I'm not sure if this is a dumb question, since they seem to be very basic building blocks of quantum information theory; however, I can't seem to wrap my head around the difference between the two. As I understand it, both quantum gates and quantum channels are operations that you can pass a quantum state into. Additionally, a quantum gate is represented by a unitary operator, whereas quantum channels are sums of probabilistic distributions of unitary operators and their inverses... Could anyone help me understand the intuitive distinction between the two, and their relation, if any?
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$\begingroup$ Note that sometimes papers designing quantum circuits will use "gate" to refer to any basic quantum operation, including measurements, even though measurement is not a unitary operator. This is sloppy notation and Condo's answer is the technically correct difference. $\endgroup$– Sam JaquesFeb 16 at 10:50
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2 Answers
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A quantum gate is a unitary operator on a Hilbert space, where typically this Hilbert space is associated with a system of qubits. In the case of a single qubit a quantum gate is a $2\times 2$ unitary matrix, where $|0\rangle$ and $|1\rangle$ are the computational basis states. For example, the quantum NOT gate is the unitary matrix $X$ that maps $|0\rangle\mapsto|1\rangle$ and $|1\rangle \to |0\rangle$. Of course, unitary operators can act on superpositions (i.e. linear combinations) $\alpha|0\rangle+\beta|1\rangle$ of these basis states, which is required to perform quantum computation. For systems of $n$ qubits, quantum gates are unitary matrices that act on the Hilbert space $\mathbb{C}^{2^n}$.
Quantum channels are a little bit more complicated mathematically. A density matrix is a positive semi-definite matrix $\rho$ with $Tr(\rho)=1$. Density matrices represent the information about a quantum state, i.e. the probabilities and outcomes of the state that can arise upon measurement. A quantum channel is a linear map that maps density matrices to density matrices by preserving the fundamental properties, such as the positivity and trace condition of the density matrix. However, any quantum channel must also have an additional property known as complete positivity to ensure that it is compatible with the other axioms of quantum mechanics. A quantum channel is a completely positive trace-preserving (CPTP) map.
Distinctions aside, in the context of physics, quantum gates and quantum channels arise from the unitary evolution of a quantum system according to the Schrodinger equation. The Hamiltonian (a mathematical object governing these dynamics via encoding the interactions of the quantum system) evolves in time, and the result is a unitary transformation of the quantum state of the system. Thus to enact a gate on a quantum computer, we find a Hamiltonian that we can evolve to produce the desired effect of the quantum gate. Although a quantum system evolves unitarily, perhaps we only have access to part of the whole system; in this case, the effect is not unitary on this portion of the system but is described by a quantum channel (i.e. a CPTP map). If the whole system is accessible, then the quantum channel is described by unitary conjugation (also called a unitary channel).
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$\begingroup$ "quantum NOT gate is the invertible matrix" for this to be unique, it has to be the invertible unitary matrix, otherwise it is under specified. I know you just said they have to be unitary, but I think being redundant in the first paragraph is worth it. $\endgroup$ Feb 16 at 15:58
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$\begingroup$ @Yakk-AdamNevraumont, now that I think of it, if you want to be precise about the uniqueness of the matrices, then you need to define a quantum gate to be an element of the projective unitary group. $\endgroup$– CondoFeb 16 at 20:30
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When you model your quantum system as closed and look at transformations the whole system can go through, those are described by unitary evolutions.
But, what if you want to describe local transformations? That's where the notion of channels are introduced; They describe local transformations (There's a dilation theorem stating that given any channel you can always find extended systems and unitary evolutions of those systems such that the channel you have in your hand describes the local evolution). As such, channels describe the most general transformations where states are preserved.
Unitary evolutions are then dubbed unitary channels. To wind up, quantum channels are the most general evolutions, unitary channels being a special case. And, you just call unitary channels as gates when you see them in circuits. From an information-theoretic point of view, you can see unitary channels as describing noiseless evolutions where (non-unitary) channels as describing noisy ones (the loss happening due to the appended system).
