# 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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### 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 ...
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### 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{...
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### 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{...
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### 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 ...
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### 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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### 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 ... 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 $\sigma_y(... 4 votes 1 answer 255 views ### Simulate Hamiltonians with Pauli operations (controlled time evolution) I had a question last week regarding the simulation of Hamiltonians composed of the sum of Pauli products: How can I simulate Hamiltonians composed of Pauli matrices? I'm having a follow-up question: ... 2 votes 2 answers 130 views ### Transforming$|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 $$|\phi \rangle = \dfrac{|01 \rangle - |10 \rangle + |11 \rangle}{\... 4 votes 1 answer 1k views ### How can I simulate Hamiltonians composed of Pauli matrices? Suppose I want to perform the time-evolution simulation on the following Hamiltonians:$$ H_{1} = X_1+ Y_2 + Z_1\otimes Z_2 \\ H_{2} = X_1\otimes Y_2 + Z_1\otimes Z_2 $$Where X,Y,Z are Pauli ... 4 votes 1 answer 131 views ### How does adding an identity to an Hamiltonian affect the corresponding time-evolution in the Bloch sphere? For the Hadarmard Hamiltonian, \hat H = (\hat X+\hat Z)/\sqrt 2, where \hat X and \hat Z are Pauli matrices. The time evolution of a state under this Hamiltonian could be visualized by a ... 3 votes 2 answers 584 views ### Benefit of phase shift in quantum computing I am new to quantum computing. I compare Pauli-X gate and Pauli-Y gate as equivalent to NOT gate in classical computers. Though I am not very sure when to use Pauli-X and Pauli-Y gates as the result ... 5 votes 1 answer 434 views ### Can QAOA be considered as simulation of a quantum annealer on a gate-based quantum computer? Quantum annealers are single purpose machines allowing to solve quadratic unconstrained binary optimization (QUBO) problems. QUBO problems have following objective function:$$ F=-\sum_{i<j}J_{ij}... 1 vote 1 answer 284 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 ... 2 votes 1 answer 178 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. ... 4 votes 1 answer 2k views ### CNOT expressed with CZ and H gates by taking into account HZH =X From this link: Where equation 1 is: I can probably brute-force this by explicitly calculating this quantum circuit's effective 4x4 matrix and seeing that its equivalent to this teleportation ... 2 votes 1 answer 543 views ### Principal square root of Pauli Y gate in Qiskit? I've seen a similar question asked (How do I compute the square root of the$Y$gate?) but I'm trying to understand how I can use the gates$Y^{\frac{1}{2}}$or$Y^{\frac{1}{4}}$in Qiskit in terms of ... 5 votes 2 answers 1k views ### controlled-Z rotation gates in symmetrical fashion I was going through the qiskit textbook and in this chapter I came across a statement under the topic "Kickback with the T-gate" related to the Controlled-Z gate that the controlled-Z ... 1 vote 0 answers 207 views ### Changing the Basis I am attempting to use a VQE algorithm to find the ground state of a deuterium nucleus by applying a constructed hamiltonian to an ansatz state with one parameter created by a circuit. While I am ... 4 votes 1 answer 429 views ### Is there a function in Qiskit to measure tensor products of Pauli operators? Is there a function in Qiskit similar to this function in Q# which measures strings of Pauli operators applied to different qubits? 3 votes 1 answer 179 views ### Can you take infinitely many square roots of Pauli-X? I am trying to find the cost for a n-bit Toffoli gate based on the recurrent circuit presented on Barenco's Work in Lemma 7.5 (Elementary gates for quantum computation) The construction requires that ... 2 votes 1 answer 248 views ### Why do we transform a Boolean variable into a a Pauli Z matrix Under Qiskit QAOA's tutorial (https://qiskit.org/textbook/ch-applications/qaoa.html), the authors specify that a cost function$C(x)$representing the optimization objective of a Binary Combinatorial ... 6 votes 2 answers 853 views ### What are boost and shift operators and why are they called so? In some texts I see$X$and$Z$Pauli operators as being said as boost and shift operators respectively. But I came across some text that defines its own operators, namely: $$X \vert j\rangle = \... 3 votes 1 answer 120 views ### What do coupling coefficients mean in terms of Pauli gates, and why are they time dependent? I am reading this error mitigation paper by the IBM team and I am slightly confused about the meaning of "coupling coefficients" when describing multi-qubit Hamiltonian. I have only seen ... 2 votes 0 answers 49 views ### Generalized set of Pauli elements for a basis for the linear transformations on the vector space [duplicate] I have been doing some practice problems from "Gentle introduction to Quantum Computing". I am a little bit lost with this one: The generalized Pauli group \mathcal G_n is defined by all elements ... 2 votes 1 answer 126 views ### What would the CHSH inequality be equal to if the two qubits were separable? I am writing an Excel spreadsheet to work through the matrix algebra for a simple Bell's game with the following parameters.$$\newcommand{\bra}{\langle #1|}\newcommand{\ket}{|#1\rangle}A_\pm = ... 5 votes 2 answers 134 views ### Pauli Identity Using Tensor Network Notation I am trying to understand the meaning of the equation shown in the above image taken from this paper, but I am unfamiliar with the tensor network notation. My current strategy is trying to write down ... 3 votes 1 answer 288 views ### VQE: Can I build a non-hermitian Hamiltonian with just Pauli matrices? From the VQE paper they claim that a Hamiltonian can be expressed as a polynomial series of pauli operators (equation 1). While coding up VQE from scratch I made a function which would allow me to ... 3 votes 2 answers 515 views ### What is the state of a qubit after measuring some a Pauli operator? I'm confused about the state of a system after a measurement. Say we have a particle$v$in the state:$ |\psi\rangle= \sqrt{1/4} \ |0\rangle + \sqrt{3/4} \ |1\rangle $. From my understanding, if one ... 1 vote 1 answer 424 views ### How to build a circuit for simulation of a simple Hamiltonian? Consider very simple Hamiltonian$\mathcal{H} = Z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$. It has eigenvalues 1 and -1 with coresponding eigenstates$|0\rangle$and$|1\rangle$, ... 0 votes 1 answer 123 views ### Cannot interpret transformations on the bloch sphere as matrix multiplications I understand that X,Y and Z gates are rotations around the axes with the respective letters, but I cannot understand how can Y gate multiply the amplitude of 0 with unreal number and have it landing ... 1 vote 1 answer 188 views ### What kind of transformation does the Y-gate do on the bloch sphere? I'm going through "Quantum Computation & Quantum Information" by Michael A. Nielsen and Isaac L. Chuang, and as a high school student with no previous knowledge, I cannot understand some things ... 1 vote 1 answer 507 views ### Effect of Pauli X gate on minus state using bloch sphere As I understood, the X gate flips the state around :$X(|0\rangle) = |1\rangle$. It can also be visualized with a$\pi$rotation around the$x$axis in the Bloch sphere. I have no problem with that. ... 10 votes 1 answer 3k views ### How can I decompose a matrix in terms of Pauli matrices? I need to see an example of how Hamiltonian, i.e. any Hermitian matrix, can be decomposed into a linear combination of Pauli matrices. I would prefer an option to do this in larger than 2 dimensions, ... 3 votes 2 answers 1k views ### Why can I apply$HS^\dagger$and then measure in the computational basis to measure$Y$? I come from a CS background I was reading Neven and Farhi's paper ("Classification with Quantum Neural Networks on near Term Processors"), and I am trying to implement the subset parity problem using ... 2 votes 1 answer 461 views ### Qiskit flipped representation of qubits in CNOT gate? The conventional CNOT gate is shown on the right, and the Qiskit version is on the left. Since Qiskit defines it has a flipped representation kindly explain what is happening to the 11 position? 3 votes 1 answer 395 views ### Question Regarding Simulating Hamiltonian With Quantum Circuit There have been a few other questions about this section of Nielsen and Chuang, but when working through the output of the circuit, there are some inconsistencies that are probably due to some mistep/... 6 votes 2 answers 4k views ### How is the ground state of a Hamiltonian defined? I'm studying VQE, but there is something I don't get. We know (I think) that for a given Hamiltonian the minimum eigenvalue is associated with the ground state. But if we take the Hamiltonian to be ... 2 votes 2 answers 147 views ### Gate Y returns wrong phase in IBM's circuit composer One can check that, with IBM's circuit composer,$Y$gate acted on$|0\rangle$or on$|1\rangle$returns the same phase of$\pi/2$. Is this a bug? 2 votes 1 answer 244 views ### What does the notation$\sigma_j^z\$ mean for Pauli matrices?
In multiples papers or online article on the QAOA algorithm (such as this one), I found notation for the Hamiltonian similar to this one : $$\sum_{ij} \frac{1}{2} (I-\sigma_i^z \sigma_j^z)$$ I don'...