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?
LeWoody's user avatar
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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(|...
nippon's user avatar
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10 votes
4 answers
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How is quantum teleportation generalized to qudits?

In an answer to a previous question, Generalization for n quantum teleportations, Craig Gidney states: The more complicated way to generalize teleportation is figuring out how to make it work on ...
user820789's user avatar
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28 votes
3 answers
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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) ...
Norrius's user avatar
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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 $\theta$ ...
Ba. Taj's user avatar
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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 ...
Sanchayan Dutta's user avatar
21 votes
2 answers
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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&#...
Sanchayan Dutta's user avatar
4 votes
3 answers
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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 ...
Sanchayan Dutta's user avatar
8 votes
1 answer
3k views

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 ...
Martin Vesely's user avatar
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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 ...
chetan waghmare's user avatar
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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 ...
Koder101's user avatar
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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 | ...
wanderingmathematician's user avatar
7 votes
3 answers
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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 ...
agiri's user avatar
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2 answers
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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 ...
Eara Shahirah's user avatar
35 votes
4 answers
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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?
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22 votes
5 answers
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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 $4$ ...
weekens's user avatar
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How to show whether a bipartite high-dimensional 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, ...
user820789's user avatar
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6 votes
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Can we rotate Bloch vectors for qudits like we do with qubits in the Bloch sphere?

I have been looking into the Bloch vectors for qudits and have been wondering if we can do rotations that are similar to the rotations in the qubit Bloch sphere. Like, once we create a Bloch vector ...
Parmeet Singh EP 066's user avatar
6 votes
1 answer
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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 ...
BlackHat18's user avatar
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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 ...
rkoni's user avatar
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1 answer
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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 ...
Alex's user avatar
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28 votes
3 answers
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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 ...
auden's user avatar
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25 votes
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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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6 answers
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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 ...
Mithical's user avatar
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19 votes
1 answer
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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,...
TheAmazingKitchen's user avatar
19 votes
4 answers
2k views

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\...
incud's user avatar
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7 votes
1 answer
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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 ...
BlackHat18's user avatar
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6 votes
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490 views

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 $2^n$. ...
Nikita Nemkov's user avatar
5 votes
1 answer
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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 ...
Ba. Taj's user avatar
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14 votes
5 answers
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What is the difference between a qudit system with d=4 and a two-qubit system?

I understand that a qudit is a quantum $d$-state system. If $d=4$, is this exactly the same as a two-qubit system, which also presents $4$ quantum states? The Hilbert space is the same, right? Are ...
Daniel Tordera's user avatar
10 votes
2 answers
3k views

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 ...
Aman's user avatar
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8 votes
1 answer
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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?
Archil Zhvania's user avatar
7 votes
3 answers
447 views

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 ...
Sam Jaques's user avatar
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7 votes
1 answer
682 views

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 ...
Quantum Guy 123's user avatar
6 votes
5 answers
1k views

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.:...
Jared's user avatar
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6 votes
3 answers
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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 ...
Anurag Vohra's user avatar
5 votes
1 answer
1k views

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) & ...
Martin Vesely's user avatar
4 votes
1 answer
279 views

Prove that uniformly random states have moments ${\bf E}_\psi|\langle x|\psi\rangle|^{2t}\sim1/\binom d t$

Im looking for the moments of Haar random states. Is it true that $\textbf{E}_{\psi\sim \text{Haar}}|\langle x| \psi\rangle|^{2t}\sim \frac{1}{\binom{d}{t}}?$ How does one prove this?
postasguest's user avatar
4 votes
1 answer
331 views

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's user avatar
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4 votes
4 answers
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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 ...
Muhammad Ikhwan Perwira's user avatar
4 votes
1 answer
2k views

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(\...
Isaac Khor's user avatar
3 votes
1 answer
361 views

Averaging over a single Haar-random unitary applied $t$ times

I'm trying to compute the following integral: $$\int U^{\otimes t}\left|x_1,\cdots,x_t\middle\rangle\middle\langle x'_1,\cdots,x_n'\right|\left(U^\dagger\right)^{\otimes t}\,\mathrm{d}\mu(U)$$ Where $\...
Tristan Nemoz's user avatar
3 votes
2 answers
1k views

Is quantum computing limited to a superposition of only two states?

From Wikipedia: A qubit is a two-state quantum system [...] There are two possible outcomes for the measurement of a qubit — usually $0$ and $1$, like a bit. The difference is that whereas the state ...
Sanchayan Dutta's user avatar
2 votes
1 answer
526 views

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 ...
Quantum Guy 123's user avatar
1 vote
1 answer
1k views

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 ...
M. Al Jumaily's user avatar
27 votes
3 answers
8k views

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.
김동민's user avatar
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15 votes
3 answers
855 views

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 $n$ ...
Kiro's user avatar
  • 1,955
15 votes
1 answer
12k views

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 ...
Archil Zhvania's user avatar
14 votes
2 answers
4k views

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 ...
Hamza's user avatar
  • 291
14 votes
3 answers
853 views

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 ...
Quantum spaghettification's user avatar

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