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- qbit: superposition of two states
$$ | \Psi \rangle = a | 0 \rangle + b | 1 \rangle, \; |a|^2 + |b|^2 = 1 $$
- N-qbit state: direct product of N qbits forms a basis
- Can be represented as state vector of size
\(2^N\)
$$ | \Psi \rangle = \sum a_x | x \rangle, \; \sum |a_x|^2 = 1 $$
- Quantum gate: performs unitary transformation of state vector
- Input data encoded as state vector
- Quantum computation as a series of quantum gates (reversible)
- Output state vector
- Do measurement on state vector to get actual result (very important point)
- Final result may come with an uncertainty
- May need to repeat calculation to get improved statistics/accuracy
- No-cloning theorem: No unitary tranformation such that
$$ |0\rangle |\Psi\rangle \rightarrow |\Psi\rangle |\Psi\rangle $$
- Effective encoding of input data into state vector
- Do calculations "in parallel through superposition" (but this might not be so easy in practise)
- Sometimes exponential speed up compared to classical algorithms
- The number of parameters needed to specify one particular quantum state is proportional to the number of all possible classical states
- Quantum algorithms usually look quite different from classical ones
- Example: Shor's algorithm for integer factorization
- Quantum supremacy
- As of today limited to about 50 qbits
- Trapped Ions
- Optical
- NMR
- Others
- Decoherence (this is the big one)
- Typically, needs cooling to mK temp
- Non-trivial to prepare system in arbitrary initial state
- Non-trivial to apply arbitrary unitary transformation
- Non-trivial to measure
- Expensive
- Adiabatic quantum computer: based on quantum annealing
- Minimizes an Ising-type Hamiltonian
- Array of flux qubits (superconducting, Josephson junctions)
- Physical realization with thousands of qubits
- Not universal, can't do Shor's algorithm
- Can still do integer factorization by alternative algorithms
- Can do a large class of problems that can be formulated as discrete optimization with constraints (binary integer linear programming)
- Traveling salesman, four colour
- Is useful already today
- Development is going quickly
- Probably will never replace regular computers
- You will not have one at home
- Will be available at big compute sites
- Use "QPU" as accelerator for part of your computation