Circuits and Qubits

A Circuit contains a QubitRegister and a sequence of operations. By default the register contains enough zero-state qubits for the highest operation index: Circuit(H(5)) creates six qubits.

import scotty.quantum._
import scotty.quantum.gate.StandardGate._
import scotty.quantum.math.Complex
import scotty.simulator.QuantumSimulator

implicit val simulator: QuantumContext = QuantumSimulator()
val circuit = Circuit(H(0)).combine(Circuit(CNOT(0, 1)))
val indices = circuit.indices
assert(indices == (0 until 2))

combine concatenates operations and generates a new default register. It does not preserve a custom register from either circuit. Apply withRegister after combining:

val initialized = circuit.withRegister("10")
val sameBasisState = circuit.withRegister(Qubit.one, Qubit.zero)

Qubit states

A qubit has complex amplitudes $\alpha$ and $\beta$:

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩,|\alpha {|}^2+|\beta {|}^2=1.$$

The amplitudes are Qubit.a and Qubit.b. Their squared magnitudes give measurement probabilities.

val message = Qubit(Complex(0.8), Complex(0.6), "message")
val register = QubitRegister(Qubit.one, Qubit.fiftyFifty, Qubit.zero)
val basisFromString = QubitRegister("1001")
val basisFromInteger = QubitRegister(9)
assert(basisFromString == basisFromInteger)

Integer initialization uses the integer's binary representation without leading zeros. Use a string such as "0001" when register width matters. A custom register must be large enough for every operation index.

Register ordering

Displayed basis strings use the usual binary notation: gate index zero is the rightmost bit. For example:

val rightmostFlip = simulator.runAndMeasure(Circuit(X(0)).withRegister("00"))
assert(rightmostFlip.toBinary == "01")
val initializedBits = simulator.runAndMeasure(Circuit(I(0)).withRegister("10"))
assert(initializedBits.toBinary == "10")

In 0.6.0, register construction and gate indexing use different orders: QubitRegister.values and explicit qubit arguments are stored left to right, while gates index the state vector's bits right to left. In a two-qubit register, QubitRegister(Qubit.one, Qubit.zero) initializes "10", so the first constructor argument is the initial state acted on by gate index 1.

Qubit labels must be unique, but reader label lookup uses the register's sequence index. For asymmetric multi-qubit registers, labels can therefore appear attached to the opposite bit from their initialization position. Prefer numeric gate indices and toBinary when verifying ordering; labels do not change gate addressing. See State Readers.

Circuits and qubits are immutable containers. A Superposition, however, contains a mutable array; its applyGate method changes that array in place.