they must ...
2
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1
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83
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Constructing an arbitrary 2-Qbit state
I am reading a book on quantum computing. The author is constructing an arbitrary 2-Qbit state from unitary transformations. I need help understanding on step in his logic.
He starts by noting that ...
- 121
2
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2
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704
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Which phase shift gate form is correct?
I am trying to figure out the matrix for a gate I'm going to be implementing - a mirror, i.e., a $180^{\circ}$ phase shift. Quantiki gives
$$R(\theta)=\begin{bmatrix}1&0\\0&e^{2\pi i\theta}\...
- 7,087
0
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1
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496
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How to produce the basis states of a quantum computer?
Suppose you have a one qubit system; the (traditional) basis states are $|0\rangle$ and $|1\rangle$, and any state of the qubit can be described by a linear combination of these two. Now suppose you ...
- 7,087
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1
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637
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Equation For a CNOT Gate Matrix, that works on multiple qubits [closed]
So Say I have 3 qubits, $\lvert000\rangle$, And I want to apply a Pauli-X Gate to the second qubit.
I know that I can create the matrix that will act on those qubits, using this equation:
$$
X_{2,3} =...
- 13
16
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2
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37k
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Trace of an operator matrix (Quantum computation and quantum information)
I'm reading the book Quantum computation and quantum information by Mike & Ike and I'm stuck at 2.60/2.61. There, the author says that, given the operator $A|ψ⟩⟨ψ|$, its trace is:
$${\rm tr}(A|\...
- 279
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339
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When is an operator subspace the span of Kraus operators?
Let $A$ and $B$ be finite dimensional Hilbert spaces, and let $\mathcal{L}(A \to B)$ be the space of linear operators from $A$ to $B$. Say that a subspace $K \subseteq \mathcal{L}(A \to B)$ is a span ...
- 809
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3
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Question about orthonormal decompositions over unitary operators
I'm teaching myself quantum information theory using Nielsen and Chuang's "Quantum Computation and Quantum Information" and I'm at a point in the book where the formalism is starting to make ...
user756