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0

I wrote a small package for these kinds of operations, QM, that is available on GitHub here. Here is a couple of examples (the code above will import directly the main file from GitHub without installing it, this is enough here but some other things will require that you install the full package): Get["https://raw.githubusercontent.com/lucainnocenti/QM/...


2

Let's define the kets, ket0 = {{1},{0}};ket1 = {{0},{1}}; This function produces input from a string, f[x_?StringQ] := ToExpression[StringJoin["ket",x]]; This function produces the diagonal matrix corresponding to a string "000", matrixFunc[x_]:= KroneckerProduct@@ f/@ StringPartition[x,1] . ConjugateTranspose[KroneckerProduct@@ f/@ StringPartition[x,...


6

The point is that free parallel computation or cloning of your existence is a wholesale misinterpretation of the concept of quantum superposition. Quantum states are analogous to probability distributions. If you might wash the dishes or you might wash the floor and you flip a coin to decide which one, then no one takes that to mean that you will wash ...


3

The statement is meant to get in front of any misconceptions, for example by the science press, about how quantum computers operate. It's not a "no-go" in the sense of a theorem, nor do I believe many researchers have spent much time considering a possible algorithm that "simply tries all possible solutions at once." I believe it's meant to say that ...


2

After using the simulator, I am very impressed! From what I can tell, it has everything necessary to be universal. I will likely be using this quite a bit. To test drive it, I implemented a simple 3-qubit Fourier transform and applied it to a set of random initial states, then compared the result to the well known 3-qubit unitary DFT (dft) applied to the ...


2

Deutsch's algorithm is not faster on a quantum computer, Deutsch's algorithm is only possibe on a quantum computer. A classical computer cannot perform Deutsch's algorithm, a classical computer can only simulate Deutsch's algorithm. Forget about speed, more fundamental is that we are performing a type of computation that can never be performed, in any ...


2

On a classical computer, you can recycle memory, so there is no need for storing all intermediate results. The main source of the higher performance of quantum computers is the possibility to do an operation with all different values you can store in n q-bits register at once. However, this is not a case always. For example, the evaluation of Boolean ...


2

CNOT is more simple gate than Toffoli (CCNOT). Actually, CCNOT is realized with six CNOT gates (see Wikipedia link above for a CCNOT circuit). Additionally, CNOT is relized physically on a quantum computer. To sum up, direct usage of CNOT is better because it enable the quantum circuit to be more simple.


2

Any "person" (x or y) can obtain the state of z using the quantum circuit consisting of CNOT and Hadamard gates and $Z$ gate; the circuit is actually a simplified version of quantum teleportation; let x be in the state $|0\rangle$, that is he wants to go to the floor 2 (state $|1\rangle$ means floor 3). Applying CNOT to z (control qubit) and x entangles z ...


2

Some algorithms do have deterministic answers or a high probability of getting the correct answer when implemented. For example, Deutsch's algorithm (Deutsch-Jozsa) which tells us with certainty whether a function is constant or balanced. On the other hand, however, you have Grover's search. Grover's has a high probability of giving you the correct answer ...


-1

Technically, there is no way to determine with perfect accuracy, and the same applies to any event. Everything that happens in our universe involves interactions between quantum states, and therefore is probabilistic. When we do a simple math calculation in our minds or on a calculator, there is always a small chance that the "wrong" answer will be produced ...


2

CNOT is a Boolean function $f:\{0,1\}^2\to\{0,1\}^2$ that eats two input bits and spits two output bits. As Wikipedia explains, Toffoli gates require some ancillary (extra) qubits to do emulate any Boolean function. In this case, just one extra qubit will do the job. As MarkS mentioned in the comments, set one of the control qubits of the Toffoli gate to $|1\...


2

I see maybe three questions here: Given an efficient (quantum) algorithm, how is it that the efficient algorithm will give the correct answer with high probability? Given an efficient (quantum) algorithm, how do we know that the efficient algorithm will give the correct answer with high probability? Upon executing our efficient quantum algorithm, how can we ...


2

Initially let's consider some registers and operators. The register $|A\rangle$, which encodes superpositions of states of the cube (e.g. a permutation of the cube $G$); The register $|B\rangle=|b_1\rangle|b_2\rangle\cdots|b_r\rangle$, which encodes superpositions of a set of Singmaster moves to be applied to a given position (e.g., superpositions of words ...


2

Another answer already addressed how the specific expression in the OP means and how to derive it. Here I'll show another way to derive the same result (very similar to the derivation in this other answer on another question, though with different notation). Using the notation $\mathbb P_\psi\equiv|\psi\rangle\!\langle\psi|$, we have $U_s\equiv 2\mathbb P_s-...


4

I had forwarded this question to Dr. Lov Grover and received the following response. I guess inversion about average is a better name for the $\mathrm{W}\mathbb I_0\mathrm{W}$ transformation. When I initially did the algorithm, I called this the diffusion transform because I was familiar with classical diffusion and this is what this transform ...


3

The mapping scheme set up by the author in this section is different from that of normal unitary operations. In the context of this equation, the author is referring to the (non-unitary) action of $U_s U_\omega$ on a particular two dimensional subspace of the full Hilbert space, specifically on the plane defined by the (non-orthogonal, non-complete) basis ...


2

QFT (more specifically QED) is already at the heart of quantum computation. Second quantization formalism is used even in constructing toy models for quantum computers (see e.g., Section 7.3.2 of Nielsen and Chuang). Many of the rapid advances in superconducting quantum processors over the past decade can be attributed to the introduction of circuit ...


2

The operator was named "diffusion transform" in the original Grover's paper (see second column of pag. 3) but no explanation is given for the terminology there (and I don't know whether it was "common" at the time). You can think of Grover's algorithm as a repeated application of an operator $\mathcal U=-\mathcal S_i\mathcal S_t$ that is the product of two ...


6

I view QAOA as an algorithm for solving (approximately) a special class of problems, namely combinatorial problems and VQE as a possible subroutine to QAOA (but not necessarily as in the case of MaxCut). Let me explain The VQE - Variational Quantum Eigensolver - solves the problem of approximating the smallest eigenvalue of some Hermitian operator $H$ which ...


3

Paraphrasing some tweets on the matter earlier, the result is rather underwhelming because it plays on a discrepancy between what they mean by quantum supremacy (QS) and what people tend to think QS means. What I find most people think QS is supposed to mean, and what I assumed it meant until a month or so ago, was that there exists a computable problem (in ...


15

Google's paper/results are kind of sideways to questions in computational complexity about the relation between $\mathrm{BPP}$ and $\mathrm{BQP}$ (and even further from questions about whether $\mathrm{P}\ne\mathrm{NP}$). It's more as if Google relies on the hypothesis that $\mathrm{BPP}\ne\mathrm{BQP}$ as evidence that their quantum computer performs a ...


2

This is not much of an answer, but is probably too long for a comment... I don't believe that there's a canonical way of doing this. You'd be best off understanding why you're asking the question, and what you want to get out of it. From there, you tailor how you're going to measure it. But multipartite entanglement is a really messy problem, even just for ...


3

The basic idea is to multiply $U$ on the left with $2\times 2$ unitaries until the identity is obtained. This method provides a sequence of gates $U_k$ such that $U_1\cdots U_n U=I$, which then gives you the decomposition of $U$ in terms of $2\times2$ unitaries: $U=U_1^\dagger \cdots U_n^\dagger$. For example, suppose you start with $$U=\begin{pmatrix}1/2&...


3

Their approach is more advanced than the simple one, described in the book "Quantum Computation and Quantum Information" by M. Nielsen and I. Chuang, section 4.5.1. It's better to understand it first. Basically we are just making zeros under diagonal step by step, where each step is the multiplication by some two-level unitary. Hence there are only $d(d-1)/2$...


5

The key to understanding many quantum protocols and circuits is in the following circuit: This is especially true in the case where $U^2=I$, such that $U$ has eigenvalues $\pm1$. You can readily calculate that if the input, $|\psi\rangle$, of the second qubit has an amplitude $\alpha_+$ for being supported on the $+1$ eigenspace, then at the end of the ...


4

What that cSWAP test does (and doesn't) do The important thing about the controlled-SWAP test is that what it does isn't just to SWAP, or to not SWAP, the two inputs. The controlled-SWAP test involves a control qubit which is in a superposition of $\def\ket#1{\lvert#1\rangle}\def\bra#1{\langle#1\rvert}\ket{0}$ and $\ket{1}$: that is, we measure the first ...


0

Perhaps you can build a device that simply demonstrates one of the characteristics of a single qubit, and that is the ability to program its 'superposition' probability amplitudes. An easy way to visualize this is by a simple coin toss or a coin spin on a table. Think of the coin as being in 'superposition' while it's in the air (or when it's spinning on ...


0

Grover's algorithm can be used to solve any numerical optimization problem faster than brute-force search, because any optimization problem can be formulated as a search problem (where you are searching for a function output greater/less than some fixed $M$ within each run, and you repeat for a logarithmic number of runs using binary search to home into the ...


2

I'll reiterate on my earlier answer to What can be a mini research project based on Grover's algorithm or the Deutsch Jozsa algorithm?: I think "Applying Grover's search algorithm to solve problem X" is a great topic for a small (or not-so-small) project. It is a very well-known algorithm (well, at least it is featured in the writings about quantum ...


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