Reducing QPU Calls in Variational Quantum Algorithms via Classical Offloading of Gradient Computation

Rahul Bhowmick; Harsh Wadhwa; Avinash Singh; Tania Sidana; Quoc Hoan Tran; Krishna Kumar Sabapathy

quant-phMar 20, 2025

Reducing QPU Calls in Variational Quantum Algorithms via Classical Offloading of Gradient Computation

Authors: Rahul Bhowmick, Harsh Wadhwa, Avinash Singh, Tania Sidana, Quoc Hoan Tran, Krishna Kumar Sabapathy

Organizations: Quantum Lab, Fujitsu Research of India · Quantum Laboratory, Fujitsu Research, Fujitsu Limited Japan

Abstract

Variational quantum algorithms (VQAs) are a promising approach for leveraging near-term quantum computers, but trainability and quantum-resource costs remain major challenges. Here we address these challenges with the Hardware Efficient and dynamical LIe algebra supported Ansatz (HELIA) and two training methods combining the classical-enhanced \g method with the quantum-based Parameter-Shift Rule (PSR). This distributes gradient estimation and training across classical and quantum hardware. We evaluate ground-state estimation of 6- to 18-qubit Hamiltonians using the Variational Quantum Eigensolver (VQE) and quantum phase classification for Hamiltonians up to 12 qubits. For VQE, our methods improve accuracy and success rates, with a mean reduction in estimated logical quantum evaluations of up to $60\%$ relative to Full-PSR training of the same ansatz. For classification, the largest improvement in mean peak test accuracy relative to Full-PSR is $2.8$ percentage points. We also use these methods as a warm-start pretraining strategy for Hamiltonians with exponential-dimensional dynamical Lie algebras. Gradient diagnostics for the studied Hamiltonian configurations show slower gradient-variance decay than the selected deep hardware-efficient reference over the simulated system sizes, potentially paving the way for training large-qubit quantum models.

Figures & tables

Figure 1: Overview of our main contributions (a) We propose H ardware E fficient and dynamical LI e algebra supported A nsatz (HELIA) composed of two blocks of gates $U_{q}$ and $U_{\mathfrak{g}}$ whose gradients can be obtained using parameter-shift rule and $\mathfrak{g}$ -sim respectively. The resources required for gradient evaluation of the full ansatz can hence be delegated to both quantum and classical hardware. We further propose two training methods: Alternate and Simultaneous which benefit from the hybrid gradient of HELIA. (b) To evaluate the gradients of $U_{\mathfrak{g}}$ , we use a $\mathfrak{g}$ -sim based method. The operators of a dynamical Lie algebra are measured after applying $U_{q}$ (using QPU in the leftmost block), which is then passed on to classical hardware that evaluates the cost function and gradients using $\mathfrak{g}$ -sim (using CPU in the middle block). Further details of our contribution are elaborated in Sec. III . The numerical experiments in Sec. IV evaluate logical gradient-evaluation costs, optimization outcomes, and finite-size gradient behavior for the proposed ansatz and training procedures. The proposed procedure can also be used as a pretraining strategy for Hamiltonians with exponential-dimensional DLAs: training is first performed using a polynomial-DLA restriction and then continued using the complete Hamiltonian, as schematically illustrated in Fig. 7 .

Panel of Figure 2 — Figure 2: Quantum circuit block for YZ linear (HEA) ansatz. The parametrized $R_{y}$ and $R_{z}$ rotations are applied on each qubit followed by $CNOT$ between neighboring qubits in a linear fashion. The blocks are often repeated to increase entanglement and expressivity.

Figure 3: Workflow of Alternate training method based on Alg. 1

Experiment type	Hamiltonian/ model	Qubits	Main improvements or observations	Comments
XY VQE benchmark	XY spin Hamiltonian; polynomial DLA	6–18	Optimization: Up to approximately $60\%$ fewer QPU calls, a 26.57-percentage-point increase in success, and an order-of-magnitude reduction in relative error. Gradients: HELIA gradient variance decays more slowly than the HEA50 reference over the tested sizes and depths.	Improvements depend on the configuration. The XY model is classically tractable, and the gradient comparison is finite-size and not complexity-matched.
TFIM VQE benchmark	Transverse-field Ising model; polynomial DLA	8–14	Optimization: Alt+Sim generally provides comparable or lower error and higher success, with up to $52.11\%$ fewer QPU calls. Gradients: HELIA maintains larger gradient variance than HEA50 over the displayed system sizes.	Improvements depend on the configuration; although generally more efficient, one setting increases QPU calls. No asymptotic gradient-scaling claim is made.
LTFIM trajectory pretraining	Longitudinal-transverse-field Ising model; exponential DLA	10, 12	QPU-call reductions of $10.71\%$ – $16.18\%$ , together with lower average relative error.	Uses fixed pretraining schedules with comparable total iteration counts.
LTFIM finite-size warm start	LTFIM with $\lambda\in\{0.5,1,2\}$	6–12	Mean QPU-call reductions of $26.29\%$ – $29.94\%$ across couplings, with a maximum of $46.2\%$ ; slightly improved success and median-error ratios. Gradients: The response is block-specific and heterogeneous.	Includes pretraining calls and conditions resource comparisons on paired successful trials. No barren-plateau mitigation claim is made.
LiH molecular VQE	LiH in the STO-3G basis; exponential DLA	12	QPU calls are reduced by $9.09\%$ ; median relative error decreases from $0.02586$ to $0.02179$ , approximately $16\%$ .	Interpreted as comparable accuracy with reduced QPU usage because the trial-to-trial spread is substantial.
Quantum-phase classification	Bond-alternating Heisenberg chain	12	The best Alt+Sim accuracy is $0.911\pm 0.008$ for the ZX DLA, 2.8 percentage points above Full-PSR and 3.8 points above $\mathfrak{g}$ -sim.	XY and YZ results are comparable to Full-PSR.

Table 1: Finite-size benchmark suite for HELIA and hybrid training protocols. Overview of the numerical experiments used to assess HELIA and the proposed training protocols. All results are finite-size numerical studies, and no claim of quantum advantage is made. For each benchmark, the table reports the physical model, simulated system sizes, principal observations relative to Full-PSR or the relevant reference, and the qualifications governing their interpretation. QPU-call reductions count logical shifted-cost evaluations and required intermediate-state observable expectations up to the best iterate; they exclude shot budgets, observable grouping, basis changes, circuit compilation, and classical processing. The gradient observations compare the displayed variances with the HEA50 reference and do not imply asymptotic barren-plateau avoidance.

Figure 4: VQE performed for a 6-qubit TFIM Hamiltonian. All coefficients of the Hamiltonian terms are either

1

-1

chosen randomly. Only using HEA with PSR (shown in orange) or multi-qubit Pauli rotation gates with

\mathfrak{g}

-sim (shown in blue) is not enough to reach exact ground state (shown in red). But combining both methods into an Alternate training procedure (in green) allows us to reach the correct solution.

Figure 5: XY Hamiltonian VQE (1 YZ linear layer) : The success (Row 1), relative error (Row 2) and QPU calls reduction (Row 3) are plotted for configurations with 1 layer of YZ linear ansatz in

U_{q}

(referred to as Block Q) and Hamiltonian DLA gates in

U_{\mathfrak{g}}

(referred to as Block G) for 6 to 18 qubits. For each qubit count and metric, PSR (orange), Alternate (Alt, blue) and Alternate+Simultaneous (Alt+Sim, purple) training methods are compared. On the first row , we show that Alt+Sim attains equal or higher success rate as Full-PSR for every tested qubit count in this configuration, whereas Alternate improves over Full-PSR for several, but not all, qubit counts. For relative error (Row 2) , we plot the median and the

25\%

and

75\%

quantiles (in black). The Alt method performs well for Layer 1 configuration, which is improved by Alt+Sim, giving lower relative error than PSR for all qubits. Most importantly, on the lowermost row we plot the percentage of reduction in QPU calls for Alt and Alt+Sim compared to PSR. The spread in standard deviation is shown in black. Except for 6 qubit case, we consistently see reduction in QPU calls using our methods. Alt+Sim generally lowers successful-trial relative error, while its success rate generally improves relative to Full-PSR.

Figure 6: Gradients for XY Hamiltonian VQE: Variance of the first parameter from

U_{q}

(red) and

U_{\mathfrak{g}}

(blue) are plotted with increasing qubits for both Full-PSR and our Alternate method for finding the ground state of the XY Hamiltonian using VQE. The diagrams show the plots for 1 YZ linear layer in

U_{q}

. The same task is performed using a deep HEA (referred to as HEA50) with standard PSR and the variance of its first partial derivative is plotted in green, to serve as a reference for how BP shows up in practice. In both

U_{q}

and

U_{\mathfrak{g}}

, we notice a slower decay of gradients as compared to HEA50 over the displayed system sizes.

Figure 7: Warm-start pretraining workflow for exponential-DLA Hamiltonians. The HELIA ansatz is first optimized on a polynomial-DLA restriction of the target Hamiltonian using Alternate and Simultaneous training. The resulting parameters initialize optimization of the complete exponential-DLA Hamiltonian, first by updating

U_{q}

and subsequently by training the full circuit with PSR. The circuit architecture remains unchanged throughout the procedure; only the Hamiltonian and gradient-evaluation protocol change. For the tested LTFIM configurations, this warm start achieves lower average relative error and reduces logical QPU calls by

11.9\%

–

46.2\%

relative to Full-PSR, including the pretraining cost. Detailed finite-size results across system sizes, circuit depths, and longitudinal coupling strengths are provided in Appendix J .

$\lambda$	$R_{\mathrm{succ}}$	$R_{\mathrm{err}}$	QPU Calls reduction
0.5	$1.0093\pm 0.0272$	$0.9085\pm 0.1921$	$29.9430\%\pm 10.2848\%$
1.0	$1.0110\pm 0.0405$	$0.8883\pm 0.2122$	$28.2384\%\pm 8.9714\%$
2.0	$1.0175\pm 0.0418$	$0.8852\pm 0.2249$	$26.2868\%\pm 6.7164\%$

Table 2: Coupling-wise summary over all qubit-depth configurations. Values are mean

\pm

sample standard deviation across configurations. The success and relative-error ratios are

R_{\mathrm{succ}}=p_{\text{pre-trained}}/p_{\mathrm{Full\text{-}PSR}}

and

R_{\mathrm{err}}=\tilde{\epsilon}_{\text{pre-trained}}/\tilde{\epsilon}_{\mathrm{Full\text{-}PSR}}

, respectively. QPU reduction is with respect to full PSR on paired successful trials, and includes pre-training QPU calls as well.

Panel of Figure 8 — Figure 8: Demonstration of PSR (a) and our method (b) to run VQE for a 12-qubit LTFIM Hamiltonian (with an exponential -DLA) using 3 YZ-linear layers in $U_{q}$ . The orange and green regions show Alternate and Simultaneous training, respectively, for the restricted Hamiltonian, followed by training $U_{q}$ and the complete circuit ( $U_{q}$ and $U_{\mathfrak{g}}$ ) in the blue and pink regions, respectively, for the full LTFIM Hamiltonian. Our training methods in the initial phase allow VQE to reach much closer to the target ground state during the final training than using PSR for the full training, for similar total iteration counts. For the overall training, the QPU calls are reduced by approximately $16.18\%$ .

		Full PSR	Our Proposal
Relative Error	$q_{25}$	0.01009	0.00503
	Median	0.02586	0.02179
	$q_{75}$	0.03770	0.02573
QPU Reduction		-	9.09 %

Table 3: Relative error quartiles and QPU reduction for Full-PSR vs our proposal.

DLA	g-sim	Full PSR	Alt+Sim
XY	0.863 $\pm$ 0.005	0.884 $\pm$ 0.016	0.888 $\pm$ 0.021
YZ	0.858 $\pm$ 0.006	0.887 $\pm$ 0.025	0.889 $\pm$ 0.020
ZX	0.873 $\pm$ 0.020	0.883 $\pm$ 0.015	0.911 $\pm$ 0.008

Table 4: Classification task using

\mathfrak{g}

-sim, Full-PSR and Alt+Sim training (with 9 layers in

U_{q}

) for 12 qubits. Three different choices of poly -DLA (referred to as

\mathfrak{g}_{XY},\mathfrak{g}_{YZ}

and

\mathfrak{g}_{ZX}

) are explored for

\mathfrak{g}

-sim, and also to construct

U_{\mathfrak{g}}

for Alternate and Full-PSR method. Average peak test accuracy is reported for each of the configuration. Alt+Sim attains performance comparable to Full-PSR for the

\mathfrak{g}_{XY}

and

\mathfrak{g}_{YZ}

choices, while the most distinct improvement is observed for

\mathfrak{g}_{ZX}

of up to 3.8 and 2.8 percentage points over the accuracy of

\mathfrak{g}

-sim and Full-PSR respectively.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

C(\vec{\theta})=\langle 0^{\otimes q}|U^{\dagger}(\vec{\theta})HU(\vec{\theta})|0^{\otimes q}\rangle

Appendix

Algorithm 3 g-sim Implementation for VQE

(a) Success, Relative Error and QPU Call Reduction for 3 YZ linear layers in

U_{q}

(a) Success, Relative Error and QPU Call Reduction for 6 YZ linear layers in

U_{q}

Figure 11: XY Hamiltonian VQE (3, 6, and 9 YZ linear layers) : Success (Row 1), relative error (Row 2), and QPU-call reduction (Row 3) are plotted for configurations with 3 (a), 6 (b), and 9 (c) layers of the YZ linear ansatz in

U_{q}

(referred to as Block Q) and Hamiltonian DLA gates in

U_{\mathfrak{g}}

(referred to as Block G) for 6 to 18 qubits. For each qubit count and metric, PSR (orange), Alternate (blue), and Alternate+Simultaneous (purple) training methods are compared. On the first row , we show that the Alternate and Alternate+Simultaneous protocols generally have better success rates than PSR, except in some depth–qubit configurations. For relative error (Row 2) , we plot the median and the

25\%

and

75\%

quantiles (in black). The Alternate method struggles in these examples, but Alternate+Simultaneous gives lower relative error than PSR for all qubits. Most importantly, in the lowermost row , QPU-call reduction is plotted only for Alternate and Alternate+Simultaneous because it is measured relative to PSR (which coincides with 0% QPU-call reduction). The standard deviation is shown in black. Logical-evaluation reductions are positive in most configurations, but success-rate improvements are configuration-dependent.

Qubits Success(%) Full-PSR Alternate Alt+Sim 6 81.25 70.31 90.62 8 70.31 59.38 75.00 10 29.69 48.44 50.00 12 65.62 70.31 90.62 14 51.56 51.56 51.56 16 21.88 43.75 48.44 18 35.94 50.00 54.69 (a) Success rate for 1 layer YZ linear ansatz in

U_{q}

and XY Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits Success(%) Full-PSR Alternate Alt+Sim 6 75.00 81.25 75.00 8 68.75 65.62 68.75 10 37.50 40.62 48.44 12 59.38 70.31 73.44 14 40.62 40.62 43.75 16 29.69 34.38 34.38 18 25.00 34.38 42.19 (b) Success rate for 3 layer YZ linear ansatz in

U_{q}

and XY Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits Success(%) Full-PSR Alternate Alt+Sim 6 82.81 79.69 81.25 8 79.69 81.25 79.69 10 43.75 45.31 54.69 12 64.06 75.00 81.25 14 40.62 35.94 46.88 16 23.44 32.81 37.50 18 28.12 37.50 37.50 (c) Success rate for 6 layer YZ linear ansatz in

U_{q}

and XY Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits Success(%) Full-PSR Alternate Alt+Sim 6 82.81 81.25 98.44 8 54.69 57.81 60.94 10 43.75 43.75 54.69 12 71.88 85.94 90.62 14 57.81 48.44 53.12 16 42.19 43.75 32.81 18 28.12 29.69 54.69 (d) Success rate for 9 layer YZ linear ansatz in

U_{q}

and XY Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits QPU Reduction (%) Alternate Alt+Sim 6 6.24

\pm

4.12

-33.27

\pm

17.07 8 22.30

\pm

11.77 27.53

\pm

13.20 10 3.07

\pm

1.66 41.88

\pm

21.81 12 20.81

\pm

10.40 45.25

\pm

21.95 14 16.70

\pm

9.60 32.07

\pm

15.24 16 38.39

\pm

17.18 60.00

\pm

25.11 18 16.71

\pm

7.34 46.81

\pm

26.52 (e) QPU Reduction for 1 layer YZ linear ansatz in

U_{q}

and XY Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits QPU Reduction (%) Alternate Alt+Sim 6 12.87

\pm

6.92 22.62

\pm

12.21 8 20.84

\pm

8.16 36.58

\pm

15.15 10 17.42

\pm

3.64 49.32

\pm

18.98 12 23.94

\pm

7.51 43.24

\pm

16.37 14 34.77

\pm

13.93 47.99

\pm

20.32 16 33.62

\pm

9.45 56.12

\pm

21.31 18 37.31

\pm

10.55 52.63

\pm

22.83 (f) QPU Reduction for 3 layer YZ linear ansatz in

U_{q}

and XY Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits QPU Reduction (%) Alternate Alt+Sim 6 20.65

\pm

6.97 7.93

\pm

3.25 8 14.76

\pm

4.21 22.87

\pm

8.77 10 26.49

\pm

6.88 32.82

\pm

14.19 12 27.05

\pm

8.22 22.63

\pm

8.37 14 34.59

\pm

13.91 46.14

\pm

17.34 16 32.50

\pm

10.90 50.88

\pm

19.61 18 32.84

\pm

12.20 45.45

\pm

20.12 (g) QPU Reduction for 6 layer YZ linear ansatz in

U_{q}

and XY Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits QPU Reduction (%) Alternate Alt+Sim 6

-4.49

\pm

1.42

-4.70

\pm

2.13 8 18.62

\pm

5.19 17.64

\pm

7.73 10 14.12

\pm

3.01 22.70

\pm

10.14 12 26.85

\pm

8.70 13.74

\pm

3.93 14 23.44

\pm

6.56 38.17

\pm

14.61 16 20.59

\pm

4.85 44.47

\pm

19.66 18 20.32

\pm

4.83 39.07

\pm

15.71 (h) QPU Reduction for 9 layer YZ linear ansatz in

U_{q}

and XY Hamiltonian DLA gates in

U_{\mathfrak{g}}

Appendix

Table 5: XY Hamiltonian (Success and QPU Reduction): Success rates and QPU-call reductions are shown for configurations with 1, 3, 6, and 9 YZ-linear layers in

U_{q}

and XY Hamiltonian DLA gates in

U_{\mathfrak{g}}

using Full-PSR, Alternate, and Alternate followed by Simultaneous training (Alt+Sim). QPU-call reductions are reported as the mean

\pm

sample standard deviation over successful trials only, where success is defined by a relative-error threshold of

10^{-3}

. Reductions are measured relative to Full-PSR. The corresponding successful-trial relative-error statistics are reported in Table 6 .

Table 6: XY Hamiltonian VQE (Relative Error) : The relative error median, 25% quantile (Q25) and 75% quantile (Q75) for Full-PSR, Alternate and Alternate + Simultaneous (referred to as Alt+Sim) for the configurations with 1, 3, 6, and 9 layers of YZ linear ansatz in

U_{q}

and XY Hamiltonian DLA gates in

U_{\mathfrak{g}}

. The relative error for

\mathfrak{g}

-sim with only XY Hamiltonian DLA gates for each qubit count is shown for comparison, which is several orders of magnitude higher than the other methods. Alt+Sim has lower reported median relative error than Full-PSR in the displayed configurations. Alternate alone is less consistent and has higher median error in several deeper configurations.

Panel of Figure 12 — Figure 12: Gradients for XY Hamiltonian VQE: Variance of the first parameter of $U_{q}$ (red) and $U_{\mathfrak{g}}$ (blue) are plotted with increasing qubits for both Full-PSR and our Alternate method for finding the ground state of XY Hamiltonian using VQE. The diagrams show the plots for 3(a), 6(b) and 9 (c) YZ linear layer in $U_{q}$ . For comparison, the same task is performed using a deep HEA (referred to as HEA50) with standard PSR and variance of its first partial derivative is plotted in green. In both $U_{q}$ and $U_{\mathfrak{g}}$ , we notice a slower decay of gradients as compared to HEA50 over the displayed system sizes.

Figure 13: Average

\mathfrak{g}

-Purity of HEA for

\mathfrak{g}_{XY}

DLA (defined in Eq. ( 32 )) with increasing qubits and varying circuit depths. For constant (

O(1)

) and logarithmic depth in qubits (

O(log(n))

), the decay is less than exponential (shown in blue and purple respectively). For linear depth (

O(n)

in green),

\mathfrak{g}

-Purity decreases exponentially. Each point in the plot corresponds to an average of 10000 independent parameter samples.

Panel of Figure 14 — Figure 14: Relative-error-versus-iteration curve for finding the ground state of an XY Hamiltonian using only $\mathfrak{g}$ -sim . For 13 ( 14(a) ) and 17 ( 14(b) ) qubits, $\mathfrak{g}$ -sim is enough to approach the ground state and does not require our protocol.

(a) Success, Relative Error and QPU Call Reduction for 1 YZ linear layer in

U_{q}

(a) Success, Relative Error and QPU Call Reduction for 3 YZ linear layers in

U_{q}

(a) Success, Relative Error and QPU Call Reduction for 6 YZ linear layers in

U_{q}

Figure 18: TFIM VQE results for

n=8,10,12,14

qubits using (a) one, (b) three, (c) six, and (d) nine YZ-linear layers in

U_{q}

(referred to as Block Q), with TFIM Lie-algebra-supported gates in

U_{\mathfrak{g}}

(referred to as Block G). Full-PSR (orange), Alternate (blue), and Alternate followed by Simultaneous training (Alt+Sim, purple) are compared. On the first row , we report success rate at the relative-error threshold

10^{-3}

. The middle row shows successful-trial median relative errors, with black intervals indicating the 25th–75th percentiles The Alt+Sim method gives comparable or lower relative error than PSR for all qubits. The bottom row shows mean QPU Calls reduction relative to Full-PSR, with black error bars indicating one standard deviation. Positive reductions indicate fewer logical evaluations; negative reductions indicate more. Resource reductions are positive in all displayed settings except Alt+Sim at 14 qubits and nine layers.

Panel of Figure 19 — Figure 19: Gradients for TFIM Hamiltonian VQE: Variance of the first parameter from $U_{q}$ (red) and $U_{\mathfrak{g}}$ (blue) are plotted with increasing qubits for both Full-PSR and our Alternate method for finding the ground state of TFIM Hamiltonian using VQE. The diagrams also show the plots for 1 (a), 3 (b), 6(c), 9 (d) YZ linear layers in $U_{q}$ . For comparison, the same task is performed using a deep HEA (referred to as HEA50) with standard PSR and variance of its first partial derivative is plotted in green. The green curve is the same in all of the plots and provided as a reference for our improved gradients. The plot shows that our chosen ansatz is able to preserve higher magnitude of gradients as compared to HEA50 over the displayed system sizes.

Qubits Success (%) Full-PSR Alternate Alt+Sim 8 48.44 50.00 45.31 10 18.75 9.38 26.56 12 25.00 28.12 32.81 14 100.00 100.00 100.00 (a) Success rate for 1 layer YZ linear ansatz in

U_{q}

and TFIM Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits Success (%) Full-PSR Alternate Alt+Sim 8 84.38 82.81 89.06 10 20.31 37.50 59.38 12 53.12 57.81 70.31 14 98.44 98.44 98.44 (b) Success rate for 3 layer YZ linear ansatz in

U_{q}

and TFIM Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits Success (%) Full-PSR Alternate Alt+Sim 8 93.75 95.31 100 10 43.75 45.31 64.06 12 75.00 76.56 76.56 14 100.00 100.00 100.00 (c) Success rate for 6 layer YZ linear ansatz in

U_{q}

and TFIM Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits Success (%) Full-PSR Alternate Alt+Sim 8 57.81 60.94 73.44 10 31.25 43.75 70.31 12 82.81 85.94 79.69 14 98.44 100.00 100.00 (d) Success rate for 9 layer YZ linear ansatz in

U_{q}

and TFIM Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits QPU Reduction (%) Alternate Alt+Sim 8 29.78 +/- 15.08 48.09 +/- 22.71 10 11.52 +/- 4.17 52.11 +/- 26.46 12 29.46 +/- 16.81 51.80 +/- 25.37 14 40.91 +/- 15.93 50.66 +/- 17.60 (e) QPU Reduction for 1 layer YZ linear ansatz in

U_{q}

and TFIM Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits QPU Reduction (%) Alternate Alt+Sim 8 17.45 +/- 5.79 30.08 +/- 13.40 10 17.97 +/- 1.97 38.72 +/- 16.46 12 25.66 +/- 6.49 35.29 +/- 15.03 14 48.79 +/- 15.07 25.09 +/- 8.19 (f) QPU Reduction for 3 layer YZ linear ansatz in

U_{q}

and TFIM Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits QPU Reduction (%) Alternate Alt+Sim 8 24.52 +/- 5.93 0.64 +/- 0.15 10 24.27 +/- 5.04 21.82 +/- 9.20 12 32.43 +/- 9.54 25.35 +/- 8.84 14 49.64 +/- 21.50 9.29 +/- 2.21 (g) QPU Reduction for 6 layer YZ linear ansatz in

U_{q}

and TFIM Hamiltonian DLA gates in

U_{\mathfrak{g}}

Qubits QPU Reduction (%) Alternate Alt+Sim 8 0.68 +/- 0.11 1.39 +/- 0.54 10 9.33 +/- 1.70 6.55 +/- 2.57 12 27.27 +/- 8.83 14.90 +/- 5.16 14 45.79 +/- 22.98 -1.98 +/- 0.37 (h) QPU Reduction for 9 layer YZ linear ansatz in

U_{q}

and TFIM Hamiltonian DLA gates in

U_{\mathfrak{g}}

Appendix

Table 7: TFIM Hamiltonian (Success and QPU Reduction) : The success and QPU calls reduction are shown for the configurations with 1, 3, 6, and 9 layers of YZ linear ansatz in

U_{q}

and TFIM Hamiltonian DLA gates in

U_{\mathfrak{g}}

using Full-PSR, Alternate and Alternate + Simultaneous training (referred to as Alt+Sim). The QPU call reduction is measured relative to the Full-PSR method where all parameters are trained by standard PSR. The metrics show that our methods are generally better at reaching higher success rates and lower relative error (in Table 8 ), while QPU calls are reduced significantly.

Table 8: TFIM Hamiltonian (Relative Error) : The relative error median, 25% quantile (Q25) and 75% quantile (Q75) for Full-PSR, Alternate and Alternate + Simultaneous (referred to as Alt+Sim) for the configurations with 1, 3, 6, and 9 layers of YZ linear ansatz in

U_{q}

and TFIM Hamiltonian DLA gates in

U_{\mathfrak{g}}

. The

\mathfrak{g}

-sim results are included as a separate restricted-ansatz comparison. Their accuracy depends strongly on the instance, including an accurate 12-qubit example. The relative performance of Alternate, Alt+Sim, and Full-PSR varies across the displayed qubit-depth configurations.

Panel of Figure 20 — Figure 20: Demonstration of PSR in (a), (c), and (e) and our method (b),(d),(f) to run VQE for 10- and 12-qubit LTFIM Hamiltonians (with an exponential DLA) using 3 and 6 YZ-linear layers in $U_{q}$ . The orange and green regions show Alternate and Simultaneous training, respectively, for the restricted Hamiltonian, followed by training $U_{q}$ and the full circuit in the blue and pink regions, respectively, for the full LTFIM Hamiltonian. Our training methods in the initial phase allow VQE to reach much closer to the target ground state during the final training than using PSR for the full training, for similar total iteration counts. Across the displayed cases, QPU calls are reduced by approximately $10.71\%$ – $15.00\%$ .

Figure 21: LTFIM warm-start performance at

\lambda=0.5

using the protocol and metrics defined in Fig. 22 .

Figure 22: LTFIM warm-start performance at

\lambda=1

using 64 paired seeds for

n=6

12

. The rows report the success fraction at relative error

10^{-3}

, the median best relative error among successful trials with Q25-Q75 intervals, and the mean paired QPU Calls reduction at the best iterate with one-standard-deviation error bars. The QPU-call estimate includes HELIA pretraining, and both methods use the same final circuit.

Figure 23: LTFIM warm-start performance at

\lambda=2

using the protocol and metrics defined in Fig. 22 .

Figure 24: Population variance of the first gradient component for the LTFIM study. Rows correspond to

\lambda=0.5

1

, and

2

, and columns show constant, logarithmic, and linear depth families. Random-initialization and warm-start variances are shown separately for

U_{q}

and

U_{\mathfrak{g}}

over the 64 trials in each setting.

Panel of Figure 25 — Figure 25: Collective-spin orbit-algebra benchmark at one and three layers for six and eight qubits, with 64 trials per setting and success threshold $10^{-2}$ . The rows show the success fraction, success-filtered median relative error with Q25-Q75 intervals, and QPU Calls reduction relative to Full-PSR for successful trials. Reductions are accumulated to each successful trial’s best iterate; error bars show the corresponding standard deviation. Missing bars indicate that none of the runs reached below the success threshold.

Panel of Figure 26 — Figure 26: Collective-spin orbit-algebra benchmark at six and nine layers using the protocol and metrics defined in Fig. 25 .

Variational Quantum Algorithms

Variational quantum algorithms use parameterized quantum circuits and classical optimization to approximate solutions to quantum and optimization problems.

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May 22, 2026

Classical State Preparation for Variational Quantum Algorithms via Reinforcement Learning

Variational Quantum Algorithms (VQAs) potentially offer a pathway to practical quantum advantage, but their optimization is heavily hindered by barren plateaus and numerous local minima. While classically simulable Clifford circuits can warm-start VQAs to accelerate convergence, existing heuristic-based initialization methods struggle to scale within vast combinatorial search spaces. To overcome this bottleneck, we propose CRiSP (a Clifford Reinforcement Learning agent for State Preparation), a framework that formulates discrete prefix selection as a sequential decision-making problem. CRiSP utilizes Neural-Guided Monte Carlo Tree Search, driven by a Transformer-based policy trained via self-play, to insert learned Clifford gates before fixed parameterized rotations. This enables the construction of high-quality initial states entirely through polynomial-time classical stabilizer simulation without altering the underlying circuit architecture. By integrating a curriculum learning strategy that progressively expands the search horizon, the agent efficiently scales to deep circuits. Evaluated on QAOA benchmarks of up to

22

qubits and

1{,}370

parameters, CRiSP outperforms state-of-the-art Clifford initialization methods by a mean of

3.17\times

(max

45.02\times

) in average energy accuracy and

2.44\times

(max

16.01\times

) in best-achieved energy accuracy. Assessments on VQE tasks further demonstrate the framework's robustness and generalizability.

Gino Kwun, Dhanvi Bharadwaj, Gokul Subramanian Ravi

Computer Science and Engineering University of Michigan Ann Arbor, MI 48109, USA

Quantum State Preparation Quantum Approximate Optimization Algorithm Variational Quantum Algorithms

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Apr 7, 2026

Soft-Quantum Algorithms

Quantum operations on pure states can be fully represented by unitary matrices. Variational quantum circuits, also known as quantum neural networks, embed data and trainable parameters into gate-based operations and optimize the parameters via gradient descent. The high cost of training and low fidelity of current quantum devices, however, restricts much of quantum machine learning to classical simulation. For few-qubit problems with large datasets, training the matrix elements directly, as is done with weight matrices in classical neural networks, can be faster than decomposing data and parameters into gates. We propose a method that trains matrices directly while maintaining unitarity through a single regularization term added to the loss function. A second training step, circuit alignment, then recovers a gate-based architecture from the resulting soft-unitary. On a five-qubit supervised classification task with 1000 datapoints, this two-step process produces a trained variational circuit in under four minutes, compared to over two hours for direct circuit training, while achieving lower binary cross-entropy loss. In a second experiment, soft-unitaries are embedded in a hybrid quantum-classical network for a reinforcement learning cartpole task, where the hybrid agent outperforms a purely classical baseline of comparable size.

Basil Kyriacou, Mo Kordzanganeh, Maniraman Periyasamy +1

Terra Quantum AG, 9000 St. Gallen, Switzerland

Variational Quantum Circuits Quantum Circuit Trainability Hybrid Quantum-Classical ML

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Jul 23, 2026

An Analytically Trained Variational Surrogate for Quantum Phase Estimation on NISQ Hardware

Quantum Phase Estimation (QPE) is a foundational algorithm for molecular ground-state energy estimation, but its deep circuit requirements make direct hardware execution impractical on Noisy Intermediate-Scale Quantum (NISQ) devices. We present an analytically grounded variational surrogate framework in which a shallow Variational Quantum Circuit (VQC) is trained to reproduce the QPE measurement distribution without any quantum circuit simulation. The training target is computed entirely classically via the Dirichlet kernel, evaluated directly from the Full Configuration Interaction (FCI) ground-state energy, the ancilla qubit count, and the time evolution parameter, eliminating the exponentially scaling simulation bottleneck of prior surrogate approaches. We apply this framework to the hydrogen molecule (H

_2

) with a symmetry-tapered Hamiltonian, conducting a four-stage experimental investigation on IBM Quantum hardware. Stage 1 compares linear and full entangler topologies for the

R_Y

R_Z

CZ

ansatz, with and without XpXm Dynamical Decoupling (DD), across four distributional metrics (Hellinger distance, fidelity error, total variation distance, Jensen-Shannon divergence), identifying the linear entangler as optimal. Stage 2 varies VQC layers (

p=1

5

) for the linear-entangler ansatz, identifying single-layer depth as optimal under hardware noise. Stage 3 applies this configuration to the reduced

R_Y

CZ

ansatz, comparing ideal and noisy simulator-trained parameters. A supplementary noise analysis at

p \in \{8,64\}

characterizes the depth-dependent interplay between circuit depth and DD effectiveness. The framework enables faithful QPE mimicry using a linearly scaling VQC, recovering the ground-state energy within the chemical accuracy threshold (1 kcal/mol), constituting a scalable, hardware-efficient paradigm for QPE-based molecular energy estimation on NISQ devices.

Mousumi Kundu, Ashish Kumar Patra, Anurag K. S. V. +4

Qclairvoyance Quantum Labs, Secunderabad, TG 500094, India. · Department of CSE(AI & ML), School of Engineering, Dayananda Sagar University, Bengaluru, KA 562112, India. · The University of Arizona, Tucson, AZ 85721, USA. +1

Quantum Chemistry Variational Quantum Circuits

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Abstract

Variational Quantum Algorithms

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Classical State Preparation for Variational Quantum Algorithms via Reinforcement Learning

Soft-Quantum Algorithms

An Analytically Trained Variational Surrogate for Quantum Phase Estimation on NISQ Hardware