Qsvt

Functions:

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qsvt_step

qsvt_step(
phase1: CReal,
phase2: CReal,
proj_cnot_1: QCallable[QBit],
proj_cnot_2: QCallable[QBit],
u: QCallable,
aux: QBit
) -> None

[Qmod Classiq-library function] Applies a single QSVT step, composed of 2 projector-controlled-phase rotations, and applications of the block encoding unitary u and its inverse: $$\Pi_{\phi_2}U^{\dagger}\tilde{\Pi}_{\phi_{1}}U$$ Parameters:

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qsvt

qsvt(
phase_seq: CArray[CReal],
proj_cnot_1: QCallable[QBit],
proj_cnot_2: QCallable[QBit],
u: QCallable,
aux: QBit
) -> None

[Qmod Classiq-library function] Implements the Quantum Singular Value Transformation (QSVT) - an algorithmic framework, used to apply polynomial transformations of degree d on the singular values of a block encoded matrix, given as the unitary u. Given a unitary $U$, a list of phase angles $\phi_1, \phi_2, ..., \phi_\{d+1\}$ and 2 projector-controlled-not operands $C_\{\Pi\}NOT,C_\{\tilde\{\Pi\}\}NOT$, the QSVT sequence is as follows: Given a unitary $U$, a list of phase angles $\phi_1, \phi_2, ..., \phi_\{d+1\}$ and 2 projector-controlled-not operands $C_\{\Pi\}NOT,C_\{\tilde\{\Pi\}\}NOT$, the QSVT sequence is as follows: $$\tilde{\Pi}_{\phi_{d+1}}U \prod_{k=1}^{(d-1)/2} (\Pi_{\phi_{d-2k}} U^{\dagger}\tilde{\Pi}_{\phi_{d - (2k+1)}}U)\Pi_{\phi_{1}}$$ for odd $d$, and: $$\prod_{k=1}^{d/2} (\Pi_{\phi_{d-(2k-1)}} U^{\dagger}\tilde{\Pi}_{\phi_{d-2k}}U)\Pi_{\phi_{1}}$$ for even $d$. Each of the $\Pi$s is a projector-controlled-phase unitary, according to the given projectors. Parameters:

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projector_controlled_phase

projector_controlled_phase(
phase: CReal,
proj_cnot: QCallable[QBit],
aux: QBit
) -> None

[Qmod Classiq-library function] Assigns a phase to the entire subspace determined by the given projector. Corresponds to the operation: \Pi_{\phi} = (C_{\Pi}NOT) e^{-irac{\phi}{2}Z}(C_{\Pi}NOT) Parameters:

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qsvt_inversion

qsvt_inversion(
phase_seq: CArray[CReal],
block_encoding_cnot: QCallable[QBit],
u: QCallable,
aux: QBit
) -> None

[Qmod Classiq-library function] Implements matrix inversion on a given block-encoding of a square matrix, using the QSVT framework. Applies a polynomial approximation of the inverse of the singular values of the matrix encoded in u. The phases for the polynomial should be pre-calculated and passed into the function. Parameters:

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projector_controlled_double_phase

projector_controlled_double_phase(
phase_even: CReal,
phase_odd: CReal,
proj_cnot: QCallable[QBit],
aux: QBit,
lcu: QBit
) -> None

[Qmod Classiq-library function] Assigns 2 phases to the entire subspace determined by the given projector, each one is controlled differentely on a given lcu qvar. Used in the context of the qsvt_lcu function. Corresponds to the operation: $$\Pi_{\phi_{odd}, \phi_{even}} = (C_{\Pi}NOT) (C_{lcu=1}e^{-i\frac{\phi_{even}}{2}Z}) (C_{lcu=0}e^{-i\frac{\phi_{odd}}{2}Z}) (C_{\Pi}NOT)$$ Parameters:

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qsvt_lcu_step

qsvt_lcu_step(
phases_even: CArray[CReal],
phases_odd: CArray[CReal],
proj_cnot_1: QCallable[QBit],
proj_cnot_2: QCallable[QBit],
u: QCallable,
aux: QBit,
lcu: QBit
) -> None

[Qmod Classiq-library function] Applies a single QSVT-lcu step, composed of 2 double phase projector-controlled-phase rotations, and applications of the block encoding unitary u and its inverse: $$(C_{lcu=1}\Pi^{even}_{\phi_2})(C_{lcu=0}\Pi^{odd}_{\phi_2})U^{\dagger}(C_{lcu=1}\tilde{\Pi}^{even}_{\phi_1})(C_{lcu=0}\tilde{\Pi}^{odd}_{\phi_1})U$$ Parameters:

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qsvt_lcu

qsvt_lcu(
phase_seq_even: CArray[CReal],
phase_seq_odd: CArray[CReal],
proj_cnot_1: QCallable[QBit],
proj_cnot_2: QCallable[QBit],
u: QCallable,
aux: QBit,
lcu: QBit
) -> None

[Qmod Classiq-library function] Implements the Quantum Singular Value Transformation (QSVT) for a linear combination of odd and even polynomials, so that it is possible to encode a polynomial of indefinite parity, such as approximation to exp(i*A) or exp(A). Should work for Hermitian block encodings. The function is equivalent to applying the qsvt function for odd and even polynomials with a LCU function, but is more efficient as the two polynomials share the same applications of the given unitary. The function is intended to be called within a context of LCU, where it is called as the SELECT operator, and wrapped with initialization of the lcu qubit to get the desired combination coefficients. The even polynomial corresponds to the case where the $lcu=|0\rangle$, while the odd to $lcu=|1\rangle$. Note: the two polynomials should have the same degree up to a difference of 1. Parameters:

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gqsp

gqsp(
u: QCallable,
aux: QBit,
phases: CArray[CArray[CReal, Literal[3]]],
negative_power: CInt
) -> None

Implements Generalized Quantum Signal Processing (GQSP), which realizes a (Laurent) polynomial transformation of degree d on the eigenvalues of the given signal unitary u. The protocol is according to https://arxiv.org/abs/2308.01501 Fig.2. Notes:

• The user is encouraged to use the function gqsp_phases to find phases that correspond to the wanted polynomial transformation.
• Feasibility: the target polynomial must satisfy $|P(e^\{i*theta\})|$ <= 1 for all theta in $[0, 2*pi)$. This ensures a unitary completion exists.
• Using negative_power = m (m >= 0) you can realize Laurent polynomials with negative exponents: the implemented transform is equivalent to applying $z^\{-m\} * P(z)$ (i.e., shift the minimal degree to -m). For ordinary (non-Laurent) polynomials, set negative_power = 0.

Parameters: