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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61 views

Whad does $ A - \langle A \rangle $ mean?

I've seen the uncertainty of $A$ written as $$ (\Delta A)^2 = \langle (A - \langle A \rangle)^2 \rangle. $$ But what does this even mean since $ A $ is an operator and $ \langle A \rangle $ is a ...
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3answers
90 views

How to show that the QFT satisfies $\frac1{\sqrt N}\sum_j\prod_le^{2\pi i j_l k/2^l}|j_1...j_n⟩=\bigotimes_l \frac1{\sqrt2}(|0⟩+e^{2\pi i k/2^l}|1⟩)$?

I'm reading Ronald de Wolf's lecture notes, and in chapter 4.5 he writes that $$ \frac{1}{\sqrt N}\sum\limits_{j=0}^{N-1}\prod\limits_{l=1}^{n}e^{2\pi i j_l k / 2^l}|j_1...j_n\rangle = \bigotimes\...
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2answers
248 views

Map a 4-body Ising Hamiltonian to a 2-body Ising Hamiltonian

I wonder if there exists a way to map the square of a 2-body Ising Hamtiltonian (which will make it 4-body) back to a 2-body Hamiltonian that has the same ground state? Let me explain what I mean by ...
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1answer
28 views

how to obtain partial transpose of a Tripartite operator?

i know for a bipartite system with elements |ij><kl| elements of its partial transpose are |kj><il| now suppose a ...
2
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1answer
44 views

Expectation value of an observable containing a single projector vs Born rule for the projector

Suppose I have a state $|\psi\rangle$ and I want to estimate the probability of obtaining a computational basis state $|x\rangle$. Then by Born rule: $$ p(x) = |\langle x|\psi\rangle|^2 = Tr[|x\rangle ...
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3answers
380 views

Why do we need Hilbert spaces when talking about qubits and quantum computation?

I was just curious to know why do we need Hilbert Spaces when talking about the qubits and quantum computation in general. I mean why can't we just work with inner product spaces, rather than going ...
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3answers
325 views

Are complex amplitudes really needed?

Qubit amplitudes are defined as complex numbers. But in all tutorials I have recently read, only real numbers are used and everything works. So, if I completely forget the official 'complex' ...
1
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1answer
61 views

Two-qubit Bell measurement matrix where the two qubits are not contiguouis

In the answer here, it is explained that where the measurement operates on only a subset of the qubits of the system (for example qubits 2 and 3 out of five), the matrix can be constructed using the ...
5
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1answer
76 views

Prove that $|(\langle \psi|_{A} \otimes \langle \phi|_{B})|\theta\rangle_{AB}|^{2}<1$ for entangled $|\theta\rangle_{AB}$

I am trying to show that $|\langle \psi|_{A} \otimes \langle \phi|_{B}|\theta\rangle_{AB}|^{2}<1$ given $|\theta\rangle$ is an entangled state, and as such has schmidt rank >1. Decomposing it, ...
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1answer
83 views

How to decompose Bloch sphere rotations $e^{\frac{i\theta}{2}(\cos(\phi)\sigma_x + \sin(\phi)\sigma_y)}$ in terms of $R_x,R_y,R_z$?

I learned a formula to represent the rotation around bloch sphere: $\theta_{\phi} = e^{\frac{i\theta}{2}(\cos(\phi)\sigma_x + \sin(\phi)\sigma_y)}$ So that $\pi_0$ is the gate $X$ and $\pi_{\frac{\pi}{...
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1answer
61 views

How to compute derivatives of partial traces of the form $\frac{\partial \operatorname{Tr}_B(F(\mathbf{X}))}{\partial \mathbf{X}}$?

The Matrix Cookbook says that for any differentiable matrix function $F(\cdot)$, it holds that $$\frac{\partial \operatorname{Tr}(F(\mathbf{X}))}{\partial \mathbf{X}}=f(\mathbf{X})^{T},$$ where $f(\...
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1answer
92 views

How to find the eigenstates of a general $2\times 2$ Hermitian matrix?

Given a measurement operator in the general Hemitian form $$ M = \begin{pmatrix} z_1 & x+iy \\ x-iy & z_2\end{pmatrix}, $$ where $x,y,z_1,z_2 \in \mathbb{R}$, show that the eigenvalues are $$ ...
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1answer
67 views

Why does $H^2=X^2 =I$ not imply $H=X$?

if $HH = I$ and $XX =I$, then is $H=X$? $HH = I = XX$ or, $HH = XX$ then, taking under root, is $H = X$? This is absurd but how to disprove it?
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1answer
95 views

Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?

I see here in Olivia DeMatteo's notes, she states: When we consider the action of the entire Clifford group on a single non-identity Pauli, it maps that Pauli to each of the $d^2 − 1$ other possible ...
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1answer
71 views

Why are orthogonal spins $(1,0)$ and $(0,1)$ represented as collinear vectors in the Bloch sphere?

I'm reading the book "Quantum Computing 4 real IT people" by Chris Bernhardt and I have a question about the following phrase in chapter 3 which says that An ordered orthonormal basis ...
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1answer
117 views

How to exactly implement Trotter-Suzuki formula on quantum computer

Recently, I am studying some topics related to product formula, and I am curious about how to implement such formula on real quantum devices. The $(2k)$-th order product formula can be witten as \...
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1answer
104 views

What is the structure of coefficients of $2^N\times 2^N$ unitary matrices decomposed in terms of the Pauli basis?

Any square $2^N\times 2^N$ matrix can be written as a sum of tensor products of pauli matrices. Eg a $8\times 8$ matrix can be written as $$U=\sum_{i_1,i_2,i_3}u_{i_1,i_2,i_3}\sigma_{i_1}\otimes\...
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2answers
96 views

Find a unitary to prepare state $|0\rangle$ to a specific vector

I am working with Variational Quantum Linear Solver (VQLS) algorithm, where it needs to prepare a control_b circuit. Assume b is 1d with $ 2^n $ elements in it. $$ {\bf Ax = b} \tag{1}$$ I need to ...
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2answers
382 views

Minimum number of 2 qubit gates to build any unitary

Any unitary $U$ acting on $N$ qubits can be decomposed in a finite product $U=U_1U_2...U_n$ where every $U_i$ acts on only 2 qubits, for example through decomposition in CNOT, phase shifts and 1 qubit ...
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1answer
70 views

Showing that $e^{i \sigma_z \otimes \sigma_z t} = \text{CNOT}(I \otimes e^{i \sigma_zt})\text{CNOT}$

While working on circuit construction for Hamiltonian simulation using this answer as reference, I'm unable to see how the following equation is true: $$ e^{i \sigma_z \otimes \sigma_z t} = \text{CNOT}...
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1answer
29 views

How to take the limits of the sandwiched Renyi divergences?

The sandwiched Renyi divergence is defined as $$\begin{equation} \tilde{D}_{\alpha}(\rho \| \sigma):=\frac{1}{\alpha-1} \log \operatorname{tr}\left[\left(\sigma^{\frac{1-\alpha}{2 \alpha}} \rho \...
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1answer
99 views

Can I use the Lie product formula to simulate the Hamiltonian of an adjacency matrix by using the QPE to take Nth roots of permutation matrices?

I have gotten some great help recently on Hamiltonian simulation, and am interested in using Hamiltonian simulation to explore (classical) random walks on large graphs, but I'm running up against ...
6
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1answer
416 views

Are nearly all pure two-qubit state entangled?

I am using the code below, utilizing QETLAB's RandomStateVector(4) and IsPPT, to generate a random state and to judge whether the state is entangled or separable: ...
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1answer
59 views

RZZ calculation: Why does the equation and the circuit correspond?

i found this tutorial about MaxCut and QAOA from pennylane and i do not understand how the equation and the circuit should be equal. When i do the math i come to this conclusion: (result of CNOT - ...
4
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0answers
94 views

Is there a way to write down the eigenstates of this two-qubit density matrix?

I am considering the density matrix which represents an arbitrary state for a pair of qubits. When written out in terms of the Pauli operators, this is as follows (certain terms vanish for another ...
1
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1answer
94 views

Wick rotation of the Schrödinger equation

Studying the following paper: https://www.nature.com/articles/s41534-019-0187-2.pdf Trying to figure out how $ E_T$ shows up from (1) and (2). Any suggestion or guidance would be appreciated. We ...
5
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1answer
108 views

Prove that rank one projectors have the same partial trace iff they differ by a local unitary operation

In nielsen and chuang's QCQI book, there is a theorem called Unitary freedom in the ensemble for density matrices, which states that the sets $|\psi_i\rangle$ and $|\phi_i\rangle$ generate the same ...
6
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2answers
295 views

How universal is the Toffoli gate for classical reversible computing?

It is easy to see that no finite set of classical reversible gates can be strictly universal (without ancilla) for classical reversible computation: for any reversible gate on $n$ bits, in its action ...
8
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1answer
162 views

Can we simulate the Hamiltonian for the Rubik's Cube with "nth-root of SWAP" gates?

I'm interested in, but confused about, local Hamiltonian simulation. I don't yet have enough intuition regarding even the approach set forth by Lloyd in 1997. I think Lloyd's recipe is to repeatedly ...
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1answer
65 views

How to know state coresponding to -1 or 1 for multi qubits VQE?

Can help to explain how to get -1 and 1 for 2 qubits VQE with $\langle XY\rangle$ since we have 4 states $|00\rangle,|01\rangle,|10\rangle,|11\rangle$? For the case of 1 qubit, it is straight forward ...
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1answer
78 views

What happens if a Pauli $X$ gate is applied to part of a Bell state?

I have started to learn about the mathematics behind ebits and I have a question. Assume $\color{red}{\text{Alice}}$ and $\color{blue}{\text{Bob}}$ share the following ebit: $\begin{align}\vert\Phi^+ \...
3
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2answers
145 views

Confusion regarding the tensor product usage in book

I have recently started with quantum computing, and I've found great book about it - Learn Quantum Computing with IBM Quantum Experience, which explains a lot of things in quite a simple language. ...
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0answers
87 views

What is the justification for the use of this arbitrary post-processing vector?

I am reading through this paper which describes the use of a post-processing vector $\vec{c}$ with elements having values of $(-1,0,1)$. In equation 3 they give their solution as a linear combination ...
1
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1answer
35 views

Vector math of applying an X-gate on an $|i\rangle$ basis state

It is well known that the X-gate will apply a rotation about the x-axis on the bloch sphere. Knowing this, the $|i\rangle$ state should be converted to the $|-i\rangle$ state on the application of ...
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2answers
93 views

Tensor product and Dirac notation

Can someone shows me how to proof this equality: $\frac{1}{\sqrt2}(\alpha|000⟩+\alpha|011⟩ + \beta|100⟩ + \beta|111⟩ )$ = $ \frac{1}{2\sqrt2}[(|00⟩+|11⟩) \otimes (\alpha|0⟩+\beta|1⟩) + (|01⟩+|10⟩) \...
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2answers
58 views

Is ($|+⟩$$⟨0|$ + $|-⟩$$⟨1|$ ) similar to ($|0⟩$$⟨+|$ + $|1⟩$$⟨-|$ )?

Is ($|+⟩$$⟨0|$ + $|-⟩$$⟨1|$ ) similar to ($|0⟩$$⟨+|$ + $|1⟩$$⟨-|$ ) ? Can we just reversed it this way when doing Dirac manipulation? I try to calculate HZH = X and i need to reverse the second H
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1answer
82 views

Steps to apply Hadamard gate to $n$ qubits

Can someone shows me, step by step, how to apply Hadamard and output the result?
5
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2answers
363 views

What are the conditions ensuring a two-qubit density matrix is positive semidefinite?

I've seen some papers writing $$\rho=\frac{1}{4}\left(\mathbb{I} \otimes \mathbb{I}+\sum_{k=1}^{3} a_{k} \sigma_{k} \otimes \mathbb{I}+\sum_{l=1}^{3} b_{l} \mathbb{I} \otimes \sigma_{l}+\sum_{k, l=1}^{...
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0answers
32 views

Simplifying equation for two qubit syndrome extraction code

In the paper Quantum Error Correction: An Introductory Guide, the author gives the following formula for a simple two qubit code (eq. 19 on the paper). $$ E|\psi\rangle_L|0\rangle_A \xrightarrow{\text{...
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2answers
189 views

How to understand combination states vs pure/mixed states?

I've learned that representing a combination of two states, I simply need to take the tensor product of the states. For example: $$\left|\Psi\right>=\alpha_0\left|0\right>+\beta_0\left|1\right&...
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1answer
65 views

Finding the measurement basis for single qubit with given probability of outcome $0$

I have the general state of a single qubit $|\psi \rangle = \alpha|0\rangle + \beta|1\rangle $. Assume I am given a probability $p$ such that $0 < p <1$. Now I need to find the basis in which ...
2
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3answers
160 views

Is every single-qubit unitary just a rotation around some unit vector on the Bloch sphere?

I remember reading this somewhere... Is there an elegant proof for this?
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0answers
25 views

Penalty Function for XOR gate

I was reading a paper on Gates for Adiabatic Quantum Computer. In the paper, there were different penalty functions already given in the form of the following table: I do not quite understand the ...
1
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1answer
79 views

Strange binomial formula for operators?

Does the binomial formula $(a+b)^n=\sum_i C_n^ia^ib^{n-i}$ still work when $n$ is replaced by operator $\hat{n}$(an operator), where $a$ and $b$ are numbers? Since it's not the normal binomial formula ...
2
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0answers
81 views

Is decomposing high-dimensional states in terms of Pauli matrices impossible?

I've been trying to decompose a 3x3 density matrix with 3-dimensional Pauli matrices but it doesn't work for all matrices. For example, the density matrix of the state $|0\rangle + |1\rangle + |2\...
0
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1answer
31 views

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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1answer
160 views

What is the eigenvalue distribution of arbitrary unitary matrices?

I had a question regarding the nature of the eigenvalue distribution of unitary matrices. Searching for the answer I found that the unitary matrices which are sampled randomly have a defined ...
2
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1answer
104 views

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 ...
2
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1answer
50 views

How to sample vectors close to the minimum eigenvector of a unitary matrix?

Say that we have an unknown $2^{n}\times2^{n}$ unitary matrix $U$ with eigenvectors $|v_{i}\rangle$ and eigenvalues $e^{2\pi j \theta_{i}}$and we want to sample a vector, say $|\phi \rangle$. Since ...
2
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0answers
40 views

Marginal output probability of first bit for constant-depth circuits

Consider a constant depth $1\text{D}$ quantum circuit, which is applied to the input state $|0^{n}\rangle$, and whose output is measured in the standard basis. You can assume that the gates of the ...

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