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Arithmetic Expressions

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This tutorial demonstrates automatic arithmetic operation management by the synthesis engine. It synthesizes a complex arithmetic expression, where uncomputation procedure, together with initialization and reuse of auxiliary qubits, are all automated. Given different global width or depth constraints results in different circuits.

Define a quantum model that applies some quantum arithmetic operation on QNum variables.

from classiq import *
from classiq.qmod.symbolic import max


@qfunc
def main(z: Output[QNum]):
    x = QNum("x")
    y = QNum("y")
    x |= 2
    y |= 1
    z |= (2 * x + y + max(3 * y, 2)) > 4


qmod = create_model(main)
qmod = set_preferences(qmod, random_seed=424788457)

You can try different optimization scenarios, below we introduce two examples:

  1. Optimizing over depth and constraining the maximal width to 9 qubits.

  2. Optimizing over depth and constraining the maximal width to 12 qubits.

Optimizing over depth and constraining the maximal width to 9 qubits

NUM_QUBITS_1 = 9
qmod_1 = set_constraints(qmod, optimization_parameter="depth", max_width=NUM_QUBITS_1)
write_qmod(qmod_1, f"arithmetic_demo_{NUM_QUBITS_1}_qubits")

qprog_1 = synthesize(qmod_1)
show(qprog_1)

result = execute(qprog_1).result_value()
print("The result of the arithmetic calculation: ", result.parsed_counts)

The result of the arithmetic calculation:  [{'z': 1.0}: 2048]

Change the quantum model constraint to treat the second scenario for optimizing over depth and constraining the maximal width to 12 qubits: 

NUM_QUBITS_2 = 12
qmod_2 = set_constraints(qmod, optimization_parameter="depth", max_width=NUM_QUBITS_2)
write_qmod(qmod_2, f"arithmetic_demo_{NUM_QUBITS_2}_qubits")

qprog_2 = synthesize(qmod_2)
show(qprog_2)

result = execute(qprog_2).result_value()
print("The result of the arithmetic calculation: ", result.parsed_counts)

The result of the arithmetic calculation:  [{'z': 1.0}: 2048]

### Mathematical Background

The given mathematical expression:

\[z = (2 \cdot x + y + \max(3 \cdot y, 2)) > 4\]

is solved by the automatic arithmetic operation management, optimizing over depth and constraining the maximal width to 9 and 12 qubits, which outputs the value for z.

The parsed_counts result:

  • Optimizing over depth and constraining the maximal width to 9 qubits: [{'z': 1.0}: 2048]

  • Optimizing over depth and constraining the maximal width to 12 qubits: [{'z': 1.0}: 2048]

Both the result are same which verifies the arithmetic expression to be True.

The expected result:

Allocated values:

\[x = 2, \,\,\, y = 1\]

Expression Calculation:

\[2 \cdot x + y = 2 \cdot 2 + 1 = 4 + 1 = 5 \,\,\, 3 \cdot y = 3 \cdot 1 = 3 \,\,\, \max(3 \cdot y, 2) = \max(3, 2) = 3\]

Therefore:

\[2 \cdot x + y + \max(3 \cdot y, 2) = 5 + 3 = 8\]

As \(\(8 > 4 \implies \text{True}\)\)

z is assigned the value True : \(z \implies 1\)