Phase in Quantum Computing

Main concepts of computing illustrated with simple examples

Quantum Theory Made Easy 0 1 Classical p0p0 p1p1 probabilities Quantum a0a0 a1a1 0 1 amplitudes p 0 +p 1 =1|a 0 | 2 +|a 1 | 2 =1 bit qubit p i is a real numbera i is a complex number Prob(i)=p i Prob(i)=|a i | 2

Quantum Theory Made Easy Classical Evolution Quantum Evolution stochastic matrix transition probabilitiestransition amplitudes unitary matrix

Interference measure 0 50% 1 measure 0 50% 1 100% 0% 100% 0% qubit input

Interfering Pathways 100% H 50% 50% C 50% H 10% 90% 20% 80% 15% H 85% C 1.0 H C H H 1.0 C Always addition!Subtraction! Classical Quantum

Superposition Qubits a0a0 a1a1 0 1 amplitudes a i is a complex number 1 √2√2 ( |  + |  ) Schrödinger’s Cat

Classical versus quantum computers

Some differences between classical and quantum computers superposition Hidden properties of oracles

Randomised Classical Computation versus Quantum Computation Deterministic Turing machine Probabilistic Turing machine

Probabilities of reaching states

Formulas for reaching states

Relative phase, destructive and constructive inferences Destructive interference Constructive interference

Most quantum algorithms can be viewed as big interferometry experiments Equivalent circuits

The “eigenvalue kick-back” concept

There are also some other ways to introduce a relative phase

The “eigenvalue kick-back” concept Now we know that the eigenvalue is the same as relative phase

The “eigenvalue kick- back” concept illustrated for DEUTSCH

The “shift operation” as a generalization to Deutsch’s Tricks

Change of controlled gate in Deutsch with Controlled-Ushift gate

Now we deal with new types of eigenvalues and eigenvectors

The general concept of the answer encoded in phase

Shift operator allows to solve Deutsch’s problem with certainty

Controlling amplitude versus controlling phase

Exercise for students

Dave Bacon Lawrence Ioannou Sources used