Questions tagged [pauli-gates]

For questions about Pauli matrices in general or Pauli gates in particular, as relevant to quantum computing and/or quantum information theory. The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. The three Pauli gates are: Pauli-X gate, Pauli-Y gate & Pauli-Z gate. X = {{0,1},{1,0}}; Y = {{0,-i},{i,0}}; Z = {{1,0},{0,-1}}.

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4
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2answers
77 views

Why isn't $Ry(\pi/2)$ gate equivalent to Hadamard gate?

I've been experimenting with quantum circuits and can't quite fathom how the difference between states comes together. Speaking in terms of simulations using qiskit,...
4
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1answer
80 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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1answer
39 views

Why can the Hamiltonian $H=P_x(t)X+P_y(t)Y$ make an arbitrary unitary $U=R_x(b)R_y(c)R_x(d)$?

p.281 of Nielsen and Chuang's book says that A single spin might evolve under the Hamiltonian $H = P_x(t)X + P_y(t)Y$, where $P_{\{xy\}}$ are classically controllable parameters. From Exercise 4.10, ...
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0answers
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How are Pauli operators measured experimentally?

We know the theory of Random Pauli measurements: Consider a system of $n$ qubits, and let $d=2^n$, the Pauli matrix set is $P=\otimes_{i=1}^n \sigma_i$, where $\sigma_i\in \{ I,\sigma_x,\sigma_y,\...
4
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2answers
181 views

Proof for Cardinality of the Clifford Group

In this article: (http://home.lu.lv/~sd20008/papers/essays/Clifford%20group%20[paper].pdf) a proof is given for the cardinality of the Clifford group. I understand all the parts of it except for how ...
2
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1answer
57 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}...
3
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1answer
260 views

Controlled Z gate using Pauli rotation operators and Z tensor product Z

I am trying to construct a controlled Z gate using elementary gates. This is what I have so far: \begin{pmatrix} -i & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 1 & 0\\ ...
4
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1answer
112 views

Qutrit analogues of controlled Z and cc-Z gates

I am trying to look for the qutrit analogues of a controlled-Z, and a cc-Z (Z gate with two controls) for qubits. There is a previous answer that gives a qutrit analogue of a CNOT gate, but does not ...
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3answers
78 views

Is there a convention for denoting $Y$ eigenstates?

Two common shorthands for eigenstates of the $Z$ operator are $\{|0\rangle,|1\rangle\}$ and $\{|1\rangle,|-1\rangle\}$, where in the first case we have $Z|z\rangle=(-1)^z|z\rangle$ and in the second ...
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1answer
96 views

Measuring tensor products of Pauli operators

Is there a neat way to derive and efficiently implement a measurement circuit for tensor products of arbitrary Pauli operators like $XZZXZ$ in Qiskit ? I tried using the ...
2
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2answers
328 views

Qiskit CNOT-gate matrix mixup?

In the qiskit textbook chapter 1.3.1 "The CNOT-Gate" it says that the matrix representation on the right is the own corresponding to the circuit shown above, with q_0 being the control and ...
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1answer
39 views

In what sense are Pauli matrices measurement operators?

Neilson and Chuang's textbook shows a nice example of measuring in the $Z$ basis on page 89 in section 2.2.5. The Hermitians for measuring in the $Z$ basis, $|0\rangle\langle 0|$ and $|1\rangle\langle ...
3
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1answer
135 views

Prove the fidelity can be written in terms of Pauli expectation values as ${\rm tr}(\rho\sigma)=\sum_k \chi_\rho(k)\chi_\sigma(\rho)$

I am reading through "Direct Fidelity Estimation from Few Pauli Measurements" and it states that the measure of fidelity between a desired pure state $\rho$ and an arbitrary state $\sigma$ ...
2
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0answers
73 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\...
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61 views

What does it mean that a qubit is a triple $(H,X,Z)$ with $H$ Hilbert space and $X,Z$ Pauli operators?

In this paper, http://users.cms.caltech.edu/~vidick/teaching/fsmp/fsmp.pdf, it gives the definition of a qubit as follows: A qubit is a triple $(H, X, Z)$ consisting of a separable Hilbert space H and ...
3
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1answer
160 views

A question from Aaronson 2004 paper

In Aaronson's paper about the efficient simulation of a stabilizer circuit (https://journals.aps.org/pra/pdf/10.1103/PhysRevA.70.052328), I have a problem with finding the reason why the following ...
2
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0answers
15 views

Measurement on a specific basis and proof of circuit output

I am trying to understand a proof from Practical optimization for hybrid quantum-classical algorithms. In particular, I need clarifications on how do you perform the measurement on a different basis ...
3
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41 views

How does Spin Measurement correspond to quantum NOT gate?

Newbie in quantum computing (and stack overflow) here. I am confused regarding the relation between spin measurement in quantum mechanics and the quantum NOT gate. I have a Bloch sphere picture of a ...
2
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1answer
86 views

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 ...
2
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1answer
32 views

Gate Cost to Transform Superposition of Hamming weight 1 states to superposition of arbitrary basis states?

Say you have something like a general-coefficient $n$-qubit W-state, i.e., $$ |\psi\rangle\equiv\sum_{j=1}^n a_j X_{j}|0\rangle^{\otimes n} \ , $$ where $a_j$ are normalized complex coefficients. ...
4
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1answer
104 views

How to prove the fundamental equation in the theory of angular momentum $\sum_{l=x,y,z}\langle J_l^2\rangle\le\frac{N(N+2)}{4}$?

How to prove the inequality$$\sum_{l=x,y,z}\langle J_l^2\rangle\le\frac{N(N+2)}{4}$$ where $J_l = \mathop{\Sigma}_{i=1}^N \frac{1}{2}\sigma_l^{i}$, and $\sigma_l^i$ is pauli matrix acting on the $i$th ...
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1answer
62 views

How can I compose the Ising XXPOW,YYPOW and ZZPOW gate in single qubit gates and CNOT,...?

I am a bit stuck in decomposing these gates in single qubit gates, in the Cirq documentation it is written, for example that XX is for example the tensor product of Rx gates. But when I calculate ...
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3answers
404 views

Can we write Pauli-Y gate without even complex part?

I was just curious, why is the quantum gate Y-gate (Pauli-Y gate) written in terms of complex numbers? We can actually write Pauli-Y gate as $$ Y = i * \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{...
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1answer
49 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 $i$...
5
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2answers
160 views

How to construct the two qubit gate generated by the Hamiltonian $H= X\otimes X + Y \otimes Y + Z \otimes Z $?

I know that the two qubit gate generated by $H=X\otimes X$ is $\exp\{-\text{i}\theta X\otimes X\}=\cos{\theta} \mathbb1 \otimes \mathbb1 - \text{i} \sin{\theta} X \otimes X$, where $X$ is the $\...
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1answer
83 views

Is there a way to present conjugate transpose of a Y Pauli rotation as a Cirq Operator?

Given: Ry(theta) acting on one qubit I'm trying to use existing Cirq Operators to build the conjugate transpose of the above gate. I need the operator to produce the exact unitary of the given gate ...
2
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1answer
82 views

Why can any density operator be written this way? (quantum tomography)

From page 24 of the thesis "Random Quantum States and Operators", where $(A,B)$ is the Hilbert-Schmidt inner product: \begin{aligned} \rho &=\left(\frac{1}{\sqrt{2}} I, \rho\right) \frac{...
2
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1answer
29 views

Expanding two rotation operators

I have the following operators acting on two qubits, denoted as $(1)$ and $(2)$: $$T_1=\displaystyle\exp\left(-i\frac{\pi}{4}Z\otimes Z\right)\cdot R_z^{(1)}\left(\frac{\pi}{2}\right)\cdot R_z^{(2)}\...
3
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1answer
205 views

What is the procedure of finding z-y decomposition of unitary matrices?

The title explains it all. Suppose one needs to find z-y decomposition of unitary matrix H or T. What is the step by step process to find it?
5
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2answers
151 views

How do I decompose the given $4\times 4$ matrix in terms of Pauli matrices? [duplicate]

I have been working on a question where I have to decompose this matrix in terms of Pauli Matrices: \begin{bmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\...
2
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1answer
126 views

Is there a different way to represent Pauli gates in X basis?

It's easy to see that in computational basis, Pauli matrices could be represented in the outer product form: $$ X=|0\rangle\langle1|+|1\rangle\langle0|\\ Y=-i|0\rangle\langle1|+i|1\rangle\langle0|\\ Z=...
4
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2answers
220 views

If the eigenvalues of $Z$ are $\pm1$, why are the computational basis states labeled with "$0$" and "$1$"?

The computational basis is also known as the $Z$-basis as the kets $|0\rangle,|1\rangle$ are chosen as the eigenstates of the Pauli gate \begin{equation} Z=\begin{pmatrix}1 & 0 \\ 0 & -1\end{...
6
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2answers
510 views

How to generalize the relationship HXH = Z for higher dimensions

Concerning the Hadamard gate and the Pauli $X$ and $Z$ gates for qubits, it is straightforward to show the following relationship via direct substitution: $$ HXH = Z.\tag{1}$$ And I would like to ...
6
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3answers
106 views

How to create the state $\vert 0 \rangle+i \vert 1 \rangle$ using elementary gates?

I am trying to write $|0\rangle+i|1\rangle$ in terms of elementary gates like H, CNOT, Pauli Y, using the IBM QE circuit composer. I was thinking some kind of combination of H and Y since $Y|0\rangle=...
4
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1answer
116 views

Single-qubit rotations on a subspace within two-qubit unitary

I would like to implement the operation $$ U(a,b) = \exp\left(i \frac{a}{2} (XX + YY) + i \frac{b}{2} (XY - YX) \right) $$ ($a,b \in \mathbb{R}$) without using Baker-Campbell-Hausdorf expansion, ...
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2answers
70 views

How to find the normalization factor of the eigenvectors of the $\sigma_x$ Pauli gate?

I'm trying to calcaute the eigenstates for the $\sigma_x$ gate, and I can follow the process up to finding eigenvalues $\pm 1$, but I don't understand where the $\frac{1}{\sqrt{2}}$ coefficient comes ...
2
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1answer
38 views

Qiskit PauliWeightedOperator in the matrix representation?

Suppose we have a PauliWeightedOperator object from Qiskit. Is there any built-in method to convert it to the matrix representation in the computational basis? My search in the docs was not successful....
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0answers
100 views

Understanding the EPR argument with a simple description using Pauli matrices

Can someone explain the EPR argument with a simple description using Pauli matrices? Two non-commuting physical quantity are being discussed philosophically whether there is an element of reality ...
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2answers
348 views

How do I create an inverse identity gate?

Is it possible for me to construct a gate that inverse everything ($|0\rangle \rightarrow -|0\rangle, |1\rangle \rightarrow -|1\rangle$, etc. basically like a $-I$ gate) from the basic $X, Y, Z, CX,......
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1answer
113 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
157 views

Quick question about Two-qubit SWAP gate from the Exchange interaction

I am reading the following paper: Optimal two-qubit quantum circuits using exchange interactions. I have a problem with the calculation of the unitary evolution operator $U$ (Maybe it is stupid): I ...
4
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1answer
40 views

Definitions of $D_y$ gate in Hamiltonian simulation: are they the same?

I'm reading a Hamiltonian simulation example proposed in this paper. From their notation, the operator $D_y$ (sometimes it's called $H_y$) serves the function to diagonalize the Pauli matrix $\sigma_y(...
4
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1answer
129 views

Simulate Hamiltonians with Pauli operations (controlled time evolution)

I had a question last week regarding the simulation of Hamiltonians composed of the sum of Pauli products: How can I simulate Hamiltonians composed of Pauli matrices? I'm having a follow-up question: ...
2
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2answers
124 views

Transforming $|01 \rangle + |10 \rangle - |11 \rangle \to |01 \rangle - |10 \rangle + |11 \rangle$

How to convert from current state: $$|\psi \rangle =\dfrac{ |01 \rangle + |10 \rangle - |11 \rangle}{\sqrt{3}}$$ into a target state $$|\phi \rangle = \dfrac{|01 \rangle - |10 \rangle + |11 \rangle}{\...
4
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1answer
537 views

How can I simulate Hamiltonians composed of Pauli matrices?

Suppose I want to perform the time-evolution simulation on the following Hamiltonians: $$ H_{1} = X_1+ Y_2 + Z_1\otimes Z_2 \\ H_{2} = X_1\otimes Y_2 + Z_1\otimes Z_2 $$ Where $X,Y,Z$ are Pauli ...
4
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1answer
57 views

How does adding an identity to an Hamiltonian affect the corresponding time-evolution in the Bloch sphere?

For the Hadarmard Hamiltonian, $\hat H = (\hat X+\hat Z)/\sqrt 2$, where $\hat X$ and $\hat Z$ are Pauli matrices. The time evolution of a state under this Hamiltonian could be visualized by a ...
3
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2answers
155 views

Benefit of phase shift in quantum computing

I am new to quantum computing. I compare Pauli-X gate and Pauli-Y gate as equivalent to NOT gate in classical computers. Though I am not very sure when to use Pauli-X and Pauli-Y gates as the result ...
4
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1answer
184 views

Can QAOA be considered as simulation of a quantum annealer on a gate-based quantum computer?

Quantum annealers are single purpose machines allowing to solve quadratic unconstrained binary optimization (QUBO) problems. QUBO problems have following objective function: $$ F=-\sum_{i<j}J_{ij}...
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1answer
78 views

How do I visualize the action of Pauli $X$ and $Y$ gates on $|0\rangle$?

I visualize Pauli Gates making rotations by π radians about the x, y, and z axes on the Bloch sphere. If this is the case, then how Pauli-X and Pauli-Y gates applied to $|0\rangle$ differ? As both ...
1
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1answer
86 views

In Variational Quantum Eigensolvers, what does "grouping Pauli operators into tensor products requiring the same post-rotations" mean?

In this paper (nature version), the authors state We group the Pauli operators into tensor product basis sets that require the same post-rotations. As a result, they have the table S2 in the suppl. ...