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(x: Output[QNum], y: Output[QNum], z: Output[QNum]):
    allocate(2, x)
    allocate(1, y)
    hadamard_transform(x)
    hadamard_transform(y)
    z |= (2 * x + y + max(3 * y, 2)) > 4

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

Optimizing over depth and constraining the maximal width to 10 qubits.
Optimizing over depth and constraining the maximal width to 12 qubits.

Optimizing over depth and constraining the maximal width to 10 qubits

NUM_QUBITS_1 = 10
qprog_1 = synthesize(
    main,
    constraints=Constraints(optimization_parameter="depth", max_width=NUM_QUBITS_1),
)
show(qprog_1)
sample(qprog_1)

Output:

Quantum program link: https://platform.classiq.io/circuit/3DcBR5bi1Kg3ZdQpUAakewAKcSh
  

Output:

Submitting job to simulator
  Job: https://platform.classiq.io/jobs/6c500169-a27b-4b80-863f-be1996c1c08c
  

	x	y	z	counts	probability	bitstring
0	3	0	1	275	0.134277	1011
1	1	1	1	274	0.133789	1101
2	3	1	1	261	0.127441	1111
3	0	1	0	255	0.124512	0100
4	2	0	1	253	0.123535	1010
5	2	1	1	252	0.123047	1110
6	0	0	0	242	0.118164	0000
7	1	0	0	236	0.115234	0001

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
qprog_2 = synthesize(
    main,
    constraints=Constraints(optimization_parameter="depth", max_width=NUM_QUBITS_2),
)
show(qprog_2)
sample(qprog_2)

Output:

Quantum program link: https://platform.classiq.io/circuit/3DcBe5FXGO6GThlRywJnBjDrEma
  

Output:

Submitting job to simulator
  Job: https://platform.classiq.io/jobs/7eafa5de-83a2-460a-ace2-e2e7332600a6
  

	x	y	z	counts	probability	bitstring
0	2	0	1	271	0.132324	1010
1	0	1	0	266	0.129883	0100
2	0	0	0	260	0.126953	0000
3	3	1	1	258	0.125977	1111
4	1	0	0	254	0.124023	0001
5	1	1	1	254	0.124023	1101
6	2	1	1	245	0.119629	1110
7	3	0	1	240	0.117188	1011

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