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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Parallel run of qiskit circuits

I am trying to run simulation instances of a parametrized circuit in parallel but my algorithm is extremely slow. Excuse my ignorance, but I just want to be sure that the circuits are indeed running ...
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Quantum Signal Operator and the unitary state preparation oracle?

I am looking into IL Chuang and GH Low's Hamiltonian Simulation with Qubitization paper. I am very confused on the terminology and motivation behind definition 1. I do not understand what the unitary ...
2 votes
1 answer
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How to find a circuit for a unitary operator $e^{-i s |v \rangle \langle v| t }$?

Let $|v \rangle$ be an eigenstate of an $n$-qubit and $2$-local Hamiltonian $$H = \sum_{i=1}^n \left (X_i + a_i Z_i \right) + \sum_{(i,j)} b_{i,j} Z_i Z_j,$$ where $\sigma_i = I \otimes \cdots \...
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Sample random unitary at a given distance from a given unitary

Is it possible? I.e., what is the most natural procedure of such sampling? The sampling has to be 'uniform' in a vicinity (of radius $\epsilon$) of given $U$ (can I say "according to Haar measure ...
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2 answers
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Understanding different forms of an arbitrary Unitary transformation in $\mathcal{H}_2$

I'm working to have a greater understanding of the arbitrary unitary transformation matrix when working in the context of the Bloch sphere. At this time I have found several equivalent ...
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2 votes
1 answer
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What is the tensor product expression for the following quantum circuit? [duplicate]

Qiskit generates the following matrix for this 3-qubit CNOT circuit. Can anyone explain how do we get this mathematically ? This is the Quantum Circuit This is the Output of Unitary Simulator
1 vote
1 answer
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Check that a channel implements a particular unitary

Consider a channel $C$ with Kraus operators $\{K_k\}$ and a unitary U. How can I check that $C$ implements $U$ ? One can write that their Choi matrices are equal i.e: \begin{equation} \sum_{i,j}|i\...
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1 answer
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Is there an inverse for Stinespring dilation?

Given a set of Kraus operators we can find a unitary that does the equivalent map on an extended space including the environment using Stinespring dilation. My question is how do we go about doing the ...
0 votes
1 answer
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Implementing Odd Permutations Without Ancilla Bit

The paper says that The inversion $\alpha \mapsto \alpha^{-1} $ (where 0 is mapped to 0) can be seen as a permutation on $\mathbb F_{256}$. This permutation is odd, while quantum circuits with NOT, ...
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6 votes
3 answers
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Is effective quantum cloning possible, given that any classical function can be implemented as a quantum circuit?

As in Compiling a classical function to a quantum circuit in practice, as far as my understanding goes, it is known that any classical function can be implemented as a quantum circuit. So given $f(x)=...
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1 answer
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What are rotation angle and axis corresponding to a higher-dimensional unitary?

We know that a single-qubit Unitary can be defined as a single rotation of angle $\theta$ around some axis $\hat{n}$, together with a global phase $\alpha$ (see Nielsen & Chuang Eq. 4.9): $$ U = e^...
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Simulation of non hermitian operators with Qiskit

I am trying to simulate on a quantum computer a wavepaket evolution with a non unitary evolution operator (Hamiltonian with an absorbing (imaginary) potential for instance) and I found this post : ...
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Is there a general way to parametrize 2-qubit unitaries?

So in the single-qubit case, we can write any unitary operation as an instance of the following parametrized unitary: $$U(\theta, \phi, \lambda) = \begin{bmatrix} \cos(\theta) & -e^{i\lambda}\sin(...
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Circuit for pre-factors +-i, -1

Setting I have a (6 qubit) circuit which implements a unitary $U$. Goal I need the circuits which implement $-U, iU, -iU$. Phase matters, because I later embed a controlled version of $\pm i U $ into ...
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Do unitary matrices acting on entangled states always give a quantum state?

I'm trying to understand what happens when Alice(Bob) apply a unitary to her(his) part of an entangled state. Let us consider the following unitary transformations: $$U_1 = \frac{1}{\sqrt{2}} \...
1 vote
3 answers
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How are black-box oracles implemented in Hamiltonian simulation?

I am currently trying to decompose a hessian to a sum of unitaries $H=\sum a_i U_i$. The papers VQLS and Black-box Hamiltonian Simulation state that it can be done, but requires the use of an oracle ...
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How to convert between little/big-endian unitary forms in Braket?

As noted in this post, the Amazon Braket unitary calculation method as_unitary has been deprecated (#325) as it uses little-endian qubit order. The new, big-endian method is to_unitary. Here's a code ...
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Mapping $| y \rangle$ to $(-1)^{x \cdot y}| y \rangle$

I was checking some QC lecture notes by Ronald de Wolf and I came across this exercise that I can't solve. Page 27 (pdf page 35), question 5, part b link: https://homepages.cwi.nl/~rdewolf/qcnotes.pdf ...
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Find unitary $U$ that transforms state $|\phi \rangle$ to state $|\phi' \rangle$ knowing those states differ only by relative phase

Given: $|\phi \rangle = \sum_n c_n |a_n \rangle $ $|\phi' \rangle = \sum_n c'_n |a_n \rangle $ such that $\forall n: c_n = c'_n \lor c_n = -c'_n $, where $|a_n \rangle $ is a canonical basis, and ...
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Rotation angles of unitary operator

Given a complex unitary $2*2$ matrix $A$ that represents some quantum gate on a single qubit. What is the formula to extract to $\theta_X, \theta_Y, \theta_Z $ rotations around each one of the axes in ...
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What role does Landauer's principle play in quantum reversibility?

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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1 answer
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How to generate statevector evolution?

I have created the following 2-qubit circuit in qiskit: ...
12 votes
1 answer
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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\...
2 votes
1 answer
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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 ...
4 votes
1 answer
222 views

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 ...
5 votes
2 answers
319 views

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 ...
5 votes
2 answers
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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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4 votes
1 answer
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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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2 votes
1 answer
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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 $\rho$, ...
4 votes
1 answer
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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 ...
1 vote
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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 ...
6 votes
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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 ...
3 votes
1 answer
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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 ...
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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2 votes
1 answer
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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 $$\Phi(\rho)=\text{tr}_E[U_{SE}(\rho_S\otimes|0\rangle\langle 0|_E)U_{SE}^\dagger]$$ ...
7 votes
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 ...