Quantum Computing Stack Exchange
Quantum Computing Stack Exchange

Questions tagged [gottesman-knill]

Questions directed to the Gottemsman-Knill theorem, which states that quantum circuits consisting of elements from the Clifford group are classically efficient to simulate.

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Twirling Quantum Channels: Pauli and Clifford Twirling

I am currently working through some papers related with approximations of more general quantum channels such as amplitude and phase damping channels to Pauli channels. The reason to do so is so that the Gottesman-Knill theorem is fulfilled and…
Josu Etxezarreta Martinez
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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. Non-Clifford gate is for example $T$ gate and…
Martin Vesely
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Why doesn't the Gottesman-Knill theorem render quantum computing almost useless?

The Gottesman-Knill theore states (from Nielsen and Chuang) Suppose a quantum computation is performed which involves only the following elements: state preparations in the computational basis, Hadamard gates, phase gates, controlled-NOT gates,…
user2723984
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Understanding Knill-Gottesman theorem: what role does the Toffoli gate play?

I’m trying to understand the relationship between the Toffoli gate, Clifford gates, and the classical simulation of quantum circuits. -I know that the Toffoli gate is universal for classical computation, since it can simulate gates like NAND. -I’m…
Tom_tomato
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Does Gottesman-Knill theorem apply with any computational basis input?

On Wikipedia, the Gottesman-Knill theorem is said to state the following: A quantum circuit using only the following elements can be simulated efficiently on a classical computer. Preparation of qubits in computational basis states, Clifford…
trillianhaze
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The construction of every element of the Clifford group using H,S and CNOT circuits

I am trying to understand the following theorem: Every element $U\in C_n$ of the Clifford group can be constructed using $H, S, CNOT$ gates. In Nielsen and Chuang's book this is left as an exercise (10.40, page 462), so I tried reading Gottesman's…
Gadi A
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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 $k$ $T$-gates (that is, we modify the definition of $BQP$ so…
Haim
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Non-universal and non classically simulatable gate set?

The Gottesman-Knill theorem says that many circuits, including all Clifford curcuits can be simulated classically in polynomial time. On the other hand it is believed that there is no polynomial time classical simulation of universal quantum…
Frederik Ravn Klausen
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How do magic states circumvent the Eastin-Knill theorem?

I'm trying to understand magic states and how they circumvent the Eastin-Knill theorem. I understand that these magic states are used to implement non-Clifford gates but how are these magic states generated in the first place? From Bravyi and…
Karim
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Understanding the Gottesman-Knill Theorem

I come from a theoretical CS background, and I am trying to gain a better appreciation of the exact formal statement of the Gottesman-Knill theorem in terms that I am more familiar with. My question in this post clarified that the input need not be…
trillianhaze
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Gottesman Knill theorem: why $O(n^2)$ classical operation to keep track of a Clifford gate

Starting from a state stabilized by Pauli matrices, and using only Clifford operations Gottesman Knill theorem ensures us that such algorithm can be classically simulated. Indeed, if I call my initial state $|\psi \rangle$, I can define a Stabilizer…
Marco Fellous-Asiani
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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 search can be given mostly by the oracle…
Martin Vesely
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Speed up in Bernstein-Vazirani algorithm and Gottesman-Knill theorem

The Bernstein-Vazirani problem: Let $f$ be a function from bit strings of length $n$ to a single bit, $$f: \{ 0, 1\}^n \to \{0, 1\} $$ thus all input bit strings $x \in \{0,1\}^n$. There exists a secret string $s \in \{0,1\}^n$ such that $$ f(x)…
KAJ226
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Gottesman-Knill simulation and Bell states

I have some problems to grasp the interpretation of the Gottesman-Knill theorem. If the first qubit is measured, since $\mathcal{Z} \otimes \mathcal{I}$ does not commute with all the stabilizers, the first qubit should $0$ or $1$ with equal…
JMark
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Are there non-stabilizer multi-qubit states that are easy to simulate?

The Gottesman-Knill theorem states that the following process is efficiently simulatable on a classical computer: start of with a set of qubits in a computational basis apply any amount of $H, S$ and $CNOT$ gates in any order measure all the qubits…
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