# The Qmod Workshop - Part 2: Higher-Level Concepts

This is the second part of the Qmod workshop, covering exercises 6 through 10. Make sure to go through Part 1 before continuing with this notebook.

### Exercise 6 - Exponentiation and Pauli Operators

The Qmod language supports different classical types: scalars, arrays, and structs. Structs are objects with member variables, or fields.

The builtin struct type PauliTerm is defined as follows:

from dataclasses import dataclass

from classiq import *

@dataclass
class PauliTerm:
pauli: CArray[Pauli]
coefficient: CReal


Note that Pauli is an enum for all the Pauli matrices (I, X, Y, Z).

Pauli based hamiltonian can be represented as a list of PauliTerms. A Pauli operator defined this way is the argument to a hamiltonian evolution functions.

In this exercise we will use the Suzuki-Trotter function to find the evolution of H=0.5XZXX + 0.25YIZI + 0.3 XIZY (captured as a literal value for the pauli-operator), with the evolution coefficient being 3, the order being 2, and use 4 repetitions.

The declaration of the suzuki_trotter function is:

@qfunc(external=True)
def suzuki_trotter(
pauli_operator: CArray[PauliTerm],
evolution_coefficient: CReal,
order: CInt,
repetitions: CInt,
qbv: QArray[QBit],
) -> None:
pass


Allocate q and invoke the suzuki_trotter quantum function to complete this exercise:

HINT suzuki_trotter(
...,
evolution_coefficient=3,
repetitions=4,
order=2,
qbv=q,
)
from classiq import *

@qfunc
def main(q: Output[QArray[QBit]]) -> None:
allocate(4, q)

qmod = create_model(main)
qprog = synthesize(qmod)
show(qprog)

Opening: https://platform.classiq.io/circuit/1bb9845d-ce2f-40ff-b47b-b0e1dcc53137?version=0.42.1


### Exercise 7 - Basic Arithmetics

#### Exercise 7a

In this exercise we will use quantum numeric variables and calculate expressions over them.

See details on the syntax of numeric types under Quantum types. See more on quantum expressions under Numeric assignment.

Create the following quantum programs: 1. Initialize variables x=2, y=7 and computes res = x + y. 2. Initialize variables x=2, y=7 and computes res = x * y. 3. Initialize variables x=2, y=7, z=1 and computes res = x * y - z.

Guidance: * Use the operator |= to perform out-of-place assignment of arithmetic expression. * To initialize the variables, use the function prepare_int.

from classiq import *

# Your code here:

qmod = create_model(main)
qprog = synthesize(qmod)
show(qprog)


#### Exercise 7b

Declare x to be a 2-qubit variable and y to be 3-qubit variable.

We will perform an addition of two superposition states: x is an equal superposition of 0 and 2, and y is an equal superposition of 1, 2, 3, and 6.

1. Use prepare_state to initialize x and y. Note that prepare_state works with probabilities, not amplitudes. The declaration of the prepare_state function is:
@qfunc(external=True)
def prepare_state(
probabilities: CArray[CReal],
bound: CReal,
out: Output[QArray[QBit]],
) -> None:
pass

HintSet the bound to 0 in your code
1. Compute res = x + y. Execute the resulting circuit. What did you get?
from classiq import *

# Your code here:

qmod = create_model(main)
qprog = synthesize(qmod)
show(qprog)


### Exercise 8 - Within-Apply

The within-apply statement applies the pattern U_dagger V U that appears frequently in quantum computing. It allows you to compute some function V within the context of another function U, and afterward uncompute U in order to release auxiliary qubits storing intermediate results.

#### Exercise 8a

In this exercise, we will use within-apply to compute an arithmetic expression in steps.

Use the within_apply operation to calculate res = x + y + z from a two-variable addition building block with the following steps: 1. Add x and y 2. Add the result to z 3. Uncompute the result of the first operation

For simplicity, initialize the registers to simple integers: x=3, y=5, z=2.

Hints:

• Use a temporary variable.
• Wrap the arithmetic operation in a function.

Execute the circuit and make sure you obtain the expected result.

from classiq import *

# Your code here:

qmod = create_model(main)
qprog = synthesize(qmod)
show(qprog)


#### Exercise 8b

Why should we use within-apply and not just write three concatenated functions? To understand the motivation, we will create another arithmetic circuit. This time, however, we will also set Classiq’s synthesis engine to optimize on the circuit’s number of qubits, i.e., its width.

Setting constraints can be done via the set_constraints operation - see here.

Perform the operation res = w + x + y + z, where w is initialized to 4 and the rest as before:

1. Add x and y (as part of the within_apply operation)
2. Add the result to z (as part of the within_apply operation)
3. Uncompute the result of the first operation (as part of the within_apply operation)
4. Add the result of the second operation to w. There’s no need to perform another uncomputation, as this brings our calculation to an end.

Create the model, optimize on the circuit’s width, and run the circuit. Can you identify where qubits have been released and reused?

from classiq import *

# Your code here:

qmod = create_model(main)
qprog = synthesize(qmod)
show(qprog)

Opening: https://platform.classiq.io/circuit/c2147d5f-03da-46f9-abb9-9f224cf494b5?version=0.42.1


#### Bonus: Use a Single Arithmetic Expression

What happens when we don't manually decompose this expression?

Use Classiq’s arithmetic engine to calculate res |= x + y + z + w and optimize for width. Look at the resulting quantum program - can you identify the computation and uncomputation blocks? What else did you notice?

from classiq import *

# Your code here:

qmod = create_model(main)
qprog = synthesize(qmod)
show(qprog)


### Exercise 9 - In-place Arithmetics

For the following exercise we will use numeric quantum variables that represent fixed-point reals.

Arithmetic expressions can be calculated in-place into a target variable, without allocating new qubits to store the result. This is done using the in-place-xor operator.

In-place assignment is often used to nest arithmetic expressions under quantum operators. Note that out-of-place assignment requires its left-value variable to be un-initialized, and therefore cannot be used under an operator if the variable is declared outside its scope. Applying operators to arithmetic expressions is required in many algorithms. One example is the piecewise evaluation of mathematical functions - calculating different expressions over x depending on the subdomain where x falls.

For this exercise, replace the missing parts in the code snippet below to evaluate the result of:

$f(x) = \begin{cases} 2x + 1 & \text{ if } 0 \leq x < 0.5 \\ x + 0.5 & \text{ if } 0.5 \leq x < 1 \end{cases}$

Notes:

• We cannot use x directly as the control variable in a constrol operator, because it also occurs in the nested scope. to determine if x is in the lower or higher half of the domain we duplicate the most significant bit onto a separate variable called label.

• In Python assignment operators cannot be used in lambda expressions, so the computation of the function needs to be factored out to a named Python function (but not necessarily a Qmod function).

Hint dup_msb(x, label)
control(label, ...) # 0.5 <= x < 1.0
X(label)
control(label, ...) # 0.0 <= x < 0.5
from classiq import *

@qfunc
def dup_msb(qba: QArray[QBit], msb: QBit) -> None:
CX(qba[qba.len - 1], msb)

@qfunc
def main(x: Output[QNum[3, False, 3]], res: Output[QNum[5, False, 3]]) -> None:
allocate(5, res)
allocate(3, x)

label = QArray("label")
allocate(1, label)
# Your code here:

qmod = create_model(main)
qprog = synthesize(qmod)
show(qprog)


### Exercise 10 - State-preparation Algorithm using Quantum-if

#### Binding

The bind operation allows to convert smoothly between different quantum types and split or slice bits when necessary. Here’s an example:

from classiq import *

@qfunc
def main(res: Output[QArray[QBit]]) -> None:
x: QArray[QBit] = QArray("x")
allocate(3, x)
...
lsb = QBit("lsb")
msb = QNum("msb", 2, False, 0)
bind(x, [lsb, msb])
...
bind([lsb, msb], res)

model = create_model(main)
qprog = synthesize(model)
show(qprog)

Opening: https://platform.classiq.io/circuit/ac3ae5f1-7af8-46d6-ac00-aead9e95ecbd?version=0.42.1


The first bind operation splits the 3-qubit register x into the 2-qubit and single-qubit registers lsb and msb, respectively.

After the bind operation: 1. The registers lsb and msb can be operated on as separated registers. 2. The registerx is consumed and can no longer be used.

The second bind operation concatenates the registers to the output register res.

For this exercise, fill in the missing code parts in the above snippet and use the control statement to manually generate the following lovely 3-qubit probability distribution: [1/8, 1/8, 1/8, -sqrt(3)/16, 1/8 + sqrt(3)/16, 1/8, 1/8, 1/8, 1/8].

The following series of gates generate it:

Perform the Hadamard transform on all three qubits.

Apply a rotation on the LSB (least-significant bit) conditioned by the MSB being |0> and the second to last MSB being |1>. How would you write this condition using a QNum?

The following series of gates generate it: 1. Perform the Hadamard transform on all three qubits. 2. Apply a pi/3 rotation on the LSB (least-significant bit) conditioned by the MSB being |0> and the second to last MSB being |1>. How would you write this condition using a QNum?

If you want to validate your results without looking at the full solution, compare them to running using Classiq’s built-in prepare_state function.

from classiq import *
from classiq.qmod.symbolic import sqrt

@qfunc
def pre_prepared_state(q: QArray[QBit]) -> None:
prepare_state(
[
1 / 8,
1 / 8,
1 / 8,
-sqrt(3) / 16,
1 / 8 + sqrt(3) / 16,
1 / 8,
1 / 8,
1 / 8,
1 / 8,
],
0.0,
q,
)

# Your code here:


## Solutions

### Exercise 6

# Solution to exercise 6:

from classiq import *

@qfunc
def main(q: Output[QArray[QBit]]) -> None:
allocate(4, q)
suzuki_trotter(
[
PauliTerm([Pauli.X, Pauli.Z, Pauli.X, Pauli.X], 0.5),
PauliTerm([Pauli.Y, Pauli.I, Pauli.Z, Pauli.I], 0.25),
PauliTerm([Pauli.X, Pauli.I, Pauli.Z, Pauli.Y], 0.3),
],
evolution_coefficient=3,
repetitions=4,
order=2,
qbv=q,
)


### Exercise 7

# Solution to exercise 7a:

from classiq import *

@qfunc
def main(res: Output[QNum]) -> None:
x = QNum("x")
y = QNum("y")
z = QNum("z")
prepare_int(2, x)
prepare_int(7, y)
prepare_int(1, z)
res |= x + y
# res |= x * y
# res |= x * y - z

# Solution to exercise 7b:

from classiq import *

@qfunc
def main(res: Output[QNum]) -> None:
x = QNum("x")
y = QNum("y")
prepare_state([0.5, 0, 0.5, 0.0], 0.0, x)
prepare_state([0, 0.25, 0.25, 0.25, 0.0, 0.0, 0.25, 0.0], 0.0, y)
res |= x + y

model = create_model(main)
qprog = synthesize(model)
show(qprog)


### Exercise 8

# Solution to exercise 8:

from classiq import *

@qfunc
def my_add(x: QNum, y: QNum, res: Output[QNum]) -> None:
res |= x + y

@qfunc
def main(res: Output[QNum]) -> None:
x = QNum("x")
y = QNum("y")
z = QNum("z")
prepare_int(3, x)
prepare_int(5, y)
prepare_int(2, z)

temp = QNum("temp")
within_apply(
compute=lambda: my_add(x, y, temp), action=lambda: my_add(temp, z, res)
)

model = create_model(main)
qprog = synthesize(model)
show(qprog)

# Solution to the advanced part of exercise 8:

from classiq import *

@qfunc
def my_add(x: QNum, y: QNum, res: Output[QNum]) -> None:
res |= x + y

@qfunc
def main(res: Output[QNum]) -> None:
x = QNum("x")
y = QNum("y")
z = QNum("z")
w = QNum("w")
prepare_int(3, x)
prepare_int(5, y)
prepare_int(2, z)
prepare_int(4, w)

temp_xy = QNum("temp_xy")
xyz = QNum("xyz")
within_apply(
compute=lambda: my_add(x, y, temp_xy), action=lambda: my_add(temp_xy, z, xyz)
)
res |= xyz + w

model = create_model(main)
model = set_constraints(
model, Constraints(optimization_parameter=OptimizationParameter.WIDTH)
)
qprog = synthesize(model)
show(qprog)


### Exercise 9

# Solution to exercise 9:

from classiq import *

def linear_func(a: float, b: float, x: QNum, res: QNum) -> None:
res ^= a * x + b

@qfunc
def dup_msb(qba: QArray[QBit], msb: QBit) -> None:
CX(qba[qba.len - 1], msb)

@qfunc
def main(x: Output[QNum[3, False, 3]], res: Output[QNum[5, False, 3]]) -> None:
allocate(5, res)
allocate(3, x)

label = QArray("label")
allocate(1, label)

dup_msb(x, label)
control(label, lambda: linear_func(1.0, 0.5, x, res))  # 0.5 <= x < 1.0
X(label)
control(label, lambda: linear_func(2.0, 1.0, x, res))  # 0.0 <= x < 0.5


### Exercise 10

# Solution to exercise 10:

from classiq import *
from classiq.qmod.symbolic import pi

@qfunc
def main(res: Output[QArray[QBit]]) -> None:
x: QArray[QBit] = QArray("x")
allocate(3, x)

lsb = QBit("lsb")
msb = QNum("msb", 2, False, 0)
bind(x, [lsb, msb])

control(msb == 1, lambda: RY(pi / 3, lsb))

bind([lsb, msb], res)

model = create_model(main)
qprog = synthesize(model)
show(qprog)