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Random permutation of strings in Trotterisation #702

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7b8860a
Implement PauliStrSum random permutations
vaferreiQMT Nov 10, 2025
141d56e
Add randomisation to Trotter functions
vaferreiQMT Nov 10, 2025
021aef3
Document random Pauli permutations for Trotterisation
vaferreiQMT Dec 3, 2025
eacf599
Add test for Trotter randomisation
vaferreiQMT Dec 3, 2025
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5 changes: 3 additions & 2 deletions examples/extended/dynamics.c
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Original file line number Diff line number Diff line change
Expand Up @@ -153,11 +153,12 @@ int main() {
int order = 4;
int reps = 5;
int steps = 20;
bool randomise = true;

for (int i=0; i<steps; i++) {

// evolve qureg under (approx) exp(-i dt H)
applyTrotterizedUnitaryTimeEvolution(qureg, hamil, dt, order, reps);
applyTrotterizedUnitaryTimeEvolution(qureg, hamil, dt, order, reps, randomise);

// calculate and report <O>
qreal time = dt * (i+1);
Expand Down Expand Up @@ -187,7 +188,7 @@ int main() {

// verify results by uninterrupted higher-order simulation to target time
initPlusState(qureg);
applyTrotterizedUnitaryTimeEvolution(qureg, hamil, dt*steps, order+2, reps*steps);
applyTrotterizedUnitaryTimeEvolution(qureg, hamil, dt*steps, order+2, reps*steps, randomise);
reportScalar("final <O>", calcExpecPauliStrSum(qureg, observ));

// clean up
Expand Down
5 changes: 3 additions & 2 deletions examples/extended/dynamics.cpp
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Original file line number Diff line number Diff line change
Expand Up @@ -150,11 +150,12 @@ int main() {
int order = 4;
int reps = 5;
int steps = 20;
bool randomise = true;

for (int i=0; i<steps; i++) {

// evolve qureg under (approx) exp(-i dt H)
applyTrotterizedUnitaryTimeEvolution(qureg, hamil, dt, order, reps);
applyTrotterizedUnitaryTimeEvolution(qureg, hamil, dt, order, reps, randomise);

// calculate and report <O>
qreal time = dt * (i+1);
Expand All @@ -181,7 +182,7 @@ int main() {

// verify results by uninterrupted higher-order simulation to target time
initPlusState(qureg);
applyTrotterizedUnitaryTimeEvolution(qureg, hamil, dt*steps, order+2, reps*steps);
applyTrotterizedUnitaryTimeEvolution(qureg, hamil, dt*steps, order+2, reps*steps, randomise);
reportScalar("final <O>", calcExpecPauliStrSum(qureg, observ));

// clean up
Expand Down
98 changes: 59 additions & 39 deletions quest/include/trotterisation.h
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Original file line number Diff line number Diff line change
Expand Up @@ -10,9 +10,12 @@
* @{
*/


#ifndef TROTTERISATION_H
#define TROTTERISATION_H

#include <stdbool.h>

#include "quest/include/qureg.h"
#include "quest/include/paulis.h"
#include "quest/include/matrices.h"
Expand Down Expand Up @@ -40,9 +43,12 @@ extern "C" {
*
* Effects an approximation to the exponential of @p sum, weighted by @p angle times @f$ i @f$, upon @p qureg,
* via the symmetrized Trotter-Suzuki decomposition (<a href="https://arxiv.org/abs/math-ph/0506007">arXiv</a>).
*
* Increasing @p reps (the number of Trotter repetitions) or @p order (an even, positive integer or one)
* improves the accuracy of the approximation by reducing the "Trotter error" due to non-commuting
* terms of @p sum, though increases the runtime linearly and exponentially respectively.
* terms of @p sum, though increases the runtime linearly and exponentially respectively.
* Using @p permutePaulis the ordering of terms in the sum can also be randomised, which generally
* improves the accuracy of the approximation for low order decompositions (<a href="https://arxiv.org/abs/1805.08385">arXiv</a>).
*
* @formulae
*
Expand Down Expand Up @@ -103,6 +109,15 @@ extern "C" {
* > These formulations are taken from 'Finding Exponential Product Formulas
* > of Higher Orders', Naomichi Hatano and Masuo Suzuki (2005) (<a href="https://arxiv.org/abs/math-ph/0506007">arXiv</a>).
*
* When @p permutePaulis=true the terms of @p sum are effected in a random order at each repetition. That is, each repetition of the Trotter-Suzuki decomposition is evaluated with the sum
* @f[
\hat{H} = \sum\limits_j^T c_{\pi(j)} \, \hat{\sigma}_{\pi(j)}
* @f]
* where @f$ \pi @f$ is a randomly selected permutation.
*
* @important
* Using @p permutePaulis=true will cause @p sum to be mutated by the Trotterisation.
*
* @equivalences
*
* - By passing @f$ \theta = - \Delta t / \hbar @f$, this function approximates unitary time evolution of a closed
Expand Down Expand Up @@ -137,11 +152,12 @@ extern "C" {
* - This function only ever effects @f$ \exp \left(\iu \, \theta \, \hat{H} \right) @f$ exactly
* when all PauliStr in @p sum = @f$ \hat{H} @f$ commute, or @p reps @f$ \rightarrow \infty @f$.
*
* @param[in,out] qureg the state to modify.
* @param[in] sum a weighted sum of Pauli strings to approximately exponentiate.
* @param[in] angle the prefactor of @p sum times @f$ i @f$ in the exponent.
* @param[in] order the order of the Trotter-Suzuki decomposition (e.g. @p 1, @p 2, @p 4, ...).
* @param[in] reps the number of Trotter repetitions.
* @param[in,out] qureg the state to modify.
* @param[in] sum a weighted sum of Pauli strings to approximately exponentiate.
* @param[in] angle the prefactor of @p sum times @f$ i @f$ in the exponent.
* @param[in] order the order of the Trotter-Suzuki decomposition (e.g. @p 1, @p 2, @p 4, ...).
* @param[in] reps the number of Trotter repetitions.
* @param[in] permutePaulis whether to randomly reorder Pauli strings at each repetition.
*
* @throws @validationerror
* - if @p qureg or @p sum are uninitialised.
Expand All @@ -157,31 +173,31 @@ extern "C" {
*
* @author Tyson Jones
*/
void applyTrotterizedPauliStrSumGadget(Qureg qureg, PauliStrSum sum, qreal angle, int order, int reps);
void applyTrotterizedPauliStrSumGadget(Qureg qureg, PauliStrSum sum, qreal angle, int order, int reps, bool permutePaulis);


/// @notyetdoced
/// @notyettested
/// @see
/// - applyTrotterizedPauliStrSumGadget()
/// - applyControlledCompMatr1()
void applyTrotterizedControlledPauliStrSumGadget(Qureg qureg, int control, PauliStrSum sum, qreal angle, int order, int reps);
void applyTrotterizedControlledPauliStrSumGadget(Qureg qureg, int control, PauliStrSum sum, qreal angle, int order, int reps, bool permutePaulis);


/// @notyetdoced
/// @notyettested
/// @see
/// - applyTrotterizedPauliStrSumGadget()
/// - applyMultiControlledCompMatr1()
void applyTrotterizedMultiControlledPauliStrSumGadget(Qureg qureg, int* controls, int numControls, PauliStrSum sum, qreal angle, int order, int reps);
void applyTrotterizedMultiControlledPauliStrSumGadget(Qureg qureg, int* controls, int numControls, PauliStrSum sum, qreal angle, int order, int reps, bool permutePaulis);


/// @notyetdoced
/// @notyettested
/// @see
/// - applyTrotterizedPauliStrSumGadget()
/// - applyMultiStateControlledCompMatr1()
void applyTrotterizedMultiStateControlledPauliStrSumGadget(Qureg qureg, int* controls, int* states, int numControls, PauliStrSum sum, qreal angle, int order, int reps);
void applyTrotterizedMultiStateControlledPauliStrSumGadget(Qureg qureg, int* controls, int* states, int numControls, PauliStrSum sum, qreal angle, int order, int reps, bool permutePaulis);


/** @notyettested
Expand Down Expand Up @@ -218,11 +234,12 @@ void applyTrotterizedMultiStateControlledPauliStrSumGadget(Qureg qureg, int* con
* - This function only ever effects @f$ \exp \left(\iu \, \theta \, \hat{H} \right) @f$ exactly
* when all PauliStr in @p sum = @f$ \hat{H} @f$ commute.
*
* @param[in,out] qureg the state to modify.
* @param[in] sum a weighted sum of Pauli strings to approximately exponentiate.
* @param[in] angle an effective prefactor of @p sum in the exponent.
* @param[in] order the order of the Trotter-Suzuki decomposition (e.g. @p 1, @p 2, @p 4, ...).
* @param[in] reps the number of Trotter repetitions.
* @param[in,out] qureg the state to modify.
* @param[in] sum a weighted sum of Pauli strings to approximately exponentiate.
* @param[in] angle an effective prefactor of @p sum in the exponent.
* @param[in] order the order of the Trotter-Suzuki decomposition (e.g. @p 1, @p 2, @p 4, ...).
* @param[in] reps the number of Trotter repetitions.
* @param[in] permutePaulis whether to randomly reorder Pauli strings at each repetition.
*
* @throws @validationerror
* - if @p qureg or @p sum are uninitialised.
Expand All @@ -232,7 +249,7 @@ void applyTrotterizedMultiStateControlledPauliStrSumGadget(Qureg qureg, int* con
*
* @author Tyson Jones
*/
void applyTrotterizedNonUnitaryPauliStrSumGadget(Qureg qureg, PauliStrSum sum, qcomp angle, int order, int reps);
void applyTrotterizedNonUnitaryPauliStrSumGadget(Qureg qureg, PauliStrSum sum, qcomp angle, int order, int reps, bool permutePaulis);


// end de-mangler
Expand All @@ -248,15 +265,15 @@ void applyTrotterizedNonUnitaryPauliStrSumGadget(Qureg qureg, PauliStrSum sum, q
/// @notyetdoced
/// @cppvectoroverload
/// @see applyTrotterizedMultiControlledPauliStrSumGadget()
void applyTrotterizedMultiControlledPauliStrSumGadget(Qureg qureg, std::vector<int> controls, PauliStrSum sum, qreal angle, int order, int reps);
void applyTrotterizedMultiControlledPauliStrSumGadget(Qureg qureg, std::vector<int> controls, PauliStrSum sum, qreal angle, int order, int reps, bool permutePaulis);


/// @notyettested
/// @notyetvalidated
/// @notyetdoced
/// @cppvectoroverload
/// @see applyTrotterizedMultiStateControlledPauliStrSumGadget()
void applyTrotterizedMultiStateControlledPauliStrSumGadget(Qureg qureg, std::vector<int> controls, std::vector<int> states, PauliStrSum sum, qreal angle, int order, int reps);
void applyTrotterizedMultiStateControlledPauliStrSumGadget(Qureg qureg, std::vector<int> controls, std::vector<int> states, PauliStrSum sum, qreal angle, int order, int reps, bool permutePaulis);


#endif // __cplusplus
Expand Down Expand Up @@ -353,11 +370,12 @@ extern "C" {
* - applyTrotterizedNoisyTimeEvolution()
* - applyTrotterizedNonUnitaryPauliStrSumGadget()
*
* @param[in,out] qureg the state to modify.
* @param[in] hamil the Hamiltonian as a a weighted sum of Pauli strings.
* @param[in] time the duration over which to simulate evolution.
* @param[in] order the order of the Trotter-Suzuki decomposition (e.g. @p 1, @p 2, @p 4, ...).
* @param[in] reps the number of Trotter repetitions.
* @param[in,out] qureg the state to modify.
* @param[in] hamil the Hamiltonian as a a weighted sum of Pauli strings.
* @param[in] time the duration over which to simulate evolution.
* @param[in] order the order of the Trotter-Suzuki decomposition (e.g. @p 1, @p 2, @p 4, ...).
* @param[in] reps the number of Trotter repetitions.
* @param[in] permutePaulis whether to randomly reorder Pauli strings at each repetition.
*
* @throws @validationerror
* - if @p qureg or @p hamil are uninitialised.
Expand All @@ -368,7 +386,7 @@ extern "C" {
*
* @author Tyson Jones
*/
void applyTrotterizedUnitaryTimeEvolution(Qureg qureg, PauliStrSum hamil, qreal time, int order, int reps);
void applyTrotterizedUnitaryTimeEvolution(Qureg qureg, PauliStrSum hamil, qreal time, int order, int reps, bool permutePaulis);


/** @notyettested
Expand Down Expand Up @@ -484,11 +502,12 @@ void applyTrotterizedUnitaryTimeEvolution(Qureg qureg, PauliStrSum hamil, qreal
* - applyTrotterizedUnitaryTimeEvolution()
* - applyTrotterizedNonUnitaryPauliStrSumGadget()
*
* @param[in,out] qureg the state to modify.
* @param[in] hamil the Hamiltonian as a a weighted sum of Pauli strings.
* @param[in] tau the duration over which to simulate imaginary-time evolution.
* @param[in] order the order of the Trotter-Suzuki decomposition (e.g. @p 1, @p 2, @p 4, ...).
* @param[in] reps the number of Trotter repetitions.
* @param[in,out] qureg the state to modify.
* @param[in] hamil the Hamiltonian as a a weighted sum of Pauli strings.
* @param[in] tau the duration over which to simulate imaginary-time evolution.
* @param[in] order the order of the Trotter-Suzuki decomposition (e.g. @p 1, @p 2, @p 4, ...).
* @param[in] reps the number of Trotter repetitions.
* @param[in] permutePaulis whether to randomly reorder Pauli strings at each repetition.
*
* @throws @validationerror
* - if @p qureg or @p hamil are uninitialised.
Expand All @@ -499,7 +518,7 @@ void applyTrotterizedUnitaryTimeEvolution(Qureg qureg, PauliStrSum hamil, qreal
*
* @author Tyson Jones
*/
void applyTrotterizedImaginaryTimeEvolution(Qureg qureg, PauliStrSum hamil, qreal tau, int order, int reps);
void applyTrotterizedImaginaryTimeEvolution(Qureg qureg, PauliStrSum hamil, qreal tau, int order, int reps, bool permutePaulis);


/** @notyettested
Expand Down Expand Up @@ -622,14 +641,15 @@ void applyTrotterizedImaginaryTimeEvolution(Qureg qureg, PauliStrSum hamil, qrea
* - applyTrotterizedUnitaryTimeEvolution()
* - applyTrotterizedImaginaryTimeEvolution()
*
* @param[in,out] qureg the density-matrix state to evolve and modify.
* @param[in] hamil the Hamiltonian of the qubit system (excludes any environment).
* @param[in] damps the damping rates of each jump operator in @p jumps.
* @param[in] jumps the jump operators specified as PauliStrSum.
* @param[in] numJumps the length of list @p jumps (and @p damps).
* @param[in] time the duration through which to evolve the state.
* @param[in] order the order of the Trotter-Suzuki decomposition (e.g. @p 1, @p 2, @p 4, ...).
* @param[in] reps the number of Trotter repetitions.
* @param[in,out] qureg the density-matrix state to evolve and modify.
* @param[in] hamil the Hamiltonian of the qubit system (excludes any environment).
* @param[in] damps the damping rates of each jump operator in @p jumps.
* @param[in] jumps the jump operators specified as PauliStrSum.
* @param[in] numJumps the length of list @p jumps (and @p damps).
* @param[in] time the duration through which to evolve the state.
* @param[in] order the order of the Trotter-Suzuki decomposition (e.g. @p 1, @p 2, @p 4, ...).
* @param[in] reps the number of Trotter repetitions.
* @param[in] permutePaulis whether to randomly reorder Pauli strings at each repetition.
*
* @throws @validationerror
* - if @p qureg, @p hamil or any element of @p jumps are uninitialised.
Expand All @@ -646,7 +666,7 @@ void applyTrotterizedImaginaryTimeEvolution(Qureg qureg, PauliStrSum hamil, qrea
*
* @author Tyson Jones
*/
void applyTrotterizedNoisyTimeEvolution(Qureg qureg, PauliStrSum hamil, qreal* damps, PauliStrSum* jumps, int numJumps, qreal time, int order, int reps);
void applyTrotterizedNoisyTimeEvolution(Qureg qureg, PauliStrSum hamil, qreal* damps, PauliStrSum* jumps, int numJumps, qreal time, int order, int reps, bool permutePaulis);


// end de-mangler
Expand Down
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