12
votes
Accepted
How to perform quantum state tomography on two qubits?
Preliminary
I would like to rewrite the equation that you have in a slightly different manner. Since a density matrix can be written as a matrix, we can also write it down as a linear combination of ...
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11
votes
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What is the relationship between Choi and Chi matrix in Qiskit?
( I copied some text from a previous answer of mine)
Defining the Choi and $\chi$ matrix
The Choi matrix is a direct result of the Choi-Jamiolkowski isomorphism. Some intuition on what this is can be ...
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8
votes
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How to perform Quantum Process Tomography for three qubit gates?
I am sure that since you are asking this question you probably already understand this, but for future & other's reference let me give a quick recap of what we are trying to achieve.
Quantum ...
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8
votes
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Why is the complexity of $n$-qubit state tomography not upper bounded as $O(3^n)$?
Let's say you have a magical machine that gives you $\langle P_{p} \rangle$ (which are expectation values and therefor, well, numbers) and only the $\langle P_{p} \rangle$. It does this for all the $3$...
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7
votes
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Is there a circuit to compare two quantum states?
The task that you describe in your question — a circuit which flips a single qubit, if and only if the two input states are different — is not possible. We can show this as follows.
First, there is no ...
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7
votes
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In shadow tomography, how is the state reconstructed from its shadows?
Under the assumption that the ensemble $\mathcal{U}$ faithfully produces the Haar expectations at least to the second moment, the inversion can be performed as suggested in the last paragraph of the ...
- 2,485
7
votes
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Using Classical Shadow to predict quantum state's fidelity has nothing to do with the dimension of the density matrix?
Without really checking your arguments, there is a fundamental reason why the scaling could be fine, but it is still not strange at all.
The point is that you estimate with additive precision, but ...
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6
votes
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Does computing the quantum mutual information $I(\rho^{AB})$ require full tomographic information of $\rho^{AB}$?
The mutual information can be written in terms of the relative entropy, please see
Nielsen and Chuang (the entropy Venn diagram figure 11.2). I am writing the equation in the question's notation:
$$I(...
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6
votes
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Can quantum state tomography break bb84?
No, weak measurement and quantum tomography don't break BB84.
I recommend that you create an explicit quantum circuit that implements the weak measurement or the quantum tomography, and check for ...
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6
votes
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Why can any density operator be written this way? (quantum tomography)
From linear algebra, if $v_1, \dots, v_n$ is a basis of the vector space $V$ then every vector $v\in V$ can be written as a linear combination
$$
v = a_1 v_1 + \dots + a_n v_n\tag1
$$
where the ...
- 20k
6
votes
Accepted
What does the POVM corresponding to single-qubit state tomography look like?
Quantum state tomography owes its power and flexibility to the fact that it supports a wide class of measurements. Any informationally complete POVM, i.e. one whose elements span the space $L_H(\...
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6
votes
What are the problems of linear inversion quantum state tomography?
The lack of positive semidefiniteness is very easy to see. Suppose your quantum state is $|0\rangle\langle 0|$, and you do tomography by measuring in the $X, Y,$ and $Z$ bases. Furthermore, assume ...
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5
votes
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Unknown quantum circuit symbol
The picture has two parts: The first goes until the dots. It is simply three $|0\rangle$ states. (The ground state.)
You will recognize that the same picture -- but only until the dot -- is used in ...
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5
votes
How to measure the sign of quantum amplitudes
An empirical solution could be to use the Grover's Diffusion Operator $D$.
Lets say the qubits are in an initial state $|\psi\rangle = \sum_{0}^{2^n-1}\alpha_i|i\rangle$. Since global phase/sign is ...
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5
votes
How to distinguish two states with same density matrix using a quantum state tomography?
The question presupposes a misconception that the vector form of a state $|\psi\rangle$ exists independently of its density operator form $|\psi\rangle\langle\psi|$, which is often described as ...
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5
votes
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Lower bounds on the number of measurements outcomes required for quantum state tomography
I apologise in advance. This is a rough and hand-waivy answer.
You can give "information-theoretic" lower bounds by noting that the measurements can be seen as a linear map $M$ from quantum ...
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5
votes
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Implement the classical shadow coding error?
There's a few bugs in your code as well as a slight misunderstanding about the guarantees of the protocol.
First to clarify some details: The protocol you implement samples $U \in \text{Cl}(2)^{\...
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5
votes
Are SIC-POVMs optimal for quantum state reconstruction?
First of all, here's a short disclaimer: I'm not an in-depth expert in this field, I'm just currently getting in contact with tomography more and more often :) So take the following with a grain of ...
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5
votes
How to sample from a unitary 2-design?
Unitary 2-designs without efficient sampling access are arguably not very useful. Indeed, if you go by the textbook definition, then a unitary design is nothing but a set of unitaries and there's ...
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4
votes
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Why does full state reconstruction require at least $N+1$ MUBs?
Denote the projections onto basis elements by $P_j^{(k)}=|u_j^{(k)}\rangle\langle u_j^{(k)}|$, where superscript indexes different bases.
Tomography of a density matrix $\rho$ gives us probabilities $\...
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votes
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How to do state tomography when using sampling in Qiskit?
I would suggest you use the code from the tutorial about quantum state tomography, adapting it to a real device of your choice. You can find the updated tutorial here
Caveat: as state tomography ...
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4
votes
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Calculating bipartite state from joint probability distribution
This would not be enough information to reconstruct the bi-partite state.
Single-qubit case
For the one-qubit case, reconstruction of the state (which we describe as $\rho$) works, because the single-...
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4
votes
Implementation of tomography on IBM Q
Density matrix of single qubit state can be estimated based on this formula
\begin{equation}
\rho = \frac{\text{tr}(\rho)I+\text{tr}(X\rho)X+\text{tr}(Y\rho)Y+\text{tr}(Z\rho)Z}{2},
\end{equation}
...
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4
votes
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How to calculate the fidelity of a certain gate of a IBMQ device in Qiskit using randomized benchmarking/tomography?
Fidelity is a single-number measure of how good a gate is. Since there are many ways that a gate can go wrong, there are multiple ways that the fidelity can be defined. The exact answer to your ...
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4
votes
Quantum algorithm for linear system of equations (HHL) - Final Step: How can I find my vector of solution $|x\rangle$?
Quick answer: You will not be able to fully recover $x$.
Explanations:
By design, the HHL algorithm stores $x$ in the amplitudes of a quantum state. Because of how quantum mechanics works, the ...
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4
votes
Give an explicit example of a $d = 4$ SIC-POVM
As indicated by Danylo in his anwser, eq. (32) in arXiv: 1103.2030
presents the sixteen vectors ("ignoring overall phases and normalisation")
\begin{equation}
\left(
\begin{array}{cccc}
x & 1 &...
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4
votes
Accepted
Is it possible to get the $\alpha$ and $\beta$ (amplitudes) of a qubit in Qiskit?
This can be done using the statevector_simulator provided with Qiskit Aer. It will return the statevector that describes the quantum state at the end of your ...
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votes
Can quantum state tomography break bb84?
Tomography generally speaking uses a collection of measurements to reproduce an underlying state. So you experimentally reproduce the same situation over and over, collect statistics and find the most ...
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4
votes
Is there a circuit to compare two quantum states?
The technical term is "quantum state discrimination". One has to carefully formulate the problem, because it is generally hard to identify an arbitrary state (tomography) as you noticed.
...
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4
votes
What does the POVM corresponding to single-qubit state tomography look like?
POVM for standard QST in the Pauli bases
In standard single-qubit QST one measures in the Pauli bases, each with equal probability $\frac{1}{3}$. As @Rammus has pointed out, this corresponds to the ...
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