# Tag Info

13

There are a continuous set of possible states for $n$ qubits, each of which can be expressed as a superposition of the $2^n$ basis states. Mostly of these states are highly entangled, and would require highly complex circuits to create (assuming the standard gate set of single qubit rotations and two or three qubit entangling gates). These circuits would ...

6

The Solovay-Kitaev algorithm is not practical. It is very useful theoretically because it proves that once you have a "dense" set of quantum gates (i.e. a set with which you can approximate any other quantum gate) you can approximate up to an arbitrary precision and quickly any quantum gate. In practice, the Solovay-Kitaev works as follow: Fill the space ...

5

There are a couple variants of the HOG test. "Old HOG" computed the proportion of unique samples whose probability is larger than the median probability of the distribution. It then compares that proportion to a threshold, e.g. 2/3. If you have enough larger-than-median outputs, you pass the test. "New HOG" instead computes the mean of the probabilities of ...

5

What does "obtaining samples" mean in this context? The same thing it means in a more classical context. Consider the probability distribution of the possible outcomes of a (possibly biased) coin flip. Sampling from this probability distributions means to flip the coin once and record the result (head or tail). If you sample many times, you can retrieve ...

5

TL/DR: The two-qubit gates are going by the moniker "Sycamore gates" in the paper, and it appears that they would ideally want to explore more of the $(\phi, \theta)$ phase-space but for their purposes (of quantum supremacy) their current Sycamore gate is sufficient. The pattern of gates $\mathrm{ABCDCDAB}$ was chosen to avoid "wedges" and maximize/optimize ...

4

A computational task doesn't have to have or be an application in order to be part of a valid model. If you claim that you can run a mile faster than I can, your four-minute mile doesn't have to be profitable employment in order to count. On the other hand, the random sampling demonstration with Sycamore certainly is an action of some kind performed by a ...

3

While a follow-up question asks for the motivation behind the two-qubit gates used in Sycamore, this question focuses on the random nature of the single qubit operations used in Sycamore, that is, the gates $\{\sqrt{X},\sqrt{Y},\sqrt{W}=(X+Y)/\sqrt{2}\}$ applied to each of the $53$ qubits between each of the two-qubit gates. Although I agree with @Marsl ...

2

In fact, you would need an astronomical circuit depth in order to get close to a uniformly random state, or even close to a randomly chosen probability distribution on the $2^{53}$ outputs. As a first estimate, consider how many different distributions you need in order to be within 1/8 of the total variation distance of any distribution on $N$ outputs. ...

2

This answer only addresses the part about the necessity of the randomness of the circuit because I am by no means familiar with the physical implementation of the qubits at Google and what kind of constraints these impose on the implementation of certain gates. Now, for the randomness: Consider the problem of sampling from the output distribution of a ...

2

Generally speaking, to prove quantum supremacy, you don't need to sample several times from the same unitary/circuit/output probability distribution. If you extract even a single sample from the output probability distribution of a circuit which you know is extremely hard to simulate classically, then you already achieved something that you couldn't do (...

2

In the Sycamore paper linked in the comments, in the description of FIG. 4, the authors state: ...For each $n$, each instance is sampled with $N_s$ between 0.5 M and 2.5 M... For $m=20$, obtaining 1M samples on the quantum processor takes 200 seconds, while an equal fidelity classical sampling would take 10,000 years on 1M cores, and verifying the fidelity ...

1

In essence you are asking could it be more efficient to use non-uniform distribution (instead of uniform) to pick numbers $r_i$ from $[0,S]$ for testing. Quantum circuit here just encodes the distribution, essentially it has no other use. Well, it depends on how we model our probability space for all polynomials. In some models it could be better to pick ...

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