Calculate the product state/quantum register back into its tensor product
So let's asume I have a product state/quantum register as a result of a tensor product of two qubits.Lets take a "hard" product state matrix like:$$\frac{1}{\sqrt{2}}\begin{bmatrix}\frac12 +...
View ArticleIn general, what is feasible quantum computation?
I don't really understand what is feasible quantum computation, in my book (Lipton and Regan's Quantum Algorithms via Linear Algebra) they said that:A quantum computation $C$ on s qubits is feasible...
View ArticleDetect if a given binary number belongs to a certain subset with an unitary...
I want to create an operator $A$ which, given three binary numbers, $a_1$, $a_2,a_3$, will detect whether $a_1a_2a_3$ (as a binary number) is in certain set of numbers (for example, detect whether...
View ArticleIs it possible to get the "symbolic" matrix operator associated with a...
Qiskit provides the qiskit.quantum_info.Operator class to get the unitary matrix operator from the corresponding quantum circuit, as in the following example:from qiskit import QuantumCircuitfrom...
View ArticleHow to convert a simple matrix into circuit? [duplicate]
Suppose you have an invertible matrix. How do you convert it into a circuit?Matrices have dimensions $2^n \times 2^n$, so a circuit representation is desirable.For example, the matrix below is a simple...
View ArticleHow to convert a basic matrix into a quantum circuit?
Classical gates are not invertible, but larger expressions made from those gates can be invertible. One example of an invertible function is the function $f(A,B,C) = X,Y,Z$:$X = A \ B \ | \ \neg B \...
View ArticleWhat is the general unitary matrix for two- and three-qubit states?
As pointed out in the QisKit tutorial here, for one qubit there exists a general unitary (see the expression for it in the previous link). I wonder if there exists equally unambiguous expressions for...
View ArticleExistence of Hamiltonians such that the time evolution unitary becomes identity
Can we always find a set of coefficients ${k_i}$ (where not every $k_i = 0$) for a given Hamiltonian $H = \sum k_i H_i$, such that the unitary operator becomes the identity operation: $e^{-iH} =...
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Toffoli Gate Matrices
Here are the different toffolis (or maybe one of them is toffoli and the others are very similar to toffoli gates)My question is:we know the matrix of the number 1 Toffoli:What are the matrices for...
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Show how the Bell state arises from the circuit with Hadamard and CNOT, using...
I understand that starting with,we can get to $\vert \Phi^+ \rangle$. First, we start with $\vert Q_1 \rangle \otimes \vert Q_2 \rangle = \vert 0 \rangle \otimes \vert 0 \rangle$ and then applying $H$...
View ArticleApplication of transformation $U_d$ that maps any qudit state to $|d-1\rangle$
When giving examples of universal gate sets in the paper Qudits and High-Dimensional Quantum Computing, the authors first define the transformation that maps any given qudit state to...
View ArticleGeneric circuit for signature matrix
Consider a set $ A = \{a_0,a_2,\ldots,a_{k-1}\} \subset [N] := \{0,1,\ldots,N-1\}$.Consider the diagonal matrix\begin{equation}R := I - 2 \sum_{a\in A} |a\rangle\langle a|,\end{equation}which is just a...
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How to prove that these equations are correct for $CZ$ and $CX$?
How do I prove that the equation on the right is $CX$ and $CZ$ gate? I don't think that reaching the matrix of the CX or CZ is possible with the given equation.For (b) I keep getting $I \otimes I$...
View ArticleQuantum Linear Algebra [closed]
Find a 4 x 4 unitary matrix U such that U = eiA. (Possibly up to multiplying by a unit scalar, U is a matrix seen in the course.) Verify your calculation by showing how if U were given, one can obtain...
View ArticleProve...
Define the Pauli-Liouville representation of a (linear) map $\mathcal{G}$ as $\mathcal{A_G}$, which has components\begin{equation}\label{2}...
View ArticleEasy way to look at matrix in computational and Hadamard bases?
Given a $2^n \times 2^n$ matrix $M$ of classical data (so, just a bunch of numbers), is there any way to query that matrix in both the computational basis (basically, $M$) and the Hadamard basis, i.e....
View ArticleCNOT circuit synthesis with Gauss elimination. Explanation and beyond?
This paper introduces to the synthesis of a (optimal) circuit of CNOTs only; starting from a parity map encoded into a matrix.It is based on Gaussian Elimination.This is an important result, which find...
View ArticleI have two Choi matrix I suspect be equivalent. Can I manipulate them?
I am performing a process tomography over a protocol I suspect to be equivalent to the $CS$ gate.To compare basic operators I usually compute the Choi matrix of the target gate -- which in this case...
View ArticleHow to get parity check matrix from a circuit in stim
I am working on QECC and, differently from classical ECC where everything is generally described by the parity-check matrices, QECC generally involves the low-level description of the circuit instead,...
View ArticleHow to compute the gate matrix for an operation on qubits not next to each other
I have a quantum circuit with 4 input qubits, A, B, C, and D. A is at the top, D is at the bottom.If I wanted to do a CNOT between B and C and leave A and D alone, I know the gate matrix for this would...
View ArticleExpectation values using qiskit
Expectation values can be calculated using$\bf{Matrix}$$\bf{mechanics}:$$A$ has eigenvalues $\lambda_j$ and eigenstates $\Phi_j$. Then the expectation value of $A$ with respect to a state$\Psi=\sum_j...
View ArticleWhy is the matrix obtained from the coefficients of orthogonal states unitary?
I'm having troubles in understanding a statement in Box 2.7 at page 113 in the Nielsen & Chuang.Firstly, it assumed to be working with a two-qubits quantum system in state $|\psi\rangle =...
View ArticleClarification on Matrix Representation of a Quantum Gate
I came across a matrix representation in my quantum computing studies and I'm seeking clarification on its interpretation. The matrix I encountered is:$$\left[\begin{matrix}1 - i & 0 & 0 &...
View ArticleShow that a $CZ$ gate can be implemented using a $CNOT$ gate and Hadamard gates
Show that a $CZ$ gate can be implemented using a $CNOT$ gate and Hadamard gates and write down the corresponding circuit.Recall from Quantum Information Theory that $Z=HXH$. As $CNOT$ is a...
View ArticleWhat is the formula for the matrix representation of a general controlled gate?
Suppose I have $n$-qubit circuit. I have a single-qubit gate (e.g. a Pauli gate) at qubit $a$ and it is controlled by the qubit $b$. What is the matrix representation for this controlled gate? The...
View ArticleLeft-canonical matrix product state
A pure quantum state $$\tag{1}|\Psi\rangle=\sum_{j_1,\ldots,j_N=1}^{d}\psi_{j_1j_2\ldots j_N} |j_1, \dots, j_N\rangle\,,$$ can be represented exactly in the MPS form\begin{equation}\tag{2} |\Psi\rangle...
View ArticleIn Schur-Weyl's duality, why is the commutant of $\pi_k(S_k)$ spanned by...
I'm reading this tutorial paper about quantum state certification. However, I'm confused about the concept of Schur-Weyl duality, explicitly Theorem 35 of the paper. Let $S_k$ denotes the symmetric...
View Articleqml.StronglyEntanglingLayers custom CNOT placement
The qml.StronglyEntanglingLayers function works great for what I need. However, I'd like to modify so that for each layer, only the first qubit is the control and the rest are targets of the control...
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Trying to use matrices for Hadamard and Controlled Not gates
I have the following simple quantum circuit:This outputs are 00 and 11 for the two qubits. Using matrices, I have applied the H gate to the first qubit (ket...
View ArticleIs there value in developing a 'physical/relativstic' QIT (discussion)?
My motivation for asking this question is that I've recently been captivated by representation theory and I am incredibly interested in studying the symmetries behind different the operators in...
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