Questions tagged [clifford-group]

For questions related to the Clifford also known as the Pauli group, as relevant to quantum computing.

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Computing specific observable for a Clifford circuit

Given a clifford circuit, how can I simulate it efficiently to get the 2-RDM (reduced density matrix) $D^2=\langle a_i^\dagger a_j^\dagger a_k a_l\rangle$ in the presence of deplorizing error? The ...
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What is the largest number of stabilizers a pure state can have?

What is the largest number of stabilizers a pure state can have? Elaborately put: Let $P(n)$ denote the Pauli group. Given an arbitrary pure state $|\psi\rangle$, what is the upper limit on how many ...
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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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Cliffords to Transform into Common Eigenbasis

Say I have the following Hamiltonian (given in terms of Pauli operators): \begin{equation} H=aX_1Z_2+bZ_1X_2. \end{equation} Both Pauli terms commute with each other. I want to make a measurement of $\...
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Is the SWAP gate a Clifford Gate? How would I express it using the Clifford Gate generators?

By my calculations, it looks like the SWAP gate is a Clifford Gate. See the following table: I follow the same method as in this paper for showing a gate is a Clifford Gate. I got the above table by ...
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1answer
62 views

Can you measure sums of Paulis in the stabilizer formalism?

Suppose we wanted to measure the observable $Z_{1} + Z_{2} + \cdots + Z_{N}$ in a stabilizer state. Is it possible to do this using only Clifford operations, and possibly adding some auxiliary qubits? ...
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What scheme should be used in case of applying non-Cliffords to estimate probability of success?

For Clifford gates (when performing randomized benchmarking and starting from ground state) the final state is always ground. It is acquired by applying at the end recovery gate, which transfers the ...
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50 views

Why do we care about the number of $T$ gates in a quantum circuit?

When reading this question and quickly reading some of the linked papers, I wondered why was the number of $T$ gates specified along the number of controlled-$X$ gates. I've often read that ...
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How can I get fidelity of a gate from randomized benchmarking?

I found (page 33) the method of finding fidelity from fit by "interleaved and reference decay" according to the sequence fidelity formula: $$F_{ref}=Ap_{ref}^{m}+B,$$ where $p_{ref}^{m}$ is ...
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Review paper on depth, qubits and $T$ gates number on Clifford+T decomposition for various "typical" algorithms

My question I am looking for some review paper, or a list of different papers providing concrete numbers about the depth, number of qubits and number of $T$ gates required on the Clifford+T basis for ...
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1answer
150 views

Getting non-Clifford after performing several Clifford gates in qiskit

I'm trying to test Clifford gates in qiskit according to the table in Fault-tolerant SQ, page 101. I tried 4 Cliffords in the test $$-X/2 - X -X/2,Y/2,X/2 - -X/2,Y/2,-X/2$$ using the following code <...
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195 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 ...
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4answers
470 views

How is a Toffoli gate built without using T gates?

Can someone tell me how to make a Toffoli gate without using T gates? Can we use $R_x$ and $R_y$. If yes, then how? I tried many circuits but I was unable to create the CCNOT gate out of $R_x$, $R_y$ ...
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How to use Clifford Data Regression for the MaxCut Problem

i read about Clifford Data Regression in https://arxiv.org/pdf/2005.10189.pdf. If I have understood this correctly, then one receives the mitigated expected value of an observable from CDR. For the ...
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Writing circuits in Qiskit using only Clifford and T gates

Is there a way in Qiskit to write my circuit using only Clifford and T gates (CX, S, H, T and I think also $S^\dagger$ and $T^\dagger$)? With the function compile (with aer simulator) it gives me some ...
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145 views

What is the set of generators for the qutrit Clifford group?

According to this article, any Clifford gate, acting on $n$ qubits, can be generated by Hadamard, CNOT, and S gates. What are the set of generators for qutrit Cliffords?
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118 views

How powerful are boundedly many $T$-gates?

For a natural number $k$ (0 is a natural number), let $T_k$ be the collection of all languages that can be efficiently decided by quantum circuits consisting of Clifford gates and at most $k$ $T$-...
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65 views

Getting exponential sequence of coefficients with not so many $T$-gates

Let $\Psi \in (\mathbb{C}^2)^{\otimes n}$ be a $n$-qubit quantum state. In the computational basis, we can write $\Psi$ as $$\Psi = \sum_{(i_1, \dots, i_n) \in \mathbb{F}_2^n} \Psi_{i_1, \dots, i_n} |...
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Explicit states with high $T$ count

It is well known, that the Clifford $+T$ gate set consisting of the gates $\lbrace H, S, CNOT, T \rbrace$ is universal for quantum computation, that is, for any n-qubit unitary $U:\left( \mathbb{C}^2\...
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Calculating symplectic dual of a code

Stabilizer codes can be treated as symplectic codes over $\mathbb{F}_2$ (or over $\mathbb{F}_p$ when taking about q-dits). While treating error class, symplectic dual of the code plays a crucial part (...
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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 ...
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Proof that encoder for a stabilizer code is in the Clifford group

Given a stabilizer code on $n$ qubits defined by a set of stabilizers $S_1,\cdots S_m$; The encoder $E$ is a matrix in $U(2^n)$ (unitary group) such that $S_i E v = E v$. I'm pretty sure that $E$ is ...
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93 views

When and how do we use basis embedding?

It is suggested in various sources that a possible approach to representing classical data as a quantum state is simply to take the binary sequence $x$ and turn it to $|x\rangle$ (i.e., "basis ...
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100 views

If $C$ is a Clifford circuit, is there necessarily a Clifford circuit $C'$ such that $CT=TC'$?

Let $C$ be a Clifford circuit, is there necessarily a Clifford circuit $C'$ such that $CT=TC'$ (where $T$ is taken as applying the $T$ gate to the same qubit on both sides)?
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Do we need to use an ancillary qubit when decomposing arbitrary $U(2^n)$ gates using Clifford+T universal gate sets?

As I know, we can decompose $U$ without ancilla if it's from special unitary group $SU(2^n)$. Do we need to use ancilla qubit on decomposing arbitrary $n$-qubit $U$ using Clifford+T universal gates ...
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81 views

How can I construct a universal transformation using Clifford+T gates? [duplicate]

How can I construct, using Pauli, Hadamard and $T$ gate, a universal transformation $U$ such that $U|0\rangle$ has a less than $\frac{\pi}{4}$ complementary angle with $|0\rangle$?
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148 views

Which states can reached using single-qubit Clifford gates?

Starting with the qubit state $|0\rangle$, which single-qubit states can be obtained by applying single-qubit Clifford gates, i.e. Pauli + Hadamard + $S$ gates?
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566 views

How can one argue that the $S$-gate is Clifford while $T$-gate is not?

How can one argue that $S$-gate is Clifford while $T$-gate is not?
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184 views

Are almost-Clifford circuits almost easy to simulate?

Circuits consisting entirely of Clifford operations in $\{X, Y, Z, H, S, \text{CNOT} \}$ are "easy" to simulate classically since there is a method that can fully compute such circuits over $...
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What are the simplest examples of codes with transversal non-Clifford gates?

There are certain codes which have transversal non-Clifford gates. My question is, what are the simplest examples of such codes and how does one check for a non-Clifford gate? Is it a computationally ...
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Understanding transversal gates for the 7 qubit steane code

How can one derive the complete list of transversal operators for the 7-qubit Steane code? I can derive the Clifford operators that are transversal, but I do not understand an easy way to check for ...
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128 views

Prepare state $|00\rangle+|1+\rangle$ using Clifford gates and the T-gate

I am looking for a quantum circuit which maps state $|00\rangle$ to $|\psi\rangle=\frac{1}{\sqrt{2}} |00\rangle+\frac{1}{\sqrt{2}}|1+\rangle$. The circuit should only apply quantum gates from the ...
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98 views

Is it possible to construct Grover search from Clifford gates only?

In the article Is Quantum Search Pratical the authors emphasized that a complexity of an oracle is often neglected when advantages of Grover search are discussed. In the end, a total complexity of the ...
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Is there a good reason to use T-count minimization for circuits executed on current IBM open quantum systems (real hardware)?

As far as I understood from a series of papers, minimizing the T-count in Clifford+T circuits is essential for fault-tolerant ...
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2answers
359 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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Average execution time for T gate

Using the Q# resource estimator, I found out that my program, meant to do graph coloring using Grover's algorithm, could be decomposed into ~500-1000*x T gates, where x in the number of iterations, ...
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Is there a CSS code for which the controlled phase gate and all Clifford gates are transversal?

In a paper that I have recently read, a protocol was given that required a CSS code with certain properties for which all logical Clifford gates and standard measurements have a transversal ...
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1answer
643 views

Is the Clifford group finite?

For any fixed $n$, how do we prove that the $n$-qubit Clifford group (subgroup of $ U(2^n) $ generated by Hadmard gates, phase-shift gates and CNOT gates) is finte or not? I know that for the single ...
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How do I check if a gate represented by Unitary $U$ is a Clifford Gate?

The Gottesman–Knill theorem states that stabilizer circuits, circuits that only consist of gates from Clifford group, can be perfectly simulated in polynomial time on a probabilistic classical ...
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Defining a Grassmann Algebra in Python

I am trying to implement a Grassmann algebra in Python and was wondering if anyone could recommend any packages or suggest how to do so? I want to define the following multiplication rules over $\Bbb{...
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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 = \...
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237 views

Is it possible to demonstrate a quadratic speed-up of a quantum algorithm on a classical computer?

In article Quantum computational finance: Monte Carlo pricing of financial derivatives the authors said that: Firstly: While a practical quantum computer has yet to become a reality, we can ...
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Why are non-Clifford gates more complex than Clifford gates?

There are two groups of quantum gates - Clifford gates and non-Clifford gates. Representatives of Clifford gates are Pauli matrices $I$, $X$, $Y$ and $Z$, Hadamard gate $H$, $S$ gate and $CNOT$ gate. ...
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Intuitive link between clifford group and gottesman-knill theorem

Elements of the Pauli group are the n-Pauli matrices with $\pm 1$ or $\pm i$ on front of them. They all commute or anti-commute between them. The Clifford group are element that preserve the n-Pauli ...
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Definition of the Pauli group and the Clifford group

There seem to be two definitions of the Pauli group. In Nielsen and Chuang, the Pauli group on 1 qubit is defined as \begin{align*} \mathcal{P}_1 = \{\pm I, \pm iI, \pm X, \pm iX, \pm Y, \pm iY, \pm Z,...
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Efficient implementation of the Clifford group for $n$ qubits

I'm looking for an efficient implementation of the Clifford group $\mathcal{C}_n$ of $n$ qubits. The Clifford group $\mathcal{C}_n$ has stucture $(2_+^{1+2n} \circ C_8).Sp(2,n)$, where $2_+^{1+2n}$ ...
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307 views

Isomorphism between the Clifford group and the quaternions

How do I find an explicit isomorphism between the elements of the Clifford group and some 24 quaternions? The easy part: The multiplication of matrices should correspond to multiplication of ...
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476 views

Is there a simple rule for the inverse of a Clifford circuit's stabilizer table?

In Improved Simulation of Stabilizer Circuits by Aaronson and Gottesman, it is explained how to compute a table describing which Pauli tensor products the X and Z observable of each qubit get mapped ...