An Efficient Quantum Circuit for Flow Model Execution Using Quantum Neural Networks

Rui Che; Ludvig af Klinteberg

quant-phOct 8, 2026

An Efficient Quantum Circuit for Flow Model Execution Using Quantum Neural Networks

Authors: Rui Che, Ludvig af Klinteberg

Organizations: Department of Business and Mathematics, Mälardalen University, Universitetsplan 1, Väster˚as, 721 23, Sweden.

Abstract

Flow models generate trajectories from an initial distribution to a target distribution by solving an ordinary differential equation defined by a velocity field. Flow matching learns this velocity field by modeling the transport dynamics between the two distributions. Wavefunction flow establishes a formal connection between flow models and quantum dynamics by introducing a continuity Hamiltonian, which drives the Schrödinger evolution of quantum states. In this paper, we investigate accurate and efficient quantum simulation of the wavefunction flow, thereby realizing the efficient implementation of flow models on quantum computers. We first leverage a quantum read-only memory (QROM)-based phase kickback framework for the wavefunction flow simulation, generating probability densities that closely match those produced by the corresponding conventional flow model. To address the high circuit-resource cost, we further incorporate a trained quantum neural network (QNN) into the phase kickback framework, replacing QROM for data encoding. Numerical experiments demonstrate that our proposed method implements flow models on quantum computers more efficiently, since it maintains the accuracy of wavefunction flow simulation compared with the QROM-based framework, and significantly reduces the circuit resources.

Figures & tables

Figure 1: QROM circuit for

n=2

and

m=2

, where add denotes the address qubits, and anc denotes the ancilla qubits. Multi-controlled operators

O_{i}

are implemented on ancilla qubits and controlled by address qubits by iterating over all

2^{2}=4

address states. Black and white control nodes represent controls on

|0\rangle

and

|1\rangle

, respectively, which select the different address states. Therefore, the controlled states of four

O_{i}

operators are

|00\rangle

|01\rangle

|10\rangle

and

|11\rangle

, respectively

Figure 2: Quantum simulation of

W(t)

using the 4-step Trotter product formula, where

\text{add}_{j}

denotes the address qubits corresponding to the

j

-th spatial dimension, and anc denotes the ancilla qubits

Figure 3: The QROM-based phase kickback framework for the simulation of

e^{i\beta D_{V_{t}}}

, and the framework for the simulation of

e^{i\alpha K}

in the

j

-th spatial dimension, where

\text{add}_{j}

denotes the address qubits corresponding to the

j

-th spatial dimension,

\text{anc}_{i}

denotes the

i

-th ancilla qubit, and

\gamma=\alpha/2

Figure 4: The QNN-based phase kickback framework for the simulation of

e^{i\beta D_{V_{t}}}

, and the framework for the simulation of

e^{i\alpha K}

corresponding to the

j

-th spatial dimension, where

\gamma=\alpha/2

Figure 5: The QNN circuit in the phase kickback framework for the simulation of

e^{i\alpha K}

corresponding to the

j

-th spatial dimension

Figure 6: The QNN layer circuit in the phase kickback framework for the simulation of

e^{i\alpha{K}}

corresponding to the

j

-th spatial dimension, where

\text{add}_{j,i}

denotes the

i

-th address qubit corresponding to the

j

-th spatial dimension

Figure 7: Circuits of Rot and controlled-Rot oracles,

\phi

\omega

and

\psi

denote three angle parameters in each oracle

Figure 8: The time-dependent QNN circuit in the phase kickback framework for the simulation of

e^{i\beta D_{V_{t_{h}}}}

Figure 9: The QNN layer circuit in the phase kickback framework for the implementation of

e^{i\beta D_{V_{t_{h}}}}

Figure 10: Examples of the Time oracles

Figure 11: The training losses for the potential and kinetic evolution QNN circuits with different numbers of layers

l

Number of layers	QNN circuit for potential evolution	QNN circuit for kinetic evolution
1	0.1712	0.1499
2	0.0421	0.0279
3	0.0039	0.0073
4	0.0001	0.0001

Table 1: Final training loss values under different depths of QNN circuits

Figure 12: The training losses for the potential and kinetic evolution QNN circuits with different loss functions

Figure 13: The comparison between the binary representations of the exact potential diagonal entries of $D_{V_{t_{0}}}$ and the corresponding probabilities learned by the QNN circuit under different loss functions. The $y$ -axis represents the states of the input, where each state corresponds to a diagonal entry. The $x$ -axis represents the ancilla qubit indices, where the $i$ -th qubit index corresponds to the $i$ -th binary bit of the diagonal entry

Loss function	Mean distance	Standard deviation
Loss function 1	0.0001	0.0001
Loss function 2	0.3726	0.3645
Loss function 3	0.4027	0.3286

Table 2: The mean distances and standard deviations across all probabilities under different loss functions

Figure 14: Comparison of final probability densities generated by wavefunction flow simulation using QNN- and QROM-based phase kickback frameworks, and reference final densities under

V_{1}(x,t)

Figure 15: Comparison of final probability densities generated by wavefunction flow simulation using QNN and QROM-based phase kickback frameworks, and reference final densities under

V_{2}(x,t)

Figure 16: Comparison of final probability densities generated by wavefunction flow simulation using QNN- and QROM-based phase kickback frameworks, and reference final densities under $V_{3}(x,y,t)$

Figure 17: Comparison of final probability densities generated by wavefunction flow simulation using QNN- and QROM-based phase kickback frameworks, and reference final densities under $V_{4}(x,y,t)$

	QNN-based phase kickback				QROM-based phase kickback
$V$	RMSE	RME	$D_{\text{KL}}$	$W_{1}$	RMSE	RME	$D_{\text{KL}}$	$W_{1}$
$V_{1}$	5.965e-3	0.011	6.475e-5	1.119e-3	6.141e-3	0.012	6.677e-5	9.107e-4
$V_{2}$	6.424e-3	0.014	6.443e-5	9.353e-4	6.381e-3	0.014	6.386e-5	9.191e-4
$V_{3}$	1.580e-4	0.013	6.028e-5	7.397e-3	1.550e-4	0.011	5.306e-5	7.202e-3
$V_{4}$	1.572e-4	0.011	5.548e-5	6.819e-3	1.568e-4	0.011	5.499e-5	6.818e-3

Table 3: Errors between final probability densities generated by wavefunction flow simulation using QNN- and QROM-based phase kickback frameworks, and reference final probability densities under the uniform initial distribution

	QNN-based phase kickback				QROM-based phase kickback
$V$	RMSE	RME	$D_{\text{KL}}$	$W_{1}$	RMSE	RME	$D_{\text{KL}}$	$W_{1}$
$V_{1}$	6.339e-3	0.012	7.010e-5	8.843e-4	6.234e-3	0.013	6.724e-5	7.669e-4
$V_{2}$	7.154e-3	0.013	5.382e-5	1.444e-3	7.852e-3	0.013	6.704e-5	1.382e-3
$V_{3}$	2.344e-4	0.013	9.852e-5	7.870e-3	2.346e-4	0.013	9.826e-5	7.870e-3
$V_{4}$	3.465e-4	0.021	1.691e-4	1.051e-2	3.525e-4	0.022	1.720e-4	1.036e-2

Table 4: Errors between final probability densities generated by wavefunction flow simulation using QNN- and QROM-based phase kickback frameworks, and reference final densities under the periodic exponential initial distribution

Figure 18: The number of CNOT gates in QNN and QROM circuits obtained from decomposition tools in PennyLane for the simulation of

e^{-i\beta D_{V_{t}}}

corresponding to

V_{1}(x,t)

	QNN		QROM
$n$	QNN circuit	Phase kickback framework	QROM circuit	Phase kickback framework
2	192	384	24	48
3	432	864	576	1152
4	960	1920	6080	12160
5	1680	3360	39424	78848

Table 5: The number of CNOT gates in QNN and QROM circuits, and the corresponding phase kickback frameworks from decomposition tools in PennyLane

Figure 19: Errors between the final probability densities generated by wavefunction flow simulation and the reference final probability densities with different numbers of ancilla qubits

m

Figure 20: Errors between the final probability densities generated by wavefunction flow simulation and reference final probability densities with different

\Delta x

Figure 21: Errors between the final probability densities generated by the wavefunction flow simulation and reference final probability densities with different

N_{t}

under different numbers of address qubits

Figure 22: The left figure shows the initial and the target particle plots. The right figure shows the final particle plot obtained by the flow model, in which the particles are transported under the velocity field generated by the learned potential $V_{t}$ through flow matching, together with the exact target particle plot

Figure 23: Comparison of the final probability density obtained by wavefunction flow simulation and the target distribution. The first figure represents the initial probability density, the second figure denotes the final probability density obtained by the wavefunction flow simulation, the third figure shows the the corresponding reference final density, and the last figure shows the target probability density consisting of four Gaussian balls

Method	RMSE	Wasserstein-1 distance	KL divergence
Wavefunction flow simulation	2.248e-2	8.338e-2	2.935e-1
Reference method	3.291e-2	9.213e-2	2.492e+0

Table 6: Comparison of final probability density errors between wavefunction flow simulation and the reference method

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The flow matching has rapidly become a dominant paradigm in classical generative modeling, offering an efficient way to interpolate between two complex distributions. We extend this idea to the quantum realm and introduce the Quantum Flow Matching (QFM), a quantum-circuit realization that offers efficient interpolation between two density matrices. QFM offers systematic preparation of density matrices and generation of samples for accurately estimating observables, and can be realized on quantum computers without the need for costly circuit redesigns. We validate its versatility on a set of applications: (i) generating target states with prescribed magnetization and entanglement entropy, (ii) estimating nonequilibrium free-energy differences to test the quantum Jarzynski equality, and (iii) expediting the study on superdiffusion. These results position QFM as a unifying and promising framework for generative modeling across quantum systems.

Abstract

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