Quantum computing and algorithms
Gates, Clifford circuits, error correction, teleportation, and the algorithms of Grover and Shor.
Global phase
Overall phase has no observable consequence; rays, not vectors, label pure states.
Computing ↗Relative phase
Relative phases govern interference, so $|+\rangle$ and $|-\rangle$ are distinguishable via an $X$-basis measurement.
Computing ↗Single-qubit synthesis
Rotations about two nonparallel axes generate all of $SU(2)$, so any single-qubit gate is a product of such rotations.
Computing ↗Universal gates
Arbitrary one-qubit gates plus CNOT approximate any unitary; a finite set such as $\{H,T,\mathrm{CNOT}\}$ approximates to arbitrary precision.
Computing ↗Clifford-Pauli normalization
Clifford gates ($H$, $S$, CNOT) map Paulis to Paulis up to phase, and are the normalizer of the Pauli group.
Computing ↗Gottesman-Knill theorem
Stabilizer circuits can be simulated classically in polynomial time despite generating entanglement.
Computing ↗Quantum error correction is not cloning
Codes entangle information across qubits, e.g. the three-qubit code, rather than copying the unknown state.
Computing ↗Knill-Laflamme condition
A code with projector $P$ corrects error set $\{E_a\}$ iff this holds with $c$ Hermitian: errors must not reveal logical information.
Computing ↗Quantum teleportation
Alice's Bell measurement collapses Bob's qubit to the state up to a Pauli, corrected using her two bits.
Computing ↗Superdense coding
Alice applies one of $I,X,Z,XZ$ to her half, and Bob's Bell measurement reads two bits.
Computing ↗Grover search
Quadratic speedup for unstructured search, using about $\tfrac\pi4\sqrt N$ rounds of amplitude amplification; optimal.
Computing ↗Shor factoring
Factoring reduces to finding the order of $a$ mod $N$, solved efficiently with quantum phase estimation and the QFT, in polynomial time.
Computing ↗Quantum phase estimation
Controlled-$U^{2^k}$ gates and an inverse QFT extract an eigenphase; the key subroutine behind many algorithms.
Computing ↗No-fast-forwarding
For generic black-box Hamiltonians, simulating time $t$ needs resources linear in $t$; special structure can circumvent this.
Computing ↗Barren plateaus
Under certain assumptions, such as random deep circuits, gradients vanish exponentially in qubit number, hampering variational training.
Computing ↗