Questions tagged [quantum-advantage]

"Quantum advantage" or "quantum supremacy" is the potential ability of quantum computing devices to solve problems that classical computers practically cannot.

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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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What is the role of choosing the single-qubits randomly in Google quantum supremacy experiment?

In supremacy paper and part D of section VII of supplementary information (below), it is said that there is a pseudo-random number generator that is initialized with a seed called $s$; And then the ...
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Reduced density matrix of a Haar random state and its Schmidt decomposition

Consider a Haar random quantum state $|\psi\rangle$. Note that $$\rho =\mathbb{E}[|\psi\rangle\langle \psi|] = \frac{\mathbb{I}_{n}}{2^{n}}.$$ $\mathbb{I}_n$ is the identity operator on $n$ qubits. ...
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If a hybrid classical+quantum algorithm can achieve quantum advantage, does this mean that the quantum part alone can?

Take for example a variational algorithm which has a classical optimization part and a quantum sampling part. In principle, the quantum part can be simulated by another classical computer given ...
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Feynman method and polynomial time algorithm for XQUATH

Consider the Feynman algorithm for simulating quantum circuits, as given here. Consider the XQUATH conjecture for random quantum circuits from here, given by (XQUATH, or Linear Cross-Entropy Quantum ...
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Boson Sampling with a single beamsplitter

In Boson Sampling as first proposed by Aaronson and Arkhipov in arxiv.org/abs/1011.3245, the interferometer is made up of phase shifters and beamsplitters. As these gates are universal, drawing the ...
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Generally speaking, are quantum speedups always due to parallelization of a given problem?

We know that quantum computers use the wave-like nature of quantum mechanics to perform interference. Sometimes we can use this interference to perform specific algorithms that will cause enough ...
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Interpretations of quantum computing power [closed]

Over the years I encountered different explanations of quantum computing advantage over classical computers. But I am not sure which explanations are in fact valid and which are not. Quantum ...
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At what depth and for what architecture are random quantum circuits $1$-designs?

I was confused about something related to quantum $1$ designs. Let us recap two facts we know about random circuit ensembles that form a $1$ design. $1$ design, for a quantum circuit over $n$ qubits, ...
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What is the problem in demonstrating quantum supremacy?

I am new to the quantum world and it's computing. But it accidentally hit in my mind that DWave built a quantum computer with 2000 qubits which can be use to simulate the whole observable universe or ...
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How does successfully sampling from a random quantum circuit invalidate the Extended Church-Turing Thesis?

According to these lecture notes from Berkeley, the Extended Church-Turing Thesis (ECT) asserts that: ...any "reasonable" model of computation can be efficiently simulated on a standard model such ...
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Quantum hardness of XQUATH conjecture

Consider the XQUATH conjectures, as defined here (https://arxiv.org/abs/1910.12085, Definition 1). (XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There is no polynomial-time ...
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Confusion about the output distribution of Haar random quantum states

Consider a Haar random quantum state $|\psi \rangle$. I was confused between two facts about $|\psi \rangle$, which appear related: Consider the output distribution of a particular $n$-qubit $|\psi \...
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How do quantum bits increase computational power?

I'm new to quantum computing, I'm learning how to use Qiskit. I'm trying to understand better how exactly the quantum characteristics of quantum computer help to increase its computational power. I ...
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Random quantum circuits and general efficient POVM measurement

Let's consider a random quantum circuit $C$, applied to the $n$ qubit initial state $|0^{n}\rangle$, producing the state $|\psi\rangle$. Consider a general efficiently implementable $m$-outcome POVM ...
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Average output state of random quantum circuits

Let $|\psi\rangle = C_1 |0^{n}\rangle$ be a quantum state, such that $C_1$ is a Haar random unitary circuit. Consider a density matrix $\rho$ as follows \begin{equation} \rho_1 = \mathbb{E}[|\psi\...
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Can I use Grover's algorithm on overlapping sets of qubits?

Let's say I have 3 qubits: $q_1,q_2,q_3$. I want to apply Grover's algorithm on q1,q2, such that q1,q2 $\neq$ 10 and do the same for q2,q3, so that q2,q3 $\neq$ 11. The final possible combinations of ...
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Grover Algorithm vs Classical Search Algorithms

If Grover algorithm has a better speed than classical search algorithms, would it be an example of where Quantum computers outruns classical computers? Can we use Grover Algorithm in real world ...
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In Google's Quantum Supremacy experiment, what if we use $\theta=45°$ for two-qubit $f_{sim}$ gates?

In Google's Quantum Supremacy experiment, they use $f_{sim}$(fermionic-simulation) gates with $\theta=90°$ and $\phi=30°$ as their two-qubit gates. What if we use $\theta=45°$ for the two-qubit $f_{...
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What is the explicit best known quantum algorithm for LWE?

Consider the learning with errors(LWE) problem which is known to be hard for quantum computers. Let $q \geq 2$ be a prime integer. Consider us being given (polynomially many samples of) either: $$A, ...
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Spreading of entanglement with depth for Haar random states

Consider a Haar random quantum state of depth $d$. Consider any bipartition of this state. According to this paper (page $2$): Haar-random states on $n$ qudits are nearly maximally entangled across ...
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Relation between approximate counting and sampling

Consider the following statement of Stockmeyer counting theorem. Given as input a function $f:\{0, 1\}^{n} \rightarrow \{0, 1\}^{m}$ and $y \in \{0, 1\}^{m}$, there is a procedure that runs in ...
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When is a Quantum Computer Slower Than a Classical Computer?

Someone offhandedly mentioned to me that quantum computers are sometimes significantly (I guess they meant asymptotically) slower than classical computers. Unfortunately, I didn't get any arguments ...
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Can quantum computers speed up parsing?

Can quantum computers offer Grover-like speed ups in parsing of context-free languages? For instance, general CFLs can be parsed in $O(n^3)$ with standard algorithm like https://en.wikipedia.org/wiki/...
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When we do a linear fit, what is the correlation coefficient of the estimated parameters?

In Google's quantum supremacy experiment, supplementary Section VIIIH, they calculate the correlation coefficient of the linear fit coefficients $p_0$,$p_1$. I can't figure out the definition of this ...
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Would quantum computers be more efficient at solving circular reference problems than classical computers?

A circular reference is when a certain value either refers to itself or a value refers to a value that refers to it. An example of a circular reference problem would be $x=f(x)$. One way to solve ...
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How does a quantum computer execute a process by leveraging superposition?

I understand in plain terms superposition and entanglement, but I'm very unclear how either of these could work as a means to increase computation power. A helpful metaphor is that of the maze. A ...
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Dirichlet distribution: posteriors and priors of distribution

Let $|\psi\rangle \in \mathbb{C}^{2n}$ be a random quantum state such that $ |\langle z| \psi \rangle|^{2} $ is distributed according to a $\text{Dirichlet}(1, 1, \ldots, 1)$ distribution, for $z \in \...
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What is the difference between classical and quantum computers as well as computing (permuting) itself?

Theoretically and for sure physically (I know the quantum physics behind it) I know something about it. But not that much. That's why I ask the question here. I'm very interested. The only answer to ...
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Optimality of the SWAP test versus weak Schur sampling for testing unitarily invariant properties

Consider the following setting. I am either given the density matrix $|\psi\rangle \langle \psi|^{\otimes k}$ or the density matrix $\frac{\mathbb{I}^{\otimes k}}{2^{nk}}$, where $\mathbb{I}$ is the $...
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Has any analogue quantum simulator showed quantum advantage yet?

Quantum advantage/supremacy was achieved by Google using a quantum computer and more recently by Pan Jianwei's group using photons. So I was wondering, has any analog quantum simulator showed quantum ...
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What is the reason for the exponential speed-up of quantum computers? [duplicate]

In quantum computers the following two effects should be seen: If an operator acts on an arbitrary qubit $Q_n$ of a quantum system $S$ consisting of several qubits than we get a new quantum system $S'...
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Quantum supremacy: shallow depth Haar random circuits and unitary designs

I had a confusion about shallow depth Haar random quantum circuits. In this paper, in Section B (related works), it is mentioned that Haar random quantum circuits form approximate $2$-designs only ...
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What is the main advantage of using the Variational Quantum Eigensolver over a classical algorithm?

What is the main advantage of using the Variational Quantum Eigensolver (quantum computing) over a classical algorithm? I know a key fact is the speed-up, but how is this speed-up quantised.
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Is it possible to design a Quantum Computing Advantage to deploy an application on the web?

I need to understand the frontier and practical applications of quantum computing. Is it possible to design a Quantum Computing Advantage to deploy an application on the web, such as a browser, ...
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How does Deutsch Oracle shows quantum supremacy?

I learnt from this lecture (at 33:20) that Deutsch Oracle is way faster on quantum computers than on classical computers. However, it seems to me that this is just due to smart structuring of input ...
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Comparing QSVM & Classic SVM on BigData. Quantum Supremacy

I work on comparing QSVM and Classic SVM (SKlearnSVM) with using Qiskit. I have to show quantum supremacy at 400000-500000 samples but I don't get good results. I have problem with long time training ...
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How are quantum computers more powerful than classical computers? [duplicate]

I feel the answer to this question is just out of reach - I "understand" the implication that a quantum computer uses all combinations of bits simultaneously compared to a classic computer, ...

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