Skip to main content

Questions tagged [quantum-state]

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

Filter by
Sorted by
Tagged with
43 votes
1 answer
2k views

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
  • 543
37 votes
4 answers
19k views

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?
user avatar
32 votes
2 answers
20k views

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
  • 854
31 votes
3 answers
7k views

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
  • 687
31 votes
4 answers
41k views

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
  • 1,007
30 votes
3 answers
10k 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
  • 937
30 votes
3 answers
4k views

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
  • 3,459
26 votes
2 answers
7k views

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
25 votes
4 answers
10k views

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?
C-Roux's user avatar
  • 888
23 votes
2 answers
3k views

Significance of The Church of the Higher Hilbert space

The term "Church of the Higher Hilbert Space" is used in quantum information frequently when analysing quantum channels and quantum states. What does this term mean (or, alternately, what does the ...
user3483902's user avatar
22 votes
5 answers
7k views

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
  • 323
22 votes
6 answers
6k views

Quantum states are unit vectors... with respect to which norm?

The most general definition of a quantum state I found is (rephrasing the definition from Wikipedia) Quantum states are represented by a ray in a finite- or infinite-dimensional Hilbert space over ...
Adrien Suau's user avatar
  • 4,987
22 votes
2 answers
21k views

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
21 votes
4 answers
4k views

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
20 votes
6 answers
1k views

Counterexamples in quantum information theory

As was already asked about in this phys.SE question many years ago—which, sadly, got closed and never received an answer—is there a collection of counterexamples in quantum information theory, "...
Frederik vom Ende's user avatar
20 votes
6 answers
2k views

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
  • 309
20 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
  • 731
20 votes
1 answer
2k views

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
18 votes
3 answers
7k views

What do the off-diagonal elements of a density matrix physically represent?

For simplicity, let's take a density matrix for a single qubit, written in the $\{|0\rangle,|1\rangle\}$ basis: $$ \rho = \begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10}^* & 1-\rho_{00} \...
KnightShuffler's user avatar
17 votes
1 answer
4k views

How can we keep Schrödinger's cat alive?

We know, Schrödinger's cat inside the box is in the equal superposition state of both alive and dead. We can express its state as $$|\text{cat}_\phi\rangle= \frac{|\text{alive}\rangle+e^{i\phi}|\text{...
Syed Emad Uddin's user avatar
17 votes
4 answers
2k views

How do the probabilities of each state change after a transformation of a quantum gate?

Quantum gates are represented by matrices, which represent the transformations applied to qubits (states). Suppose we have some quantum gate which operates on $2$ qubits. How does the quantum gate ...
ItamarG3's user avatar
  • 705
16 votes
1 answer
13k 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
16 votes
2 answers
3k views

Representation of real numbers in quantum computers

In classical binary computers, real numbers are often represented using the IEEE 754 standard. With quantum computers you can of course do this as well - and for measurements this (or a similar ...
blalasaadri's user avatar
  • 1,152
15 votes
4 answers
5k views

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
  • 1,517
15 votes
3 answers
891 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
  • 1,985
15 votes
2 answers
2k views

Is it true to say that one qubit in an entangled state can instantaneously affect all others?

When a qubit is measured, there is a ‘collapse of the wave-function’ as a result is randomly chosen. If the qubit is entangled with others, this collapse will also effect them. And the way it affects ...
James Wootton's user avatar
14 votes
5 answers
3k views

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
14 votes
2 answers
5k 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
1 answer
6k views

What is a stabilizer state?

I am reading through the paper "Direct Fidelity Estimation from Few Pauli Measurements" (arXiv:1104.4695) and it mentions 'stabilizer state'. "The number of repetitions depends on the ...
Quantum Guy 123's user avatar
14 votes
2 answers
759 views

What makes quantum computations different from randomized classical computations?

One of the many thing that confuse me in the field of QC is what makes the measurement of a qubit in a quantum computer any different than just choosing at random (in a classical computer) (that's not ...
ItamarG3's user avatar
  • 705
14 votes
2 answers
7k views

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.
Shweta Aggrawal's user avatar
14 votes
3 answers
900 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
13 votes
2 answers
2k views

Does quantum computing already possess the level of abstraction to be explicable even without knowledge of physics?

Currently, quantum computer science (in contrast to classical computer science) can mostly only be understood if one has a good inside knowledge of physics, or more precisely quantum physics. Only ...
Tetragrammaton's user avatar
13 votes
1 answer
702 views

What are min and max overlaps of a maximally entangled state with a separable state?

Let $A,B$ be Hilbert spaces of dimension $d$. Let $\rho$ be some separable quantum state of the composite system $AB$. Given a maximally entangled state: $$\vert\phi\rangle = \frac{1}{\sqrt{d}}\sum_{i=...
Jules's user avatar
  • 133
13 votes
1 answer
1k views

Does the trace distance have a geometric interpretation?

Consider the trace distance between two quantum states $\rho,\sigma$, defined via $$D(\rho,\sigma)=\frac12\operatorname{Tr}|\rho-\sigma|,$$ where $|A|\equiv\sqrt{A^\dagger A}$. When $\rho$ and $\sigma$...
glS's user avatar
  • 25.5k
13 votes
3 answers
870 views

What are the possible ways to visualise large, entangled states?

What are the prominent visualizations used to depict large, entangled states and in what context are they most commonly applied? What are their advantages and disadvantages?
SLesslyTall's user avatar
  • 1,646
13 votes
2 answers
1k views

How do you rotate a Fock state qubit?

I read that a qubit can be encoded in a Fock state, such as the presence or absence of a photon. How do you perform single qubit rotations on Fock states?
Daniel Tordera's user avatar
13 votes
3 answers
7k views

Why does a Hamiltonian have to be Hermitian?

Starting from: $$ -i\hbar \frac{d|\psi⟩}{dt} = H|\psi⟩ $$ I was able to do some working to prove that $U$ in the corresponding discrete representation $$ U(t_1,t_2) = exp\frac{-iH(t_2-t_1)}{\hbar} $...
Alexander Soare's user avatar
12 votes
5 answers
798 views

Advances on imperfect quantum copying

It is known by the no-cloning theorem that constructing a machine that is able to clone an arbitrary quantum state is impossible. However, if the copying is assumed not to be perfect, then universal ...
Josu Etxezarreta Martinez's user avatar
12 votes
3 answers
2k views

Why do we have to uncompute rather than simply set registers to zero?

In implementing a quantum subroutine it is important to uncompute temporary registers after use, to ensure the output state of the subroutine is not entangled with them (which would affect its ...
Sideshow Bob's user avatar
12 votes
1 answer
510 views

Prove that the trace distance is upper-bounded by the Hilbert-Schmidt distance

In (Haah et al. 2015), in the third page, second column, the authors use the following result: given a pair of states $\rho,\sigma$, we have $$ \|\rho-\sigma\|_1 \le 2\sqrt{\min(\operatorname{rank}(\...
glS's user avatar
  • 25.5k
12 votes
1 answer
362 views

Sampling random circuits vs Solovay-Kitaev compiler

Suppose I want to obtain a gate sequence representing a particular 1 qubit unitary matrix. The gate set is represented by a discrete universal set, e.g. Clifford+T gates or $\{T,H\}$ gates. A well ...
Yaroslav Kharkov's user avatar
12 votes
1 answer
868 views

Embedding classical information into norm of a quantum state

According to An introduction to quantum machine learning (Schuld, Sinayskiy & Petruccione, 2014), Seth Lloyd et al. say in their paper: Quantum algorithms for supervised and unsupervised machine ...
Sanchayan Dutta's user avatar
11 votes
2 answers
2k views

State produced by spontaneous parametric down-conversion (SPDC)

I'm researching SPDC's efficacy for use in an optical quantum computing model and I've been trying to figure out exactly what state the photons are in when they come out (as represented by a vector, ...
auden's user avatar
  • 3,459
11 votes
2 answers
811 views

What does it mean for a density matrix to "act on a Hilbert space $\mathcal{H}"$?

For a Hilbert space $\mathcal{H}_A$, I have seen the phrase density matrices acting on $\mathcal{H}_A$ multiple times, e.g. here. It is clear to me that if $\mathcal{H}_A$ has finite Hilbert ...
Peter's user avatar
  • 519
11 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
  • 503
11 votes
2 answers
1k views

What are useful resources about the geometric of qutrits and its relation with Gell-Mann matrices?

I need some useful sources about the geometry of qutrit. Specifically related to the Gell-Mann matrix representation.
Azadeh Zohrabi's user avatar
11 votes
2 answers
773 views

What is the complexity of determining if a state is entangled?

I have been looking around for an answer to this question but can't really come up with anything. Given some oracle, $U$, that maps $| 0 \rangle$ to $| \psi \rangle$, is there some algorithm that ...
Loic Stoic's user avatar
11 votes
2 answers
767 views

In shadow tomography, how is the state reconstructed from its shadows?

I'm reading Huang et al. (2020) (nature physics), where the authors present a version of Aaronson's shadow tomography scheme as follows (see page 11 in the arXiv version): We want to estimate a state $...
glS's user avatar
  • 25.5k
11 votes
1 answer
326 views

Are qudit graph states well-defined for non-prime dimension?

Qudit graph states are $d$-dimension generalisations of qubit graph states such that each state is represented by a weighted graph $G$ (with no self-loops) such that each edge $(i, j)$ is assigned a ...
SLesslyTall's user avatar
  • 1,646

1
2 3 4 5
34