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22

When we talk about quantum computers, we usually mean fault-tolerant devices. These will be able to run Shor's algorithm for factoring, as well as all the other algorithms that have been developed over the years. But the power comes at a cost: to solve a factoring problem that is not feasible for a classical computer, we will require millions of qubits. This ...

22

An oracle (at least in this context) is simply an operation that has some property that you don't know, and are trying to find out. The term "black box" is used equivalently, to convey the idea that it's just a box that you can't see inside, and hence you don't know what it's doing. All you know is that you can supply inputs and receive outputs. In the ...

21

The church of the larger (or higher, or greater) Hilbert space is just a trick that some people like (myself included) for rewriting some operations. The most general operations that you can write down for a system are described by completely positive maps, while we like describing things with unitaries, which you can always do by moving from the original ...

16

"Postselection" refers to the process of conditioning on the outcome of a measurement on some other qubit. (This is something that you can think of for classical probability distributions and statistical analysis as well: it is not a concept special to quantum computation.) Postselection has featured quite often (up to this point) in quantum mechanics ...

13

There are a few things to distinguish here, which are often conflated by experts because we're using these terms quickly and informally to convey intuitions rather than in the way that would be most transparent to novices. A "qubit" can refer to a small system, which has a quantum mechanical state. The states of a quantum mechanical system form a vector ...

13

The terminology of 'surface code' is a little bit variable. It might refer to a whole class of things, variants of the Toric code on different lattices, or it might refer to the Planar code, the specific variant on a square lattice with open boundary conditions. The Toric Code I'll summarise some of the basic properties of the Toric code. Imagine a square ...

11

Code spaces and code-words A quantum error correcting code is often identified with the code-space (Nielsen & Chuang certainly seem to do so). The code space $\mathcal C$ of e.g. an $n$-qubit quantum error correction code is a vector subspace $\mathcal C \subseteq \mathcal H_2^{\otimes n}$. A code word (terminology which was borrowed from the ...

11

The qsphere is a way of representing multi-qubit states. So it could be used for 5 qubit states, but it could also be used for any other number. It could also be used for just one qubit. But in this case it is important to note that the single qubit qsphere is not the same as the Bloch sphere, which is our standard way of representing single qubit states. ...

10

The surface codes are a family of quantum error correcting codes defined on a 2D lattice of qubits. Each code within this family has stabilizers that are defined equivalently in the bulk, but differ from one another in their boundary conditions. The members of the surface code family are sometimes also described by more specific names: The toric code is a ...

8

Quantum computing deals (mostly) with finite-dimensional quantum systems called qubits. If you know basic quantum mechanics then you know that the Hilbert space of a qubit is $\mathbb{C}^2$, i.e., the two-dimensional complex Hilbert space over $\mathbb{C}$ (for the more technical people, the Hilbert space is actually $\mathbb{C}P^1$). Therefore, to ...

7

The difficulty with explaining quantum computing is that quantum objects and processes have no direct classical analogue; they're an entirely new ontological category. For example, you might have learned in high school physics that light "is both a particle and a wave" in an attempt to relate it to two classical objects you can intuitively understand. In ...

6

When we have just one qubit, there's nothing particularly special about the computational basis; it's just nice to have a canonical basis. In practice you could think that first you implement a gate $Z$ with $Z^2 = I$ and $Z\neq I$, and then you say that the computational basis is the eigenbasis of this gate. However, when we talk about multi-qubit systems, ...

6

Is qsphere an actual term representing 5 qubits? If it is, it is not widely used. I claim this because I looked around in arXiv, a repository of electronic preprints of research articles, and found nothing. There are many other units of quantum information than just qubit though. All of the following appear at least occasionally in the relevant literature....

6

If you think of a electronic spin $S=1/2$, imagine measuring it on the z-axis to obtain $S_z=+1/2$ (or $S_z=-1/2$). This (the z projection of the spin magnetic moment) is a possible basis for the measurement. Or you could measure the spin on the x-axis, and they you will obtain $S_x=+1/2$ (or $S_x=-1/2$). This is a different basis. The measurements on Bell ...

6

When translating a classical circuit into a quantum circuit, you often need to introduce extra qubits simply because quantum computers only implement reversible logic. Such extra qubits are ancilla (or ancillary qubits). One way to spot which qubits are ancilla is to look for those qubits that typically need to be "uncomputed" when using the quantum circuit ...

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Preliminary - The DiVincenzo criteria for a 'normal' quantum computer The DiVincenzo criteria, as originally proposed by DiVincenzo, are $5$ criteria that he originally proposed in his 2000 paper. In this paper, he proposed five criteria, which are widely considered to be the five (sufficient and necessary) criteria that any physical quantum computer should ...

5

Talking about bases such as $\left|0\rangle\langle0\right|$ and $\left|1\rangle\langle1\right|$ (or the equivalent vector notation $\left|0\right>$ and $\left|1\right>$, which I'll use in this answer) at the same time as 'horizontal' and 'vertical' are, to a fair extent (pardon the pun) orthogonal concepts. On a Bloch sphere, there are 3 different ...

5

There is no standard name for a qudit for $d>3$. The community has mostly settled on the term qudit (but you will still find qunit or quNit, for example, using $n$ or $N$ instead of $d$ in some older papers). You will find the odd paper where an individual author will pick a name for the $d=4$ case. I’ve certainly seen ququad and ququart. But I think ...

5

The general meaning of ancilla in ancilla qubit is auxiliary. In particular, when people write about "constant input" what they mean is that, for a given algorithm -which has a purpose, such as finding the prime factors of an input number, or effecting a simple arithmetic operation between two input numbers the value of the ancilla qubits will be independent ...

5

In a quantum error correcting code, you store a number of logical qubits, $k$, in a state of many physical qubits, $n$. A code word is a state of the physical qubits that is associated with a specific logical state. So, for example, however you store the $|0\rangle$ state for one of your logical qubits is a code word. The code space is the Hilbert space ...

5

A code word (for a quantum code) is a quantum state that is typically associated with a state in the logical basis. So, you’ll have some state $|\psi_0\rangle$ that corresponds to the 0 state of the qubit to be encoded (you don’t have to use qubits, but you probably are), and you’ll have another that’s $|\psi_1\rangle$ that corresponds to the 1 state of the ...

5

Qubits are essentially quantum objects from which you can extract a bit. But there are different ways that this can be done, and the answer you get depends on the measurement you choose. If you qubit is an electron spin, the measurement basis corresponds to measuring spin in a particular direction. We use that picture more generally in the form of the Bloch ...

5

In the computational $\left(Z\right)$ basis, the parity of a (classical) bit string is $0$ if the number of $1$s in the string is even (i.e. 'even parity'), or $1$ if the number of $1$s in the string is odd (i.e. 'odd parity'). The parity can be measured by applying CNOT gates from each qubit that you want to measure (the control qubits) to an ancilla qubit ...

5

"Church of the higher hilbert space" is a term coined by John Smolin. According to quantiki it is: for the dilation constructions of channels and states, which [...] provide a neat characterization of the set of permissible quantum operations and to quote wikipedia, it: describe[s] the habit of regarding every mixed state of a quantum system as a pure ...

4

No, the computational basis does not have any special meaning, it is just the basis that is "most natural" in a given context, and is conventionally denoted with $|0\rangle$ and $|1\rangle$ in the case of qubits. To give a few examples: If the qubits are encoded into the polarization of single photons, the computational basis is typically the basis formed ...

4

As the other answer conveyed (and to which I am just trying to provide some clarification), post-selection is about just looking at a subset of possible measurement outcomes. To my mind, this falls into two different cases, as below. Yes, they are different aspects of the same thing, but they are used very differently by two different communities. ...

4

Forget about quantum mechanics for a second and consider two people predicting a coin flip. Alice flips a coin, covers it with her hand, and asks Bob to predict the result. Alice knows the coin is heads, but Bob is unsure if it is heads or tails. They will describe the state of the coin using different probability distributions. The same situation can apply ...

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

For the diagonal basis, the measurement operators are the $|0\rangle\langle 0|$ and $|1\rangle\langle 1|$, as stated in the question. For the other basis, any mutually unbiased basis will do, but people usually go for the two operators $(|0\rangle+|1\rangle)(\langle 0|+\langle 1|)/2$ and $(|0\rangle-|1\rangle)(\langle 0|-\langle 1|)/2$. The labels of which ...

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