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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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7 votes
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what is square root of a density matrix power two?

If $\rho$ is a density matrix, then $\sqrt{\rho^2} = \rho$. To see why this is, let's start with the definition of the square root of a matrix. Ordinarily, if $A$ is a square matrix, there may be ...
John Watrous's user avatar
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6 votes
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Why is the orbit of a unitary t design a complex projective t design?

One of the equivalent definitions of a unitary t-design $\{U_i\} \subset \mathbb{U}(d)$ is that $$ \frac{1}{n}\sum_{i=1}^n (U_i^{\otimes t})M(U_i^{\otimes t})^\dagger = \int_{\mathbb{U}(d)} (U^{\...
Danylo Y's user avatar
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5 votes
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Definition of quantum junta is not basis independent: isn't this a problem?

The definition that you give of a quantum junta is in relation to a tensor product structure. Such structures (and things like entanglement) only make sense if you have some sort of definitive ...
DaftWullie's user avatar
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5 votes
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Existence of Hamiltonians such that the time evolution unitary becomes identity

I don't believe that this is always possible. For instance, what if my set of $\{H_i\}$s comprise a single term that I can construct to be arbitrarily awkward? The key feature will be gaps between ...
DaftWullie's user avatar
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5 votes
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Which Clifford groups are 2-designs?

The confusion stems from the existence of incompatible definitions of the "Clifford group" in dimensions which are not prime. With your definition, the Clifford group is indeed a unitary 2-...
Markus Heinrich's user avatar
4 votes

Can every unitary be approximated by gates from the Clifford Hierarchy?

This is too long for a comment but not an answer. First we must define some notion of closeness of unitaries. Other metrics should be fine, but operator norm seems like a good one. Now I think your ...
Jonas Anderson's user avatar
4 votes
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Given an observable $O$, what's the achievable maximum value of $\operatorname{Tr}(O\rho)$?

Expectation values are bounded by the spectrum of the observable $O$. This follows essentially from the definition of eigenvalues and the fact that Hermitian operators always have real eigenvalues: $$\...
Banach space fan's user avatar
4 votes
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Upper bound on $\Vert U_1 \otimes U_2 \otimes \cdots \otimes U_k - V_1 \otimes V_2 \otimes \cdots \otimes V_k \Vert$

Let's write $$G_i=U_1\otimes U_2\otimes\ldots U_{i-1}\otimes V_i\otimes V_{i+1}\otimes\ldots\otimes V_k.$$ So, we have $$ \|G_{k+1}-G_1\|=\|G_{k+1}+(G_k-G_k)+\ldots (G_2-G_2)-G_1\|\leq \sum_{i=2}^{k+1}...
DaftWullie's user avatar
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4 votes
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Bounds on local expectation values for two states close in trace distance

We first have: $$|\mathrm{tr}(A(\rho-\sigma))|\leqslant\mathrm{tr}(|A(\rho-\sigma)|)=\|A(\rho-\sigma)\|_1$$ We can then use Hölder's inequality: $$\|A(\rho-\sigma)\|_1\leqslant\|\rho-\sigma\|_1\|A\|_{\...
Tristan Nemoz's user avatar
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3 votes
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Are quantum channels bounded linear maps?

A linear map defined on density operators is uniquely extendible on the set of all linear operators (in finite-dimensional case). Simply because any linear operator is a linear combination of density ...
Danylo Y's user avatar
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3 votes
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Decomposition of a $4 \times 4$ unitary matrix

That paper appears to do their rotations in a very strange order. The method you're interested in is how to use Givens rotations to perform a QR decomposition (see, e.g. https://en.wikipedia.org/wiki/...
DaftWullie's user avatar
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3 votes
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$\mathbb{C}^2 \otimes \mathbb{C}^2$ vs $\mathbb{C}^4$

You don't have to directly express the basis of one space in another to know they are the "same". We say two Hilbert spaces $A$ and $B$ are isomorphic if there exists a linear isomorphism ...
Rammus's user avatar
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3 votes
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A conceptual Query regarding measurement during a Quantum Algorithm

Quantum gates (which I assume you mean by "quantum operations") are reversible: applying a gate to different "input" states results in different "output" states. This ...
Mariia Mykhailova's user avatar
3 votes

Are permutations of the Pauli strings unitary operations?

No. This fails because the operation $U_{g}$ is not necessarily trace-preserving. Suppose $N = 1$ and $g(1) = 0$, i.e. the Permutation that maps $X$ to $\mathbb{I}$. We thus have $\mathbb{I} = \tau_{0}...
JSdJ's user avatar
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3 votes
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What does Pauli's $Y$ matrix represent?

$Y = iXZ = -iZX$, so it can be thought of as both a bit flip and a phase flip, plus an overall $i$ phase.
Abdullah Khalid's user avatar
3 votes
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Truncated Qumode States and Support

Your guess is correct that this is not possible. A $d\times d$ matrix times an $d\times 1$ column vector gives you another $d\times 1$ column vector, so your truncated unitary cannot take the vector ...
Quantum Mechanic's user avatar
3 votes

conditions for two hermitians operators same up to unitary

It holds precisely if they have the same spectrum: You already argue for necessity. For sufficiency: $A=VDV^\dagger$, $B=WDW^\dagger$, then $D=W^\dagger BW$ and thus $A=VW^\dagger DV^\dagger W$. Thus, ...
Norbert Schuch's user avatar
2 votes

Weakly transversal gates for the $ [[5,1,3]] $ code

I suppose, for the 5-qubit code, you could brute-force it. Define any single-qubit unitary that you might want: $$ U=\begin{bmatrix} U_{00} & U_{01} \\ U_{10} & U_{11} \end{bmatrix}. $$ We ...
DaftWullie's user avatar
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2 votes

what is square root of a density matrix power two?

Take any spectral decomposition of a density operator $$\rho=\sum_n \rho_n |n\rangle\langle n|.$$ The square is defined unambiguously as $$\rho^2=\sum_n \rho_n^2 |n\rangle\langle n|.$$ By inspection, ...
Quantum Mechanic's user avatar
2 votes

Derivative of cost function with respect to the unitary matrix

If I take your questions literally, then the answer is fairly straightforward: $$ \frac{dU^\dagger}{dU}=-1. $$ You should think about it this way: we know $UU^\dagger=I$. If $U$ changes by $\delta U$ (...
DaftWullie's user avatar
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2 votes

References for homology, suitable as background for quantum codes

Here you can find a nice tutorial video by Dan Browne on the concepts of Homology and how we can use them to understand surface codes. And here is the slide deck.
Egretta.Thula's user avatar
2 votes

References for homology, suitable as background for quantum codes

There is a discussion of these topics in Fujii's book: https://arxiv.org/pdf/1504.01444.pdf, and in this paper of Hector Bombin: https://arxiv.org/pdf/1311.0277.pdf You might also find interesting ...
Yaron Jarach's user avatar
2 votes
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References for homology, suitable as background for quantum codes

Part 3 and 4 of Dan Brown's lecture notes gives an introduction to the basics of homology including worked out examples.
Peter-Jan's user avatar
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2 votes

Are quantum channels bounded linear maps?

To complement Danylo's great answer, let me go into a bit more details about the infinite-dimensional case and point out that the correct extension of a channel $\mathcal{N}:D(\mathcal{H}_A) \...
Frederik vom Ende's user avatar
2 votes
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How to calculate the Haar measure for the middle SU(2), in an SU(3) factorization?

Let us consider the $SU(3)$ decomposition as such: Assuming the matrix we create is Haar-random, it means that when it is applied to $|0\rangle$ it should yield a Haar-random state. So let's see how ...
Tristan Nemoz's user avatar
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2 votes
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closeness between two unitaries on the bloch sphere

A unitary $U$ corresponds to a rotation on the Bloch sphere. Technically, there is a smooth map from $2 \times 2$ unitaries with determinant 1 (which form the group SU(2)) to $3\times 3$ rotation ...
Danylo Y's user avatar
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2 votes

closeness between two unitaries on the bloch sphere

There are several ways to quantify the closeness of unitary operators. The best approach will be heavily dependent on context. The approach you laid out is close to an operator norm, which would look ...
Jonathan Trousdale's user avatar
2 votes
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unitary that transforms one Hilbert space to another Hilbert space

No. Suppose $A$ and $B'$ are qutrits, $A'$ and $B$ are qubits and $U$ is the SWAP unitary that sends $|ab\rangle$ to $|ba\rangle$ for every $a\in\{0,1,2\}$ and $b\in\{0,1\}$. Clearly, SWAP transforms $...
Adam Zalcman's user avatar
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2 votes
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What is the relationship between gate fidelity and norm?

The gate fidelity can be expressed with the Frobenius norm as follows: \begin{align*} \min_{\varphi\in\mathbb R}\frac1{2^n}\|U-e^{i\varphi}V\|_2^2&=\min_{\varphi\in\mathbb R} \frac1{2^n}{\rm tr}((...
Frederik vom Ende's user avatar

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