Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach

Markus Gross; Hans-Martin Rieser

	Feature $\phi_{a_{k}}^{(k)}(u_{k})$
Factor in observable $P$	Amplitude on $[0,1]$	Amplitude on $[-1,1]$	Rotational
$\cdots\otimes X_{k}\otimes\cdots$	$2\sqrt{(1-u_{k})u_{k}}$	$2u_{k}\sqrt{1-u_{k}^{2}}$	$\sin(2\pi u_{k})$
$\cdots\otimes Y_{k}\otimes\cdots$	$0$	$0$	$0$
$\cdots\otimes Z_{k}\otimes\cdots$	$2u_{k}-1$	$2u_{k}^{2}-1$	$\cos(2\pi u_{k})$
$\cdots\otimes\tfrac{1}{2}(\mathbbm{I}+Z_{k})\otimes\cdots$	$u_{k}$	$u_{k}^{2}$	$\cos^{2}(\pi u_{k})$

Table 1: Pauli–basis projections

\langle\psi_{\text{enc}}|P|\psi_{\text{enc}}\rangle

of an encoded product state

\rho_{\text{enc}}(\mathbf{u})

for amplitude encoding ( 8 ) or ( 9 ) as well as rotational encoding ( 10 ). Here

P=\bigotimes_{k=1}^{n}\sigma_{a_{k}}

denotes a Pauli operator (

\sigma_{a_{k}}\in\{\mathbbm{I},X,Y,Z\}

) and the classical datum entering qubit

k

is

u_{k}

. For convenience, we have included the projection operator

\frac{1}{2}(\mathbbm{I}+Z_{k})

. One observes that each Pauli factor

\sigma_{a_{k}}

(1st column) in an observable

P

produces the corresponding single-qubit feature

\phi_{a_{k}}^{(k)}(u_{k}):=\mathrm{tr}\!\big(\sigma_{a_{k}}\,\rho_{k}(u_{k})\big)

(2nd–4th columns). For instance, the

X_{k}

row gives

\phi_{x}^{(k)}(u_{k})

and the

Z_{k}

row gives

\phi_{z}^{(k)}(u_{k})

.

Figure 1: Time evolution of the Pauli weight average

\bar{\nu}_{k}

[ Eq. 36 ] of a single-site Pauli

\sigma_{k}

in the

h

–

J

coupling parameter space of the TFIM ( 5a ) for

n=3

qubits.

Figure 2: Visualization of the operator spreading ( 35 ) via the effective PTM

R

[ Eq. 24 ] constructed in the temporal multiplexing scheme for the TFIM (a) and random unitary (b). The selected observable set

\mathcal{S}

are all weight-1 Pauli operators of a

n=3

qubit system. With increasing evolution time (from top to bottom), these initial operators develop nonzero projection onto the full

d^{2}

-dimensional Pauli basis [ Eq. 35 ]. In the TFIM case, the supporting basis exhibits a sparse pattern, whereas it is dense for a random unitary.

Figure 3: Projector

R^{+}R

of the effective PTM [ Eq. 24 ] for different unitary types and a complete weight-1 Pauli observable set. Panels (a,b) directly correspond to the

R

shown in Fig. 2 , which are constructed by temporal multiplexing. Panel (c) shows the projector based on

R=V(t=0.5)

(no temporal multiplexing). The dashed white lines indicate features for which

\gamma_{j}^{2}>0.5

, while the black lines indicate Pauli weight sector boundaries. Due to operator spreading over a time

t=0.5

, decodability in (c) has significantly decreased.

Figure 4: Feature decodability score

\bar{\gamma}_{j}^{2}

obtained from the effective PTM

R

[ Eq. 41 ], averaged over each Pauli weight sector, of a

n=3

qubit model as a function of temporal multiplexing iteration length

{L}

[ Eq. 24 ]. The unitary evolution time is given by

t_{L}={L}

(i.e., initial time

t_{1}=1

). The corresponding rank and number of rows of

R

, representing the measurement budget (

{B}={m}{L}

), are shown on the top axis (as a fraction of the total operator space dimension

d^{2}=64

). We consider the TFIM ( 5a ) in (a,c,e), the TFIM ( 5b ) in (g), and random Hamiltonians in (b,d,f). The observable set

\mathcal{S}

is either only

Z

-Paulis (a,b),

Z

- and

ZZ

-Paulis (c,d,g), or all weight-1 Paulis (e,f).

Hamiltonian	Observables	Asymptotic saturated rank $r_{\infty}$
$H_{\text{TFIM}}^{\text{xx-z}}$	$Z_{j}$	$\mathcal{O}(n^{2})$
$H_{\text{TFIM}}^{\text{xx-z}}$	$Z_{j}$ , $Z_{i}Z_{j}$	$\mathcal{O}(n^{4})$
$H_{\text{TFIM}}^{\text{xx-z}}$	$X_{j},Y_{j},Z_{j}$	$\mathcal{O}(4^{n})$ (up to poly. factors)
$H_{\text{TFIM}}^{\text{zz-x}}$	$Z_{j}$	$\mathcal{O}(4^{n})$ (up to poly. factors)
$H_{\text{TFIM}}^{\text{zz-x}}$	$Z_{j}$ , $Z_{i}Z_{j}$	$\mathcal{O}(4^{n})$ (near full)
$H_{\text{TFIM}}^{\text{zz-x}}$	$X_{j},Y_{j},Z_{j}$	$\mathcal{O}(4^{n})$ (up to poly. factors)

Table 2: Asymptotic saturated rank

r_{\infty}:=\lim_{L\to\infty}\mathrm{rank}(R_{L})

(see Section E.2 ) of the temporally multiplexed PTM ( 24 ) for the two TFIM variants [ Eq. 5 ] and different observable families

\mathcal{S}

. For temporal multiplexing of pure Pauli

Z

-type observables, the Hamiltonian

H^{\text{xx-z}}_{\text{TFIM}}

leads to exponentially lower expressivity, reflected by polynomial-in-

n

ranks

\ll 4^{n}

.

Figure 5: Rank of the effective PTM

R

[ Eq. 24 ] obtain by temporal multiplexing an observable set

\mathcal{S}

(one and two-site Pauli

Z

) under unitary evolution with a TFIM Hamiltonian [ Eq. 5 ]. (a) Evolution of

\mathrm{rank}(R)

with temporal multiplexing steps

{L}

for the TFIM ( 5a ) and

\mathcal{S}=\{Z_{i},Z_{i}Z_{j}\}

observables. (b) Asymptotic PTM rank,

\lim_{L\to\infty}\mathrm{rank}(R)

, obtained numerically (solid curves), compared to the theoretical scaling predictions from Table 2 (broken lines).

Figure 6: Ideal nonlinear capacity score

\mathcal{R}^{2}(k)

[ Eq. 44 ] for amplitude and rotational encodings [ Eqs. 8 and 10 ], indicating the ability of an ideal QELM

f(\mathbf{u})=\mathbf{w}^{\top}\boldsymbol{\upphi}(\mathbf{u})

to generate degree-

k

monomials of the input

u_{\alpha}

(

\alpha=1,\ldots,D

and

n=D

). Unless indicated, only Pauli-

Z

features are measured in (a). Nonlinear capacity is significantly enhanced when measuring over all Paulis (b). This is also demonstrated by the score for amplitude encoding with Pauli-

X

readout included in (a). The dash-dotted curve represents the theoretical score for

\mathcal{R}^{2}_{Z}(k)

with amplitude encoding and

n=5

input qubits [ Eq. 45 ], which agrees exactly with the numerical results.

\Sigma

denotes the integrated capacity over all

k

.

Figure 7: Nonlinear capacity score

\mathcal{R}^{2}(k)

[ Eq. 44 ] as a function of the nonlinear order

k

and the temporal multiplexing iteration length

{L}

[ Eq. 24 ]. The unitary evolution time is given by

t_{L}=L

. We consider a

n=3

qubit model in the same setup as in Fig. 4 . The unitary is based on the TFIM ( 5a ) (a,c) or a random Hamiltonian (b,d), while the observable set

\mathcal{S}

consists of either only

Z

-Paulis (a,b) or

Z

- and

ZZ

-Paulis (c,d).

Figure 8: Relative nonlinear capacity score

\mathcal{R}^{2}(k)/(\mathcal{R}^{2}_{\text{all}}(k)+\delta)

[ Eq. 44 ] as a function of the nonlinear order

k

and the temporal multiplexing iteration length

{L}

[ Eq. 24 ].

\mathcal{R}^{2}_{\text{all}}

is the nonlinear capacity score obtained for a complete Pauli observable set, and we choose

\delta=0.1

as a cut-off since scores

\mathcal{R}^{2}_{\text{all}}\lesssim 0.1

are not considered meaningful. We use the same setup as in Fig. 7 . The unitary is based on the TFIM ( 5a ) (a,c) or a random Hamiltonian (b,d), while the observable set

\mathcal{S}

consists of either only

Z

-Paulis (a,b) or

Z

- and

ZZ

-Paulis (c,d).

Figure 9: Ability to construct monomials of the input

\{x,y,z\}

for amplitude encoding ( 8 ) and rotational encoding ( 10 ) with

n=D=3

. (a) Decodability score

\kappa_{j}^{2}

according to Eq. 46 . (b) Projector matrix

A^{+}A

, where nonzero off-diagonals correspond to

\kappa_{j}^{2}<1

and indicate decoding is possible only within a certain accuracy. The underlying Taylor expansion is performed up to order

r=4

in each

u_{j}=\{x,y,z\}

and changes are mild upon further increasing

r

. The scores and matrices are averaged over a uniform grid of expansion points

\mathbf{u}_{0}\in[0.1,0.9]^{3}

for amplitude encoding (with some margin to the singular boundaries) and

\mathbf{u}_{0}\in[0,1]^{3}

for rotational encoding. For readability, monomial labels are incompletely displayed in (a) and replaced by column/row indices in (b).

Figure 10: QELM vs. its classical representation: The forecast obtained from a QELM [ Eq. 1 ] trained on the rescaled Lorenz-63 model is matched exactly by the forecast of its iterated classical (surrogate) representation [ Eq. 25 ]. Both eventually diverge from the true trajectory due to the Lyapunov instability. The QELM employs amplitude encoding ( 8 ) with

n=3

qubits. Ground-truth trajectories are generated with a resolution of

\Delta t=0.01

. Time is given in model time units. For the scaling transformation in Eq. 53 , we use parameters

\alpha_{\kappa}=0.004

and

\mathbf{m}=[-30,-30,-5]

.

System	type	QELM setup	$\alpha$	$\lambda$	$\epsilon_{\text{train}}$	$T_{\text{fch}}/T_{L}$
Lorenz-63	std.	ampl. enc. $[0,0.3]$ , Eq. 8	-	$10^{-6}$	$10^{-5}$	$4.3\pm 1.7$
Lorenz-63	resc.	ampl. enc. $[0,1]$ , Eq. 8	0.005	$10^{-8}$	$10^{-6}$	$6.2\pm 1.4$
Halvorsen	std.	ampl. enc. $[0.1,0.8]$ , Eq. 8	-	$10^{-8}$	$10^{-4}$	$2.3\pm 0.8$
Halvorsen	std.	…+ add squares to readout vector	-	$10^{-8}$	$10^{-5}$	$5.8\pm 1.7$
Halvorsen	resc.	ampl. enc. $[0,1]$ , Eq. 8	0.027	$10^{-7}$	$10^{-4}$	$3.4\pm 1.0$
Halvorsen	resc.	…+ add squares to readout vector	0.027	$10^{-7}$	$10^{-6}$	$6.8\pm 1.7$

Table 3: Training RMS errors

\epsilon_{\text{train}}

[ Eq. 55 ] and forecast horizons

T_{\text{fch}}

[ Eq. 56 ] for the benchmark systems and (unitary-free) QELM setups discussed in the text. We generally use

n=3

input qubits (no delay states) and a time resolution of

\Delta t=0.01

.

T_{\text{fch}}

with its standard deviation (not error of the mean) is given in units of the Lyapunov time

T_{L}

. The column ‘type’ indicates whether we use the standard [ Eq. 51 ] or rescaled dynamical system. In the latter case,

\alpha

gives the rescaling factor in Eqs. 53 and 54 (same for all coordinates).

Figure 11: Prediction performance (forecast horizon and RMS training error) under temporal multiplexing of a QELM trained on the Halvorsen model ( 54 ). We use amplitude encoding ( 9 ) with

n=3

qubits, single-qubit Pauli-

Z

observables, and (a) the TFIM Hamiltonian

H_{\text{TFIM}}^{\text{zz-x}}

[ Eq. 5a ] or (b) a random Hamiltonian. The top horizontal axis shows the numerically obtained rank of the readout matrix

G

[ Eq. 28 ] relative to the maximum achievable rank

q=3^{n}

[ Eq. 30 ]. Forecast horizon is given in units of the Lyapunov time

T_{L}

; no additional features are added to the readout. For the TFIM variant

H_{\text{TFIM}}^{\text{xx-z}}

[ Eq. 5b ], we obtain a maximum rank of

\mathrm{rank}(G)/3^{n}\simeq 0.22

, RMSE

\epsilon_{\text{train}}\simeq 7\times 10^{-4}

, and forecast horizon

T_{\text{fch}}\simeq 0

.

Figure 12: Illustration of the flow map learned by a QELM, obtained from the Taylor-approximation approach Eq. 58 . The QELM is trained on (a) the Lorenz-63 model ( 53 ) and (c) the Halvorsen model ( 54 ), using the amplitude encoding variants ( 8 ) and ( 9 ), respectively. The rescaling parameters are here

\mathbf{m}=[-30,-30,5],\alpha_{\kappa}=0.004

(Lorenz-63) and

\mathbf{m}=[-10,-10,-10],\alpha_{\kappa}=0.001

(Halvorsen). The corresponding true flow maps

\boldsymbol{\uppsi}_{\mathbf{g}}

[ Eq. 48 ], expanded to third order in the state variables, are shown in (b,d) for comparison (terms identically zero are omitted).

Figure 13: Illustration of the flow map learned by a QELM, obtained from the Taylor-approximation approach Eq. 58 . The QELM uses rotational encoding [ Eq. 10 ] and is trained on (a) the Lorenz-63 model ( 53 ) and (c) the Halvorsen model ( 54 ). The rescaling parameters are here

\mathbf{m}=[1,1,20],\alpha_{\kappa}=0.002

(Lorenz-63) and

\mathbf{m}=[-3,-3,-3],\alpha_{\kappa}=0.002

(Halvorsen). The corresponding true flow maps

\boldsymbol{\uppsi}_{\mathbf{g}}

[ Eq. 48 ], expanded to third order in the state variables, are shown in (b,d) for comparison. The coefficients differ from Fig. 12 due to the different expansion points.

Figure 14: Pauli weight average (a) of Heisenberg-evolved measurement observables can predict the training error (b) and forecast horizon (c) of a QELM. We use a TFIM Hamiltonian (varying coupling parameters

J,h

, fixed

t=6

) with

n=3

qubits and amplitude encoding ( 8 ) (no temporal multiplexing). The Pauli weight average is data-independent, while the QELM in (b,c) is trained on the Lorenz-63 model and employs 1-qubit Pauli-

Z

observables in the readout, enhanced by monomials up to degree 2 to increase sensitivity to nonlinear terms.

Symbol	Meaning
$\mathcal{H}$	Hilbert space
$\mathrm{Herm}(\mathcal{H})$	Space of Hermitian operators on $\mathcal{H}$
$n$	Number of qubits
$d$	Hilbert space dimension ( $d=2^{n}$ )
$D$	Input data dimension
$P$	Number of training samples

Table 4: Summary of notation used in this work. A trained quantity is indicated by a hat (e.g.,

\hat{\mathbf{w}}

).

Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach

Abstract

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