Questions tagged [unitarity]

For questions related to the unitarity (unitary evolution) of quantum systems, as applicable to quantum computing or quantum information.

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Is the column vector of a uniformly sampled random unitary matrix a uniformly sampled random state vector?

I am wondering if a random unitary matrix taken from a Haar measure (as in, it is uniformly sampled at random) can yield a uniformly sampled random state vector. In section 3 of this paper it says &...
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40 views

How to find the output state after evolution through a unitary?

I was reading about quantum postulates, and I have a few questions about the second postulate that describes the evolution of a quantum system. For a system S, we describe the evolution after applying ...
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67 views

Unitary over bipartite states that can turn a non-product state into a product state

Consider a bipartite quantum state $\rho_{AB}$ over a product of finite-dimensional Hilbert spaces $\mathcal{H}_A \otimes \mathcal{H}_B$. Does there exists a unitary $U$ over $\mathcal{H}_A \otimes \...
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A question about Grover's paper “Fixed-Point Quantum Search”

I am reading Grover's paper "Fixed-Point Quantum Search," (arXiv version with a different name) which improves on his earlier quantum research algorithm. However, I'm having difficulty in ...
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Creating a parameterized Operator in Qiskit

I'm trying to run a VQE for a specific custom Anzats. The Anzats is built up of an unitary matrix $U_H$, which I'm trying to created in this way: ...
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What is $\sum_{i}\langle i \vert U \vert j\rangle$ for unitary $U$?

The question is basically the title but given a unitary operator $U$ and a computational basis, can we say anything about the complex number below? $$c = \sum_{i}\langle i \vert U \vert j\rangle$$ I ...
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1answer
37 views

How to apply a operator to qubit system on the basis of current state of system?

Suppose I have three different operators $U_1, U_2,U_3$. Now, these three operators will be applied if my current state of the system is $|\psi_0\rangle,|\psi_1\rangle $ and $|\psi_2\rangle$ ...
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73 views

How to perform the unitary transformation $U|i,j_1\rangle = 1/\sqrt{k}(|i,j_1\rangle+|i,j_2\rangle+|i,j_3\rangle…+|i,j_k\rangle)$?

Is the following unitary transformation possible? If so, what will be the value of $U$? $$U|i,j_1\rangle = 1/\sqrt{k}(|i,j_1\rangle+|i,j_2\rangle+|i,j_3\rangle...+|i,j_k\rangle)$$ Here, $i$ is a node ...
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46 views

What is the effect of the reset gate on the matrix form of a gate/circuit?

From what I understand, any circuit can be combined to make a gate, which has a square, unitary matrix form that acts on the $2^n$ row of the qubits state column vector. For example, the circuit ...
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103 views

Can the (universal) state inversion operator be physically realized?

I was trying to solve an exercise from Vazirani's course "Qubits, Quantum Mechanics and Computers": A mathematically nice, but unphysical, way to detect entanglement is to use the state ...
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1answer
36 views

What are the physical meanings of the outer product when writing expressions for unitary gates?

I'm really confused with the interpretation of those equations: $1.$ The evolution of states under unitary operations can be expressed as $$ U = \sum_k\exp(i\phi_k)|\psi_k\rangle\langle\psi_k| $$ $2.$ ...
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53 views

How to force a matrix to be unitary given constraints on some of the elements? [duplicate]

I am working with a matrix of the following form: $$ A =\begin{pmatrix} a_{11} & Q & \ldots & Q\\ a_{21} & Q & \ldots & Q\\ \vdots & \vdots & \ddots & \vdots\\ ...
2
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1answer
59 views

Hadamard gate for three qubits; inconsistency between IBM and Matlab

I am trying to build a large and quite complex three qubit quantum circuit on IBMs quantum computer. I have a specific unitary which I am trying to implement and I am building a circuit following the ...
2
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1answer
57 views

How can I convert the unitary matrix $e^\frac{i\pi}{2}$ into a quantum circuit in Qiskit?

How can I put the unitary matrix $$e^\frac{i\pi}{2}I$$ to the quantum circuit? I don't know if it is belong to $$U3(\theta, \phi, \lambda), U2(\phi, \lambda), U1(\lambda)$$ Thanks a lot.
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Is the Haar measure invariant under conjugation?

Denote the Haar measure on the unitary group $U(\mathcal X)$ by $\eta$. Does this equation hold (assuming the integral exists): $\int d\eta(U) f(U) = \int d\eta(U) f(U^\dagger)$? Intuitively this ...
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1answer
103 views

How to find the unitary operation of a depolarizing channel?

Suppose we have a depolarizing channel operation $$E(\rho)=\frac{p}{2}\textbf{1}+(1-p)\rho$$ acting on a Spin$\frac{1}{2}$ density matrix of the form $\rho=\frac{1}{2}(\textbf{1}+\textbf{s}\cdot\...
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1answer
56 views

Nielsen & Chuang Theorem 2.6 Proof

I got a problem in understanding the proof of the Theorem 2.6 (Unitary freedom in the ensenble for density matrices), 2.168 and 2.169 in the Nielsen and Chuang book Equation 2.168 Suppose $|{\tilde\...
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1answer
39 views

Can we always find a unitary operation connecting qubit states with given eigendecompositions?

Consider the density matrices $\rho_0 = |0 \rangle \langle 0|$ and $\rho_1 = |1 \rangle \langle 1|$. Let $\{p_1, p_2\}$, and $\{p_3, p_4\}$ be two probability distributions, that is, $$0 \leq p_1, p_2,...
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What is the probability $\Pr(||U-I||_{op}<\varepsilon)$ of a Haar-random unitary being close to the identity?

If one generates an $n\times n$ Haar random unitary $U$, then clearly $\Pr(U=I)=0$. However, for every $\epsilon>0$, the probability $$\Pr(||U-I||_{op}<\varepsilon)$$ should be positive. How can ...
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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 $U$?

Given the state of a system as $\rho_s$ and that of the ancilla (pointer) as $\rho_a$, the Von-Neumann measurement involves entangling a system with ancilla and then performing a projective ...
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1answer
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How do you embed a POVM matrix in a Unitary?

In QuantumKatas Measurement Task 2.3 - Peres-Wooter's Game, we are given 3 states A,B and C. We construct a POVM of these states. But how do we convert that POVM into a Unitary that we can apply. ...
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Convert a two-ququart (16 x 16) density matrix into normal form--so that the components of the Bloch vectors of the two reduced systems are all zero

The two-ququart ($16 \times 16$) "Hiesmayr-Loffler" density matrix https://iopscience.iop.org/article/10.1088/1367-2630/15/8/083036/meta, (https://arxiv.org/abs/2004.06745 eq. (13)), What ...
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1answer
64 views

Is $e^{i\beta} R_Z(-2\beta)$ equivalent to $U_1(2\beta)$?

As far as I know the single qubit gate $$ e^{i\beta\sigma_z} = \begin{bmatrix} e^{i\beta} & 0 \\ 0 & e^{-i\beta} \end{bmatrix} = e^{i\beta} \begin{bmatrix} 1 & 0 \\ 0 & e^{-i2\beta} \...
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Bipartite states whose coefficients are entries of a unitary matrix

I've been trying to solve this question It seems that in order to show it has unit length, we must show that $$ \frac{1}{d} \sum_{m, n=0}^{d=1} \lvert U_{m, n}\rvert ^2 = 1 $$ I've tried searching ...
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1answer
83 views

Initial assumption of the unitary that allows us to estimate the label function

You can find the paper here , in which they describe the architecture of a QNN that can be used to learn binary functions and correctly classify unseen data. They say that for each binary label ...
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1answer
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Find the decomposition of the following matrix into two level unitary matrices [duplicate]

Find the decomposition of the following matrix into two level unitary matrices: $$ \frac{1}{2} \begin{pmatrix} 1 & 1 & 1 & 1\\ 1 & i & -1 & -i\\ 1 & -1 & 1 & -1\\ ...
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1answer
56 views

Could someone give an example of this pic?

This is a picture from Wiki(https://en.wikipedia.org/wiki/Quantum_logic_gate). Can someone give me a simple example by using two qubits?
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How can I fill a unitary knowing only its first column?

I have a unitary matrix that I want to construct. I only care what happens to the first computational state, so the first column is specified. So far, I've been assigning each question mark to a ...
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2answers
100 views

How to prove that a matrix is an arbitrary unitary?

My goal is to prove that I can synthesise arbitrary unitary from two components. In the end, I find a matrix with the form \begin{equation} \mathbf{W}_j=\begin{pmatrix} |\alpha|2\cos{(\phi_{...
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2answers
739 views

How should I understand the change of qubit's basis as a rotation?

I have a little difficulty with understanding. How do I properly visualize the change of qubit's basis as a rotation? Let's say that we have classical basis vectors, $|0\rangle$ and $|1\rangle$. Now, ...
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2answers
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Explicit form for composition of Choi representation quantum channels

Let $|\Omega \rangle$ be the maximally entangled state over a bipartite system whose parts are each dimension $d$, i.e. $$ | \Omega \rangle \equiv \sum_i^{d}| ii \rangle $$ Then one way of writing ...
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What is the correct sign in the unitary evolution operator of a beam splitter?

I'm a bit confused about which is the correct sign in the unitary evolution operator of a beam splitter. In paper Digital quantum simulation of linear and nonlinear optical elements author uses the ...
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Circuit of a very trivial thing

I am writing to double check that if have a hamiltonian of the form $H = I_1 \otimes I_2$, when I seek to find the unitary, $e^{-i\gamma I_1 \otimes I_2}$, there really is no need to convert this into ...
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1answer
107 views

How can I build up an arbitrary quantum circuit given a certain unitary matrix operation?

Suppose I want to put a qubit whose initial state is $|0\rangle$ to the final state $\frac{1}{\sqrt{3}}|0\rangle + \sqrt{\frac{2}{3}}|1\rangle$. Well, in that case, the unitary matrix that performs ...
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Are there different orderings of the fifteen SU(4) generators in common use?

I've recently performed certain analyses (Archipelagos of Total Bound and Free Entanglement) pertaining to eq. (50) in Separable Decompositions of Bipartite Mixed States , that is \begin{equation} ...
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142 views

What unitary gate produces these quantum states from the computational basis?

Suppose that we have one-qubit unitary $U$ that maps $$ \left| 0 \right> \longmapsto \frac{1}{\sqrt{2}} \left| 0 \right> + {\frac{1+i}{2}} \left| 1\right> $$ and $$ \left| 1 \right> \...
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Why is a Hadamard gate unitary?

The Hadamard gate is a unitary gate, but how does the matrix times its own conjugate transpose actually result in the $I$ matrix? I am currently looking at it as a scalar, 0.707..., multiplied by the ...
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2answers
100 views

Prove that the state $\sum_{S\in P_n}(-)^{\tau(S)}|S\rangle$ is invariant up to a phase when changing the basis

I am trying to prove that the $|S_{n}\rangle$ is $n$-lateral rotationally invariant, where $|S_{n}\rangle$ is defined as $$|S_{n}\rangle=\sum_{S \in P_{n}^{n}} (-)^{\tau(S)}|S\rangle\equiv\sum_{S \in ...
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1answer
102 views

Trying to make irreversable operation in the quantum circuit

I want to make a 2 qubit circuit such that the non-unitary program will transform the regular basis in the way that: $|0 0\rangle \to |00\rangle$ $|0 1\rangle \to |01\rangle$ $|10\rangle \to |01\...
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Is there a quantum operation whose output is always orthogonal to the input?

I'm trying to show/convince myself the following statement is correct (I haven't been able to find any similar posts): "There is no reversible quantum operation that transforms any input state to a ...
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1answer
60 views

How can classical computations be non-unitary?

Given that classical physics emerges from quantum physics on a macroscopic scale, and all quantum operators are unitary, how are we able to perform non-unitary operations (such as setting a register ...
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1answer
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Should a Pauli $X$ matrix equal the identity matrix to be unitary?

My understanding is that any unitary matrix must have its inverse be equal to its conjugate transpose. Looking at the pauli x gate as shown here: $$\begin{bmatrix}0&1\\1&0\end{bmatrix}$$ It ...
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1answer
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Find the local unitary that takes the bell state to a state phi that has an extractable bell state

I have a state $|p\rangle$ that has an extractable Bell state and I want to write it as a Bell state, $|b\rangle$, with a local unitary acting on one side. Basically I am trying to find a local ...
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1answer
107 views

Does the general form of a unitary operator define strict signs for the second column?

As per IBM's documentation for quantum circuits, the general unitary operator is defined as: $$\hat{U}=\begin{bmatrix}\cos(\frac{\theta}{2})&-e^{i\lambda}\sin(\frac{\theta}{2})\\e^{i\phi}\sin(\...
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1answer
340 views

Can the Kraus decomposition always be chosen to be a statistical mixture of unitary evolutions?

If $\mathcal{E}$ is a CPTP map between hermitian operators on two Hilbert spaces, then we can find a set of operators $\{K_j\}_j$ such that $$\mathcal{E}(\rho)=\sum_j K_j\rho K_j^\dagger $$ in the ...
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1answer
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What is the unitary operator realizing a given CPTP operator

Complete Positive Trace Preserving Map (CPTP) operator is the most general operation that can be performed on a quantum system. This post mentioned that a CPTP operator is nothing but a unitary ...
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Are CPTP operators and unitary operators the same thing?

I am reading some quantum papers (In particular, this one page 34) . One of the theorem statement reads, "For every CPTP operator M, we have that .... " I ...
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1answer
135 views

Finding a global phase that transform the Hadamard gate to an element of $SU(2)$ and propose an evoultion operator which implents the operation

I was looking back over an old assignment and I came across a question I wasn't quite sure how to do the problem statement is as follows: The Hadamard rotation is an element of the group $U(2)$. (i)...
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Implementing “Classical AND Gate” and “Classical OR Gate” with a quantum circuit

Quantum cNOT Gate (Classical XOR Gate) A "Controlled NOT (cNOT) Gate" flips the 2nd qubit if the 1st qubit is $\left|1\right>$, and returns the 2nd qubit as-is if the 1st qubit is $\left|0\right&...
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798 views

How to prove that the query oracle is unitary?

The query oracle: $O_{x}|i\rangle|b\rangle = |i\rangle|b \oplus x_{i}\rangle$ used in algorithms like Deutsch Jozsa is unitary. How do I prove it is unitary?