# Tag Info

### What exactly is "Random Circuit Sampling"?

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 ...
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### Multiplication by a Haar random unitary two times

There is an explicit formula for the integral with respect to the Haar measure of any polynomial in the entries of a unitary and its conjugate, due to Collins and Śniady: Benoît Collins and Piotr ...
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### Understanding Google's “Quantum supremacy using a programmable superconducting processor” (Part 2): simplifiable and intractable tilings

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 ...

### Sampling random circuits vs Solovay-Kitaev compiler

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 ...
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### Understanding Google's “Quantum supremacy using a programmable superconducting processor” (Part 3): sampling

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 ...
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### What is the HOG test and how would it help proving quantum supremacy?

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 ...

### Help Identifying a Gate In Nielsen and Chuang

Here are the corresponding decompositions of each gate with use of Toffoli and $X$ gates: The first one applies $X$ gate on the third qubit if control qubits are in the $|00\rangle$ state. The second ...
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### Randomness using simple parallel Hadamard circuit

The issue is that you are using noisy hardware with imperfect operations and measurements. In particular, the most likely problem here is that after you prepare a qubit it immediately begins decaying ...
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### Random circuits with google cirq

Cirq does have some methods for generating random circuits, such as cirq.testing.random_circuit and cirq.random_rotations_between_grid_interaction_layers_circuit. That being said, in my experience, ...
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### How does successfully sampling from a random quantum circuit invalidate the Extended Church-Turing Thesis?

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 ...
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### What did exactly Google do in simulating a random quantum circuit on a classical computer in supremacy experiment?

All quantum circuits can be simulated on a classical computer, but not all circuits take the same amount of time to simulate. If information about the circuit is known in advance, certain patterns may ...
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### How to make circuit for randomly selected gate?

This is very similar to an function in terra random_circuit: https://github.com/Qiskit/qiskit-terra/blob/master/qiskit/circuit/random/utils.py#L30-L113 It randomly ...
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### Understanding Google's “Quantum supremacy using a programmable superconducting processor” (Part 1): choice of gate set

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 ...
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To study unitary $t$-designs, we define the moment operator with respect to a probability measure $\nu$ as $$M_t(\nu) := \int_{U(d)} U^{\otimes t} (\cdot) (U^{\otimes t})^\dagger d\nu(U) \simeq \int_{... 3 votes ### Spoofing XQUATH with the Feynman method The paper does not specify the exact algorithm or class of distributions \mathcal{D} for which such algorithm fails to refute XQUATH, and some classes of distributions \mathcal{D} do not satisfy ... 3 votes Accepted ### Quantum supremacy: shallow depth Haar random circuits and unitary designs First of all, that does not imply anything for shorter (constant/logarithmic) depths. Moreover, the 2-design property does not imply that the outcome distribution is the same as for Haar-random ... 3 votes ### Understanding Google's “Quantum supremacy using a programmable superconducting processor” (Part 1): choice of gate set 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 ... 3 votes Accepted ### Do quantum supremacy experiments repeatedly apply the same random unitary? 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 ... 3 votes ### How exactly is solving the random circuit sampling problem a computation in the Church-Turing thesis sense? In the framing of the question (which I believe to be asked in good faith), there seems to be at least two objections. Sampling from a set of strings is not clearly a function, and Sampling is a ... 3 votes Accepted ### How exactly is solving the random circuit sampling problem a computation in the Church-Turing thesis sense? The Church-Turing thesis is not in and of itself a rigorous concept, but rather a judgment on rigorous concepts of computability. As such, it's negotiable. The language in Rosser's 1939 expository ... 2 votes ### What is the role of choosing the single-qubits randomly in Google quantum supremacy experiment? It's my understanding that in Google's quantum computational supremacy experiment, they have executed exactly the same random circuit up to 1M times, e.g up to 1M instances. They must perform that ... 2 votes Accepted ### Are 20 repetitions of Sycamore's one- and 2-qubit gates sufficient to produce a uniformly random state? 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 ... 2 votes ### Do quantum supremacy experiments repeatedly apply the same random unitary? 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 ... 2 votes ### If two reduced density matrices are equal, does that mean that the two subsystems are the same? It depends what you mean in wanting to say that the states "are the same". You probably mean all observable consequences on just that subsystem (i.e. measurements in arbitrary bases etc, ... 2 votes Accepted ### If two reduced density matrices are equal, does that mean that the two subsystems are the same? Density matrix completely describes the state of a system (or a subsystem). So if two density matrices are equal then the two states are equal (and vice versa). But you should not forget that The ... 2 votes ### How to create a Quantum circuit to implement the generation of 3-qubit uniform superposition wavefunction A uniformly distributed superposition can be prepared by Hadamard gate. If you apply a Hadamard on single qubit in state |0\rangle you get state \frac{1}{\sqrt{2}}(|0\rangle+|1\rangle). Both ... 2 votes ### Could random quantum circuits be efficiently approximately simulated? I can't answer all your questions and I certainly am not an expert, but I have something to say about your first point. According to the first paper linked in my comment (by Aaronson and Chen), the ... 2 votes ### Implementing a circuit that returns |01\rangle and |10\rangle with equal probability One potential combo is  RY(\theta) on qubit 1, CX from qubit 1 to qubit 2, then RX(\pi) on qubit 2. This would be the following transformation:$$ |0\rangle |0\rangle \mapsto (\cos \frac{\theta}{...
To erase a HDD with a random numbers generated by quantum computer is in theory possible. Lets imagine you want to generate bit strings of length $n$, then you can simply put Hadamard gates on $n$ ...
In the absence of additional assumptions, $\mathbb{E}[p_i]$ can be any real number in $[0, 1]$. For example, let $a\in[0,1]$ and define the POVM as $M_0=aI$ and $M_1=(1-a)I$. Then  \mathbb{E}[p_0] = ...