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}}.

Filter by
Sorted by
Tagged with
3 votes
0 answers
68 views
+50

Almost perfect quantum encryption of a mixed state using only $n + O(\log(n) +\log(\frac{1}{\epsilon}))$ shared bits

Alice holds a state $\psi$ of $n$-qubits, and wants to send it to Bob using a single quantum message. Bob and Alice share only $n + O(log(n) +log(\frac{1}{\epsilon}))$ random bits, for some value $\...
3 votes
1 answer
54 views

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}(\...
0 votes
1 answer
42 views

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 ...
1 vote
1 answer
57 views

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 \...
6 votes
2 answers
553 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 votes
1 answer
208 views

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 ...
5 votes
1 answer
331 views

Regarding the inductive proof that any Clifford gate can be made of Hadamard, phase and c-not

In Exercise 10.40 of Nielsen and Chunang's textbook, the reader is supposed to construct an inductive proof of Theorem 10.6 that any Clifford gate can be made of Hadamard, phase and c-not. There it is ...
2 votes
1 answer
1k views

construction of Y gate from X,Z and H gates

As a part of textbook exercise, Y gate is to be constructed using H,Z and X-gates, just like we have $X = HZH$. is there some way/process/intuition to find such combinations or it is just like we need ...
1 vote
1 answer
69 views

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 $\...
1 vote
2 answers
72 views

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 \...
0 votes
1 answer
70 views

The Output of Transversal Bell Measurement in Knill's Method of Fault-Tolerant Error Correction (FTEC)

On page 26 of arXiv:quant-ph/0504218, it is written that in Knill's method of fault-tolerant error correction (FTEC), the output of the transversal bell measurement becomes $(P_m \otimes I) | \Phi_0 \...
3 votes
0 answers
42 views

Trying to prove Theorem 4.1 from Neilsen and Chuang algebraically

Background Theorem 4.1 of Neilsen 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 ...
2 votes
0 answers
19 views

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 ...
2 votes
3 answers
250 views

Why is the error propagation by the CNOT gate considered without taking into account the state?

In the syndrome measurement circuit of a stabilizer code, I think you would consider that Pauli errors propagate through the CNOT gates. I don't understand why one usually considers the propagation of ...
8 votes
3 answers
261 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 ...
3 votes
2 answers
47 views

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 = \...
0 votes
1 answer
85 views

Commutation relationship and measurement results

There are things I do not understand about the following circuit, and I would appreciate it if you could explain. ...
1 vote
0 answers
15 views

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 ...
3 votes
1 answer
42 views

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 ...
3 votes
2 answers
102 views

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: $$\...
5 votes
2 answers
197 views

Is there an efficient algorithm for decomposing an arbitrary Hamiltonian into Pauli strings?

Basically the title. If I have a $2^N\times 2^N$ Hamiltonian $H$ of random numbers (we can take the Hamiltonian as normalized if we want) and $N$ is an integer, is there an efficient way of writing $$ ...
-2 votes
1 answer
53 views

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{...
1 vote
1 answer
89 views

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 ...
0 votes
0 answers
24 views

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-...
2 votes
1 answer
97 views

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 ...
1 vote
0 answers
33 views

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^{...
1 vote
1 answer
185 views

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{...
0 votes
1 answer
40 views

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 $(...
1 vote
1 answer
239 views

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 ...
0 votes
1 answer
100 views

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 ...
1 vote
2 answers
61 views

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\...
1 vote
0 answers
41 views

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 ...
1 vote
1 answer
56 views

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?
5 votes
2 answers
789 views

How to perform a controlled Pauli string rotation gate?

I would like to know some circuit decomposition for an arbitrary controlled Pauli string rotation: \begin{equation} |0\rangle\langle 0| \otimes e^{i \theta (P_1\otimes...\otimes P_n)}+ |1\rangle\...
1 vote
0 answers
176 views

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}, \...
0 votes
1 answer
54 views

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 ...
1 vote
1 answer
76 views

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 ...
0 votes
1 answer
69 views

Phase estimation of the Pauli-Y matrix

I'm trying to use the phase estimation algorithm to extract the eigen value for both eigen vectors of the Pauli-Y matrix using the ibm quantum experiance. So far I have this for the possitive state |+&...
1 vote
1 answer
111 views

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 $...
2 votes
2 answers
92 views

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 $I \otimes I$ ...
4 votes
1 answer
262 views

Efficient way to calculate trace of product of Pauli string and matrix?

Basically the title, but more formally: is there a way to efficiently calculate the trace of the product of a Pauli string $P$ and a $2^n \times 2^n$ matrix $M$? That is, is there a way to calculate ...
1 vote
2 answers
70 views

tricks to finding possible stabilisers for $|GHZ_{3} \rangle$

The famous 3 - qubit Greenberger, Horne and Zeilinger state: $|GHZ_{3} \rangle = \frac{1}{\sqrt{2}}[|000\rangle + |111\rangle]$. A stabiliser for $|GHZ_{3} \rangle$ is the 3 - tensor product X Pauli ...
0 votes
2 answers
50 views

Visualizing Y-gate operation to achieve quantum state

In the below snippet how qc.y(1) helps to achieve the quantum state $i|10\rangle$ ? ...
6 votes
1 answer
229 views

Getting intuition on the state-injection relations for the generalized $\exp(-iP \pi/8)$ $T$-gates (ideally using ZX calculus)

In Litinsky's paper, there are many circuits relations, like the one below. The left handside represents the "rotation" $\exp(-i P \phi)$ with $\phi=\pi/8$ with similar definitions for the ...
1 vote
2 answers
77 views

Notation: Hamiltonian Simulation of Pauli Gates

Let $\sigma^j_x$ describe the following unitary over $n$ qubits: on the $j$-th qubit, it acts as the Pauli $x$ operator; instead, on any other qubit, it acts as the identity. A paper states now that \...
2 votes
1 answer
194 views

Half Adder using CNOT Gates

As per this schematic of qubits, how this explanation is correct --"If you look again at the four possible sums, you’ll notice that there is only one case for which this is 1 instead of 0: 1+1=10....
3 votes
2 answers
74 views

Do the operators $\Phi(\sigma_k)$ form a basis if $\sigma_k$ do?

Assume we have a quantum channel $\Phi$. The single qubit Pauli basis is $\sigma_0, \sigma _1, \sigma_2, \sigma_3$. Now we apply $\Phi$ to Pauli basis and get $\gamma_0=\Phi(\sigma_0), \gamma_1 = \Phi(...
0 votes
2 answers
427 views

How to prove the matrix identities $HXH = Z$ and $HZH = X$?

As we know Hadamard gates are used to bring quantum bits into superposition states. I’m trying to understand how identities $HXH = Z$ & $HZH = X$ w.r.t rotation.
1 vote
2 answers
412 views

Can any Qiskit circuit be converted to a gate?

I am trying to convert the following qiskit QuantumCircuit to a gate using to_gate() method. ...
1 vote
1 answer
124 views

A question on the structure of the Clifford group

Let $P_n$ be the Pauli group on $n$ qubits defined by $P_n=\{i^k \sigma_1...\sigma_n: k=0, 1, 2 \; \mathrm{or}\;3, \sigma_j\in \{I, X, Y, Z\}\}$, where $I, X, Y, Z$ are Pauli matrices. The Clifford ...

1
2 3 4 5