Fully quantum training of parametric circuits with Quantum Hamiltonian Descent

Marco Burato; Leonardo Banchi

quant-phOct 23, 2026

Fully quantum training of parametric circuits with Quantum Hamiltonian Descent

Authors: Marco Burato, Leonardo Banchi

Organizations: Department of Physics and Astronomy, University of Florence, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy · INFN Sezione di Firenze, via G. Sansone 1, I-50019, Sesto Fiorentino (FI), Italy

Abstract

Parametric quantum circuits are usually trained by classical, local optimizers, which use the quantum computer only to evaluate the cost function, or its gradient, one point at a time. Quantum Hamiltonian Descent (QHD) takes a different route: in this quantum algorithm for continuous optimization, the parameters become the position of a quantum particle, with the cost function as its potential, and two time schedules control kinetic and potential energy so that the wavefunction explores the whole landscape and tunnels towards the global minimum. QHD, however, assumes a classically computable cost function, and its digital implementation is poorly understood: standard error bounds grow exponentially with the number of qubits per parameter, and the schedules are borrowed from classical optimization. Here we propose to train parametric circuits with QHD entirely on the quantum computer, through an online step in which the qubits encoding the parameters coherently control the circuit on a second register. We prove that, for cost functions with a smooth periodic extension, including those of parametric circuits, the simulation error admits a bound independent of the grid resolution, and that for circuits the bound grows at most linearly with the number of parameters. We also show that the ideal dynamics depends on the two schedules only through a single curve, which controls the adiabaticity of the search, and that the remaining freedom can be used to minimize the gate count. Simulations on benchmark functions and parametric circuits show that, at equal cost, schedules designed in this way markedly increase the probability of finding the global minimum.

Figures & tables

		$d=1$		$d=2$
Landscape	Function	$P_{\mathrm{suc}}$	$\mathbb{E}$	$P_{\mathrm{suc}}$	$\mathbb{E}$
Valley shaped	csendes †	1.00	0.029	1.00	0.029
	alpine1	1.00	0.039	1.00	0.039
	hosaki	/	/	1.00	0.047
	camel3 †	/	/	0.98	0.010
	styblinski tang	0.89	0.049	0.79	0.049

Table 1: Final success probability

P_{\mathrm{suc}}

and final coefficient

\mathbb{E}

for the benchmark functions in

d=1

and

d=2

, obtained with the same protocol for all functions and dimensions, in the native units of each function:

q=8

qubits per dimension, curve

r(\tau)=r_{0}\,e^{k\tau}

of eq. 26 with rate

k=0.1

, from

r_{0}=10^{-4}

r_{\mathrm{f}}=10^{2}

, i.e.

\tau_{\mathrm{f}}\simeq 138

, run through in the optimal gauge

a\propto g(r)^{-1/2}

of section 2.2 , with

c_{1}

and

c_{2}

given by eq. 19 with the maxima of

\left|\nabla f\right|

and

\left|\Delta f\right|

over the domain, and

R=2^{15}

Trotter steps. Each case is repeated with

2R

steps as a check. Symbols: “/”, the function is defined only for

d=2

; † , non-negligible Trotterization error; ‡ , incomplete convergence:

P_{\mathrm{suc}}

and

\mathbb{E}

are still changing at the end of the curve. Unmarked values are converged both in

R

and in

r

, so that

P_{\mathrm{suc}}<1

is due to the lack of adiabaticity of the evolution at the rate

k

Figure 1: Evolution of QHD along the curve $r(\tau)$ for four benchmark functions in $d=1$ , with the protocol of table 1 ( $R=2^{15}$ ). In each panel, left: cost function $f(x)$ , with the basin of the global minimum shaded; top: probability distribution $p(x)$ of the state as a function of $\log_{10}r$ (power-law color scale, common to all panels); bottom: success probability $P_{\mathrm{suc}}$ (left axis) and coefficient $\mathbb{E}$ (right axis, logarithmic scale), with the least-squares fit $\mathbb{E}_{\infty}+c\,r^{-1/2}$ over $10\leq r\leq r_{\mathrm{f}}$ (dashed line).

Figure 2: Evolution of QHD for alpine2 in $d=1$ with two choices of the schedules, at the same number of qubits $q=8$ and Trotter steps $R=2^{15}$ , and over the same range of $r$ : (a) schedules $a(t)=2/\left(t^{3}+10^{-3}\right)$ , $b(t)=2t^{3}$ of [ 23 ] , with Trotter steps uniform in $t$ , up to $T\simeq 2.15$ , where $r=b/a$ reaches $r_{\mathrm{f}}=10^{2}$ ; (b) protocol of table 1 , i.e. curve $r(\tau)=r_{0}\,e^{k\tau}$ in the optimal gauge of section 2.2 . Left: cost function $f(x)$ , with the basin of the global minimum shaded; top: probability distribution $p(x)$ of the state as a function of $\log_{10}r$ (power-law color scale, common to both panels); bottom: success probability $P_{\mathrm{suc}}$ (left axis) and coefficient $\mathbb{E}$ (right axis, logarithmic scale), with the least-squares fit $\mathbb{E}_{\infty}+c\,r^{-1/2}$ over $10\leq r\leq r_{\mathrm{f}}$ (dashed line).

Figure 3: QHD with the online procedure on the quantum functions of table 2 , with the same protocol. (a), (b) Examples 1 and 2 ( $d=1$ ), as in fig. 1 : cost function $f(x)$ , with the basin of the global minimum shaded (left), and probability distribution of the state as a function of $\log_{10}r$ (power-law color scale). (c)–(e) Examples 3–5 ( $d=2$ ): probability distribution of the final state, at $r=r_{\mathrm{f}}$ , on the simulation grid (height), colored according to the value of $f$ ; stars and circles mark the global and local minima, and the black line the boundary of the basin of the global minimum. Bottom of each panel: success probability $P_{\mathrm{suc}}$ (left axis) and coefficient $\mathbb{E}$ (right axis, logarithmic scale) along the curve, with the online procedure (solid lines) and with exact phases (dashed lines).

					exact phases		online
Instance	$d$	$q$	$m$	minima	$P_{\mathrm{suc}}$	$\mathbb{E}$	$P_{\mathrm{suc}}$	$\mathbb{E}$
Example 1	1	8	3	2 (1)	0.98	0.25	0.94	0.31
Example 2	1	8	3	3 (1)	0.99	0.28	0.94	0.33
Example 3	2	5	2	4 (2)	0.92	0.28	0.86	0.34
Example 4	2	5	3	9 (1)	0.83	0.32	0.74	0.39
Example 5 ‡	2	5	3	8 (4)	0.96	0.56	0.86	0.63

Table 2: Final success probability

P_{\mathrm{suc}}

and final coefficient

\mathbb{E}

for the quantum functions of appendix F , obtained with the same protocol for all instances: curve

r(\tau)=r_{0}\,e^{k\tau}

of eq. 26 with

k=0.1

, from

r_{0}=10^{-4}

r_{\mathrm{f}}=10

, run through in the gauge with constant

b

of section 4.1 , and

R=2^{15}

Trotter steps. Columns:

d

, number of parameters;

q

, qubits per parameter;

m

, qubits of the measure register; minima, number of local minima of

f

[0,2\pi)^{d}

, with the global ones in parentheses. The Trotterized evolution with exact potential steps (exact phases) is compared with the online procedure. The results with exact phases are checked with

2R

steps:

P_{\mathrm{suc}}

and

\mathbb{E}

change by less than

1\%

, so that the Trotter error is negligible. Symbols: ‡ ,

P_{\mathrm{suc}}

is still changing at

r_{\mathrm{f}}

and reaches

1.00

r=10^{2}

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

periodic extension $F$	$\left\|\tilde{F}(\bm{\kappa})\right\|$ decay	$\Gamma_{1}^{(q)}$	$\Gamma_{2}^{(q)}$
smooth	faster than any power	$\mathcal{O}(1)$	$\mathcal{O}(1)$
continuous, with discontinuous derivative	$\sim\left\|\bm{\kappa}\right\|^{-2}$	$\mathcal{O}(q)$	$\mathcal{O}\left(2^{q}\right)$
discontinuous	$\sim\left\|\bm{\kappa}\right\|^{-1}$	$\mathcal{O}\left(2^{q}\right)$	$\mathcal{O}\left(4^{q}\right)$

Appendix

Table 3: Behavior of

\Gamma_{1}^{(q)}

and

\Gamma_{2}^{(q)}

as functions of

q

, for a fixed cost function

f

, according to the regularity of its periodic extension

F

. The constants depend on

f

, and no explicit factor of

d

appears. In the last two rows, factors polynomial in

q

can be added by the singularities at the edges and corners of the domain.

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