# 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.

497 questions
Filter by
Sorted by
Tagged with
19 views

76 views

### What are some good resources for learning quantum math?

I'm new to Quantum dynamics as a whole and everytime i read an article on arxiv.org or watch a video on youtube and they introduce an equation like Shrodinger or other equations to show the logic and ...
1 vote
38 views

### What does Pauli's $Y$ matrix represent?

It is easy to see that Pauli's $X$ matrix represents the bit flip operation, i.e. $X \lvert 0 \rangle = \lvert 1 \rangle$ and $X \lvert 1 \rangle = \lvert 0 \rangle$. Similarly, Pauli's $Z$ matrix ...
1 vote
31 views

### Improving operator norm bound on total variation distance

Let $U$ be an $n$-qubit unitary and $P_U(x) = |\langle x |U|0^n\rangle|$ the probability of measuring $x$ after acting $U$ on $|0^n\rangle$. For two $n$-qubit unitaries $U$ and $V$, one can prove that ...
27 views

### Tighter upper bound of $\operatorname{tr}[ (\mathcal{H}[L]\rho )^2] \leq \delta$

I am wondering about an upper bound of the trace function $\operatorname{tr}[ (\mathcal{H}[L]\rho )^2] \leq \delta$ (we assume that $\rho$ is the $N\times N$ density matrix representing the quantum ...
3 views

### Is the inequality $E[( tan(x))^{-1} ] \leq (tan(E[x]))^{-1}$ true? [migrated]

Given a random variable $x$ and expectation E, does the inequality $E\left[\frac{1}{tan(x)} \right] \leq \frac{1}{tan(E[x])}$ hold?
1 vote
60 views

### Is every diagonal gate whose non-zero entries are $2^k$th roots of unity in the two qubit Clifford hierarchy?

Does the two qubit Clifford hierarchy contain all diagonal gates whose entries are $2^k$ roots of unity? In particular, is it true that every $4 \times 4$ diagonal matrix whose diagonal entries ...
350 views

31 views

### What will be the new state after the outcome of the measurement of the state of the first qubit as ǀ0>? [closed]

Two qubits are prepared in a superposition state of the form: What will be the new state after the outcome of the measurement of the state of the first qubit as ǀ0>?
56 views

### How to prove that CNOT and Rz gates are permutable?

How to prove that CNOT and Rz gates are permutable? I tried to equate their switch to zero and calculate it, but for this you need to multiply the matrices. But the 4x4 and 2x2 matrices cannot be ...
30 views

### What are the expected measurement results in the diagram below? [closed]

I ask you to give a mathematical solution to this problem
113 views

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

For the $[[5,1,3]]$ code https://en.wikipedia.org/wiki/Five-qubit_error_correcting_code $X^{\otimes 5}$ implements logical $X$ and $Z^{\otimes 5}$ implements logical $Z$. Also logical $HP$ ...
79 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 ...
1 vote
72 views

72 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 $$A_j=\frac{1}{(Tr(H))^2} \sum_{E \in \mathcal{E}_j} |tr(EH)|^2$$ ...
120 views

### Link between quantum computing and Lie theory?

I know only little thing about Lie theory but I would like to learn more about its link to quantum computing. Has someone got some references explaining it well ?
195 views

### 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 ...
269 views

### Proving the inequality $|\mathrm{tr}(AU)|\le \mathrm{tr}|A|$ in Uhlmann's theorem

In Nielsen and Chuang, in the Fidelity section, (Lemma 9.5, page 410 in the 2002 edition), they prove the following. $$\mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)|$$ ...
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 $\alpha=-\measuredangle \langle\... 2 votes 0 answers 34 views ### Dimension of local operators stabilizing the code space? What is the maximum dimension of a connected group of local operators stabilizing an$ [[n,k,d]] $code with$ d \geq 2 $? Some background: Consider an$ [[n,k,d]] $quantum error correcting code with ... 6 votes 1 answer 1k views ### Understanding a quantum algorithm to estimate inner products While reading the paper "Compiling basic linear algebra subroutines for quantum computers", here, in the Appendix, the author/s have included a section on quantum inner product estimation. Consider ... 2 votes 1 answer 34 views ### Weight enumerators for Hermitian operator (wrong$ 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?... 5 votes 1 answer 129 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 ... 2 votes 1 answer 64 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 ... 1 vote 0 answers 47 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 32 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 ... 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 47 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-... 5 votes 2 answers 376 views ### Map a 4-body Ising Hamiltonian to a 2-body Ising Hamiltonian I wonder if there exists a way to map the square of a 2-body Ising Hamtiltonian (which will make it 4-body) back to a 2-body Hamiltonian that has the same ground state? Let me explain what I mean by ... 0 votes 2 answers 78 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 40 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 ... 3 votes 1 answer 105 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 87 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 ... 2 votes 1 answer 65 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 ... 4 votes 1 answer 466 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} $$for some V . Is D also a Clifford gate? Update: Filling in ... 4 votes 2 answers 119 views ### Spectral theorem for Pauli matrices Let P be a Pauli matrix. Pauli matrices are normal. So by the spectral theorem P can be written as$$ P=VDV^{-1} $$for V unitary and D diagonal (in other words P is unitarily ... 0 votes 1 answer 94 views ### How to prove that the trace of a density matrix is 1? Equation 2 gives the following proof:$$ \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 213 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 ...
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 ...