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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What role does Landauer's principle play in quantum reverisbility?

In section 3.2.5 of Nielsen and Chuang (starting page 153) they talk about Landauer’s principle, where they discuss the lower bound on the thermodynamic cost of erasing information. In irreversible ...
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Is the Eastin-Knill Theorem incorrect?

I am reading through this paper (the Eastin-Knill Theorem) and there is a step in the proof of the main theorem that I do not understand. Let $Q$ be a composite quantum system supporting a quantum ...
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Quantum channels that commute with any unitary channel

Consider a quantum channel $\Phi$ that maps from density operators $\mathcal{S}(\mathcal{H}_A)$ to itself, that commutes with any unitary channel $\mathcal{U}$ on $\mathcal{S}(\mathcal{H}_A)$, i.e. $\...
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Redundant parameters in $2\times 2$ unitary operators?

A complex $n \times n$ unitary operator has $n^2$ free real parameters. For example, a $2 \times 2$ unitary matrix can be parametrized as \begin{equation} \begin{pmatrix} e^{i(\alpha - \beta/2 - \...
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Using rotation gates instead of Grover

I have a conceptual question about Grover's algorithm. In the textbook case, we always assume to have an oracle that singles out the correct states by giving them a negative phase. Then, we use phase ...
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How to generate statevector evolution?

I have created the following 2-qubit circuit in qiskit: ...
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How can classical bits be copied if qubits cannot be copied?

The no-cloning theorem of quantum mechanics tells us there can be no general quantum circuit that can copy arbitrary qubit states, i.e. a quantum gate or circuit cannot send $|0\rangle |\psi\rangle\...
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Minimal Circuit Decomposition for a 3 qubit gate

I have this unitary matrix and i need to find the decomposition with the small number of c-not. I tried to use Quantum Shannon Decomposition but the simple form of the matrix make me think that there ...
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4 votes
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Showing that two unitary matrices are equal up to a global phase

Let $U$ and $V$ be two $d × d$ unitary matrices, representing two reversible quantum processes on a $d$-dimensional quantum system. We say that the two processes “act in the same way” on the state $|ψ\...
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Quantum Linear Algebra [closed]

[![Question][1]][1] Find a 4 x 4 unitary matrix U such that U = eiA. (Possibly up to multiplying by a unit scalar, U is a matrix seen in the course.) Verify your calculation by showing how if U were ...
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What is a Haar random quantum state?

Can somebody please explain me what is a Haar random state? I am not able to find any friendly resource to read about it.
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How can I delay the construction of the power of a gate in a parametric circuit?

I want to implement the parametric power of a gate. I know that the parametric power is impractical to implement (cf https://github.com/Qiskit/qiskit-terra/issues/4751). So, I want to delay the ...
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Does there exists an algorithm to construct a quantum circuit given an arbitrary unitary?

Suppose there exists an algorithm that takes as input an arbitrary unitary matrix and produces as output a quantum circuit representing that matrix. Then in theory that algorithm could construct any ...
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What is the form of a unitary $U$ that preserves the marginals on a given state, $\text{Tr}_A(U\rho_{AB} U^\dagger) = \rho_B$?

Suppose for some quantum state $\rho_{AB}$ and unitary $U_{AB}$, one has $$\text{Tr}_A(U\rho U^\dagger) = \rho_B$$ does this imply that $U_{AB} = U_A\otimes I_B$? Also, the same question as above, but ...
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Is there a higher dimensional Fredkin gate?

The Fredkin gate is CSWAP gate. Given a control register in $0$ or $1$, the gate does nothing or swaps two target registers respectively. Is there a higher dimensional version of this gate? I have ...
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Understanding the Quantum Hebbian algorithm

I've been reading the paper from Lloyd and al. on Quantum Hopfield Networks, but I don't understand the quantum Hebbian algorithm (page 3). I am trying to understand the mathematical development on ...
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Can every unitary on $\mathcal{H}\otimes \mathcal{K}$ be modelled by quantum operations on $\mathcal{H}$?

In section 8.2.3 of Nielsen and Chuang, they discuss how unitary dynamics of a system and environment arise from quantum operations (i.e. Kraus operators $E_k$ such that $\sum_k E_k^*E_k=I$). ...
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Does the teleportation protocol work for any dimension? [duplicate]

Suppose Alice and Bob share a maximally entangled state in $d$ dimensions i.e $$\vert\phi\rangle = \frac{1}{\sqrt{d}}\sum_{i=1}^d \vert i\rangle\vert i\rangle$$ Given a state $d$ dimensional $\rho$, ...
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Closeness of unitary dilations of CPTP maps

Let $\Phi_1,\Phi_2 \colon S(\mathcal{H}) \to S(\mathcal{H})$ be CPTP maps on the same Hilbert space $\mathcal{H}$ which are $\varepsilon$-close in diamond norm, and let $U_1,U_2$ be respective unitary ...
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$E(U_j,V_j)\leq\Delta/(2m)$ if probabilities of outcomes obtained from the approximate circuit is within a tolerance $Δ>0$

Suppose we wish to perform a quantum circuit containing $m$ gates, $U_1$ through $U_m$. Unfortunately, we are only able to approximate the gate $U_j$ by the gate $V_j$ . In order that the ...
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What is the structure of coefficients of $2^N\times 2^N$ unitary matrices decomposed in terms of the Pauli basis?

Any square $2^N\times 2^N$ matrix can be written as a sum of tensor products of pauli matrices. Eg a $8\times 8$ matrix can be written as $$U=\sum_{i_1,i_2,i_3}u_{i_1,i_2,i_3}\sigma_{i_1}\otimes\...
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How can extract reduced dynamics of a bipartite system from unitary evolution in quite

Let us assume that I have a bipartite system $A\otimes B$ and an initial product state undergoing some evolution $H^{AB} = H^A+H^B+V^{AB}$, which is time independent. I want to simulate the reduced ...
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6 votes
2 answers
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When can pairs of states be transformed into other pairs of states via unitary mapping?

The states $|+\rangle, |-\rangle$ can be mapped to $|0\rangle, |1\rangle$ by a simple rotation. But if I now have other states ($|\psi_0\rangle, |\psi_1\rangle$) which are not orthogonal, does a ...
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Proof for Cardinality of the Clifford Group

In this article: (http://home.lu.lv/~sd20008/papers/essays/Clifford%20group%20[paper].pdf) a proof is given for the cardinality of the Clifford group. I understand all the parts of it except for how ...
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Why are all the eigenvalues of a "Hermitian block-encoding" equal to $\pm1$?

I was looking at the paper : https://arxiv.org/abs/2002.11649 and the eigenvalue discussion is not clear to me. Block-encoding is a general technique to encode a nonunitary matrix on a quantum ...
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7 votes
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How large can we make the fidelity between mixed states by allowing unitaries?

For pure states, it is known that one can always find a unitary that relates the two i.e. for any choice of states $\vert\psi\rangle$ and $\vert\phi\rangle$, there exists a unitary $U$ such that $U\...
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Can a triplet be a qutrit?

Original question A triplet is a space that consist of three states that have the same total angular momentum (spin 1). If we restrict ourselves to a set of quantum gates that keep triplet states in ...
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How does the Kraus decomposition imply the Stinespring representation?

To show that the Kraus decomposition $\Phi(\rho)=\sum_{k=1}^D M_k\rho_S M_k^\dagger$ implies the Stinespring form $$\Phi(\rho)=\text{tr}_E[U_{SE}(\rho_S\otimes|0\rangle\langle 0|_E)U_{SE}^\dagger]$$ ...
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1 answer
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Are anti-unitary gates possible?

According to Wigner’s theorem, every symmetry operation must be represented in quantum mechanics by an unitary or an anti-unitary operator. To see this, we can see that given any two states $|\psi\...
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How is outer product an operator?

I was going through Qiskit online text book and came across this part. The relevant (slightly modified) paragraph is - Suppose we have two states $|\psi_0\rangle$ and $|\psi_2\rangle$. Their inner ...
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Unitary Transformations for States with Same Entanglement [duplicate]

$\newcommand{\Ket}[1]{\left|#1\right>}$ I know this has been asked before in another context (How to construct local unitary transformations mapping a pure state to another with the same ...
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2 votes
3 answers
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Why do quantum gates have to be unitary?

In my textbook, it's said the unitarity constraint is the only constraint on quantum gates. Any unitary matrix specifies a valid quantum gate! Why do quantum gates have to have to be unitary? How do ...
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How to construct local unitary transformations mapping a pure state to another with the same entanglement?

$\newcommand{\Ket}[1]{\left|#1\right>}$In Nielsen's seminal paper on entanglement transformations (https://arxiv.org/abs/quant-ph/9811053), he gives a converse proof for the entanglement ...
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1 answer
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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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1 vote
1 answer
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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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1 answer
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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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6 votes
1 answer
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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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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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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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1 answer
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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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1 vote
1 answer
217 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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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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1 vote
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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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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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3 votes
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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 $\|\rho\|_1 = \text{...
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6 votes
2 answers
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Are SU($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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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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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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2 answers
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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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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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