Questions tagged [mathematics]

Use this tag for questions about mathematics relevant to quantum computing and/or quantum information theory. DO NOT use this tag for general mathematics questions.

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2 votes
1 answer
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Does every code have transversal Pauli group?

A transversal logical gate for an $ n $ qubit code is a gate from the group of local unitaries $$ \bigotimes_{i=1}^n U(2) $$ which also preserves the codespace. For an $ ((n,K,d)) $ code we say a ...
0 votes
1 answer
26 views

Find min of a quantum state L2 norm

I have a problem minimizing this norm with respect to $\alpha$: $\min_{\alpha}||e^{i\alpha}|\psi\rangle-|\phi\rangle ||^2$ (1) The result is that this achieves min when $\alpha=-\measuredangle \langle\...
1 vote
0 answers
48 views

Same weight enumerator iff equivalent by permutations and local unitaries

A non-entangling gate on $ n $ qubits is an element of the group $$ N\Big(\bigotimes_{i=1}^n U(2)\Big)=\bigotimes_{i=1}^n U(2) \rtimes S_n $$ which is generated by $ U(2) $ acting locally on each ...
2 votes
2 answers
52 views

Weight enumerators for Hermitian operator

Let $ H $ be Hermitian operator on an $ n $ qubit Hilbert space $ \mathbb{C}^{2^n} $. Define the weight enumerator coefficients $$ A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2 $$ ...
2 votes
1 answer
33 views

Weight enumerators for Hermitian operator (wrong $ B_j $ definition)

Let $ H $ be Hermitian operator on an $ n $ qubit Hilbert space $ \mathbb{C}^{2^n} $. Define the weight enumerator coefficients $$ A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2 $$ ...
10 votes
3 answers
2k views

Why can all quantum circuits be converted into circuits that use only real matrices?

I know that you need to add an additional ancilla qubit to "keep track" of whether or not you are in real space or imaginary space, but how exactly does this work? What is the proof for this?...
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1 vote
0 answers
44 views

Eastin Knill Theorem and global phase

In quantum we don't care about global phases, but I want to ask a question about global phases anyway. The original Eastin-Knill Theorem paper https://arxiv.org/abs/0811.4262 says $$ CP = \Pi_{i=1}^k ...
0 votes
1 answer
27 views

Where am I going wrong in my understanding of qubit associativity?

I am studying the basics of quantum computing math and am confused about qubit associativity. As I understand it, in quantum math, multiple qubits are represented as the tensor product of the qubits ...
5 votes
1 answer
97 views

Building universal gate set for $SU(d^n)$ from universal gate set for $SU(d)$

Let $G$ be a universal gate set for $SU(d)$. Then the words $\langle G \rangle$ of $G$ form a dense subset of $SU(d)$ with respect to some reasonable norm, and so every element of $SU(d)$ can be ...
0 votes
1 answer
22 views

What is the difference between Gate.power() and Gate.repeat()?

Why are the gates a and b in this code not the same? a = UGate(0,0,0.9*np.pi).power(2) b = UGate(0,0,0.9*np.pi).repeat(2) I thought that unitary gates function ...
2 votes
1 answer
36 views

Bounding operator norm by total variation distance

Let $P_U(y \mid x) = |\langle y | U | x \rangle|^2$ denote the probability distribution of obtaining the bitstring $y \in \{0,1\}^n$ on a fixed input $x \in \{0,1\}^n$ w.r.t. the unitary $U$. For $n$-...
0 votes
2 answers
46 views

Modular Addition general explanation

This is an incredibly basic question, but basically I'm really struggling to understand what the "addition modulo 2" is and why is it used in quantum computing. I've tried Wikipedia, endless ...
2 votes
1 answer
33 views

time evolution of Hamiltonian to generate the Bell pair

Consider two different Hamiltonians: $H_1(t) = ZZ + \alpha(t)X_1 + \beta(t)X_2$ and $H_2(t) = XX + \alpha(t)Z_1 + \beta(t)Z_2$, where $\alpha(t)$ and $\beta(t)$ are time-dependent functions. Starting ...
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2 votes
1 answer
60 views

Codes with codewords that aren't uniform modulus superposition

All stabilizer codes and also all non stabilizer codes that I am aware of, for example the ones here, Example non-stabilizer code? have a basis of codewords which are all uniform modulus ...
4 votes
1 answer
93 views

How many $ \sqrt{X} $ are there?

I was reading Square root of Pauli operators: is there a common convention to define them uniquely? and it got me thinking about square roots. Recall the Pauli $ X $ gate $$ X=\begin{bmatrix} 0 & ...
2 votes
1 answer
68 views

Are all powers $g^m$ in the Clifford hierarchy if $g$ is?

It is already known that the Clifford hierarchy is not closed under arbitrary products, see this post which shows that the product $ THT $ is not in any level of the hierarchy. What about products of ...
4 votes
1 answer
449 views

Is every Clifford gate conjugate to a diagonal Clifford gate?

Let $ C $ be a Clifford gate. Let $ D $ be the diagonalization of $ C $. In other words $ D $ is a diagonal gate and $$ C=VDV^{-1} $$ for some $ V $. Is $ D $ also a Clifford gate? Update: Filling in ...
3 votes
1 answer
101 views

Can Clifford gates be diagonalized using a gate from the third level of the Clifford hierarchy?

Is it always possible to diagonalize a Clifford gate $ g $ using a gate $ V $ from the third level $\mathcal{C}^{(3)}$ of the Clifford hierarchy? In other words can every Clifford gate be written as $...
2 votes
1 answer
54 views

What are the elements of quotienting the Pauli group $\mathcal{P}_n := \widetilde{\mathcal{P}}_n / N$, and how to do calculations with it?

Let $\widetilde{\mathcal{P}}_n = \langle X_1,X_2,\dots,X_n,Z_1,\dots,Z_n\rangle$ together with all the phases $\{\pm 1, \pm i\}$ the regular Pauli group, and $N = \langle \pm i I\rangle $. I would ...
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4 votes
2 answers
89 views

Spectral theorem for Pauli matrices

Let $ P $ be a Pauli matrix. Pauli matrices are normal. So by the spectral theorem $ P $ can be written as $$ P=VDV^{-1} $$ for $ V $ unitary and $ D $ diagonal (in other words $ P $ is unitarily ...
0 votes
1 answer
64 views

How to prove that the trace of a density matrix is $1$?

Equation 2 gives the following proof: $$ \text{Tr}[\rho] = \sum_x \langle x\vert \rho\vert x\rangle = \sum_x \langle x\vert \sum_i p_i\vert \psi_i\rangle \langle \psi_i\vert\vert x\rangle = \sum_i ...
1 vote
1 answer
205 views

Matrix representation of any conditioned gate

Is there an algorithm explaining how to represent any gate in the matrix form? Suppose, the circuit is the following: where operator $ U = e^{iA\pi/4} = \begin{bmatrix} 0.35-0.85i & -0.35-0.15i ...
3 votes
1 answer
50 views

Clarification defining/finding the relative phase of a qubit

Let the vector $ |V\rangle = r_0 e^{i\theta_0} |0\rangle + r_1 e^{i\theta_1} |1\rangle $ correspond to the state of a qubit where $r_0,r_1,\theta_0,\theta_1 \in \mathbb{R}$. According to p. 22 of ...
1 vote
0 answers
45 views

Close in operator norm imply close in weak multiplicative sense?

Fix $\epsilon > 0$, and suppose $U$ and $S$ are $n$ qubit unitaries such that $\| U - S \| \leq \epsilon$ (operator norm). Furthermore, let $P_U(y \mid x) = |\langle y | U | x \rangle|^2$ be the ...
2 votes
1 answer
84 views

Notation for Lindblad operators

I was reading the paper Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation . The claim is that universal quantum computation can be achieved using the ...
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2 votes
1 answer
194 views

How to calculate the log of a density matrix?

In quantum information theory, calculating the log of a density operator is essential for things like the Von Neumann entropy or the entropy of entanglement. Unfortunately, this topic is considered a ...
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2 votes
0 answers
52 views

How many gates are in the $ k $ level of single qubit Clifford hierarchy?

Define the single qubit Clifford hierarchy recursively by $$ \mathcal{C}^1:=<iX,iZ> $$ the determinaint 1 subgroup of the Pauli group. Define the rest of the the hierarchy inductively by $$ \...
3 votes
1 answer
50 views

Does the real Clifford group contain all real diagonal gates? all permutation gates?

The real Pauli group is the subgroup of $ O_{2^n}(\mathbb{R}) $ generated by products and tensor products of $ X $ and $ Z $ (this deviates from the usual Pauli group in that only real Paulis are ...
6 votes
1 answer
126 views

Is the Clifford hierarchy finite?

This question is inspired by Is the Clifford group finite? Which shows that that the Clifford group (the second level of the Clifford hierarchy) is finite. (finite meaning finite mod global phases) ...
3 votes
3 answers
324 views

Realizing a swap gate using a commutator sequence and an auxiliary qudit

Say I have two qudits $1$ and $2$, each of which has Hilbert space of dimension $m$. Is it possible to introduce an auxiliary qudit $a$ (of any dimension $d_a\in \mathbb{Z}_{\geq 2}$) and find quantum ...
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0 votes
1 answer
99 views

Is Shor demonstration wrong?

in Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer by Peter W. Shor (also in Algorithms for quantum computation: discrete logarithms and factoring). In ...
0 votes
0 answers
13 views

How do we know what angle formula X1 is encoded into qml.MottonenStatePreparation?

Known that X1 is a quantum state, it is prepared by qml.MottonenStatePreparation. How do we know what angle formula X1 is encoded into qml.MottonenStatePreparation? can be interpreted in python code. ...
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3 votes
0 answers
35 views

Expectation value over random $k$-local Pauli operators for two random quantum states

Suppose we have a uniform distribution $D$ over $k$-local Pauli operators $P_{q_1}\otimes \dotsc \otimes P_{q_k} $, $P_{q_i} \in \{ X, Y, Z, I \}$. Is it possible to calculate $\mathbb{E}_{P_i \sim D} ...
2 votes
1 answer
154 views

The Clifford hierarchy and $ e^{2 \pi i/2^k} $

Could someone give me an example of a gate in the Clifford hierarchy which cannot be written as $$ e^{i \theta} V $$ for some unitary $ V $ with entries in terms of $ \zeta_{2^k} $? If no such example ...
6 votes
1 answer
131 views

Is this single qubit gate in the Clifford hierarchy?

For a single qubit, the Clifford hierarchy is defined to be $$ \mathcal C^{(k)} = \Bigl\{ U \in \mathbf U(2) \mathrel{\Big\vert} \forall P \in \mathcal C^{(1)} : U P U^\dagger \in \mathcal C^{(...
2 votes
0 answers
32 views

Dimension of local operators stabilizing the code space?

What is the maximum dimension of a connected group of local operators stabilizing an $ [[n,k,d]] $ code with $ d \geq 2 $? Some background: Consider an $ [[n,k,d]] $ quantum error correcting code with ...
3 votes
1 answer
117 views

Which monomial matrices are in the Clifford hierarchy?

This is essentially a follow-up on the very interesting answer given here Is there a closure property for the entire Clifford hierarchy? I'm interested in sufficient conditions to conclude that a ...
1 vote
2 answers
82 views

How to perform a basis change on a 2x2 density operator?

I have an ensemble described by following density operator: $$ P=3/8 |+\rangle\langle+| + 5/8 |-\rangle\langle-| $$ I am trying to write this operator in $\{|0\rangle, |1\rangle\}$ basis. I know that ...
4 votes
2 answers
275 views

Exotic transversal gate group

What are examples of interesting $ [[n,1,d]] $ or $ [[n,2,d]] $ stabilizer codes, $ d \geq 2 $, whose group of transversal gates is not isomorphic to a subgroup of the Clifford group (on 1 and 2 ...
3 votes
1 answer
261 views

Show that, averaging over uniformly random unitaries, $\mathbb{E}[UXU^\dagger]=\operatorname{Tr}(X)\frac{I}{d}$

As mentioned e.g. in this answer, if we compute the average $$\Phi(X)\equiv \mathbb{E}_U[UXU^\dagger] \equiv \int_{\mathbf U(d)} d\mu(U) UXU^\dagger,$$ where $d\mu(U)$ is the Haar measure over the ...
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5 votes
1 answer
142 views

What are well-known orthogonal 2-designs, other than the real Clifford group?

The paper Real Randomized Benchmarking https://quantum-journal.org/papers/q-2018-08-22-85/ https://arxiv.org/abs/1801.06121 makes use of the fact that the real Clifford group is an orthogonal 2-design ...
0 votes
0 answers
62 views

How to represent the following regular quantum circuits with tensor and concatenation symbols?

I am fascinated by such a quantum structure as above, how should the regular distribution for Toffoli and SWAP gates be described by the formula? Is the following correct? $\prod\limits_{n}{{{I}_{{{2}^...
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0 votes
0 answers
45 views

Understanding Shor algorithm fo Elliptic Curves Demonstration

I was reading Shor's discrete logarithm quantum algorithm for elliptic curves. And i have two questions. In page 7 they say that $x = (x0 - dy) mod q$, where $x0$ is between 0 and q-1, but then they ...
2 votes
1 answer
82 views

Is there an expression for the partial trace of a vectorized density matrix?

Is there an expression for the partial trace of vectorized density matrix? I did some literature review but didn't find not much relevant information.
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1 vote
0 answers
50 views

Simple Maths Operators implementation in Quantum

I am a newbie in quantum programming and trying to learn implementing some simple classical programs in quantum, just as a starting point. The thing I am kind of struggling in is availability of ...
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0 votes
0 answers
49 views

Equivalence check between rotational gates and Pauli gates

My question is highly related to this one. I am trying to understand the relationship between rotational gates $R_P(\theta)$, where $P \in \{X,Y,Z\}$. As stated here, $\exp(iPx)=\cos(x)I+i\sin(x)P$. ...
9 votes
2 answers
334 views

Non-entangling two-qubit gates

The non-entangling gates in $ SU_4 $ contains the entire group of gates of the form $$ SU_2 \otimes SU_2. $$ It also contains $$ \zeta_8 SWAP= \zeta_8 \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 &...
2 votes
1 answer
151 views

Heisenberg Uncertainty Principle (Nielsen and Chuang Box 2.4)

I'm trying to follow Nielsen and Chuang Book on Quantum Computation and Quantum Information. There is Box 2.4 on the Heisenberg Uncertainty Principle. I got stuck pretty fast. In that box they define: ...
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2 votes
0 answers
143 views

How to implement an unitary operator expressed as a linear combination of unitaries without qubits ancilla

Let's say that I know the decomposition of a unitary operator $\hat{A}$ in terms of other unitary operators $U_{k=0, \dots, M}$, i.e: $$ \hat{A} = \sum_k \alpha_k U_k$$ I know how to implement in ...
2 votes
1 answer
89 views

Writing a Density matrix in terms of the magnitude of the Bloch Vector

Working with the density matrix and the Bloch sphere, I have been attempting to complete an exercise in Entangled Systems; New Directions in Quantum Physics. If anyone has the book it is Question 4.3 ...
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