Category: Quantum Gates and Circuits

Show that the n-qubit Hadamard gate acts as H ⊗n |xi n = 1 √ 2 n 2 n −1 X y=1 (−1) x·y |yi. (7.28) where x · y is the bitwise product of x and y: x · y = x 0 y 0 ⊕ x 1 y 1 ⊕ . . .…

Measurement Some issues regarding measurement in quantum circuits are to be noted here, which you can prove for yourself with some thought: 1. Deferred measurement: when measurements are made in a circuit and after that further gates are implemented (whether controlled by the measurement or no), it can always be assumed that the measurement is…

gates We’ve proved that the CNOT gate along with all possible single-qubit gates form a universal set. But this set is still infinite. We’d like to do better: to get a finite set of gates as in the classical case. Of course we must realize that the set of possible single qubit gates is itself…

We now wish to push the circuit analogy further and explore the possibility of universal quantum gates. Let’s start with single-qubit gates. We’ve seen that these are 2 ×2 unitary matrices, which take a point on the Bloch sphere to another. It is easy to see that there are infinitely many possible 1-qubit gates. These…

We’ve taken the circuit analogy for quantum computation up to gates. Can we go further? Can we identify a set of universal gates, as we did for classical computation? Since a computation is essentially the evaluation of a function of the inputs, let’s first fix what we mean by a quantum function evaluation. Consider a…

Two qubits together can be represented as 4-column vectors in Hilbert space. The most general 2-qubit gate is therefore a 4×4 unitary. An operation on two qubits that acts independently on each of the two can be expressed as a direct product of two single-qubit operations as defined in Equation 3.31: O = O 1…

At the end of a computation we need to measure the output in order to read out the result of the computation. This leads to obtaining classical information (in bits) out of the quantum system. One sets up an experiment that measures an appropriate physical quantity to give one of its eigenvalues as the result…

Two successive operations are two unitary gates, say A and B, acting one after another. Algebraically, we represent the resultant by the action of the usual matrix product of the two gates: |ψi A −→ A|ψi B −→ BA|ψi. (7.13) Note that the order of the gates is important. Operators do not in general commute.…

Classically, there exists only one reversible single bit gate: the NOT gate which effects 0 → 1, 1 → 0. However, any unitary operation on the qubits |0i and |1i is a valid single qubit gate. As we will see, such a gate can always be regarded as a linear combination of the Pauli gates…

We are now ready to see how computing with qubits can be done. In this book, we will mainly use the circuit model for computation which was first introduced by Deutsch [25]. We will represent by quantum “wires,” the qubits upon which manipulations. The length of the wire is to be interpreted as the time…