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}}.
254 questions
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Pauli Twirling not increasing Circuit depth?
This paper talks about the properties of Pauli twirling a circuit. Pauli twirling is a technique that converts arbitrary quantum noise into Pauli errors by applying random Pauli gates before and after ...
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Implementing an arbitrary complex valued single qubit unitary gate using a given matrix
I’m implementing an arbitrary single-qubit gate in Q# , using pauli decompositon. The goal is to match the behavior of a specified unitary matrix acting on a single qubit. Here’s the code I’ve written ...
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What class of matrices have efficient decompositions?
Suppose we have an arbitrary matrix $A \in \mathbb{C}^{N \times N}$ where $N=2^n$. Now suppose that we can decompose $A$ into tensor products of the Pauli basis given by $A=\sum_{l=0}^Lc_lS_l$ where $...
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What's the earliest reference noting that a NOT gate is the same as a Pauli X operator?
Coming from an electrical/computer engineering background, I knew what a NOT gate is well before I learned about the Pauli matrices. But the NOT gate is operationally the same as a Pauli X matrix, ...
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CX and CZ commutation
Suppose I have control qubits $1$ and $2$ and target qubit $3$. I have the circuit element
$$E = CX_{1-> 3}CZ_{2-> 3}$$
I would like to swap the order and have
$$E' = CZ_{2-> 3}CX_{1-> 3}$$...
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Forward propogation of rotated clifford gates in lattice surgery
In A Game of Surface Codes (arXiv), I have been trying to understand the process that is described for forward propogating rotated clifford gates through other rotated clifford gates, and either I am ...
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If $g \in S$, does this necessitate that $g^{\dagger} \in S$?
In Lecture 5 of Aram Harrow's lecture series "Quantum Information Science II", he sets about proving that $\Pi_{S} = \frac{1}{|S|} \sum_{g \in S} g$ can be written as $\Pi_{S}= \prod_{i=1}^{...
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Why no $Z$'s in the $\operatorname{F} (\sum_{j=0}^{n-1} 2^j Z_j) \operatorname{F}^\dagger$ operator?
An interesting numerical observation is that an operator defined as $\phi=\sum_{j=0}^{n-1} 2^j Z_j$ upon a QFT is rotated into an operator $\pi=\operatorname{F} \phi \operatorname{F}^\dagger$ which ...
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Coding a hamiltonian in qiskit
I have a hamiltonian of the form:
$H=\sum_{i=1}^N Z_i Z_{i+1}-Z_NZ_1$
And another one as:
$H=-\sum_{i=1}^N X_i$
I need it to it for N terms.
I am a bit lost can anybody help. I tried looking for ...
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Eigenvalues of Pauli Gate and connection to measurement
Suppose I measure a qubit in the $Z$ basis. If I measure and obtain the outcome $+1$, I get the post-measurement state $\vert 0\rangle\langle 0\vert$ and if I measure and obtain the outcome $-1$, I ...
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How does one identify Pauli logicals in a hyperbolic surface code?
I understand how the Pauli X and Z logicals on the planar surface code work - they are stringlike operators between either the rough boundaries (in case of the Z logical) or between smooth boundaries (...
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What is the connection between an observable and a gate?
I am reading some introductory quantum mechanics and I don't understand the connection between an observable and a gate. I thought a gate just applies a rotation to a state while a measurement gives ...
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How can I measure qutrits in the X basis using cirq?
I attempted to create a custom measurement class which, in my case, allows us to go from the z basis to x basis using a hadamard gate transformation, and then we measure wrt that new basis. However, ...
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Deriving the number operator in a Cooper Pair Box (CPB)
The details of my question follows closely from this source (pg 10 - 12): https://clerkgroup.uchicago.edu/PDFfiles/0210247.pdf
The standard CPB hamiltonian in charge basis is written as
$$
H = 4E_{c}\...
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Is every pure 1-qubit state an eigenstate of $aX + bY + cZ$?
As stated in the question, I have seen this claim made that a pure state can be written as an eigenstate of $aX + bY + cZ$ for some $a,b,c$ where $X,Y,Z$ are Pauli matrices. Why is this true and what ...
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QuTiP tutorial: How to derive the analytical solution to the expectation value of an operator for a system evolving by Lindbladian
I am following the simple tutorial below:
(https://nbviewer.org/urls/qutip.org/qutip-tutorials/tutorials-v5/time-evolution/003_qubit-dynamics.ipynb)
In this they look at single qubit with Hamiltonian $...
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Moving pauli product rotations past measurements
I'm trying to understand how the clifford + T compiler works in "A Game of Surface codes".
How do I move a pauli product rotation block past a pauli product measurement block? More ...
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Why is the Pauli Y gate eigenstate so hard to create?
In a lot of quantum computing formalism, it is relatively easy to create $\vert 0\rangle$, $\vert 1\rangle$, $\vert +\rangle$ and $\vert -\rangle$. However, it is hard to create $\vert i\rangle$. Why ...
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Calculating number of CNOT gates in Pauli evolution gate
How to calculate the number of CNOT gates for a Pauli exponentiation for given time?
I am performing Trotterization which involves performing Pauli evolution ...
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How to modify the quantum circuit to do superdense coding with the state $|00\rangle-|11\rangle$?
Suppose I want to perform the superdense coding protocol, but instead of using the state $\beta_{00}=\frac{1}{\sqrt2}(|00\rangle+|11\rangle)$, I have to initialize it with the state $\beta_{10}=\frac{...
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Commutation of $XX$ and $ZZ$ operators
It is known that the Pauli operators $XX$ and $ZZ$ commute. Consider the state $\vert{++}\rangle$ which is an eigenstate of $XX$. But we also know that
$$ZZ\vert{++}\rangle = \vert{--}\rangle$$
so ...
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In context of stabilizer codes, are logical gates and Pauli operators the same?
I was reading this blog post and as I understood, a unitary operator is logical if and only if it's in $\mathcal{N}(\mathcal{S})$, and a Pauli operator is Logical if and only if it's in $\mathcal{C}(\...
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Connection between a Pauli measurement and the corresponding Pauli gate?
Suppose I have a qubit and the ability to act a Pauli $Z$ gate on it. This is a black box that does the phase flip and I don't know how it works on the inside. Can I use this black box to implement a ...
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Exponentiating a tensor product of operators acting on disjoint qubit registers
Consider a problem of implementing $\operatorname{e}^{i\bigotimes_j O_j}$, where all the $O_j$ terms act on disjoint sets of qubits.
Assume that efficient circuits implementing individual $\...
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What is the action of $CCZ$ on $X \times I \times I$?
Confused about the action of the $CCZ$ gate on Pauli operators:
I understand the action of the $CZ$ gate: $$CZ: XI \rightarrow XX$$ $$CZ: IX \rightarrow IX$$ $$CZ: ZI \rightarrow ZI$$ $$CZ: IZ \...
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Help with a lemma on the argument of a qubit after transformation
From:
King, R. (2023). An improved approximation algorithm for quantum max-cut on triangle-free graphs. Quantum, 7, 1180.
I have trouble understanding item 3 of the above lemma. Here $n_k \cdot \...
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Trying to prove Theorem 4.1 from Nielsen and Chuang algebraically
Background
Theorem 4.1 of Nielsen and Chuang (10th Anniversary Edition) states how a universal single-qubit unitary can be constructed from Y and Z rotations.
Suppose $U$ is a unitary operation on a ...
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Interested in software helping with projecting multi-qubit states onto irreducible components
My interest in QC comes from a problem in geometry called the Atiyah problem on configurations of points. In short, there is a nice one-to-one correspondence between quantum states of a single qubit ...
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Why is the linear combination of Pauli matrices $G =I-XX-YY-ZZ$ PSD?
Define
$$G = I \otimes I - X \otimes X - Y \otimes Y - Z \otimes Z,$$
where $X,Y$ and $Z$ denote the Pauli matrices, and $I$ the identity. I can plug this matrix in my computer and note that
$$G = \...
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Action of below circuit using heisenberg representation
Can someone please explain how the above gate affects logical operators? My understanding is that the circle indicates that we are measuring the second qubit?
My initial guess is that it is equivalent ...
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Can I postpone a Pauli gate $X$ over a conditioned measurement $Y$ or $X$?
The above circuit shows a first measurement, which is $\langle X \rangle$ or $\langle Y \rangle$, depending on the outcome of a second measurement.
Assuming now that a third measurement decides ...
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Why can a quantum code correct $t$ errors only when $d \geq 2t + 1$?
I am working from chapter 7 from notes for ph229 by J. Preskill.
The notes define the distance of a quantum code as:
The distance $d$ is the the minimum weight of a Pauli operator $E$
such that: $$\...
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In the phase flip action on standard basis, why do we consider the $-1$ phase only for the $|1\rangle$?
Prof. Watrous in the first lecture of Qiskit summer school 2023, mentions:
"....the significance of putting a minus sign in front of the $|1\rangle$
basis vector and not $|0\rangle$ will be more ...
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help understanding gate to hamiltonian and representation
So I have this question: Given an operator, find some Hamiltonian implementing this operator/gate. I have realized that this is a swap gate and I know the matrix for it. I also know that $U = \text{...
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Dimension reduction to subsystem in Pauli-Liouville basis: How to implement partial trace of Pauli-Transfer Matrices?
I have a complementary question to my previous (thankfully answered, but I can't verify for larger systems) question:
Multi-qubit quantum channels in Pauli-Liouville basis: Tensor product of Pauli-...
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scaling of error of sum of Pauli strings with number of shots
I have a question which I suppose is quite basic.
Let's say I want to measure the average of an obersvable which is the sum of non-commuting Pauli strings on $N_q$ qubits:
$$
\langle O\rangle =\sum_i^{...
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Process matrix of CNOT gate
The fig below is the process matrix of the CNOT gate from this paper:
where the legend explains that red corresponds to $\frac14$, green to $-\frac14$ and white to zero.
I know the $U_{CNOT} = \frac{...
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Multi-qubit quantum channels in Pauli-Liouville basis: Tensor product of Pauli-Transfer Matrices?
I would like to verify something, need a sanity check. Are the quantum channels for different qubits in the Pauli-Liouville basis (Pauli Transfer Matrices) also given by a tensor product?
The Kraus ...
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Understanding the error operator representation $E = i^{\lambda}X(a)Z(b)$
Question regarding exercise $27.3.2$ in "Concise Encyclopedia of Coding Theory".
The exercise states:
We write $E = X((0,1))Z((0,0))$ and $E' = iX((0,1))Z((1,1))$. We
choose the ordering $(...
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How is Quantum Computing expressed in the language of abstract algebra?
I've lately been taking further coursework in abstract algebra, and it has struck me as fairly reminiscent of quantum computing. Of course, Pauli matrices, etc. have relevant roots within abstract ...
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Expectation value of a given observable computed manually using qiskit.Sampler is different as with qiskit.Estimator
I am trying to calculate the expectation value of a given observable for a certain state $\psi$ using qiskit primitives Sampler (using Sampler requires some further calculations) and Estimator. I ...
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Digitization of errors in QEC
In Nielsen and Chuang, it is stated that any error is given by a quantum channel with Kraus operators $E_i$. A pure state $\vert\psi\rangle\langle\psi\vert$ becomes $\sum_i E_i\vert\psi\rangle\langle\...
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Measuring a single-qubit PauliZ using Qiskit's EstimatorQNN
I am currently working with the EstimatorQNN from Qiskit to construct a Quantum Neural Network using a custom Parametrized Quantum Circuit. But I want to change the ...
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How to find density matrix of 3 qubit W state?
Given a state in bra-ket notation as $|\psi\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$. What is the density matrix of this state written using Pauli's spin operator?
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How is Pauli twirling so powerful?
So the Pauli twirling approximation gives us a quantum channel $\Phi$ that transforms a density matrix $\rho$ to:
$\Phi(\rho)\mapsto\sum_{i=0}^3 \sigma^i \rho \sigma^i,$
where $\sigma^0 = \mathbb{I}, \...
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Physical description of trace of ancilla state yields a depolarising channel
Let's start with
$Tr_{\Omega}[|0,\Omega_{0}\rangle\langle0,\Omega_{0}|U^{\dagger}] = \sum_{\alpha}E_{\alpha}|0\rangle\langle0|E_{\alpha}^{\dagger}$
where $U$ be a unitary operator. The trace operator ...
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544
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Coupling map in QISKIT transpile
I have a 3-qubit unitary represented by a circuit with the following dictionary: {'cx': 30, 'h': 22, 'rz': 15, 's': 4, 'sdg': 4}. I want to use this circuit on IBM ...
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How to interpret the encoding circuit for the 5-qubit QECC
I have a question on circuit which constitutes the sydnrome measurement for the 5-qubit error correcting code. If I focus on just a portion of the circuit:
Reference for image. The full circuit can ...
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When is a block diagonal matrix a tensor product of Pauli matrices?
$U = |0\rangle\langle 0|\otimes U_1 + |1\rangle\langle 1|\otimes U_2$ is a block-diagonal unitary matrix. For this question we will assume $U$ acts on qubits. Then for some integer $N\ge 1$, $U$ is a $...
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How to prove that these equations are correct for $CZ$ and $CX$?
How do I prove that the equation on the right is $CX$ and $CZ$ gate? I don't think that reaching the matrix of the CX or CZ is possible with the given equation.
For (b) I keep getting $I \otimes I$ ...
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