What are some common conventions in quantum computing?

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In many fields of human endeavor there are often conventional practices or other "house rules" that go unstated and initially appear arbitrary but might have historical or otherwise practical reasons. Think "the tie goes to the runner" or "$0\not\in\mathbb N$."

Indeed in a recent survey from the American Mathematical Society propose that collections of various conventions can be

... quite useful for both present and future mathematical learners because ... the diversity in mathematical conventions is prevalent in current literature and will almost surely persist.

To that end quantum computing is no exception, and has a lot of pretty standard conventions that are either unstated or maybe clarified in the prologue of a paper or book.

What are some examples of conventions in quantum computing that often go unstated in the standard course-of-business?

Please consider posting an example, stating at least what the convention is, and if there are two or more practices, what is the more common one... It might be useful as a resource when debating whether to spell out or clarify more commonly used practices.

Answer 1

Often if a Hamiltonian acts locally on $n$ qubits as

$$H=\sum_{j=1}^m H_j$$

with each term $H_j$ acting on only $k$ qubits, then for each $H_j$ there is an identity gate implicitly acting on the $(n-k)$ other qubits.

Answer 2

Generally circuits are read from left-to-right; however, there are exceptions.

Indeed gates are applied from right-to-left. Thus $U_3U_2U_1|\psi\rangle$ is preferred over $U_1U_2U_3|\psi\rangle$.

Answer 3

"Unitary" is often used a a noun, meaning "unitary matrix" (or "unitary operation").

Answer 4

Hats $\hat{}$, which are used to denote quantum operators and observables, are almost always omitted without notice. This happens in many other quantum-physics subfields, but this omission is particularly prevalent in quantum computing.

Answer 5

Often the unit convention $\hbar=1$ is used, e.g., in the time-dependent Schrödinger equation

Answer 6

Global phases are usually omitted because they cannot be measured, i.e., something like $|0\rangle$ and $e^{i\varphi}|0\rangle$ describe the same state. The same is true for unitaries: $U$ and $e^{i\varphi}U$ describe the same gate. This also where the formulation "up to global phase" comes from.

For similar reasons additive constants in Hamiltonians are usually ignored (i.e., $H$ vs. $H+\lambda I$ for some $\lambda\in\mathbb R$) because they describe the same dynamics; one example of this would be $H=Z$ and $H=Z+\frac12I=|0\rangle\langle0|$.

Answer 7

The inverse temperature is essentially always denoted by the letter $\beta$. In addition, there is a critical exponent (for the order parameter) which is always denoted as $\beta$. Additionally, the Boltzmann constant is generally set to $k_B=1$, so $\beta=1/T$.

Answer 8

The term tensor product is often interchangeably used with left Kronecker product when referring to the operation between vectors/matrices.

Answer 9

The inner product is defined to be linear in the second argument (and consequently antilinear in first). In mathematics, the standard definition defines the inner product linear in the first argument.

This is true not only for quantum computing but a general quantum-mechanics convention. However, it really got me puzzling when I started to learn about quantum information and wanted to compare some results with mathematical textbooks so I'm listing it here.

Answer 10

When qubits are either spin-up or spin-down, often $\mid0\rangle=\:\mid\uparrow\rangle$ and $\mid 1\rangle=\:\mid\downarrow\rangle$.

It took me some time to figure this one out as my first guess is that $1$ is "above" or "up" from $0$, but it is quite consistent with, for example, the analysis of Boolean functions:

$$\chi (0_{\mathbb{F}_2})=+1, \:\chi (1_{\mathbb{F}_2})=-1.$$

Answer 11

The plural of ancilla is commonly ancillas; less commonly ancillae; and sometimes ancilla itself. I personally like "a plurality of ancilla registers."

Answer 12

CNOT and CX denote the same thing; while CNOT is more prevalent in theoretical contexts, CX is the more common in programming frameworks and circuit implementations.

Answer 13

Spin-1/2 operators sometimes have eigenvalues $\pm\tfrac12$ and sometimes $\pm1$.

Answer 14

$|+\rangle$ and $|-\rangle$ are reserved to denote a superposition of the computational basis states:

$$ |\pm \rangle = \frac{1}{2}\left(|0\rangle \pm |1\rangle \right).$$

This is differs from many other quantum mechanics references, where $|+\rangle$ and $|-\rangle$ are used to denote the eigenvectors of the $z$-spin operator.

Answer 15

In quantum complexity theory literature,

BQP has become a synonym for PromiseBQP.

This might not be very pleasing to some classical complexity theorists who study BPP and PromiseBPP with well-pronounced distinction.

Note: Page 26 of S. Gharibian's notes for more context.