Quantum Computing Stack Exchange
Quantum Computing Stack Exchange

Questions tagged [unitary]

17 questions
6
votes
2 answers

Is it possible to have a trace fidelity of 1 even if two unitary operations are different?

The gate fidelity of two quantum unitary operations is often described using $\frac{1}{2^n}|\text{tr}(U^\dagger V)|$. Is it ever possible that $U^\dagger V \ne I$ however $\frac{1}{2^n}|\text{tr}(U^\dagger V)| = 1$? Specifically, is it possible to…
smi
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4
votes
2 answers

Is any multi-qubit unitary operation a rotation about a specific unit vector?

I understand that any single qubit unitary operation can be expressed as a rotation around a three dimensional unit vector. Is it possible to do the same for multi-qubit unitaries? Can I express an $n$-qubit unitary operation as a rotation about…
smi
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3
votes
2 answers

On Pauli matrices being a basis for unitary transformations

I am confused about the Pauli matrices. I am trying to decipher a statement like this: "[...]these three operators form a complete basis for the set of all unitary transformations on a single qubit. That is, any quantum circuit on a single qubit can…
Matyas
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3
votes
1 answer

How to select the embedded matrix in a unitary encoding?

I embed a matrix M in a U unitary, where $M$ is $2^n \times 2^n$, and $$U = \begin{bmatrix}M & *\\ * &*\end{bmatrix}$$ with twice as many rows and columns.. I prepare a state vector with the first qubit being $|0\rangle$, that is $$ |0\rangle…
Matyas
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2
votes
1 answer

What is the circuit that simulates this unitary

What is the circuit that simulates the following unitary matrix? $$ P = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1…
Coper
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2
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0 answers

Expectation value of an ensemble composed of multiplying the same Haar random unitaries

Let $U$ be an $n$ qubit Haar random unitary and $\mathbb{I}_n$ is the $n$ qubit identity operator. I want to find the density matrix corresponding to the following: $$ \rho = \underset{U}{\mathbb{E}}\left[ (\mathbb{I}_{n-1} \otimes U \otimes…
BlackHat18
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1 answer

When performing a projective measurement on a subsystem X entangled with another system Y, can the evolution of Y be unitary?

I am interested in the evolution of a subsystem Y entangled with another subsystem X. X and Y are initially in a pure product state. They undergo some global joint evolution E (not necessarily unitary), after which they are entangled with each…
Bidule
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2
votes
1 answer

Unitary Transformation for QFT with 2 base states is giving the columns in the wrong order

I'm trying to reproduce in Qiskit the multiplicative form of the QFT for two qubits. It is similar to what is asked in Nielsen's QCQI book in Exercise 5.2 and Box 5.1. To check the results I'm printing the unitary transformation for the Qiskit…
Gustavo Mirapalheta
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1
vote
1 answer

Proof of Monotonicity and Invariance Under Local Unitary Operations for Summed Measures

Suppose we have a set of measures $A_1, A_2, ..., A_n$ that individually satisfy the following properties: 1-Monotonicity under Local Operations and Classical Communication (LOCC) 2-Invariance under Local Unitary Operations I want a prove that…
reza
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1 answer

Qiskit simulation failed and returned the following error message

I am trying to compute some unitary matrix for some big circuit on GPU using qiskit-aer, specifically a 11 qubits circuit with the following error: Simulation failed and returned the following error message: ERROR: [Experiment 0]…
LukeTheWalker
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vote
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Implementation of the CSD Decomposition for a 4x4 matrix

I am currently reading Volume I of "Principles of Quantum Computation and Information" by Benenti-Casati-Strini. There is a section in chapter 3 which covers the decomposition of any arbitrary $2^n \times 2^n $ unitary matrix into simpler 2x2…
Malaik Kabir
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1
vote
1 answer

Why does the Solovay-Kitaev theorem use the operator norm?

In this review explaining the Solovay-Kitaev theorem, it is stated that the theorem uses the operator norm to define closeness between unitaries. This is then used to determine if a particular set of gates is universal. That is, we require the…
user290109
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Computational cost of transpile 2^n X 2^n real valued unitary to quantum circuit

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.93.032318 This paper seems showing boundary of number of CNOT gates to express general unitary. How about for real-valued general unitary? Is there any paper described that? Thank you.
maar hybrid
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How to see successive contraction of gates in a DAG Circuit

I have a circuit and I want to see the DAG graph at each step of a series of contraction until I get to the unitary matrix of the circuit, I put together some code that should accomplish this, but clearly I made some mistakes as I cannot understand…
LukeTheWalker
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0
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2 answers

How to generate a unitary matrix given a permutation of the qubits

The title explains it all, let's say that I have a unitary matrix whose effect is defined when applied to qubits 0 and 1 in order, how can I create its unitary for when it is applied to 1, 0? For two qubit gate it's trivial as that's the…
LukeTheWalker
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