8 votes
Accepted

What unitary commutes with all local Pauli operators?

TL;DR: The only $U$ that commutes with all $\sigma_{X,i}$ and all $\sigma_{Z,i}$ is a scalar multiple of identity. This follows from the Schur's lemma, but can also be shown using elementary linear ...
Adam Zalcman's user avatar
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5 votes
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Is it possible to modify the QFT circuit to use only 1-qubit gates?

This is called qubit recycling (when combined with introducing the qubits-to-QFT one by one). All you have to do is take the normal circuit, measure each qubit immediately after it gets Hadamard'ed, ...
Craig Gidney's user avatar
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4 votes
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Evolution of a state vector: Why is the action of $N$ equivalent to the action of $UNU^{†}$?

I think it probably helps to understand what Gottesman is trying to do with the operator $N$ (later in the paper). He wants to start with some state $|\psi\rangle$, but instead of directly describing ...
DaftWullie's user avatar
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4 votes

Evolution of a state vector: Why is the action of $N$ equivalent to the action of $UNU^{†}$?

Maybe it's easier to see when it's presented like this: $UN\left|\psi\right> = UNU^\dagger U\left|\psi\right>$ means that $UNU^\dagger$ maps $U\cdot \left|\psi\right>$ to $U \cdot N\left|\psi\...
Vladimir Lysikov's user avatar
3 votes

Evolution of a state vector: Why is the action of $N$ equivalent to the action of $UNU^{†}$?

Indeed mathematically $N$ and $UNU\dagger$ may be interpreted as 2 matrix representations of exactly the same physical operator $n$ (mind uppercase/lowercase), but in 2 different orthonormal basis. ...
Pierre-Paul T.'s user avatar
3 votes

Algorithm to reveal properties of a random quantum permutation gate

To answer 1: Find the smallest Hamming code that acts on $n$ or more bits. Write out its parity-check matrix. Remove as many columns as you need (doesn't matter which ones) so that it only has $n$ ...
DaftWullie's user avatar
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3 votes
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Implementing cphase gate on IonQ through Amazon Braket

You'll want to write the cphaseshift gate in terms of gates that are supported by the IonQ device through Amazon Braket. For example, the circuit ...
Ryan Shaffer's user avatar
3 votes

Qiskit Runtime forbids reset gate?

Indeed, the problem is that initialize introduces a reset (see docs). ...
luciano's user avatar
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3 votes

How to create the state $(n!)^{-\frac{1}{2}} \sum_{\pi \in S_n} | \pi, \pi (G) \rangle$ when $G$ is an undirected graph and $\pi$ a permutation

Surprisingly, the way to make a uniform superposition of all permutations is by using a sorting network. You make random shuffle states by sorting! Weird, right? This was figured out in "Improved ...
Craig Gidney's user avatar
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2 votes
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How to create the state $(n!)^{-\frac{1}{2}} \sum_{\pi \in S_n} | \pi, \pi (G) \rangle$ when $G$ is an undirected graph and $\pi$ a permutation

What you're missing is that if there is an efficient classical circuit to compute $\pi(G)$ from a binary representation of $\pi$, then it means that you are able to build the oracle: $$\mathcal{O}|\pi,...
Tristan Nemoz's user avatar
2 votes
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Why is the first logical unitary gate in this example fault tolerant?

However, because the first gate is acting on disjoint pairs of physical qubits, can the 2 𝑋 errors ever propagate to more than 2 𝑋 errors? Yes, 2 $X$ errors can propagate to 4 $X$ errors via the ...
Peter-Jan's user avatar
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2 votes
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I would like to understand the meaning of applying permutation to a unitary matrix

You have to review the definition of the matrix representation of an operator in a given basis: If the matrix $U$ is the representation of some operator $u$ in the basis $\{|00\rangle,|01\rangle,|10\...
Pierre-Paul T.'s user avatar
2 votes

Evolution of a state vector: Why is the action of $N$ equivalent to the action of $UNU^{†}$?

I think you're just confused by the wording. $U N | \psi \rangle $ means apply N, then apply U. $UNU^\dagger U|𝜓 \rangle$ means apply U, then apply $UNU^\dagger$ So $U N | \psi \rangle = UNU^\dagger ...
Peter-Jan's user avatar
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2 votes
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How to figure out the action of a circuit form its effect on logical operators?

Most of the time, you don't actually need to calculate the state, but, sure, it's helpful when you're learning the topic to relate back to things you already know about. The key ingredients that you ...
DaftWullie's user avatar
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2 votes
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How does one create a long range CNOT gate on a square grid of qubits using constant depth circuits?

Delfosse is presenting the paper: "Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes" The claim is that one needs 3 CNOT gates along ...
Peter-Jan's user avatar
  • 1,378
2 votes

Evolution of a state vector: Why is the action of $N$ equivalent to the action of $UNU^{†}$?

Indeed there is some unclarity in the wording "the operator $UNU^\dagger$ acts on states in the same way that $N$ did". Better would be: "the operator $UNU^\dagger$ acts on states $U|\...
Jos Bergervoet's user avatar
1 vote

U(2) vs. SU(2) for single-qubit gates; ignoring global phases

It is true that an element from U(2) can change the global phase of a state in an arbitrary way, the U(2) matrix $$\tag{1} \begin{pmatrix} e^{i\varphi} & 0 \\ 0 & e^{i\varphi} \end{...
Jos Bergervoet's user avatar
1 vote
Accepted

How does syndrome extraction in bit-flip codes using stabilizers work?

The state is not $-\vert ++\rangle\vert 100\rangle$ after the second set of gates. That would be true if you applied $Z$ gates but you are applying controlled $Z$ gates. Let's just look at the ...
rnva's user avatar
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1 vote

Hilbert space of n qubits always has a decoherence free subspace if n is even?

TL;DR Decoherence-free subspaces (DFS) do exist for both, even and odd numbers of qubits $(N)$, for noise of the form $g^{\otimes N}$. Let me give you some examples from a different perspective than ...
FDGod's user avatar
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1 vote
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What is the explicit action of the following circuit?

I am assuming the two control symbols connecting |𝜓⟩ and |+⟩ are entangling the states somehow? Yes, this represents a CZ gate. Then I don't understand the top right measurement. It is not in a ...
Peter-Jan's user avatar
  • 1,378
1 vote

Encoding boolean function in a quantum circuit

It depends on how you have implemented the map that takes $|x\rangle$ to $(-1)^{f(x)}|x\rangle$. If you simply have a single qubit gate like that, it will be impossible to get out $|y \oplus f(x)\...
Poseidaan's user avatar
  • 111
1 vote

What unitary commutes with all local Pauli operators?

If $P$ commutes with $U$, that means $U$ conjugates $P$ into $P$. $$ \begin{aligned} &([P, U] = 0) \\\equiv& (P U = U P) \\\equiv& (U^\dagger P U = P) \end{aligned}$$ In the stabilizer ...
Craig Gidney's user avatar
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