Modular Arithmetics

Functions:

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modular_add_inplace

modular_add_inplace(
modulus: CInt,
x: Const[QNum],
y: QNum
) -> None

[Qmod Classiq-library function] Performs the transformation |x>``|y> -> |x>|(x + y mod modulus)>. Note: \|x> and \|y> should have values smaller than modulus. The modulus should satisfy 1 < modulus < 2**n, where n is the size of |x> and |y>. Implementation based on: https://arxiv.org/pdf/1706.06752 Chapter 3.2 Fig 3 Parameters:

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modular_negate_inplace

modular_negate_inplace(
modulus: CInt,
x: QNum
) -> None

[Qmod Classiq-library function] Performs the transformation |x> -> |(-x mod modulus)>. Note: \|x> should have values smaller than modulus. The modulus should satisfy 1 < modulus < 2**n, where n is the size of |x>. Parameters:

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modular_subtract_inplace

modular_subtract_inplace(
modulus: CInt,
x: Const[QNum],
y: QNum
) -> None

[Qmod Classiq-library function] Performs the transformation |x>``|y> -> |x>|(x - y mod modulus)>. Note: \|x> and \|y> should have values smaller than modulus. The modulus should satisfy 1 < modulus < 2**n, where n is the size of |x> and |y>. Parameters:

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modular_double_inplace

modular_double_inplace(
modulus: CInt,
x: QNum
) -> None

[Qmod Classiq-library function] Performs the transformation |x> -> |(2x mod modulus)>. Note: \|x> should have a value smaller than modulus. The modulus must be a constant odd integer. The modulus should satisfy 1 < modulus < 2**n, where n is the size of |x>. Implementation based on: https://arxiv.org/pdf/1706.06752 Chapter 3.2 Fig 4 Parameters:

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modular_add_constant_inplace

modular_add_constant_inplace(
modulus: CInt,
a: CInt,
x: QNum
) -> None

[Qmod Classiq-library function] Performs the transformation |x> -> |(x + a mod modulus)>. Note: \|x> and a should have values smaller than modulus. The modulus should satisfy 1 < modulus < 2**n, where n is the size of |x>. Implementation is based on the logic in: https://arxiv.org/pdf/1706.06752 Chapter 3.2 Fig 3 Parameters:

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modular_multiply

modular_multiply(
modulus: CInt,
x: Const[QArray[QBit]],
y: Const[QArray[QBit]],
z: QArray[QBit]
) -> None

[Qmod Classiq-library function] Performs the transformation |x>``|y>``|0> -> |x>``|y>|(x*y mod modulus)> Note: \|x>, \|y> should have the same size and have values smaller than modulus. The modulus must be a constant odd integer. The modulus should satisfy 1 < modulus < 2**n, where n is the size of |x> and |y>. The output register z must be pre-allocated with the same size as x and y. Implementation is based on the logic in: https://arxiv.org/pdf/1706.06752 Chapter 3.2 Fig 5 Parameters:

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modular_square

modular_square(
modulus: CInt,
x: Const[QArray[QBit]],
z: QArray[QBit]
) -> None

[Qmod Classiq-library function] Performs the transformation |x>``|0> -> |x>|(x^2 mod modulus)>. Note: \|x> should have the same size and have values smaller than modulus. The modulus must be a constant odd integer. The modulus should satisfy 1 < modulus < 2**n, where n is the size of |x>. The output register z must be pre-allocated with the same size as x. Implementation is based on: https://arxiv.org/pdf/1706.06752 Chapter 3.2 Fig 6 Parameters:

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modular_multiply_constant

modular_multiply_constant(
modulus: CInt,
x: Const[QNum],
a: CInt,
y: QNum
) -> None

[Qmod Classiq-library function] Performs the transformation |x>``|y> -> |x>|(x * a mod modulus)>. Note: \|x> and \|y> should have values smaller than modulus. The modulus must be a constant odd integer. The modulus should satisfy 1 < modulus < 2**n, where n is the size of |x> and |y>. Parameters:

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modular_multiply_constant_inplace

modular_multiply_constant_inplace(
modulus: CInt,
a: CInt,
x: QNum
) -> None

[Qmod Classiq-library function] In-place modular multiplication of x by a classical constant modulo a symbolic modulus. Performs |x> -> |(x * a mod modulus)>. Note: \|x> should have values smaller than modulus. The modulus should satisfy 1 < modulus < 2**n, where n is the size of |x>. The constant a should have an inverse modulo modulus, i.e. gcd(a, modulus) = 1. The constant a should satisfy 0 <= a < modulus. Parameters:

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modular_to_montgomery_inplace

modular_to_montgomery_inplace(
modulus: CInt,
x: QNum
) -> None

[Qmod Classiq-library function] Converts a quantum integer |x> into its Montgomery representation modulo modulus in place. The Montgomery factor is R = 2n, where n = x.size (the number of qubits in |x>). This function performs the transformation |x> -> |(x * R mod modulus)>. Note: \|x> should have values smaller than modulus. The modulus should satisfy 1 < modulus < 2n, where n is the size of |x>. The modulus must be odd so that R = 2**n is invertible modulo modulus (gcd(R, modulus) = 1). Parameters:

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modular_montgomery_to_standard_inplace

modular_montgomery_to_standard_inplace(
modulus: CInt,
x: QNum
) -> None

[Qmod Classiq-library function] Converts quantum integer |x> from Montgomery representation to standard form in place modulo modulus. The Montgomery factor is R = 2n, where n = x.size (the number of qubits in |x>). This function performs the transformation |x> -> |(x * R^-1 mod modulus)>. Note: \|x> should have values smaller than modulus. The modulus should satisfy 1 < modulus < 2n, where n is the size of |x>. The modulus must be odd so that R = 2**n is invertible modulo modulus (gcd(R, modulus) = 1). Parameters:

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modular_inverse_inplace

modular_inverse_inplace(
modulus: CInt,
v: QNum,
m: Output[QArray[QBit]]
) -> None

[Qmod Classiq-library function] Computes the modular inverse of a quantum number |v> modulo modulus in place, using the Kaliski algorithm. Performs the transformation |v> -> |(v^-1 mod modulus)>. Based on: https://arxiv.org/pdf/2302.06639 Chapter 5 Note: \|v> should have values smaller than modulus. If |v> = 0, the output will be 0 (although 0 does not have an inverse modulo modulus). The modulus should be prime OR at least gcd(v, modulus) = 1. The modulus must be a constant odd integer. The modulus should satisfy 1 < modulus < 2**n, where n is the size of |v>. The ancilla qubits m are provided as Output, will be allocated to length 2*n. Parameters:

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kaliski_iteration

kaliski_iteration(
modulus: CInt,
i: CInt,
v: QNum,
m: QArray[QBit],
u: QNum,
r: QNum,
s: QNum,
a: QBit,
b: QBit,
f: QBit
) -> None

Single iteration of the Kaliski modular inverse algorithm main loop. Based on: https://arxiv.org/pdf/2302.06639 Figure 15 Note: Assumes the global inversion constraints (odd modulus, 1 < modulus < 2**n). Called with 0 <= v < modulus; per-iteration ancilla bit is m[i]. Parameters:

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modular_rsub_inplace

modular_rsub_inplace(
modulus: CInt,
a: CInt,
x: QNum
) -> None

[Qmod Classiq-library function] Performs the in-place modular right-subtraction |x> -> |(a - x mod modulus)>. Note: \|x> should have values smaller than modulus. The modulus should satisfy 1 < modulus < 2**n, where n is the size of |x>. The classical constant a should be in the range 0 <= a < modulus. Parameters: