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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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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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
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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
93 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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3answers
196 views

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
50 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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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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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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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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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
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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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4answers
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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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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?
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112 views

Differentiate between local and global unitaries

Just like we have the PPT, NPT criteria for checking if states can be written in tensor form or not, is there any criteria, given the matrix of a unitary acting on 2 qubits, to check if it is local or ...
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3answers
352 views

Is it correct to say that we need controlled gates because unitary matrices are reversible?

I am new to quantum computing and saw this argument on this site but I don't understand it. First of all, I don't understand what is exactly meant by 'reversible'. Because even if you had a unitary ...
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1answer
733 views

General parametrisation of an arbitrary $2\times 2$ unitary matrix?

Following is an excerpt from QCQI: I can understand that this matrix satisfies a unitary matrix. Also, intuitively, I am able to understand it. However, what is the proof that any given Unitary ...
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What do you specify when you physically apply a unitary?

In the Environment and Quantum Operations in Nielsen and Chuang, section 8.2.2, they say that when you apply a unitary on a state, you expect the output as the just the state transformed by the ...
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221 views

Extending a square matrix to a unitary matrix

Suppose we have a square matrix $M$ of size $n\times n$. It is given that any element $M_{ij}$ of $M$ is a real number and satisfies $0 \leq M_{ij} \leq 1$, $\forall$ $i,j$. No other property for $M$ ...
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138 views

Understanding why the modular function part of Shor's algorithm is unitary

I've been struggling to understand the modular exponent bit of Shor's algorithm. My understanding is that it takes a register in the state $\frac{1}{\sqrt{Q}}\sum_{k=1}^{Q-1} |k\rangle |0\rangle$ to ...
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1answer
160 views

Unitary gate(s) from product of exponent

Often unitary gates are defined as a product of exponentials, with some parameter in the power-term. However, often it is not clear how to construct unitary gates from it, at least not for me that is. ...
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2answers
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Unitary acting on standard qubit basis properties

If we have a $U$ (unitary with all real entries) such that: $U|0\rangle =a|0\rangle +b|1\rangle$ What is $U|1\rangle=?$ I know: the definition of what it means to be unitary ie. $U^\dagger U=UU^\...
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What do they mean by “qubit can't be copied”?

What does it mean by ''qubit can't be copied'' ? In a note, it is saying that: Copying a qubit means $$U|\psi\rangle_A|0\rangle_B=|\psi\rangle_A|\psi\rangle_B$$ i.e; applying a unitary ...
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What is the smallest quantum circuit to produce two-qubit state (a,b,b,b)?

How can I synthesis a two-qubit quantum state of the state vector (a,b,b,b) using basic quantum-gate circuit (arbitrary single-qubit rotation and controlled $Z$ gate)? And further, can I know a given ...
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Quantum states are unit vectors… with respect to which norm?

The most general definition of a quantum state I found is (rephrasing the definition from Wikipedia) Quantum states are represented by a ray in a finite- or infinite-dimensional Hilbert space over ...
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4answers
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Why are quantum gates unitary and not special unitary?

Given that the global phases of states cannot be physically discerned, why is it that quantum circuits are phrased in terms of unitaries and not special unitaries? One answer I got was that it is just ...
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7answers
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If all quantum gates must be unitary, what about measurement?

All quantum operations must be unitary to allow reversibility, but what about measurement? Measurement can be represented as a matrix, and that matrix is applied to qubits, so that seems equivalent to ...
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If quantum gates are reversible how can they possibly perform irreversible classical AND and OR operations?

Quantum gates are said to be unitary and reversible. However, classical gates can be irreversible, like the logical AND and logical OR gates. Then, how is it possible to model irreversible classical ...