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Positive semidefinite relationship after partial trace

Let $\rho_{ABC}$ and $\sigma_{C}$ be arbitrary quantum states and $\lambda\in \mathbb{R}$ be minimal such that $$\rho_{ABC}\leq \lambda \rho_{AB}\otimes\sigma_C$$ We assume there are no issues with support in the above statement to avoid infinities.…
JRT
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Is Gaussian boson sampling (used for showing quantum advantage) a subcategory of the continuous variable approach?

I read about the photonic QC Jiŭzhāng that showed quantum advantage by Gaussian boson sampling. I read that boson sampling itself is a sub-universal technique of QC (where they use single-photon states as input states). In the paper, the scientists…
ile2N
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What is the matrix for a SWAP operation on two qubits?

Say we want to swap qubits $a$, $b$ in the same register, where $a,b \in \left \{ 0, 1,\cdots, n-1 \right \}$. What would be the corresponding matrix. For those interested, I'm curious about this matrix because I'm implementing the QFT, which by…
LPenguin
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Get state vector of a single qubit in a circuit in Qiskit

I have two quantum circuits, and I would like to compare state vector of the first qubit and check if equals, what is the best way to do that in qiskit ? Let's say I have : psi = QuantumCircuit(5) psi.ry(np.pi/4.,0) psi.x(0) psi.x(1) psi2 =…
user12910
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Qubit Connectivity of IBM Quantum Computer

When simulating a quantum circuit using the qiskit qasm_simulator, the qubits that you assign to the quantum circuit may be connected to all other qubits. In other words, if there are $n$ qubits in the quantum circuit, a single qubit can interact…
Yitian Wang
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Decomposing gates resembling exponentiated members of desired gateset

Suppose I have access to a pretty typical gate set, for example $\{\text{CNOT}, \text{SWAP}, \text{R}_{x}, \text{R}_{y}, \text{R}_{z}, \text{CR}_x, \text{CR}_y, \text{CR}_z\}$ where $\text{CR}$ is a controlled single-qubit rotation. I'm curious…
forky40
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What can I conclude about $\langle \phi|\pi_1\pi_2|\phi\rangle$ if $\langle \phi|\pi_i|\phi\rangle\ge e$?

If I have two projectors $\pi_1, \pi_2$ such that for some $|{\phi}\rangle$: $\langle {\phi}| \pi_1 |{\phi}\rangle \geq e$ and $\langle {\phi}| \pi_2 | {\phi}\rangle \geq e$ What can I conclude about the following quantity? $\langle {\phi} | \pi_1…
islamfaisal
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Shor's algorithm: initialization of second register

I am trying to understand Shor's algorithm. I am not quite sure why the initialization, indicated as $|1\rangle$ in the below image at the bottom left is chosen as it is? I understand the modular exponentiation method in principle, but I am not sure…
user823
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How to learn parameters in a quantum circuit, given an interference pattern?

Using cirq, I have the following quantum circuit, with three parameters: phi, alpha and beta: q0 = cirq.GridQubit(0,0) q1 = cirq.GridQubit(0,1) phi = sp.Symbol('phi') alpha = sp.Symbol('alpha') beta = sp.Symbol('beta') circuit = cirq.Circuit([ …
Calum Macdonald
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How are the IBM's and Google's Hadamard gates fabricated and operated?

There are thousands of articles, books and web sites describing the Hadamard Gate from a theoretical point of view. But I haven't been able to find any photo about any real implementeation of a Hadamard Gate on superconducting circuits nor any…
skan
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Could random quantum circuits be efficiently approximately simulated?

Google's landmark result last year was to compute a task with a quantum computer that a classical computer could not compute, and they chose random circuit sampling. Part of their justification was complexity-theoretic reasons that, if one can…
Sam Jaques
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Bob applies a projector - what happens to eigenvalues of Alice's reduced state?

Suppose Alice and Bob share a state $\rho_{AB}$. Let us denote the reduced states as $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$. Bob applies a projector so the new global state is $$\rho'_{AB} = (I_A\otimes…
user1936752
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What is the intuition behind "states with support on orthogonal subspaces"?

I'm sure I don't fully understand support, but I am having trouble seeing how it connects to things like density operators. I have an idea that it means, according to Wikipedia: "In mathematics, the support of a real-valued function f is the subset…
BADatThis
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Intuition behind the construction of an ansatz circuit

I'm learning about the VQE algorithm. When I looked at the declaration in Qiskit I saw you need to pass an ansatz which prepares the state. I looked at some commonly used ansatz functions, e.g. EfficientSU2 of Qiskit, and I saw many of them use…
user14092875
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Quantum relative entropy with respect to a pure state

I want to evalualte the quantum relative entropy $S(\rho|| \sigma)=-{\rm tr}(\rho {\rm log}(\sigma))-S(\rho)$, where $\sigma=|\Psi\rangle\langle\Psi|$ is a density matrix corresponding to a pure state and $\rho$ is a density matrix corresponding to…
Confinement
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