QUANTUM computing BACK TO BASICS for instance. This coupling generates a quintessentially quantum property called entanglement: the joint state of two entangled qubits cannot be described by specifying the individual state of each qubit. Hence, the state of two qubits, described by the superposition α 0 | 00
> + α 1 | 01
> + α 2 | 10
> + α 3 | 11 can in general not be factorized
>
). n qubits as( α 1 | 0
> + β 1 | 1
>)( α 2 | 0
> + β 2 | 1
> are thus described by 2 n coefficients. Conversely, N complex numbers could a priori be stored in log 2 N qubits! This exponential storage capacity can be leveraged in some algorithms, with an important caveat: reading out the information stored in the coefficients is not straightforward. Indeed, measuring a qubit in state α | 0
> > + β | 1 will only return 0 or 1 with respective probabilities given by the squared modulus of α and β. More than that, it will project the state to | 0
> > or | 1. Learning the precise value of α and β thus requires more work than meets the eye.
HOW TO PROGRAM A QUANTUM COMPUTER? A quantum program is a list of instructions that evolve the state | Ψ
> of the quantum computer from an initial state to a desired final state, which one can subject to quantum measurements in order to read off the solution to the problem at hand. From a physical perspective, these instructions essentially define a time-dependent Hamiltonian which, through Schrödinger’ s equation, completely determines the evolution of the system. The sequence of these instructions is commonly represented as a quantum circuit: a diagram whose horizontal lines represent qubits, and boxes represent the instructions, aka quantum gates that act on one, two or more qubits( lines) in a given order. For instance, the circuit used to implement a Fourier transform on a quantum computer is displayed in
Fig. 1 for five qubits. Some gates( like the so-called Hadamard gate H) act on one qubit, corresponding to physical operations that act only on one qubit. They can put the qubit in a superposition state, but do not create entanglement. Some others( like the controlled-phase gates), act on two qubits, and may create entanglement.
A major theoretical advantage of quantum computers is that their quantum properties— superposition and entanglement— should afford them a computational advantage over classical processors. We can look at the Fourier transform circuit of Fig. 1 to understand this. On a quantum computer, executing a gate corresponds to a single operation, while on a classical computer, a generic quantum gate corresponds to a matrix-vector multiplication U • | ψ
>. Since the vector in question is generally represented with size 2 n
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