Questions tagged [mathematics]

Use this tag for questions about mathematics relevant to quantum computing and/or quantum information theory. DO NOT use this tag for general mathematics questions.

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15 views

RAC for XOR functions

I need the optimal encoding protocol for 3 $\rightarrow$ 1 Classical RAC such that the receiver is able to retrieve any one of the initial bits, as well as the XOR combinations of those bits. ( If a, ...
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62 views

In Nielsen and Chuang, how can $\frac{1}{2(e-1)}$ result from $\frac12\int_{e-1}^{2^{t-1}-1}dl\frac{1}{l^2}$?

From Nielsen and Chuang's book: $\textit{Quantum computation and quantum information}$, how can (5.34) equal (5.33)? I.e. $$\dfrac{1}{2} \int_{e-1}^{2^{t-1}-1} dl \dfrac{1}{l^2} = \dfrac{1}{2(e-1)}.$$...
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1answer
62 views

Discrepancy in inner product between tensor products

I have noticed one identity in case of tensor product from this post. But I can't understand why it is true. $\langle v_i| \otimes \langle w_j| \cdot |w_k\rangle \otimes |v_m\rangle = \langle v_i|v_m\...
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1answer
47 views

Inequality in overlap of quantum states

For quantum states $\vert\psi_1\rangle, \vert\psi_2\rangle, \vert\phi\rangle$, is it true that $$\tag{1}\langle \phi\vert\psi_1\rangle\langle\psi_1\vert\phi\rangle\langle \phi\vert\psi_2\rangle\langle\...
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2answers
74 views

What is $\sum_{i}\langle i \vert U \vert j\rangle$ for unitary $U$?

The question is basically the title but given a unitary operator $U$ and a computational basis, can we say anything about the complex number below? $$c = \sum_{i}\langle i \vert U \vert j\rangle$$ I ...
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1answer
43 views

How does the CPTP constraint reflect on the matrix representation of a qubit channel in the Pauli basis?

Let us write the possible states of a qubit in the Bloch representation as $$\newcommand{\bs}[1]{{\boldsymbol{#1}}}\rho_{\bs r}\equiv \frac{I+\bs r\cdot\bs \sigma}{2},$$ where $\bs\sigma=(\sigma_1,\...
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1answer
49 views

How can I verify that the Pauli group is a group? And is it abelian? [duplicate]

So how can I verify that the Pauli Group is a Group? Then furthermore, Abelian? And then to sum it up, the order of the group. Trying to do some research into the group but I can't find much about it.
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1answer
35 views

How can we upper bound the norm of a partial trace?

Suppose we have the normalised states $|\phi_{1}\rangle,|\phi_{2}\rangle \in A \otimes B$ where $A$ and $B$ are $d$-dimensional complex vector spaces. Suppose $|\langle\phi_{2}|\phi_{1}\rangle| < ...
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1answer
39 views

How to calculate the exponential of all elements in an input array using qiskit? [closed]

How can I perform an operation similar to Numpy.exp() in qiskit?
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88 views

Is the set of two-qubit absolutely separable states convex?

Companion question on MathOverflow Let us order the four nonnegative eigenvalues, summing to 1, of a two-qubit density matrix ($\rho$) as \begin{equation} 1 \geq x \geq y \geq z \geq (1-x-y-z) \geq 0. ...
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1answer
60 views

Difference between change of basis in bra-ket notation and matrix notation

In matrix notation, say I have the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. It is currently represented in the computational basis $\{\begin{bmatrix} 1 \\ 0\end{bmatrix}, \begin{bmatrix} 0 \\ 1\...
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1answer
35 views

What are the physical meanings of the outer product when writing expressions for unitary gates?

I'm really confused with the interpretation of those equations: $1.$ The evolution of states under unitary operations can be expressed as $$ U = \sum_k\exp(i\phi_k)|\psi_k\rangle\langle\psi_k| $$ $2.$ ...
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1answer
76 views

What is the difference between the states $i|1\rangle$ and $|+i\rangle$?

I am new to Quantum computing. I see $|\mbox{+}i\rangle$ state maps to y-axis on bloch sphere ($\theta = 90$ degree and $\phi = 90$ degree) while $i|1\rangle$ maps on x-axis, $i|1\rangle$ is stated as ...
2
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1answer
36 views

Pseudoinverse of a quantum state

The max-relative entropy between two states is defined as $D_{\max}(\rho\|\sigma) = \log\lambda$, where $\lambda$ is the smallest real number that satisfies $\rho\leq \lambda\sigma$, where $A\leq B$ ...
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2answers
113 views

Prove that the trace norm is dual to the spectral norm

Suppose $A\in L(X,Y)$. $||\cdot||$ denotes spectral norm and denotes the largest singular value of a matrix, i.e. the largest eigenvalue of $\sqrt{A^*A}$. $||\cdot||_{tr}$ denotes trace norm. We have ...
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1answer
58 views

Measuring Pauli strings using generators

I am trying to find the ground state of a Hamiltonian using VQE. I have decomposed the Hamiltonian into a set of Pauli strings. To decrease the number of actual measurements that has to be done, can I ...
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0answers
28 views

How to implement the Mixer of Quantum Alternating Operator Ansatz for Max-Independent-Set

I am trying to implement the Mixer of the Max-Independent Set from The Quantum Alternating Operator Ansatz. From this paper: https://arxiv.org/pdf/1709.03489.pdf in Chapter 4.2 page 15 to 17. For ...
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56 views

Is the tensor product with the multiplication distributive or associative?

Hello is the tensor product with the multiplication distributive or associative? When having the formula $$X_{1} \prod_{i\in (2,3)}(Z_{i})$$ is the then $$X_{1} \prod_{i\in (2,3)}(Z_{i}) = (X_{1}\...
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1answer
139 views

Bob applies a projector - what happens to eigenvalues of Alice's reduced state?

Suppose Alice and Bob share a state $\rho_{AB}$. Let us denote the reduced states as $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$. Bob applies a projector so the new global ...
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1answer
33 views

Basis Change Substitution

question: "A spin right $\frac{1}{\sqrt 2}(|0\rangle + |1\rangle)$ is sent through a Hadamard gate, creating the superposition of $|+\rangle$ and $|-\rangle$, given by $\frac{1}{\sqrt 2}(|+\...
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1answer
56 views

What is the intuition behind “states with support on orthogonal subspaces”?

I'm sure I don't fully understand support, but I am having trouble seeing how it connects to things like density operators. I have an idea that it means, according to Wikipedia: "In mathematics, ...
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1answer
54 views

How do I prove that $\newcommand{\tr}{\operatorname{Tr}}\tr(A \sqrt{B} A \sqrt{B}) = \tr\Big[\Big(\sqrt{\sqrt{B}} A \sqrt{\sqrt{B}}\Big)^2\Big]$?

Let's say I have 2 density operators $A$ and $B$. Now, here is what I am trying to calculate: $$\newcommand{\tr}{\operatorname{trace}} \tr(A \sqrt{B} A \sqrt{B}). $$ I saw that this trace can be ...
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1answer
51 views

How to force a matrix to be unitary given constraints on some of the elements? [duplicate]

I am working with a matrix of the following form: $$ A =\begin{pmatrix} a_{11} & Q & \ldots & Q\\ a_{21} & Q & \ldots & Q\\ \vdots & \vdots & \ddots & \vdots\\ ...
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1answer
25 views

Why is the VQE insensitive to noise?

I was going through the Grove documentation on the Variational Quantum Eigensolver. In one of the demonstrations with noisy gates, it is seen that resulting eigenvalue is quite close to the expected ...
3
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1answer
235 views

Problem with Shor's factoring algorithm [closed]

I'm trying to figure out the Shor's factoring algorithm. References i've been using wikipedia page, the book Quantum Computer Science by David Mermin and the orignal paper(1996) By Peter Shor. I ...
3
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3answers
45 views

Prove that different purifications of a state can be mapped into one another via local unitaries

Let $\rho \in \mathfrak{D}(A)$ be a density matrix. Show that $\left|\psi^{A B}\right\rangle \in A B$ and $\left|\phi^{A C}\right\rangle \in A C$ (assuming $\left.|B| \leqslant|C|\right)$ are two ...
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2answers
87 views

How do I apply the Hadamard gate to one qubit in a two-qubit pure state?

So in lectures I see lots of these: And somehow I intuitively understand it (at least for the 1 qubit case), but I don't understand the math – especially for 2 qubits.
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1answer
142 views

How are the Pauli $X$ and $Z$ matrices expressed in bra-ket notation? [duplicate]

For example: $$\rm{X=\sigma_x=NOT=|0\rangle\langle 1|+|1\rangle\langle 0|=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}}$$ $$\rm{Z=\sigma_Z=signflip=|0\rangle\langle 0|-|1\rangle\langle 1|=\...
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3answers
60 views

How do I apply a Hadamard gate on a given qubit, in matrix formalism?

Hadamard gate matrix is: $$\frac{1}{\sqrt2}\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}$$ The matrix for $|0\rangle$ is: $$\begin{bmatrix}1 \\ 0\end{bmatrix}$$ I am unable to understand, how ...
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1answer
62 views

Does composition of two single qubit rotations yield a single rotation around a unit vector?

$ \newcommand{\coefcos}[0]{c_1 c_2 - s_1 s_2 \hat{n}_1 \cdot \hat{n}_2} \newcommand{\coefsin}[0]{s_1 c_2 \hat{n}_1 + c_1 s_2 \hat{n}_2 - s_1 s_2 \hat{n}_2 \times \hat{n}_1}$This question relates to ...
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1answer
313 views

How to translate the Hadamard gate matrix into Dirac notation?

Hadamard gate matrix is: $$\frac{1}{\sqrt 2}\begin{bmatrix}1 && 1 \\ 1 && -1\end{bmatrix}$$ The Dirac notation for it is: $$\frac{|0\rangle+|1\rangle}{\sqrt 2}\langle0|+\frac{|0\...
2
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1answer
50 views

What linear map is needed for acting on a maximally entangled state?

I was reading a textbook and I encountered this question. I was wondering why we don't consider $M^\dagger$ instead of $M^{T}$, so I didn't show this relation, could you please help me to show below ...
2
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1answer
31 views

How is the Hartree accuracy calculated between the exact and VQE results?

In the Simulating Molecules using VQE section of the Qiskit textbook it states an accuracy of $0.0016$ Hartree between the exact and ...
3
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1answer
48 views

Show that for any measurement operator $M_m$ there exists unitary $U_m$ such that $M_m=U_m\sqrt{E_m}$ with $E_m$ POVM

Exercise 2.63 of Nielsen & Chuang asks one to show that if a measurement is described by measurement operators $M_m$, there exists unitary $U_m$ such that $M_m = U_m \sqrt{E_m}$ where $E_m$ are ...
3
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0answers
54 views

Eigenvalues of a quantum state after partial tracing

I am interested in the smallest nonzero eigenvalue of a quantum state. Does this eigenvalue always increasing after a partial trace i.e. the smallest nonzero eigenvalue of $\rho_A$ is always larger ...
2
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1answer
90 views

Give an explicit derivation of the exact formula for the two-qubit absolute separability Hilbert-Schmidt probability $\approx 0.00365826$

The two-qubit eigenvalue ($\lambda_i$ >= 0, $i=1,\ldots,4$, $\lambda_4=1-\lambda_1-\lambda_2-\lambda_3$) condition of Verstraete, Audenaert, de Bie and de Moor AbsoluteSeparability (p. 6) for ...
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1answer
50 views

Obtaining hermitian matrix using Knill and Laflamme condition?

Let $E$ be the set of all correctable errors and $E_a, E_b \in E$. Let $\lbrace \vert c_1\rangle, \vert c_2\rangle, \ldots \vert c_k\rangle\rbrace$ be the basis of codewords in the codespace. It is ...
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0answers
43 views

Deriving Expression For QAOA Optimal Trial State Parameters

I am going through the QAOA section in the Qiskit Textbook - QAOA and am stuck in one of the steps. In section 5.2, the method for getting the Optimal Trial State Parameters are discussed. I do not ...
2
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2answers
75 views

Having trouble finding angles for Bloch vector

I am doing the 5th exercise on https://qiskit.org/textbook/ch-states/representing-qubit-states.html#Quick-Exercise (all the way at the bottom). Which states find the angle for the vector $\frac{1}{\...
5
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1answer
87 views

Is the Haar measure invariant under conjugation?

Denote the Haar measure on the unitary group $U(\mathcal X)$ by $\eta$. Does this equation hold (assuming the integral exists): $\int d\eta(U) f(U) = \int d\eta(U) f(U^\dagger)$? Intuitively this ...
3
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1answer
149 views

Nielsen & Chuang Exercise 2.55: Prove that $\exp \left[ -\frac{iH(t_2 - t_1)}{\hbar} \right]$ is unitary

$\newcommand{\expterm}[0]{\frac{-iH(t_2 - t_1)}{\hbar}} \newcommand{\exptermp}[0]{\frac{iH(t_2 - t_1)}{\hbar}}$Nielsen & Chuang (10th edition, page 82) states that $H$ is a fixed Hermitian ...
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1answer
39 views

Equivalence of two ways to recover a map from its Choi state

Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a quantum channel, $\Phi:\mathrm{Lin}(\mathcal X)\to\operatorname{Lin}(\mathcal Y)$. We define its Choi representation as the operator $J(\Phi)$ ...
2
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0answers
30 views

When we do a linear fit, what is the correlation coefficient of the estimated parameters?

In Google's quantum supremacy experiment, supplementary Section VIIIH, they calculate the correlation coefficient of the linear fit coefficients $p_0$,$p_1$. I can't figure out the definition of this ...
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1answer
73 views

Nielsen & Chuang Exercise 2.32: Show that the tensor product of two projectors is a projector

$\newcommand{\bra}[1]{\left<#1\right|} \newcommand{\ket}[1]{\left|#1\right>}$Here is what I tried: Given that we have two projectors: $$ A = \sum_i \ket{i} \bra{i}, \hspace{2em} B = \sum_j \ket{...
3
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1answer
65 views

Eigenvectors and eigenvalues of the gate $U_a:|s\rangle\mapsto|sa \bmod N\rangle$

I'm studying Shor algorithm. This is a demostration about the eigenvectors and eigenvalues of $U_a$ gate: Can somebody explain me from the third step to the last?
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1answer
36 views

Create this state using CIRQ Coding language [closed]

I needed help with CIRQ coding as I'm quite new to Quantum Computing. I read the tutorials on CIRQ but don't really understand it. I'd be very thankful if someone could help. A program to create the ...
2
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1answer
33 views

Asymmetry in distributing phase change across components

The quantum computing text books and theory in general seems to have added an asymmetry in the distribution of change in phase across the components in the context of a qubit. Is there any reason for ...
2
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1answer
59 views

How is $S(\rho)=H(p_{i})+\sum_{i}p_{i}S(\rho_{i})\le \log(d)$ possible if $\rho_{i}$ are not pure states?

I know how this can be proved using the quantum relative entropy. However, even with this proof, and am still confused about how this emerges. Say I have a source that produces two states $\rho_1$ and ...
3
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1answer
78 views

Deriving $\left( A | v \rangle \right)^\dagger = \langle v | A^\dagger$ without using $A^\dagger=\left(A^* \right)^T$

From Nielsen & Chuang (10th edition), page 69: Suppose $A$ is any linear operator on a Hilbert space, $V$. It turns out that there exists a unique linear operator $A^\dagger$ on $V$ such that for ...
2
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1answer
52 views

Confused about associativity of outer product notation

Consider this expression where $A$ and $B$ are matrices, $|i \rangle$ is a ket (column vector) and $\langle j |$ is a bra (row vector) : $$ A | i \rangle \langle j | B \tag1\label1 $$ Due to the ...

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