Meta-learning of Gibbs states for many-body Hamiltonians with applications to Quantum Boltzmann Machines

Ruchira V Bhat; Rahul Bhowmick; Avinash Singh; Krishna Kumar Sabapathy

quant-phJul 22, 2025

Meta-learning of Gibbs states for many-body Hamiltonians with applications to Quantum Boltzmann Machines

Authors: Ruchira V Bhat, Rahul Bhowmick, Avinash Singh, Krishna Kumar Sabapathy

Organizations: Quantum Lab, Fujitsu Research of India

Abstract

The preparation of quantum Gibbs states is a fundamental challenge in quantum computing, essential for applications ranging from modeling open quantum systems to quantum machine learning. Building on the Meta-Variational Quantum Eigensolver framework proposed by Cervera-Lierta et al. and a problem-driven ansatz design, we introduce two meta-learning algorithms: Meta-Variational Quantum Thermalizer (Meta-VQT) and Neural Network Meta-VQT (NN-Meta VQT) for efficient thermal state preparation of parametrized Hamiltonians on Noisy Intermediate-Scale Quantum (NISQ) devices. Meta-VQT utilizes a fully quantum ansatz, while NN Meta-VQT integrates a quantum--classical hybrid architecture. Both use collective optimization over a finite set of training parameter values to prepare approximate Gibbs states at held-out parameter values within the sampled parameter domain. We validate our methods on Transverse Field Ising Models (TFIM) with up to eight qubits and on a two-qubit Heisenberg model with all field terms. Within these finite-size benchmarks, the trained models accurately prepare thermal states at test points inside the training interval. For the six and eight qubit cases of TFIM, meta-learned parameters when combined with appropriately designed ansatz serve as warm-start initializations, significantly outperforming random initializations in the optimization tasks. Furthermore, a 3 qubit Kitaev ring example showcases our algorithm's effectiveness across finite-temperature crossover regimes. Finally, we apply our algorithms to train a Quantum Boltzmann Machine (QBM) on a 2-qubit Heisenberg model with all field terms, achieving enhanced training efficiency, improved Gibbs state accuracy, and a 30-fold runtime speedup over existing techniques such as variational quantum imaginary time (VarQITE)-based QBM highlighting the practicality of meta-algorithm-based QBMs.

Figures & tables

Figure 1: Overview of meta-algorithm for Gibbs state preparation :(a) Depicts the core task addressed by our meta-algorithm: preparing thermal states for a parametrized many-body spin Hamiltonian at finite temperature. Shown are illustrative profiles of the Helmholtz free energy and the purity of the corresponding Gibbs state as functions of a Hamiltonian parameter. The parameter space is split into training (green circles) and test (blue star) sets. (b) Illustrates the training process, where parameters from the training set are encoded into a variational ansatz composed of a hardware-efficient and a Hamiltonian variational ansatz, separated by a linear arrangement of CNOT gates. The global cost, evaluated across all training points, is minimized using a classical optimizer. (c) Shows inference, where test parameters are input into the trained ansatz, initialized with the optimized parameters from (b), to generate Gibbs states. These are then used to reconstruct the Helmholtz free energy profile.

Figure 2: Workflow of VQT, Meta-VQT and NN Meta-VQT : The workflow of VQT, where the PQC is trained for approximating the Gibbs state corresponding to a particular parameter value, $\vec{h}_{1}$ , of the Hamiltonian is shown in (a). For a different parameter, this workflow needs to be run again. The workflow of Meta-VQT and NN Meta-VQT is shown in (b). Unlike (a) here a set of Hamiltonian parameters $\vec{h}_{train}$ (Step 1), are encoded in the PQC (Step 3, pink colored box) or classical neural network (Step 3, green colored box), and a global loss function is evaluated (Step 7) that does collective optimization over these set of parameters. After training, the PQC is evaluated at test values $\vec{h}_{test}\notin\vec{h}_{train}$ . Our primary tests use omitted values within the training interval; performance outside that interval is assessed separately in Appendix H . In both (a) and (b), the blue boxes denote modules implemented on a classical computer, while the orange boxes indicate modules that become classically intractable as system size increases and are therefore intended for implementation on a quantum computer.

Figure 3: Schematic workflow of meta-learning algorithms : In the workflow presented here, different types of training data denoted by training data 1,..,training data N are associated with distinct learning algorithms and models, each defined by a separate loss function. The meta-learning approach optimizes a combined loss function based on $P_{i,j}$ and shared feature representations and configurations, $m_{j}~\text{and}~\lambda_{i}$ , enabling the resulting meta-learner model to perform effectively on unseen new data that was not included in the training dataset. This schematic workflow has been adapted from Bahri et al. [ 7 ] .

Figure 4: Meta-VQT ansatz : The Hamiltonian parameters,

\vec{h}

, are encoded in the angles of the single qubit parametrized gates through a linear transformation. This along with the linear implementation of CNOTs forms the encoding layer. The output state from the encoding layer is further processed by the parametrized HVA. The gates in HVA are chosen according to the Hamiltonian whose Gibbs states needs to be prepared. An example HVA for a transverse field Ising model is shown, where

XX=e^{-i\theta\sigma_{i}^{x}\sigma_{i+1}^{x}}

l

represents the number of encoding or processing layers. After the final processing layer, measurement on the system qubits yield a density matrix,

\tilde{\rho}(\vec{\omega},\vec{\phi},\vec{\theta})

, that is further optimized to approximate the true Gibbs state of the system. The number of ancillas are taken to be same as the number of system qubits.

Figure 5: NN Meta-VQT ansatz : In NN Meta-VQT, the neural network encodes the Hamiltonian parameters, $\vec{h}$ , and provides the circuit parameters, $\vec{\theta}$ , as output. The quantum circuit is constructed out of both hardware-efficient and HVA as shown above. Similar to Meta-VQT, after this final processing layer, measurement on the system qubits yield a density matrix, $\tilde{\rho}(\vec{\theta})$ , that is further optimized to approximate the true Gibbs state of the system.

Figure 6: Gibbs state preparation of TFIM using Meta-VQT : The results of Meta-VQT algorithm run for different qubit sizes of TFIM is shown. The value of

\beta

is chosen to be

1

. (a) shows the exact Helmholtz free energy (dashed) evaluated in the interval

h\in[-2,2]

, and the one calculated after training from the variationally prepared Gibbs state for the parameters which are present (open circles) and absent (cross mark) in the training set. The relative error between exact Helmholtz free energy and the Helmholtz free energy evaluated using variationally prepared Gibbs state at

h_{test}

with trained parameters of Meta-VQT is shown in (b). The low relative errors show accurate Gibbs state preparation at held-out test parameters within the domain spanned by the training set. This is also evident from (c) where the fidelity between exact Gibbs state and Gibbs state measured from the trained Meta-VQT for test parameters gives value between

0.93

1

. The number of encoding and processing layers chosen for system sizes

n=2,3,4

are

[2,2],[3,3]~\text{and}~[4,4]

respectively.

Figure 7: Gibbs state preparation of TFIM using NN Meta-VQT : The results of NN Meta-VQT algorithm for different qubit sizes of TFIM is shown. The value of

\beta

is chosen to be

1

. (a) shows the exact Helmholtz free energy (dashed) evaluated in the interval

h\in[-2,2]

h_{test}

with trained parameters of NN Meta-VQT is shown in (b). The low relative errors show accurate Gibbs state preparation at held-out test parameters within the domain spanned by the training set. This is also evident from (c) where the fidelity between exact Gibbs state and Gibbs state measured from trained NN Meta-VQT ansatz for test parameters gives value between

0.98

1

for

n=2,3

and

4

. The number of SU2 hardware efficient and HVA ansatz layers in the processing part is chosen as

[2,2],~[3,3]

and

[4,4]

for system sizes

n=2,3

and

4

respectively.

Figure 8: Scaling of Meta-VQT and NN Meta-VQT: (a) shows the mean value with standard deviation of relative error between the Helmholtz free energy evaluated for values of

h

h_{test}

from the trained meta-algorithms and the exact one for system sizes

2,4,6

and

8

qubits. The relative error increases for both Meta-VQT and NN Meta-VQT as system size increases to

8

, with NN Meta-VQT exhibiting lower value than Meta-VQT. (b) shows the mean value and standard deviation of fidelity evaluated between prepared and true Gibbs state at

h_{test}

as a function of system size. The fidelity decreases for both Meta-VQT and NN Meta-VQT as system size increases to

8

, with a slighltly higher value for NN Meta-VQT than Meta-VQT for larger system sizes. For

n

qubit Hamiltonian, the number of encoding (SU2 hardware efficient) and processing (HVA) layers considered for Meta-VQT (NN Meta-VQT) is

[n,n]

. The test set,

h_{test}

, over which the mean and standard deviation in relative error in Helmholtz free energy and fidelity is evaluated are

40

uniform points between

[-2,2]

Figure 9: VQT initialization with meta-trained ansatz: Fidelity between the exact and variationally prepared Gibbs states for system sizes (a)

n=6

and (b)

n=8

qubits, using different implementations of VQT. The meta-trained HEA+HVA ansatz with CNOTs (cyan color) consistently outperforms other approaches, demonstrating the benefit of meta-learning-based initialization in achieving higher fidelity Gibbs state preparation.

Figure 10: Finite-temperature phase diagram of Kitaev ring model : (a) is the magnetic susceptibility, $\chi$ , plotted as a function of temperature, $T$ , for different h values. For a particular value of $h$ , the temperature at which magnetic susceptibility takes maximum value gives the crossover temperature and thus the phase diagram, which is shown in (b). The calculations in both (a) and (b) correspond to system size in the thermodynamic limit. (c) plots the crossover temperature evaluated for finite system sizes $L=2,10,20,50$ along with the one in thermodynamic limit, $L=\infty$ . From (c), it is evident that smaller system sizes also gives the phase diagram close to that of the one at $L=\infty$ . Figure (a) and (b) are adapted from Shi et al. [ 57 ] and (c) is adapted from Zhang et al. [ 80 ] .

Figure 11: Preparation of Gibbs state in different regimes of Kitaev ring model : (a) shows the plots of Helmholtz free energy evaluated from the approximate Gibbs state prepared using the trained Meta-VQT and VarQITE for different

h

values of Kitaev ring Hamiltonian. For Meta-VQT the

h

values considered are

40

uniform points between

[0.7,1.2]

including

20

training points, while for VarQITE it is

20

uniform points between

[0.7,1.2]

. The black dashed lines correspond to exact Helmholtz free energy. The trained Meta-VQT generates a more accurate Helmholtz free energy profile than VarQITE at test values omitted from, but contained within, the interval

[0.7,1.2]

. Additionally, (b) shows the fidelity calculated between the exact Gibbs state and the one approximated using both Meta-VQT and VarQITE. The fidelity for Meta-VQT remains close to one while that of VarQITE depends on the parameter values as well as the regimes of the Hamiltonian. The system size considered is

N=3

. The number of encoding and processing layers for Meta-VQT is

4

SU2 hardware efficient and

1

HVA respectively. The depth of SU2 hardware efficient layers in VarQITE is considered to be

5

Figure 12: Overview of conventional and meta-algorithm-based QBM training : (a) Conventional QBM training: To model a target data distribution

\rho_{data}

, the Gibbs state of a model Hamiltonian

H(J,h)

must be prepared within each iteration of the QBM training loop. This requires a nested optimization loop where circuit parameters

\theta

are updated separately for each hamiltonian parameter set

(J,h)

. Once the Gibbs state

\tilde{\rho}(\theta)

is prepared, the QBM loss

L_{QBM}(\rho_{data},\tilde{\rho}(\theta))

is minimized by updating

(J,h)

. The dotted pink box highlights the expensive inner-loop Gibbs training step where

\theta

is optimized. (b) QBM training with meta-algorithm: Here, the variational ansatz is designed to encode the Hamiltonian parameters

(J,h)

directly to the circuit. The ansatz is initialized with pretrained

\phi_{optimized}

and

\theta_{optimized}

obtained from the meat-training stage (as described in Fig. 1 (b)). As a result, Gibbs state preparation no longer requires a separate training loop for each

(J,h)

and the optimization is performed only over

(J,h)

. This eliminates the need for the inner-loop

\theta

training shown in the dotted pink box in (a), significantly reducing the QPU calls.

Figure 13: NN Meta-VQT in QBM workflow : (a) and (b) shows the convergence of loss function and the trace distance between exact and prepared Gibbs state using our method of the NN Meta-VQT and existing method of the VarQITE respectively. The KL divergence in (a) converges to

0.0012

while that in (b) converges to

0.0067

. Additionally, the prepared Gibbs state is found to be more accurate and precise for NN Meta-VQT based QBM than VarQITE based QBM. The number of layers considered for NN Meta-VQT is

4

SU2 while VarQITE requires

32

layers of SU2 circuit to prepare Gibbs state with an average value of trace distance

0.22\pm 0.094

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Prior Variational Approaches	System Qubits	Ancilla Qubits
Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm, Verdon et al. [ 63 ]	$4$	0
Variational quantum Gibbs state preparation with a truncated Taylor series, Wang et al. [ 69 ]	11	1
Extending the Variational Quantum Eigensolver to Finite Temperatures, Selisko et al. [ 56 ]	4	0
Variational quantum simulation of the quantum critical regime, Shi et al. [ 57 ]	6	0
Training Quantum Boltzmann Machines with the $\beta$ Variational Quantum Eigensolver, Huijgen et al. [ 30 ]	10	0
Thermal state preparation of the SYK model using a variational quantum algorithm, Araz et al. [ 6 ]	12	0

Appendix

Table 1: System and ancilla qubit counts in prior variational approaches to Gibbs state preparation.

Figure 14: Restricted Boltzmann Machine : The figure illustrates the architecture of a classical restricted Boltzmann machine, where connections exist between visible and hidden neurons but not among neurons within the same layer.

a_{i}

and

b_{i}

are the biases of visible and hidden neurons, and

w_{i,j}

are the connection weights between hidden and visible neurons. The input data is mapped to the visible neurons. The hidden neurons learn the latent features in the input data by adjusting the connection weights and biases. The output is then obtained by sampling the visible layer after a round-trip of Gibbs sampling, where information is propagated from the visible to hidden layer and then back to the visible layer. This reconstructed visible state serves as the model’s output, enabling it to generate samples, reconstruct inputs, or estimate probability distributions over input data.

Hamiltonian	Commuting Blocks	NN Meta-VQT	VarQITE
$H=-J\sigma_{1}^{x}\sigma_{2}^{x}$	$1$	$\mathbf{0.04\pm 0.01}$	$0.002\pm 0.001$
$H=-J\sigma_{1}^{x}\sigma_{2}^{x}-h\sum_{i=1}^{2}\sigma_{i}^{x}$	1	$\mathbf{0.05\pm 0.02}$	$0.005\pm 0.003$
$H=-J\sigma_{1}^{x}\sigma_{2}^{x}-h\sum_{i=1}^{2}(\sigma_{i}^{x}+\sigma_{i}^{z})$	2	$\mathbf{0.06\pm 0.02}$	$0.016\pm 0.006$
$H=-J\sigma_{1}^{x}\sigma_{2}^{x}-h\sum_{i=1}^{2}(\sigma_{i}^{x}+\sigma_{i}^{z}+\sigma_{i}^{y})$	3	$\mathbf{0.07\pm 0.05}$	$0.33\pm 0.20$
$H=-J(\sigma_{1}^{x}\sigma_{2}^{x}+\sigma_{1}^{y}\sigma_{2}^{y})-h\sum_{i=1}^{2}(\sigma_{i}^{x}+\sigma_{i}^{z}+\sigma_{i}^{y})$	3	$\mathbf{0.08\pm 0.05}$	$0.30\pm 0.17$
$H=-J(\sigma_{1}^{x}\sigma_{2}^{x}+\sigma_{1}^{y}\sigma_{2}^{y}+\sigma_{1}^{z}\sigma_{2}^{z})-h\sum_{i=1}^{2}(\sigma_{i}^{x}+\sigma_{i}^{z}+\sigma_{i}^{y})$	3	$\mathbf{0.08\pm 0.03}$	$0.38\pm 0.17$

Appendix

Table 2: Robustness to Hamiltonian complexity : We compare the quality of Gibbs states prepared by NN Meta-VQT and VarQITE as a function of the complexity of a

2

-qubit Hamiltonian. The precision of NN Meta-VQT, measured by the avearge trace distance between the prepared and true Gibbs state over a range of

(J,h)

values, remains largely unaffected as the number of non-commuting terms increases, demonstrating robustness to Hamiltonian complexity. In contrast, the accuracy of VarQITE degrades noticeably when the Hamiltonian contains more than two non-commuting operator blocks.

Method	Layers	Average Trace distance
Meta-VQT	4 SU2 + 1 HVA	$0.18\pm 0.07$
Meta-VQT	4 SU2	$0.22\pm 0.09$
NN Meta-VQT	4 SU2 + 1 HVA	$0.15\pm 0.02$
NN Meta-VQT	4SU2	$0.08\pm 0.03$
VarQITE	4SU2	$0.32\pm 0.20$

Appendix

Table 3: Comparsion between Meta-VQT, NN Meta-VQT and VarQITE : For the 2-qubit Heisenberg Hamiltonian with external fields shown in Eq. ( 7 ), the Meta-VQT and NN Meta-VQT algorithms prepare the Gibbs state with much better precision than VarQITE technique with similar circuit depth.

Figure 15: Purity of TFIM Gibbs state prepared using Meta-VQT and NN Meta-VQT: The dashed lines in panels (a), (b), and (c) correspond to the purity of the exact Gibbs state of the TFIM for system sizes

n=2,3~\&~4

, respectively. The meta-algorithms are trained on

10

uniformly spaced values in the interval

[-2,2]

. The Gibbs state is evaluated on a test set consisting of

40

uniformly spaced points in

[-2,2]

. The red markers denote purity of the Gibbs state prepared using Meta-VQT, while the cyan markers indicate purity of the Gibbs state prepared using NN Meta-VQT. Both meta-algorithms are able to prepare the Gibbs state with accuracy close to that of the exact state across system sizes

n=2,3~\&~4

Figure 16: Helmholtz free energy profile from meta-algorithms outside the training interval: In (a) and (b), the dashed lines correspond to the exact Helmholtz free energy evaluated at

100

uniformly spaced points in

h\in[-2.5,0]

for a TFIM of size

n=4

and

n=6

. The meta-algorithms are trained on

5

uniformly spaced values in the interval

h\in[-2,0]

. The cyan markers in (a) and (b) denote the Helmholtz free energy obtained from the variationally prepared Gibbs state using the trained Meta-VQT algorithm for system sizes

n=4

and

6

, while the red color markers correspond to the Helmholtz free energy obtained from states prepared using trained NN Meta-VQT algorithm. For

n=4

, the NN Meta-VQT is able to generate the Helmholtz free energy accurately to the exact one across the test set. The accuracy for

n=4

Meta-VQT and

n=6

Meta and NN Meta-VQT is observed to decrease outside the training range, particularly for

h\leq-2

Figure 17: Fidelity between exact Gibbs state and the Gibbs state prepared using meta-algorithms outside the training interval: Panels (a) and (b) show the fidelity between the exact and variationally prepared Gibbs states using the trained meta-algorithms, evaluated at

100

uniformly spaced points in

h\in[-2.5,0]

for the TFIM with system sizes

n=4

and

n=6

. The meta-algorithms are trained on

5

uniformly spaced values in the interval

h\in[-2,0]

. The cyan markers in (a) and (b) correspond to fidelity of states from Meta-VQT algorithm, while the red marker correpsonds to fidelity of states prepared from NN Meta-VQT algorithm. For

n=4

, the fidelity of NN Meta-VQT algorithm is found to be between

0.95

1

. Generally, the accuracy is observed to decrease outside the training range, particularly for

h\leq-2

. However, the degradation is not severe, indicating that even outside the training domain the models predict states reasonably close to those corresponding to unseen points within the training range.

Figure 18: Fidelity under noisy simulation : (a) and (b) show the fidelity between the exact Gibbs state and the state prepared using the trained Meta-VQT for the test points under noiseless (cyan) and noisy (red) simulations for the TFIM and the Kitaev ring model, respectively. The system size considered in both cases is

3

qubits. The TFIM training set consists of

10

points sampled from

h\in[-2,2]

, while the Kitaev ring training set contains

20

points from

h\in[0.9,1.2]

. In both models,

40

test points are used to generate these plots. The noisy simulations are performed by incorporating a depolarizing channel with error probabilities

0.001

for single-qubit gates and

0.01

for two-qubit gates. As expected, the performance degrades in the presence of noise; however, the prepared state still retains the essential properties of the Gibbs state and the fidelity does not drop significantly.

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