Questions tagged [quantum-state]

Questions about or related to quantum states. Consider using the density-matrix tag when relevant.

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Helstrom Measurement when two quantum states are close

I've been reading a paper about Entangled-quantum GAN (see this PDF) and wondering why descriptions below Eq.(3) in the paper are in fact true. To summarize the description, suppose we have two ...
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3answers
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QFT circuit product state derivation

I'm reading Ronald de Wolf's lecture notes, and in chapter 4.5 he writes that $$ \frac{1}{\sqrt N}\sum\limits_{j=0}^{N-1}\prod\limits_{l=1}^{n}e^{2\pi i j_l k / 2^l}|j_1...j_n\rangle = \bigotimes\...
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How to write this in expression in python [closed]

Writing a ket $\vert{\phi}\rangle$ in position basis. Defining $\langle x\vert\phi\rangle = \phi(x)$, we can write the ket in position basis as $$\int_{-\infty}^{\infty}dx \hspace{1mm}\phi(x)\hspace{...
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Quantum GAN implementation

Can anyone provide a good link to understand how to implement qgan using pytorch in qiskit. Trying to understand this ( https://qiskit.org/documentation/machine-learning/tutorials/...
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1answer
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Multiplication using nearest neighbour quantum gates

Let $A$ be an $n \times m$ matrix, and $x$ be an $m \times 1$ vector and $q$ be a number such that $q$ is polynomial in $n$. Let us be given both $A$, $q$, and $x$ as input and let us also have a 2D ...
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1answer
25 views

What is a histogram of a job a plot of, if not the squared amplitudes of the final state-vector?

The histogram of a circuit is the result of running the circuit (with measurement) many times, right? Does this correspond to the squared amplitudes of the final state-vector? If not, why?
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1answer
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What happens to the elements of the simulated state vector when we set a qubit to $|0\rangle$?

In IBM's Qiskit online simulator, we have the (non-reversible) ability to set a specific qubit to $| 0\rangle$. This is convenient but I'm left confused as to what happens to the elements of the ...
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1answer
62 views

Density matrix after measuring Bell state in CHSH game

In these notes, the author says the following about the CHSH game Does Alice and Bob’s ability to succeed more than 75% of the time mean that they are communicating? Well, we know it’s not possible ...
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What is a non-local state?

The notion of entangled states is very neat, it simply means non-separability of the states. Another important class of states that one often hears is the non-local state. But when do we say that a ...
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1answer
37 views

How to compute the partial trace on the first register of the state $|\chi\rangle=\frac{1}{||A||}\sum_{i=0}^{m-1}||A_i|||A_i\rangle|i\rangle$?

Given the quantum state $$|\chi\rangle=\dfrac{1}{||A||}\sum_{i=0}^{m-1}||A_i|||A_i\rangle|i\rangle,$$ how can we obtain the partial trace operation on the first register, i.e., $$\begin{align}\text{tr}...
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1answer
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How to distingush between two very similar pure quantum states?

I'm trying to prove the claim that Given two pure states: $|\psi_i\rangle$ and $|\phi_i\rangle$ such that $|\,|\psi_i\rangle - |\phi_i\rangle\,|\le \delta$ then no measurement can distinguish ...
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1answer
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Expectation value of an observable containing a single projector vs Born rule for the projector

Suppose I have a state $|\psi\rangle$ and I want to estimate the probability of obtaining a computational basis state $|x\rangle$. Then by Born rule: $$ p(x) = |\langle x|\psi\rangle|^2 = Tr[|x\rangle ...
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Question regarding the output probability of a quantum circuit

Consider a quantum circuit $\text{Q}$, run on $|0^{n}\rangle$. For a specific $x \in \{0, 1\}^{n}$, let's say we are interested in the probability $$p_x = |\langle x|~\text{Q}~|0^{n}\rangle|^{2}.$$ ...
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Prove that $\rho_{AB} \leq |B|(\rho_A\otimes I_B)$ for any bipartite state $\rho_{AB}$

I'm trying to prove the following statement but am lost on how to show it. For a quantum state $\rho_{AB}$ with marginal $\rho_A$, how can one show that $$ \rho_{AB} \leq|B|(\rho_A\otimes I_B)$$ where ...
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Data processing inequality for relative entropy in the presence of an amplitude damping channel

Consider the single qubit quantum depolarizing channel, given by $$T(\rho) = (1- p)\rho + p \frac{\mathbb{I}}{2}. $$ For an $n$ qubit state $\rho$, according to Definition 6.1 of this paper, the ...
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How to write the three-qubit GHZ state in the Pauli basis?

How can one write the GHZ state defined in Ket notation as $|\psi\rangle= \frac{1}{\sqrt{2}} \left(|000\rangle + |111\rangle\right)$, in terms of Pauli matrices $\sigma_{1},\sigma_{2},\sigma_{3}$?
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Given this code fragment, what is the probability that a measurement would result in $|0\rangle$?

Trying to understand below probability how it occured? qc = QuantumCircuit(1) qc.ry(3 * math.pi/4, 0) A. 0.8536 B. 0.5 C. 0.1464 D. 1.0 And the answer is C. But I ...
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Are complex amplitudes really needed?

Qubit amplitudes are defined as complex numbers. But in all tutorials I have recently read, only real numbers are used and everything works. So, if I completely forget the official 'complex' ...
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Are the states in the convex decomposition of a density matrix necessarily orthogonal?

In Nielsen and Chuang's QC&QI, I do not see a statement one way or another. In Steeb and Hardy's Problems and Solutions, orthogonality is asserted. If the $p_i$ in $\sum_i p_i |\psi_i\rangle\...
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Separable decomposition of states near the maximally mixed state

This question concerns the neighborhood of separable states around the maximally mixed state in a bipartite system; I will restate the theorem as it appears in Watrous' The Theory of Quantum ...
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1answer
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Statistical 'process' behind quantum gates

Thinking about gate creating entangled state, it looks to me that the inputs to the gate look like marginal/independent distributions, and the output looks like their joint distribution. Does this ...
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3answers
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Does each qubit correspond to some wave?

Reading about qubits, I see a lot of names related to waves from physics (amplitude, magnitude, phase, ...). Does it mean that each qubit corresponds to some wave? If yes, what is the mapping between ...
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Entanglement entropy and depth

I wanted to verify two intuitions about the entanglement entropy of quantum states. Consider an $n$ qubit quantum state, prepared by a depth $d$ circuit acting on $|0\rangle^{\otimes n}$ and a ...
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Closest quantum state with a fixed marginal: Analytical solution?

Let $\rho_{AB}$ be a bipartite state and let $\sigma_{B}$ be another state. What state $\tilde{\rho}_{AB}$ is closest to $\rho_{AB}$ and satisfies $\tilde{\rho}_B = \sigma_B$? We can define closeness ...
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What does the $\sigma_z$ operator correspond to in the context of a Transmon qubit?

In the context of the transmon qubit and an LC circuit with a coupling capacitor and driving voltage. We can write the charge operator $Q$ as $-iQ_{ZPF}\sigma_y$ (since we defined $Q$ using the ladder ...
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1answer
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Prove that $|(\langle \psi|_{A} \otimes \langle \phi|_{B})|\theta\rangle_{AB}|^{2}<1$ for entangled $|\theta\rangle_{AB}$

I am trying to show that $|\langle \psi|_{A} \otimes \langle \phi|_{B}|\theta\rangle_{AB}|^{2}<1$ given $|\theta\rangle$ is an entangled state, and as such has schmidt rank >1. Decomposing it, ...
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Volume law spread after Hamiltonian evolution

Start with an $n \times n$ lattice, with each qubit initialized to the state $|0\rangle$. Then, apply the Hadamard gate on each qubit. Then, evolve the system under the Hamiltonian \begin{equation} ...
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Getting different sets of plots for the teleportation algorithm in Composer and Lab

I tried recreating the quantum teleportation circuit (in the Qiskit Foundation Youtube Playlist) both in Lab and Composer and getting different plots for each. Here is circuit and histogram from ...
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1answer
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Deutsch-Josza on balanced 3 bit function

I am trying to understand the Deutsch-Jozsa algorithm for general $U_f$, but the following function is causing me trouble $f(x,y,z)=x\cdot y \oplus z$. It gives rise to the following quantum gate $U_f:...
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Is it possible to build a "Quantum Karnaugh Map"?

Is it possible to build something similar to the Karnaugh Map but instead of classical bits input and output with qubits input and output?
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2answers
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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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1answer
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How to measure a correlated operator $Z_1Z_2$?

I was reading this articl and I am stuck trying to understand equation $(60)$, which reads $$\langle\psi|\Lambda_{1,2}(X)Z_1\Lambda_{1,2}(X)|\psi\rangle=\langle\psi|Z_1Z_2|\psi\rangle$$ where $\Lambda(...
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QuTiP VS RK45: Which one gives the correct results for time-dependent systems?

I am writing a code for a quantum thermal machine which includes both coherent and dissipative time evolutions in its different stages of operation. However, evolving the system with "mesolve&...
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2answers
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How to create superposition of states with fixed parity with a quantum circuit?

I'm searching for a circuit to generate, starting from the $|00\, ...\,0\rangle$ state, an arbitrary superposition of all states with either even or odd parity. The gate choice is irrelevant for now, ...
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1answer
76 views

How to create a Bell state with asymmetric amplitudes using single-qubit and CNOT gates?

Is there a systematic way - in terms of a quantum circuit with single qubit and CNOT gates - to create a bell state with asymmetric amplitudes, e.g., $$ \alpha |00\rangle + \beta|11\rangle $$ where $\...
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1answer
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Does entanglement entropy follow a volume or an area law for 2D cluster states?

Consider a 2D cluster state defined on a rectangular lattice, which is universal for one way quantum computers. For a description of the state, see for example question 2 in this problem set. Now, ...
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For two-qubit systems, do we have $\langle 01|01\rangle = \langle 0|0\rangle\langle 1|1\rangle$?

I am new to quantum computing and I want to know the following: If I have a 2 qubit system in state e.g. $\left|01\right>$ and I want to calculate the probability of measuring e.g. $\left<01\...
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1answer
53 views

Universal resource for measurement based quantum computation

Consider universal resources for measurement based quantum computation, as defined here: We are now ready to formulate the following definition. A family $\Psi$ is called a universal resource for MQC ...
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What is the largest number of stabilizers a pure state can have?

What is the largest number of stabilizers a pure state can have? Elaborately put: Let $P(n)$ denote the Pauli group. Given an arbitrary pure state $|\psi\rangle$, what is the upper limit on how many ...
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1answer
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What is the difference between quantum "system", "register" and "Hilbert space"?

As far as I can tell, these terms are interchangeable but I am not sure of this. What is meant by each of the terms "quantum system", "quantum register" and "Hilbert space&...
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1answer
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Are $(|00\rangle-|11\rangle)/\sqrt2$ and $(|11\rangle-|00\rangle)/\sqrt2$ the same quantum state?

The state $(|00\rangle-|11\rangle)/\sqrt2$ is an entangled state. If we think about the state $(|11\rangle-|00\rangle)/\sqrt2$, is this also entangled, but with maybe a phase change? The above two can ...
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I am a beginner and therefore, these three H terms are confusing. "Hemitian", "Hamiltonian", and "Hilbert"

I am a beginner and therefore, these three H terms are confusing. "Hemitian", "Hamiltonian", and "Hilbert". Could someone give proper context and explain these terms . ...
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1answer
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How many minimum Quantum Rats are needed to figure out which bottle contains poison?

For the classical Poison and Rat puzzle, we need at least $\lceil\log_2({\rm bottles})\rceil$ rats to figure out the poisoned bottle. If we have Schrödinger’s quantum rats, can we use fewer rats(...
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1answer
49 views

From the final quantum state to the quantum circuit composition

When I build a quantum circuit and my initial state is the one composed only by zeros ($|000\ldots 0\rangle$), I have a final state $|\psi\rangle$ that is the result of the application of the quantum ...
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How do I construt a mixed state for an arbitrary symmetric matrix?

A symmetric matrix can be seen as a density matrix. If I have an arbitrary symmetrical matrix, can I use a quantum random access memory to construct a corresponding mixed state? What kind of quantum ...
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3answers
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Why can't we simulate a Qubit using classical computer?

I am completely a noob in terms of quantum computing, have watched several videos to understand what Quantum computers are trying to achieve. I am a programmer of classical computers. We have a ...
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1answer
33 views

What are the typical gate times for single-qubit and 2-qubit gates for ion trap, superconducting, neutral atom, photonic, spin QC?

What are the typical gate times for single-qubit and 2-qubit gates for -- ion trap, -- superconducting, -- neutral atom, -- photonic, -- spin quantum computers based on today's technologies?
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How many physical qubits are needed to encode a logical qubit on ion trap, superconducting, neutral atom, photonic QC?

How many physical qubits are needed to encode a logical qubit on an -- ion trap, -- superconducting, -- neutral atom, -- photonic, -- spin quantum computer based on today's technologies?
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1answer
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How to determine the threshold value of the quantum error correction code

How to determine the threshold value of the quantum error correction code, what is the specific method, such as surface code, how to determine the threshold value of the color code with a decoder, I ...
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2answers
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Closeness of $\rho$ such that $\text{Tr}(|\psi\rangle\langle\psi|\rho)\le1/2^n+{\cal O}(2^{-2n} )$ for all $|\psi\rangle$ to the maximally mixed state

Consider an $n$ qubit density matrix $\rho$ such that $$\text{Tr}(|\psi\rangle\langle \psi| ~\rho) \leq \frac{1}{2^{n}} + \mathcal{O}\left(\frac{1}{2^{2n}} \right), $$ for every $n$ qubit pure state $|...

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