Tag: textbook-and-exercises

14 What does "measurement in a certain basis" mean? 2018-04-22T15:43:42.417

10 Density matrix after measurement on density matrix 2018-11-30T17:55:57.293

8 How do I prove that the Hadamard satisfies $H\equiv e^{i\pi H/2}$? 2019-10-22T21:42:34.990

8 Can arbitrary matrices be decomposed using the Pauli basis? 2019-11-09T12:43:20.377

8 What does it mean to "measure an operator"? 2020-12-09T01:20:52.837

7 How do you represent the output of a quantum gate in terms of its basis vectors? 2018-07-29T00:53:45.717

7 How to compute the measurement probability in swap test? 2020-10-19T03:54:19.127

7 What's the 'physical consistency' in the partial trace scenario? 2021-01-23T23:45:06.900

6 How to translate matrix back into Dirac notation? 2019-01-22T22:53:54.743

6 How to calculate an Expected Value of some operator acting on qubits? 2019-08-04T12:05:06.087

5 Confusion on the definition of the phase-damping channel 2018-11-26T11:40:24.430

5 How does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$? 2018-12-17T06:28:36.610

5 How to show a density matrix is in a pure/mixed state? 2019-04-22T05:29:18.183

5 Find the quantum operation corresponding to a given unitary evolution and projective measurement 2019-07-13T22:19:53.150

5 Procedures and intuition for designing simple quantum circuits? 2020-06-12T15:48:38.343

5 Nielsen&Chuang 5-qubit quantum error-correction encoding gate 2020-10-20T08:40:31.943

5 How are the eigenvalues of $\rho=\frac12(|a\rangle\!\langle a| +|b\rangle\!\langle b|)$ derived? 2020-11-28T03:04:56.547

5 In Nielsen and Chuang, how can $\frac{1}{2(e-1)}$ result from $\frac12\int_{e-1}^{2^{t-1}-1}dl\frac{1}{l^2}$? 2020-11-29T13:13:01.290

5 What does $M_m |\psi_i\rangle$ mean in the equation $p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle$? 2021-01-17T23:14:06.043

5 Derivation of the identity $\sum_j p_j \langle \psi_j|M|\psi_j \rangle = \sum_j p_j \operatorname{tr}\left(|\psi_j \rangle \langle \psi_j|M\right)$ 2021-01-28T15:51:52.357

5 Why is the transpose of a density matrix positive and trace preserving? 2021-02-08T19:50:58.767

4 Can vectorization lead to mixed states? 2018-12-02T20:26:33.073

4 How to find the operator sum representation of the depolarizing channel? 2018-12-16T20:35:45.460

4 How are arbitrary $2\times 2$ matrices decomposed in the Pauli basis? 2019-07-28T15:12:03.840

4 What is the unitary operator realizing a given CPTP operator 2019-07-31T22:59:33.860

4 How is the partial trace related to the operator sum representation? 2019-11-25T13:25:14.487

4 Prove that there exist tripartite $|\psi\rangle$ which cannot be written as $|\psi\rangle=\sum_i\lambda_i|i_A\rangle|i_B\rangle|i_C\rangle$ 2020-02-25T11:41:34.587

4 Unable to use qiskit_textbook module 2020-06-09T06:51:03.893

4 How to apply the Schmidt Decomposition to a Bell state? 2020-06-16T13:05:20.537

4 Is the tensor product of two states commutative? 2020-06-23T09:59:51.103

4 What to do when the amount of solutions is not known before applying Grovers Algorithm? 2020-08-14T12:40:50.800

4 Why does the Hadamard gate satisfy $H|x\rangle=\frac{1}{\sqrt2}\sum_{z\in\{0,1\}}(-1)^{xz}\lvert z\rangle$? 2020-08-14T17:25:06.770

4 Why is the action of controlled-Z unaltered by exchanging target control qubits? 2020-08-15T13:56:39.123

4 What is the thought process for circuit making after seeing input and output of a matrix? 2020-09-16T01:01:38.243

4 Difference between change of basis in bra-ket notation and matrix notation 2020-11-04T13:01:00.307

4 What is the probability of finding the second qubit as $0$ in the state $|\psi\rangle=\frac1{\sqrt2}|00\rangle+\frac12|10\rangle-\frac12|11\rangle $? 2020-11-21T17:05:44.387

4 Complexity of controlled operations in a two-level unitary operation 2020-12-10T22:27:08.673

4 How do I prove that $P_\pm=\frac12(1\pm U)$ if $U^2=I$? 2020-12-16T23:47:48.107

4 A question in classical and quantum information 2020-12-19T14:40:45.083

4 Show that the Hadamard gate is equivalent to a 180 degree rotation of a certain axis 2020-12-29T06:25:48.407

4 How is a qubit in superposition between $|0\rangle$ and $|1\rangle$ different from a mixture of $|0\rangle$ and $|1\rangle$? 2021-01-02T00:56:22.677

4 Writing state $ |\Psi⟩ =\frac{1}{\sqrt{2}}|00⟩+\frac{i}{\sqrt{2}}|01⟩$ as separate qubits (qiskit textbook) 2021-01-11T07:36:33.563

4 If the eigenvalues of $Z$ are $\pm1$, why are the computational basis states labeled with "$0$" and "$1$"? 2021-01-31T21:04:31.193

4 What is the difference between $|0\rangle+|1\rangle$ and a balanced mixture of $|0\rangle$ and $|1\rangle$? 2021-02-07T23:02:03.147

4 Does $\frac{\Pi_A\otimes I_B}{\text{Tr}((\Pi_A\otimes I_B)\rho_{AB})}\rho_{AB}=\rho_{AB}$ hold for a state $\rho_{AB}$ and projector $\Pi_A$? 2021-02-23T07:16:32.580

3 How do you implement a negative controlled gate using the regular controlled gate? 2019-06-21T10:50:02.903

3 Estimate of the absolute value of the probability amplitude of |0⟩ in the superposition 2019-08-23T00:54:01.040

3 More than one Schmidt coefficient implies entanglement? 2020-03-29T14:36:01.607

3 Nielsen and Chuang ex 2.73 2020-04-17T18:42:32.873

3 How to compute the tensor product of the depolarizing channel with the identity? 2020-05-19T10:20:11.083

3 Bipartite states whose coefficients are entries of a unitary matrix 2020-06-03T21:16:51.567

3 How to decompose a unitary single qubit gate by universal quantum gate set? 2020-06-04T02:26:43.703

3 Show that a $CZ$ gate can be implemented using a $CNOT$ gate and Hadamard gates 2020-06-13T11:54:43.447

3 How to spot the matrix representation of the quantum NOT operation 2020-06-13T13:53:24.087

3 Can $U(\rho_1\otimes\rho_2)U^\dagger$ be entangled if either of $\rho_1$ or $\rho_2$ is a maximally mixed state? 2020-06-14T02:50:09.460

3 Problem about entanglement swapping 2020-06-29T07:53:39.477

3 How do you embed a POVM matrix in a Unitary? 2020-07-05T08:09:11.263

3 Representing a von Neumann measurement as $[\mathcal{I} \otimes P_i] U(\rho_s \otimes \rho_a)U^{-1} [\mathcal{I} \otimes P_i]$, how do we choose $U$? 2020-07-15T14:43:17.730

3 How is it possible to guess what state the qubit was in by measuring it? 2020-07-26T09:52:19.660

3 Deriving $\left( A | v \rangle \right)^\dagger = \langle v | A^\dagger$ without using $A^\dagger=\left(A^* \right)^T$ 2020-08-23T00:15:45.970

3 Prove entanglement in the final state of the Deutsch-Jozsa circuit 2020-08-24T11:02:58.063

3 Eigenvectors and eigenvalues of the gate $U_a:|s\rangle\mapsto|sa \bmod N\rangle$ 2020-08-27T17:49:33.520

3 How to find the unitary operation of a depolarizing channel? 2020-09-01T16:18:02.667

3 Nielsen & Chuang Exercise 2.55: Prove that $\exp \left[ -\frac{iH(t_2 - t_1)}{\hbar} \right]$ is unitary 2020-09-03T18:14:06.033

3 Having trouble finding angles for Bloch vector 2020-09-10T18:58:24.800

3 Show that for any measurement operator $M_m$ there exists unitary $U_m$ such that $M_m=U_m\sqrt{E_m}$ with $E_m$ POVM 2020-09-25T02:25:30.420

3 IF statement in OpenQASM on IBM Quantum Experience 2020-09-25T06:59:29.980

3 How to translate the Hadamard gate matrix into Dirac notation? 2020-10-04T19:35:47.530

3 How are the Pauli $X$ and $Z$ matrices expressed in bra-ket notation? 2020-10-08T17:18:43.663

3 What is a basis (not necessarily orthogonal) of Herm(A) consisting of pure density matrices in D(A)?($A \cong \mathbb{C}^{n}$) 2020-10-10T19:32:23.870

3 Prove that different purifications of a state can be mapped into one another via local unitaries 2020-10-12T00:03:28.907

3 Optimizations in quantum circuits 2020-10-23T04:20:46.527

3 Nielsen & Chuang Exercise 6.13: Standard deviation of classical counting algorithm 2020-11-21T19:54:40.933

3 What does $|j\rangle$ mean, when $j > 1$? 2020-12-06T13:10:31.213

3 Mistake in using dirac notation when applying $X$ gate to vector 2020-12-21T18:11:00.050

3 Pure state ensembles achieving the Holevo $\chi$-quantity with at most $d^2$ states 2020-12-26T23:05:25.677

3 Show that the two circuits are equivalent mathematically 2020-12-30T01:32:29.290

3 How to show that Bell states are orthonormal 2020-12-30T04:08:13.890

3 Why do the eigenvectors of a spin observable not align with the direction of the spin? 2020-12-31T21:21:12.153

3 How to show that a given mixed two-qubit state is separable? 2021-01-04T10:06:58.343

3 Application of QFT to Order-finding 2021-01-12T12:51:25.223

3 What's the difference between $p(i|m)$ and $p(m|i)$ in measurement? 2021-01-18T21:32:01.877

3 How to construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift? 2021-01-28T23:27:21.927

3 Is a projective measurements over a superposition of eigenstates possible? 2021-01-31T22:14:49.520

3 How does one create the unitary sending $|0\rangle$ into a target quantum state? 2021-02-03T07:09:15.167

3 How do we find the stabilizer generators for the three-qubit bit-flip code spanned by $|000\rangle$ and $|111\rangle$? 2021-02-03T15:12:48.910

3 Shouldn't the input state of Deutsh-Jozsa's algorithm look like $|0\rangle^{\otimes n}\otimes |1\rangle$ rather than $|0\rangle^{\otimes n}|1\rangle$? 2021-02-03T18:16:36.203

3 How do I determine if a given pure two-qubit state is separable? 2021-02-05T19:21:10.553

3 Probability of measuring one qubit from the state of two qubits 2021-02-07T12:56:54.337

2 How to find the reduced density matrix of a four-qubit system? 2019-08-26T21:45:20.287

2 Trace of Hermitian Operator and Operator Function 2019-10-02T00:04:54.083

2 Can Eve perform this operation? 2019-11-25T11:56:03.647

2 Conditional version of the triangle inequality for Von Neumann entropy 2020-02-05T05:04:52.377

2 Show that $I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$ for all states $\rho$ 2020-05-28T00:03:26.877

2 Generalized set of Pauli elements for a basis for the linear transformations on the vector space 2020-05-29T21:00:15.010

2 What are the possible qubit states? 2020-06-02T23:49:00.977

2 What happens when you send a Bell state through depolarizing channel? 2020-06-04T17:40:03.033

2 How do you decompose an arbitrary quantum state into its corresponding projection subspaces such that their direct sum is the quantum state? 2020-06-12T15:15:50.223

2 Find the probability of a measurement outcome in terms of the coefficients of the state 2020-06-13T09:16:06.847