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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Unitary Transformations for Schmidt Decomposition

$\newcommand{\ket}[1]{|#1\rangle}$ Suppose a pure state $\ket{\psi}$ has a Schmidt decomposition given by $\ket{\psi^{SD}}$, which can be obtained via the diagonalization of the reduced density matrix ...
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33 views

What is a bipartite unitary?

What is a 'bipartite unitary'? I saw it appearing in a paper "Efficient verification of quantum gates with local operations" (https://arxiv.org/pdf/1910.14032.pdf) A reference to the ...
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Why is it not easy to distinguish $U|\psi\rangle$ and $U'|\psi\rangle$ if $\|U-U'\|<\epsilon$?

So I am currently working on an assignment, which is about the induced Euclidian norm $$ ||A||:= \max_{v\in\mathbb{C}^d\text{ s.t. }||v||_2=1} ||Av||_2 $$ for some $A\in\mathbb{C}^{d\times d}$. For ...
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157 views

How efficient is Qiskit's unitary decomposition?

In Qiskit's extension package we have the UnitaryGate module that you can initialize using a unitary matrix and then add it to your circuit. How efficiently is this ...
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1answer
136 views

What is the eigenvalue distribution of arbitrary unitary matrices?

I had a question regarding the nature of the eigenvalue distribution of unitary matrices. Searching for the answer I found that the unitary matrices which are sampled randomly have a defined ...
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51 views

An algorithm to perform Gram-Schmidt orthogonalization of linearly independent state vectors

In the first paragraph of the 2nd section of this article, it is stated that given a set of linearly independent $n$-qubit state vectors, Alice can perform the Gram-Schmidt procedure to obtain ...
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1answer
45 views

How to sample vectors close to the minimum eigenvector of a unitary matrix?

Say that we have an unknown $2^{n}\times2^{n}$ unitary matrix $U$ with eigenvectors $|v_{i}\rangle$ and eigenvalues $e^{2\pi j \theta_{i}}$and we want to sample a vector, say $|\phi \rangle$. Since ...
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1answer
63 views

What is the general formula for unitary rotations in terms of Pauli spin operators?

Recently I have read a paper in which they have used a unitary transformation as follows: $$U_{\frac{7\pi}{16}}=\cos\left(\frac{7\pi}{8}\right)\sigma_{z}+\sin\left(\frac{7\pi}{8}\right)\sigma_{x}$$ ...
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1answer
57 views

Manipulating the amplitude of state based on the state information

In the past, I thought I have seen quantum circuits/algorithm techniques to change the amplitude of state based on the state? $\lvert \psi \rangle = \sum_x \ C_x \lvert x \rangle$, here $C_x$ is just ...
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69 views

Fastest way to solve Gram Schmidt orthogonalization in quantum computer

Suppose we have $n$ (large) dimensional vector space and I want to orthogonalize $n$ linearly independent vectors using the Gram-Schmidt orthogonalization process. Is there some time-bound on how ...
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1answer
71 views

How to construct an $n\times n$ unitary matrix taking an arbitrary $|\psi\rangle$ to a target state $|\phi\rangle$?

I came across Lecture 12 here https://viterbi-web.usc.edu/~tbrun/Course/ that does this but I was not able to understand. An example would be very helpful
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Minmax theorem for optimization over isometries and states

I have the following minmax problem and I am wondering if the order of the minimum and maximum can be interchanged and if yes, why? Let $\|\cdot\|_1$ be the trace norm defined as $\|\rho\|_1 = \text{...
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Are SU($n$) operations enough for quantum computation?

Usually we want a quantum computer that can perform all foreseeable unitary operations U($n$). A quantum processor that can naturally perform at least 2 rotation operators $R_k(\theta)=\exp(-i\theta\...
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1answer
55 views

How to represent a sine function in the statevector $|\psi\rangle = sin(kx)$

Is there a quantum circuit that encodes the statevector so that the coefficients of the statevector $|\psi\rangle$ corresponds to a discrete representation of $sin(kx)$ in $[0,1]$? In particular, I'd ...
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132 views

How can one cheat in Mahadev's classical verification protocol if one can find a "claw''?

I was going through the seminal paper of Urmila Mahadev on Classical Verification of Quantum Computations(for an overview see this excellent talk by her). As a physicist by training, I am not very ...
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61 views

Can any “control-something unitary” be written with control and target spaces flipped?

Consider a simple two-qubit gate such as the CNOT. The typical presentation of this gate is $$\text{CNOT} = |0\rangle\!\langle0|\otimes I + |1\rangle\!\langle1|\otimes X,$$ with $X$ the Pauli $X$ gate....
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1answer
45 views

How do you compute the compiled unitary of a quantum circuit comprised of different $n$-input gates?

Given a quantum circuit consisting of two qubits, how is the compiled unitary of the circuit computed when we have different input type gates? (X-gate, H-gate are single-input gates, CNOT is a 2-input ...
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68 views

How accurate are Qiskit’s unitary and Statevector simulators for very large circuits?

Background: I wrote a code that takes a Qiskit circuit C as an input and outputs a random circuit C’ such that C and C’ have the same unitaries. I tested my code using Qiskit’s Statevector simulator ...
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1answer
30 views

Unitary interaction term of two-qubit graph state

Consider the controlled phase gate $$U_{ab}(\varphi_{ab}) := e^{-i \varphi_{ab}H_{ab}}~~~~\text{where}~~~~H_{ab} := |1 \rangle^{a} \langle 1 | \otimes |1 \rangle^{b} \langle 1 |$$ is the two-qubit ...
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How do I get the Unitary matrix of this circuit without using 'unitary_simulator'? [duplicate]

I am using jupyter notebook and qiskit. I have a simple quantum circuit and I want to know how to get the unitary matrix of the circuit without using 'get_unitary' from the Aer unitary_simulator. i.e.:...
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1answer
28 views

Why is my unitary matrix using linear algebra not matching the 'get_unitary' simulation?

I am using jupyter notebook and qiskit. I have a simple quantum circuit and I want to know how to get the unitary matrix of the circuit without using 'get_unitary' from the Aer unitary_simulator. i.e.:...
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4answers
142 views

How do I get the Unitary matrix of a circuit without using the 'unitary_simulator'?

I am using jupyter notebook and qiskit. I have a simple quantum circuit and I want to know how to get the unitary matrix of the circuit without using 'get_unitary' from the Aer unitary_simulator. i.e.:...
2
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1answer
44 views

How does measuring a value of one operator affect the probability of measuring a value for another operator?

Suppose I have two non-commuting operators, $U_1$ and $U_2$ with eigenvalues $\lambda_{1,1}, \lambda_{1,2}$ and $\lambda_{2,1}, \lambda_{2,2}$, respectively. In order to determine how measuring one ...
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306 views

If a Hamiltonian is quadratic in the ladder operator, why is its time evolution linear in the ladder operator?

How can one show that $\hat{U}^\dagger\hat{a}\hat{U}$ (with $\hat{U} =e^{-i\hat{H}t}$) involves only linear orders of the ladder operator, when $H$ is the general quadratic Hamiltonian $(\hat{H} = \...
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1answer
87 views

What is the most general quantum operation that preserves the marginal?

Suppose I have two states $\rho_{AB}$ and $\sigma_{AB}$ such that the marginals $\rho_A = \sigma_A$. What is the most general operation that could have acted on $\rho$ to output $\sigma$? For example, ...
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80 views

If you apply a unitary transformation to an entangled state, is it still entangled?

See title. If this is not true, is there a counter example? If it is not true, does it hold true for certain combinations of unitaries and entangled states?
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138 views

What's the state-of-the-art to calculate $|Ab\rangle$, given a matrix $|A\rangle$ and a vector $|b\rangle$ in QRAM encoding

Assuming that we have a matrix $A\in \mathbb{R}^{m\times n}$ stored in a quantum superposition, i.e. $$|A\rangle= \frac{1}{\|A\|_F}\sum_{i,j=0}^{n-1}{a_{ij}}|i,j\rangle$$ and a vector $b\in \mathbb{R}^...
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3answers
122 views

Find unitary such that $U:|i\rangle|0\rangle\rightarrow|i\rangle|A_i\rangle$

Let's assume I have two qubits of state $|A_0\rangle$ and $|A_1\rangle$ correspondingly stored in a quantum memory. How do I find a Unitary $U$ that acts on another register of 2-qubits such that $$U:|...
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Matrix multiplication through Block Encodings

For a project, I want to simulate a matrix multiplication on a simulated quantum circuit. Assuming that we have a matrix $A\in \mathbb{R}^{m\times n}$ stored in a quantum superposition, i.e. $$|A\...
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2answers
134 views

How do Rényi entropies act under unitary time evolution?

I am trying to find information/ help on Rényi entropies given by $$ S_n(\rho) = \frac{1}{1-n} \ln [Tr(\rho^n)] $$ and how it acts under unitary time evolution? Is the entropy independent on the state ...
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1answer
75 views

How can I construct a universal transformation using Clifford+T gates? [duplicate]

How can I construct, using Pauli, Hadamard and $T$ gate, a universal transformation $U$ such that $U|0\rangle$ has a less than $\frac{\pi}{4}$ complementary angle with $|0\rangle$?
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1answer
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Textbook 2.5 (Qiskit) - Unitary and Hermitian matrices

In section 2.5 of the Qiskit textbook, it states that $X$, $Y$, $Z$ and $H$ are examples of unitary Hermitian matrices. As I understand it, this means that the following rule applies: $$UU^\dagger=U^\...
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53 views

3-qubit phase shift gate/circuit implementation without any Ancilla qubits

Hi, I need help me with figuring out the 3-qubit phase shift circuit without any ancillas similar to the 2-qubit circuit shown in below attached picture....... Please do let me know! Thanks in advance!...
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1answer
82 views

How to prepare a random 1-qubit superposition for data encoding

Let's assume we have a normalized data vector $\vec{x}= [x_1,x_2]$. How can I prepare a state $$|\psi\rangle = x_1|0\rangle+x_2|1\rangle$$ for any $\vec{x}$. I know that this state is in general not ...
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2answers
75 views

Find the unitary implementing the transformation $|0\rangle\to\frac1{\sqrt2}(|0\rangle+|1\rangle),|1\rangle\to\frac1{\sqrt2}(|0\rangle-|1\rangle)$ [closed]

I have found a question for finding the Unitary operator for the following transformation: I found the solution as well. But I didn't understand how they got the solution!
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Compiling Quantum Circuits using the Palindrome Transform

This paper shows a way to produce optimal circuits. I haver verified most of them and they are correct except this procedure: procedure ProduceArray(n) I cannot ...
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3answers
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How does one create the unitary sending $|0\rangle$ into a target quantum state?

The Hadamard gate allows us to construct an equal superposition of states. If one wants to construct an arbitrary superposition e.g. $\alpha\vert 0\rangle + \beta\vert 1\rangle + ..$, how does one ...
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526 views

How to check if a quantum circuit can be constructed for a given matrix representation?

Let's say I have a matrix representation, e.g. $$ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$ How ...
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1answer
494 views

Does the no-hiding theorem suggest that quantum information is never destroyed?

According to Wikipedia: The no-hiding theorem proves that if information is lost from a system via decoherence, then it moves to the subspace of the environment and it cannot remain in the ...
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2answers
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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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1answer
48 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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1answer
79 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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2answers
151 views

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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2answers
82 views

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
43 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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2answers
83 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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85 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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2answers
106 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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38 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 $$ U = \sum_k\exp(i\phi_k)|\psi_k\rangle\langle\psi_k| $$ $2.$ ...