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Unitary operations are only a special case of quantum operations, which are linear, completely positive maps ("channels") that map density operators to density operators. This becomes obvious in the Kraus-representation of the channel, $$\Phi(\rho)=\sum_{i=1}^n K_i \rho K_i^\dagger,$$ where the so-called Kraus operators $K_i$ fulfill $\sum_{i=1}^n K_i^\... 16 Let's say we have a function$f$which maps$n$bits to$m$bits (where$m<n$). $$f: \{0,1\}^{n} \to \{0,1\}^{m}$$ We could of course design a classical circuit to perform this operation. Let's call it$C_f$. It takes in as input$n$-bits. Let's say it takes as input$X$and it outputs$f(X)$. Now, we would like to do the same thing using a quantum ... 13 Short Answer Quantum operations do not need to be unitary. In fact, many quantum algorithms and protocols make use of non-unitarity. Long Answer Measurements are arguably the most obvious example of non-unitary transitions being a fundamental component of algorithms (in the sense that a "measurement" is equivalent to sampling from the probability ... 11 At risk of going off-topic from quantum computing and into physics, I'll answer what I think is a relevant subquestion of this topic, and use it to inform the discussion of unitary gates in quantum computing. The question here is: Why do we want unitarity in quantum gates? The less specific answer is as above, it gives us 'reversibility', or as ... 10 What is the proof that any given unitary matrix can be converted as above? Let$U$be an arbitrary$2\times 2$unitary matrix. This is equivalent to the rows/columns of$U$forming an orthonormal system. Let us write a generic$U$as $$U=\begin{pmatrix}a&b\\c&d\end{pmatrix}.$$ The constraints imposed on the coefficients$a,b,c,d$by the requirement ... 8 The fact that quantum gates are unitary, is rooted in the fact that the evolution of (closed) quantum systems is by the Schrödiner equation. For a time interval in which we are trying to realise a particular unitary transformation at a constant rate, we use the time-independent Schrödinger equation: $$\tfrac{\mathrm d}{\mathrm dt} \lvert \psi(t) ... 8 Even if you only limit yourself to special-unitary operations, states will still accumulate global phase. For example, Z = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} is special-unitary but Z \cdot |0\rangle = i |0\rangle \neq |0\rangle. If states are going to accumulate unobservable global phase anyways, what benefit do we get out of limiting ... 8 There are several misconceptions here, most of them originate from exposure to only the pure state formalism of quantum mechanics, so let's address them one by one: All quantum operations must be unitary to allow reversibility, but what about measurement? This is false. In general, the states of a quantum system are not just vectors in a Hilbert space ... 8 All operations on quantum states are unitary operations. We don't make the rules, this is just how nature seems to work. So if you want to define an operation that copies a qbit, it has to be a unitary operation. That unitary operation would look like this: U|\psi\rangle_A|0\rangle_B=|\psi\rangle_A|\psi\rangle_B So you have the qbit you want to copy, |\... 8 Apply it twice:$$ O_xO_x|i\rangle|b\rangle=O_x|i\rangle|b\oplus x_i\rangle=|i\rangle|b\oplus x_i\oplus x_i\rangle=|i\rangle|b\rangle $$Hence, O_x is its own inverse, and therefore reversible. To prove unitarity, it makes more sense to prove that O_x has eigenvectors$$ |i\rangle(|0\rangle+|1\rangle)\quad\text{and}\quad|i\rangle(|0\rangle-|1\rangle) $$... 7 Some terminology seems a little bit jumbled here. Quantum states are represented (within a finite dimensional Hilbert space) by complex vectors of length 1, where length is measured by the Euclidean norm. They are not unitary, because unitary is a classification of a matrix, not a vector. Quantum states are changed/evolved according to some matrix. Given ... 6 This is actually a much easier problem. In the case of states, you're trying to use the PPT criterion, or others, to distinguish if \rho can be written in the form$$ \rho=\sum_ip_i\sigma^A_i\otimes\sigma^B_i, $$where \sum_ip_i=1 and the \sigma^A_i and \sigma^B_i are valid states on single sites. The difficulty actually comes from the freedom that ... 6 As already mentioned in the other answers, the crucial point is that copying means implicitly that the state of the original qubit is unknown, i.e. given a qubit in an unknown state, you want to prepare a second qubit to be in exactly the same state. To make it more intelligible, there is a less mathematical argument that this should not be possible: By the ... 5 I'll add a small bit complementing the other answers, just about the idea of measurement. Measurement is usually taken as a postulate of quantum mechanics. There's usually some preceding postulates about hilbert spaces, but following that Every measurable physical quantity A is described by an operator \hat{A} acting on a Hilbert space \mathcal{H}. ... 4 When writing gates for, for example, a quantum circuit diagram, you could always write them using the convention of having determinant one (from the special unitary group), but it's just a convention. It makes no physical difference to the circuit that you implement. As said elsewhere, whether what you naturally produce corresponds directly to the special ... 4 To answer the first part of the question (whether unitary matrix U operates on |\psi_A \rangle only): A unitary matrix can operate on an arbitrary number of qubits. Single-qubit gates, like Pauli X, Y and Z gates, operate on one qubit and are represented by 2x2 matrices; CNOT gate operates on two qubits and is represented by a 4x4 matrix, etc. In this ... 4 A particularly efficient way is the look at the Schmidt coefficients of your target state. You know that your state can be written as$$ U_1\otimes U_2(\alpha|00\rangle+\beta|11\rangle), $$and the Schmidt decomposition tells you what \alpha,\beta,U_1,U_2 are. So, obviously, the problem becomes producing$$ \alpha|00\rangle+\beta|11\rangle. $$This is ... 4 The simplest way to solve this problem is to work backwards from the output to the input. Suppose you have the state a|00\rangle + b|01\rangle + b|10\rangle + b|11\rangle. How can you reduce this to just the state |00\rangle with unitary operations? Applying the inverse of those operations in reverse order will send you from |00\rangle to the desired ... 4 An elegant argument can be derived by asking which theories can we build which are described by vectors \vec v = (v_1,\dots,v_N), where the allowed transformations are linear maps \vec v\to L\vec v, probabilities are given by some norm, and probabilities must be preserved by those maps. It turns out that there are basically only three options: ... 4 More mathematically, because \mathbb{R}^n with an L^p norm is a Hilbert space only for p=2. 4 You have U|1\rangle=e^{i\phi}(b^*|0\rangle-a^*|1\rangle) (and for real entries, e^{i\phi}=\pm1). This condition follows automatically from$$ \langle 0|U^\dagger U |1\rangle=0 $$-- this is exactly the condition you describe -- together with the fact that U|0\rangle and U|1\rangle must have the same normalization,$$ \langle k|U^\dagger U |k\rangle=1 ... 4 Controlled gates are used to create entanglement. They don't have anything to do with reversibility; all unitary gates are reversible by definition, since all unitary operators have inverses. You could come up with an alternative choice of gates that you used to write algorithms in that didn't include controlled gates. For instance, for quantum computers ... 4 Any quantum operation is basically a Hamiltonian$H$acting on an isolated Hilbert space states. Now, the requirements for a Hamiltonian is to a matrix which is Unitary and Hermitian. This intrinsically implies it has an inverse, which means it is reversible because you can always find a matrix$H^{-1}$and apply it on the state. This is where reversibility ... 4 Your construction by gueswork in this answer is OK but not really elegant. Moreover, it's a convention to start in the state$|0\rangle$; we usually don't initialize a qubit with the state$|1\rangle$. It's better to follow the general construction which I illustrate here. The idea here is to use ancillary qubits and impose unitary evolution on the larger ... 4 Simulating Classical "AND/NAND/OR/NOR/XOR/XNOR" Gates With the help of this answer from Blue, constructing a matrix for a classical gate is just a matter of following the steps. Here is the combined truth table for classical logic gates: $$\begin{array}{|c|c|c|c|c|c|c|} \hline \text{Input} & \text{AND} & \text{NAND} & \text{OR} & \text{... 4 You did more than you needed to in part i. You effectively did part ii already. Because$$ g = \frac{u}{\sqrt{\det{u}}} $$is in SU(2), the phase you need to multiply by is \frac{1}{\sqrt{\det{u}}}.$$ \det H = -1/2-1/2 = -1 $$So to get in SU(2), you need to multiply \sqrt{-1}=\pm i. That is seen as the e^{\pi i /2} you see as the prefactor ... 4 The existing answers have quite an elegant idea behind them. However, my concern is that they don't seem to allow for the introduction of an ancilla. If we introduced an ancilla in a fixed state, and applied a unitary across the two systems, then it could be that all the eigenvectors are entangled across the two subsystems, and the basic argument wouldn't ... 3 Siddhant's answer should suffice, but as you said you are new to quantum computing (QC), I am not sure whether you are new to quantum mechanics (QM) as well. I will assume you are new to QM, and if that is not the case please excuse my ignorance. Is it correct to say that we need controlled gates because unitary matrices are reversible? No, controlled ... 3 No. The rows and columns of a unitary U must have a sum-mod-square of 1.$$ \sum_{i}|U_{ij}|^2=\sum_{j}|U_{ij}|^2=1$$Your$M$, as specified, could have a 1 element along a whole row so the sum-mod-square of the corresponding row in$U$would be$n$. So, unless$n=1$, it's impossible without further constraints on$M\$.