Questions tagged [quantum-state]
Questions about or related to quantum states. Consider using the density-matrix tag when relevant.
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What is the difference between a relative phase and a global phase? In particular, what is a phase?
I know that $re^{i\theta} = x + iy$ for any complex number $x + iy$ by Euler's formula. How do you calculate relative and global phase?
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What is the difference between superpositions and mixed states?
My understanding so far is: a pure state is a basic state of a system, and a mixed state represents uncertainty about the system, i.e. the system is in one of a set of states with some (classical) ...
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Find the $\theta$ and $\phi$ values on the Bloch sphere corresponding to the state $\frac{1+i}{2}|0\rangle+\frac1{\sqrt2}|1\rangle$
If I have the following state:
$$
\left| \varphi \right>=\frac{1}{\sqrt{2}}\left(\left(\frac{1+i}{\sqrt{2}} \right)\left| 0 \right> + \left| 1\right>\right)
$$
How can I find the $\theta$ ...
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General construction of $W_n$-state
Two of the most well known entangled states are the GHZ-state $|\psi\rangle = 1/\sqrt{2}\left( |0\rangle^{\otimes n} + |1\rangle^{\otimes n}\right)$ and the $W_n$-state, with $W_3 = 1/\sqrt{3}\left(|...
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Density matrices for pure states and mixed states
What is the motivation behind density matrices? And, what is the difference between the density matrices of pure states and density matrices of mixed states?
This is a self-answered sequel to What&#...
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Purity of mixed states as a function of radial distance from origin of Bloch ball
@AHusain mentions here that the purity of a qubit state can be expressed as a function of the radius from the center of a Bloch sphere. The state corresponding to the origin is maximally mixed whereas ...
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How to perform quantum state tomography on two qubits?
I would like to do a quantum tomography on two qubit states.
Recently, I successfully did so for one qubit based on Nielsen-Chuang. They advise to use this formula for one qubit density operator ...
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Why are half angles used in the Bloch sphere representation of qubits?
Suppose we have a single qubit with state $| \psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle$. We know that $|\alpha|^2 + |\beta|^2 = 1$, so we can write $| \alpha | = \cos(\theta)$, $| \beta | ...
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How can infinite information be theoretically encoded or stored in a single qubit?
I've just gotten started with Nielsen and Chuang's text, and I'm a little stuck. They mention that theoretically, it would be possible to store an infinite amount of information in the state of a ...
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What is the result of measuring $\sigma_x$ on the state $|01\rangle+|10\rangle$?
I confused about how to calculate the probabilities and getting a certain result of measuring Bell's states with Pauli matrices as the operator. When you measure something, the state involved would be ...
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What is meant by the term "computational basis"?
What is meant by the term "computational basis" in the context of quantum computing and quantum algorithms?
user1039
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What's the difference between a pure and mixed quantum state?
As per my limited understanding, a pure state is the quantum state where we have exact information about the quantum system. And the mixed state is the combination of probabilities of the information ...
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Why does the "Phase Kickback" mechanism work in the Quantum phase estimation algorithm?
I've probably read the chapter The quantum Fourier transform and its applications from Nielsen and Chuang (10 th anniversary edition) a couple of times before and this took this thing for granted, but ...
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How can I build a circuit to generate an equal superposition of 3 outcomes for 2 qubits?
Given a $2$ qubit-system and thus $4$ possible measurements results in the basis $\{|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle\}$, how can I prepare the state, where:
only $3$ of these $4$ ...
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Preparing a quantum state from a classical probability distribution
Suppose I have a black-box unitary $U_p$ which is described as follows: given a finite probability distribution $p:\{1,\ldots,n\}\rightarrow \mathbb{R}_{\geq0}$, where $\sum_{x=1}^n p(x)=1$, the ...
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Help Identifying a Gate In Nielsen and Chuang
I am seeking help to identify the oracle gates listed in this example. I understand that the right-most one is a toffoli gate, but what are the other ones? Specifically, I do not understand what a ...
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How to implement the Circuit of Steane's code for Quantum Error Correction?
I have referred this same question here 'Circuit for implementing Steane's code for Quantum Error Correction' . But the answer discusses the circuit to compute the syndrome and not clearly the ...
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Quantum machine learning after Ewin Tang
Recently, a series of research papers have been released (this, this and this, also this) that provide classical algorithms with the same runtime as quantum machine learning algorithms for the same ...
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How does measurement of one qubit affect the others?
To represent a quantum computer's state, all the qubits contribute to one state vector (this is one of the major differences between quantum and classical computing as I understand it). My ...
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How to calculate circuit depth properly?
Is the circuit depth the longest sequence of gates applied on one of the qubits?
Or is it something more complicated?
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No-cloning theorem and distinguishing between two non-orthogonal quantum states
I'm currently reading Nielsen and Chuang's Quantum Computation and Quantum Information and I'm not sure if I correctly understand this exercise (on page 57) :
Exercise 1.2: Explain how a device which,...
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Alternative to Bloch sphere to represent a single qubit
In order to represent the single qubit $|\psi\rangle$ we use an unitary vector in a $\mathbb{C}^2$ Hilbert space whose (one of the) orthonormal base is $(|0\rangle, |1\rangle)$.
We can draw $|\psi\...
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What is a qubit?
What is a "qubit"? Google tells me that it's another term for a "quantum bit". What is a "quantum bit" physically? How is it "quantum"? What purpose does it serve in quantum computing?
Note: I'd ...
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Compute the output of the quantum teleportation circuit
Sender and receiver use the teleportation protocol, where the sender teleports a quantum state $\left| \varphi \right>=\alpha\left| 0 \right> + \beta \left|1\right>$ to the receiver.
I want ...
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What separable $\rho$ only admit separable pure decompositions with more than $\mathrm{rank}(\rho)$ terms?
As shown e.g. in Watrous' book (Proposition 6.6, page 314), a separable state $\rho$ can always be written as a convex combination of at most $\mathrm{rank}(\rho)^2$ pure, separable states.
More ...
glS♦
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How can I calculate the inner product of two quantum registers of different sizes?
I found an algorithm that can compute the distance of two quantum states. It is based on a subroutine known as swap test (a fidelity estimator or inner product of two state, btw I don't understand ...
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How to show that an n-level system is entangled?
"How do I show that a two-qubit state is an entangled state?" includes an answer which references the Peres–Horodecki criterion. This works for $2\times 2$ and $2\times3$ dimensional cases; however, ...
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How to find a density matrix of a qubit?
If we are given a state of a qubit, how do we construct its density matrix?
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Are projective measurements the only optimal measurements to discriminate between two states?
Consider two density matrices $\rho$ and $\sigma$. The task is to distinguish between these two states, given one of them --- you do not know beforehand which one.
There is an optimal measurement to ...
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Forming states of the form $\sqrt{p}\vert 0\rangle+\sqrt{1-p}\vert 1\rangle$
I'm curious about how to form arbitrary-sized uniform superpositions, i.e.,
$$\frac{1}{\sqrt{N}}\sum_{x=0}^{N-1}\vert x\rangle$$
for $N$ that is not a power of 2.
If this is possible, then one can ...
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Is it true that for a quantum algorithm to be efficient it must feature a highly entangled state at some point?
I'm wrapping my head around how and why quantum computers can provide advantage over classical. A basic and naive argument is that the dimension of the Hilbert space of $n$ qubits grows as $2^n$. ...
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How to calculate the average fidelity of an amplitude damping channel
An answer to this question shows how to calculate the average fidelity of a depolarizing channel. How would one go about calculating this for an amplitude dampening channel? I tried working out the ...
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How do I get the Unitary matrix of a circuit without using the 'unitary_simulator'?
I am using jupyter notebook and qiskit. I have a simple quantum circuit and I want to know how to get the unitary matrix of the circuit without using 'get_unitary' from the Aer unitary_simulator. i.e.:...
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What is the $\lambda$ parameter in the $U3$ gate used for?
The most general single qubit gate is $\mathrm{U3}$ given by matrix
$$
\mathrm{U3}=
\begin{pmatrix}
\cos(\theta/2) & -\mathrm{e}^{i\lambda}\sin(\theta/2) \\
\mathrm{e}^{i\phi}\sin(\theta/2) & ...
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Random quantum states and Schur-Weyl duality
Consider the following density matrix over $n$ qubits, with $C$ being a single qubit operator:
$$
\rho_{n} = \int_{C \sim \text{Haar}} \big(C|0\rangle\langle0|C^\dagger\big)^{\otimes n} dC.
$$
Let's ...
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How to get the Bloch sphere angles given an arbitrary qubit?
I understand that given a pure state $ |\psi\rangle$, we can express it in terms of two angles $\theta$ and $\varphi$ such that $|\psi\rangle = \cos(\theta/2)|0\rangle + \mathrm{e}^{i\varphi}\sin(\...
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Simulate a quantum channel with a certain fidelity
I am looking for an easy-to-use framework for simulating a quantum channel that can accept the desired average fidelity of the channel as input.
For example, if I want a channel with 98% average ...
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Why do computer scientists care about the phase of qubits?
When I design some classical register, flip-flop, binary counter, small byte of RAM, etc from scratch with classical logic gate, I never deal with such binary direction because classical bit doesn't ...
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How do two qubit states differing by a global phase relate to each other?
I have looked at the following:
What is the difference between a relative phase and a global phase? In particular, what is a phase?
Global and relative phases of kets in QM
Global phases and ...
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What are magic states?
I wonder what are magic states, and a magic state gadget. While I'm reading a paper, these terms frequently appear.
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How to input 2 qubits in 2 Hadamard gates?
Let's say we have a circuit with $2$ Hadamard gates:
Let's take the $|00\rangle$ state as input. The vector representation of $|00\rangle$ state is $[1 \ 0 \ 0 \ 0]$, but this is the representation ...
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How to compactly represent multiple qubit states?
Since access to quantum devices capable of quantum computing is still extremely limited, it is of interest to simulate quantum computations on a classical computer. Representing the state of $n$ ...
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Is acting with a positive map on a state not part of a larger system allowed?
In the comments to a question I asked recently, there is a discussion between user1271772 and myself on positive operators.
I know that for a positive trace-preserving operator $\Lambda$ (e.g. the ...
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How and why does swap test works?
I am having some trouble understanding why a SWAP test would work. I meant I read that and understood the concepts as follows:
If the two input states are equal, the output register always results
in ...
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On the distribution of the fidelity of a random product state with an arbitrary many-qubit state
Consider an arbitrary $n$-qubit state $\lvert \psi \rangle$. How much do we understand about the probability distribution of the fidelity of $\lvert \psi \rangle$ with a tensor product $\lvert \alpha \...
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Measuring in standard basis meaning
What does it mean to measure a qubit (or multiple qubits) in standard basis?
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$n$ qubit vs. a $d=2^n$ qudit states and measurements
The pure states of a qudit inhabit the $\mathbb{CP}(d-1)$ manifold.
Is it true that the pure states of $n$ qubits live on the $\mathbb{CP}(2^n-1)$ manifold?
If the answer to the first question is yes,...
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Does the circuit with qubit-wise CZ gates compute the inner product of two states? If not, is there another circuit that does?
I've been searching for a quantum algorithm to compute the the inner product between two $n$-qubit quantum states, namely $\langle\phi|\psi\rangle$, which is in general a complex number.
One can get $|...
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Is there anything practical that can be done with a single qubit?
Is there anything practical that can be done with a single qubit? And by "practical," I mean a problem that can be solved or information that can be stored.
I realize that one practical thing that ...
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How to get subspace of quantum circuit?
How can I get a subspace of a quantum circuit? More precisely, I'm dealing with quantum circuit with data qubits ('q') and ancilla qubits ('anc'), such as $(q_0,q_1,...,q_n,anc_0,..anc_m)$.
After some ...
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