Questions tagged [linear-algebra]

For questions about vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc.

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How to write the joint action of a CP map that acts on a single qubit of a bipartite state?

The question Say I have a completely-positive (CP) map $\mathcal{A}_{ij}$ defined in terms of two projectors $\Pi_i = |i\rangle \langle i |$ and $\Pi_j = |j\rangle \langle j |$ that acts on a density ...
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What is the best way to write a tridiagonal matrix as a linear combination of Pauli matrices?

I'm looking for an algorithm to write an arbitrarily sized tridiagonal matrix as a linear combination of Pauli matrices. The tridiagonal matrix has the form, for example, \begin{pmatrix} 2 & -1 &...
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Encoding a block unitary matrix in a quantum circuit

My goal would be to implement the unitary matrix $M=\begin{bmatrix}U_{1} &0\\0&U_{0}\end{bmatrix}$ as a circuit for arbitrary $N \mathrm{x} N$ unitary matrices. It is trivial to show that if ...
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How to verify a matrix-vector product with Grover search?

I am looking at the Ambainis et al. method of verifying whether $AB = C$ in $O(n^{7/4})$ queries, as described in Buhrman and Špalek. They have the following sentence: verify the matrix-vector ...
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Relation between symmetric subspaces and $n$-exchangeable density matrices

Let us consider $n$ elements, each taken from the set $\{1, 2, \ldots, d\}$ and let $S_n$ be the set of all permutations on these $n$ elements. Define a permutation operator on the set of $n$ qudits ...
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How is the probability of success for Simon's algorithm determined?

In step 3 of Simon's algorithm, we are told to "Repeat until there are enough such $y$’s that we can classically solve for $s$." It then goes on: The above are from this course notes. I am ...
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In the context of block-encoding, what does $|0\rangle\otimes I$ represent?

New to quantum and ran into the block-encoding. Having a bit of trouble understanding $|0\rangle \otimes I$. $|0\rangle$ is just a vector but $I$ is an $n$ by $n$ matrix? Not clear how vector can be ...
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Is the quantum mutual information variance bounded from above?

The relative entropy variance between two quantum states $\rho$ and $\sigma$ is defined to be $$V(\rho\|\sigma) = \text{Tr}(\rho(\log\rho - \log\sigma)^2) - D(\rho\|\sigma)^2,$$ where $D(\rho\|\sigma)$...
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Variational Quantum Linear Solver (Hadamard test): circuit question

Trying to understand the circuit/algorithm for VQLS and I found this diagram to show the high-level idea of doing the Hadamard test in this tutorial. But I am not quite sure why we need the two ...
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Clarification on Watrous' proof of Alberti's theorem on the fidelity function

I am reading John Watrous' quantum information theory book. In the proof of Theorem 3.19 (practically the Alberti's theorem on the characterization of the fidelity function) he claims the following ...
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Permutation covariant channels and their Stinespring dilations

I am interested in a quantum channel from $A^{\otimes n}$ to $B^{\otimes n}$ denoted as $N_{A^{\otimes n} \rightarrow B^{\otimes n}}(\cdot)$. Let $\pi(\cdot)$ be a permutation operation among the $n$ ...
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Prove that any Hermitian Matrix is a real linear combination of Pauli operators [duplicate]

This is an important result in Quantum Computing because it means that the Hamiltonian of a Quantum System can be encoded as a sequence of real numbers and their corresponding Pauli Operator. How do ...
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Quantum gates with respect to phase angles [duplicate]

We can say that $X (\cos \frac{\theta}{2} |0\rangle + e^{i \phi}\sin \frac{\theta}{2} |1\rangle) = \cos \frac{\pi-\theta}{2} |0\rangle + e^{-i \phi}\sin \frac{\pi-\theta}{2} |1\rangle$, a fact that ...
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HHL algorithm for linear systems with a real matrix and a real right side

HHL algorithm can be used for solving linear system $A|x\rangle=|b\rangle$. If we put $|b\rangle$ (to be precise its normalized version) into the algorithm and measuring ancilla to be $|1\rangle$ we ...
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How do you compute the compiled unitary of a quantum circuit comprised of different $n$-input gates?

Given a quantum circuit consisting of two qubits, how is the compiled unitary of the circuit computed when we have different input type gates? (X-gate, H-gate are single-input gates, CNOT is a 2-input ...
The Stinespring dilation theorem states that any CPTP map $\Lambda$ on a system with Hilbert space $\mathcal{H}$ can be represented as $$\Lambda[\rho]=tr_\mathcal{A}(U^\dagger (\rho\otimes |\phi\... 1answer 50 views Is there a matrix whose sum with the canonical Mixing Hamiltonian in Qaoa is proportional to the identity matrix? Does there exist a Hermitian matrix, K s.t B^\prime = B + K satisfies (B^\prime)^2 = c\cdot I, where B = \sum_{i=1}^{n}\sigma_x^{(i)}, \sigma_x^{(i)} is the Pauli X matrix acting on qubit i... 1answer 69 views How does measuring a value of one operator affect the probability of measuring a value for another operator? Suppose I have two non-commuting operators, U_1 and U_2 with eigenvalues \lambda_{1,1}, \lambda_{1,2} and \lambda_{2,1}, \lambda_{2,2}, respectively. In order to determine how measuring one ... 2answers 114 views Can we express \mathrm{tr}_A((A\otimes B)\rho_{AB}) in terms of A, B, \rho_A and \rho_B? For a density matrix \rho_{AB} and some operators A, B, is there a way to express$$\text{Tr}_A((A\otimes B)\rho_{AB}) using the reduced states $\rho_A$ and $\rho_B$ and operators $A$ and $B$? ...
This is probably a very obvious question, but I am going through this problem set and I don't understand why in 1b) it says that it is obvious that $|\langle\psi_1^\perp|\psi_2\rangle|=\sin\theta$ ...