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Phase

The phase statement is used to encode in the phase of a quantum state the result of some arithmetic computation. It applies a relative phase to computational-basis states of variables, which is proportionate to the value of a specified expression over these variables. phase statements are often used to compute the cost of an optimization problem.

Syntax

phase ( quantum-expression [ , coefficient ] )

def phase(expr: SymbolicExpr, coefficient: float = 1.0) -> None:
    pass

Semantics

  • quantum-expression consists of quantum scalar variables, numeric constant literals, and classical scalar variables, composed using arithmetic operators. See below the set of supported operators.
  • coefficient expression is optional, and may include an execution parameter. Note that an execution parameter cannot occur in quantum-expression. If not provided, the default coefficient value is 1.0.
  • The operation rotates each computational basis state about the Z axis by an angle equal to the value of quantum-expression, multiplied by coefficient if specified.
  • For quantum-expression over quantum variable \(x_1, x_2, ... x_n\) that computes the function \(f(x_1, x_2, ... x_n)\), and coefficient = \(\theta\) the operation performed by the statement is - \(|x\rangle \rightarrow e^{i \theta f(x_1, x_2, ... x_n)}|x\rangle\)
  • The expression must be a polynomial in the quantum variables. It is compiled into an Ising-model Hamiltonian, which is evolved per the specified coefficient.

The following operators are supported:

  • Add: +
  • Subtract: - (binary)
  • Negate: - (unary)
  • Multiply: *
  • Divide: /
  • Power: ** (quantum base, positive classical integer exponent)

Note that when the expression consists of a single one-qubit variable, phase statement is equivalent to the core-library function PHASE().

Examples

Example 1

In the following model phase \(x^2\) is applied to variable \(x\) with the coefficient \(\frac{\pi}{4}\). \(x\) is initialized to a superposition of the values 0, 1, 2, and 3. After the phase statement, state 1 is in phase \(\frac{\pi}{4}\) relative to state 0, state 2 is rotated \(\pi\) relative to state 0. State 3 is rotated \(\frac{\pi}{4}\), which is a full \(2\pi\) + \(\frac{\pi}{4}\) rotation, that is, the same phase as state 1.

qfunc main(output x: qnum) {
  allocate(2, x);
  hadamard_transform(x);
  phase (x**2, pi/4);
}

from classiq import qfunc, Output, QNum, allocate, hadamard_transform, phase
from classiq.qmod.symbolic import pi


@qfunc
def main(x: Output[QNum]):
    allocate(2, x)
    hadamard_transform(x)
    phase(x**2, pi / 4)

Visualizing the synthesized quantum program, you can see how Z-rotations and controlled Z-rotations are used to achieve the required rotation.

simple_phase_circuit.png

When executing this model using a state-vector simulator, the relative phases of the different states can be observed. In

simple_phase_exe_result.png

Example 2

The following example demonstrates the use of phase statement to encode the cost of a max-cut problem, as the implementation of the cost-layer in a QAOA ansatz. The qubit array v represents the partition of the set of vertices in a graph into two, and the expression inside the phase statement computes the number of edges that cross the partition. Executing this model with the right set of parameter values will yield with high probability an optimal solution.

qfunc main(gammas: real[3], betas: real[3], output v: qbit[]) {
  allocate(3, v);
  hadamard_transform(v);
  repeat (i: 4) {
    phase(
      (v[0] * (1 - v[1]) + v[1] * (1 - v[0]))  // edge 0-1
      + (v[0] * (1 - v[2]) + v[2] * (1 - v[0])),  // edge 0-2
      gammas[i]
    );
    apply_to_all(lambda(q) {
      RX(betas[i], q);
    }, v);
  }
}

from classiq import *


@qfunc
def main(
    gammas: CArray[CReal, 4],
    betas: CArray[CReal, 4],
    v: Output[QArray[QBit]],
):
    allocate(3, v)
    hadamard_transform(v)
    for i in range(4):
        phase(
            (v[0] * (1 - v[1]) + v[1] * (1 - v[0]))  # edge 0-1
            + (v[0] * (1 - v[2]) + v[2] * (1 - v[0])),  # edge 0-2
            gammas[i],
        )
        apply_to_all(lambda q: RX(betas[i], q), v)