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.

Filter by
Sorted by
Tagged with
4 votes
2 answers
191 views

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 (...
user avatar
  • 365
2 votes
1 answer
63 views

Group of commuting Pauli matrices doesn't permit synthesis

I am working on learning grouped measurement and I began by reading this paper by a group out of UChicago showing a method for the synthesis of circuits for the grouped measurement of a set of ...
user avatar
5 votes
1 answer
141 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 $k$ $T$-...
user avatar
  • 227
2 votes
1 answer
60 views

In quantum circuits, why does $UNU^\dagger$ act on states in the same way $N$ acts before the operation?

I understand that the Schrodinger picture changes the quantum states, while the Heisenberg picture changes the operators. In this paper The Heisenberg Representation of Quantum Computers, in equations ...
user avatar
  • 105
2 votes
1 answer
161 views

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 ...
user avatar
3 votes
1 answer
150 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 ...
user avatar
3 votes
1 answer
191 views

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 ...
user avatar
  • 12.6k
13 votes
1 answer
2k views

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. ...
user avatar
11 votes
2 answers
3k views

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, ...
user avatar
14 votes
1 answer
2k views

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 ...
user avatar