"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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### Schmidt vectors for random quantum states

Consider a random quantum circuit $U$ over $n$ qubits, drawn from the Haar measure. Consider the quantum state $$|\psi\rangle = U |0^{n}\rangle.$$ Now, partition $n$ into two and consider the Schmidt ...
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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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### Quantum advantage with only Clifford gates (Gottesman Knill theorem)

Let's say I want to solve a computational task which input can be encoded in $n$ bits of information. The look for a quantum advantage is (usually) asking to find a quantum algorithm in which there ...
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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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### 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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### How can one define contextuality within the circuit model?

It is in general believed that contextuality is one of the quantum resource that provides the quantum advantage. A context is usually defined in terms of a set of commuting observables. The quantum ...
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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 \rho_1 = \mathbb{E}[|\psi\...
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### How powerful are boundedly many $T$-gates?

For a natural number $k$ (0 is a natural number), let $T_k$ be the collection of all languages that can be efficiently decided by quantum circuits consisting of Clifford gates and at most $k$ $T$-...
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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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### Why exactly are variational algorithms considered promising?

There is obviously a great deal of work happening at the moment on variational quantum algorithms. However, I'm struggling to understand why exactly are they considered promising? Looking through some ...
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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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### 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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### Is there a practical architecture-independent benchmark suitable for adversarial proof of quantum supremacy?

Recent quantum supremacy claims rely, among other things, on extrapolation, which motivates the question in the title, where the word "adversarial" is added to exclude such extrapolation-...
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### Spoofing XQUATH with the Feynman method

Consider the XQUATH conjecture for random quantum circuits, as mentioned here. (XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There is no polynomial-time classical algorithm that ...
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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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### 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 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Q-means, QRAM and how it helps algorithmic speedup

I am trying to understand how QRAM will help improve algorithm performance. I am reading a paper on Q-means classification, but I have noticed that some other algorithms (Grovers) seem to have a ...
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### Why does QAOA achieve quantum supremacy in an algorithmic sense?

In the paper Quantum Supremacy through the Quantum Approximate Optimization Algorithm the authors claim (last sentence of page 15): "If [...] the QAOA outperforms all known classical algorithms ...
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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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### Could random quantum circuits be efficiently approximately simulated?

Google's landmark result last year was to compute a task with a quantum computer that a classical computer could not compute, and they chose random circuit sampling. Part of their justification was ...
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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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### What is the advantage of quantum machine learning over traditional machine learning?

Why exactly is machine learning on quantum computers different than classical machine learning? Is there a specific difference that allows quantum machine learning to outperform classical machine ...
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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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### Does quantum computers give any advantage over classical computers in Sudoku?

To my basic knowledge I know that solving a generalized Sudoku problems is an NP-complete problem so, is there any possible way quantum computers give an advantage over classical computers in this ...
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### Is the BB84 protocol an example of "quantum supremacy"?

This is a fairly broad question, I hope it fits here. I am wondering if the BB84 protocol is an example of "quantum supremacy", ie. something a quantum computer can do but something that is assumed a ...
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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 ...
### Why Google has used $\sqrt{X}$ and $\sqrt{Y}$ instead of $X$ and $Y$ in supremacy experiment?
In supremacy experiment Google has used $\sqrt{X}$ and $\sqrt{Y}$ as two of their single qubit gates (paper). So My questions are: Is there any specific reason for choosing these gates and not $X$...