Compensating for non-linear distortions in controlled quantum systems Juhi Singh 1 2Robert Zeier

2025-04-29 0 0 954.32KB 15 页 10玖币

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Compensating for non-linear distortions in controlled quantum systems

Juhi Singh ,1, 2, ∗Robert Zeier ,1, †Tommaso Calarco,1, 2 and Felix Motzoi1

1Forschungszentrum J¨ulich GmbH, Peter Gr¨unberg Institute,

Quantum Control (PGI-8), 52425 J¨ulich, Germany

2Institute for Theoretical Physics, University of Cologne, 50937 K¨oln, Germany

(Dated: May 16, 2023)

Predictive design and optimization methods for controlled quantum systems depend on the accu-

racy of the system model. Any distortion of the input ﬁelds in an experimental platform alters the

model accuracy and eventually disturbs the predicted dynamics. These distortions can be non-linear

with a strong frequency dependence so that the ﬁeld interacting with the microscopic quantum sys-

tem has limited resemblance to the input signal. We present an eﬀective method for estimating these

distortions which is suitable for non-linear transfer functions of arbitrary lengths and magnitudes

provided the available training data has enough spectral components. Using a quadratic estimation,

we have successfully tested our approach for a numerical example of a single Rydberg atom sys-

tem. The transfer function estimated from the presented method is incorporated into an open-loop

control optimization algorithm allowing for high-ﬁdelity operations in quantum experiments.

I. INTRODUCTION

Over the last few decades, various quantum sys-

tems, including superconducting circuits, neutral atoms,

trapped ions, and spins [1–3], have shown exciting

progress in controlling quantum eﬀects for applications

in quantum sensors [4], simulators [5], and computers [6].

In these setups, quantum operations are implemented us-

ing external ﬁelds or pulses which are generated and in-

ﬂuenced by several electronic and optical devices. For

high-ﬁdelity and uptime applications, this requires high

performance of, e.g., population transfers and quantum

gates, while suppressing interactions with the environ-

ment as well as decoherence. By shaping temporal and

spatial proﬁles of external ﬁelds and pulses, the time-

dependent system Hamiltonian steers the quantum dy-

namics towards the targeted outcome.

Experimental distortions of the applied pulses may re-

duce the eﬀectiveness and robustness of the desired quan-

tum operation [7, 8]. Methods have been developed to

characterize distortions based on the impulse response

or transfer function of the experimental system [7–15].

These approaches for estimating ﬁeld distortions work

well for distortions with a linear transfer function. This

work, however, addresses the more general case with sub-

stantial non-linear distortions originating from the exper-

imental hardware.

The description of the distortions can be challenging

without knowing the exact characteristics of the experi-

mental hardware. Also, approximating a signiﬁcant non-

linearity using a linear model will result in model coef-

ﬁcients and control pulses that are not robust against

experimental distortions and suﬀer from a loss in ﬁ-

delity. To account for this problem, we introduce a

mathematical model and an estimation method which

∗j.singh@fz-juelich.de

†r.zeier@fz-juelich.de

rely on limited experimental data and can characterize

the system behavior up to a non-linearity of ﬁnite order.

To streamline our presentation, we focus on quadratic

non-linearities, but more general non-linearities can be

treated similarly. We illustrate our estimation approach

with numerical data for a single-Rydberg atom excitation

experiment in the presence of signiﬁcant non-linearities

and we highlight how our approach can calibrate for and

suppress large distortions. We describe an eﬀective ap-

proach for estimating the coeﬃcients of this non-linear

model and correct the pulses accordingly. We empha-

size that our approach is independent of a speciﬁc ex-

perimental setup and can therefore be applied to various

(spatially or temporally) ﬁeld-tunable phenomena on dif-

ferent quantum platforms.

Our estimation method for distortions is particularly

eﬀective in combination with methods from quantum op-

timal control [16–20] and it yields optimized pulses for

highly eﬃcient gates while accounting for estimated dis-

tortions. To this end, we provide an analytical expres-

sion for estimating the Jacobian of the transfer function

for quadratic distortions, which can be further general-

ized to higher orders. We also validate this combined

approach with our Rydberg atom excitation example.

In the context of quantum control, any inaccuracy in

the system Hamiltonian can severely aﬀect the perfor-

mance of pulses produced by optimal control. Given

a reasonably accurate model, control ﬁelds might also

suﬀer from discretization eﬀects, electronic distortions,

and bandwidth limitations (mostly assumed to be lin-

ear). Accounting for these distortions by including the

linear transfer function within the dynamics, as well as

its combined gradient, has been incorporated in related

optimization work [7, 15, 21–23]. Another strategy for

minimizing non-linear pulse distortions is to avoid high

frequencies altogether in control pulses [24, 25].

Starting from initial applications [26, 27], optimal con-

trol methods have been extensively used in quantum

computing, quantum simulation, and quantum informa-

tion processing [17, 20, 28–31]. Analytic results applica-

arXiv:2210.07833v2 [quant-ph] 16 May 2023

ble to smaller quantum systems shape our understand-

ing for the limits to population transfers and quantum

gates (see [17, 20, 32–48] and references therein). In-

creasing the eﬃciency of quantum operations by numer-

ically optimizing and ﬁne-tuning control parameters can

rely on open-loop or model-based optimal control meth-

ods [7, 49–57]. Our work on the estimation of distortions

can be seen in the context of model-based approaches,

which might rely on an accurate gradient calculation of

the analytical cost function and thus on the knowledge

of the Hamiltonian of the system [7, 17]. This knowl-

edge might be available in naturally occurring qubits

(such as atomic, molecular, or optical systems), but may

also be estimated in engineered (solid-state) technologies.

Similarly, closed-loop (i.e. adaptive feed-forward) control

methods [31, 58–64] are used in situ to reduce adverse

experimental eﬀects on the control pulses, while direct

(real-time) feedback and reservoir engineering methods

can also be used where appropriate to counteract control

uncertainties [65, 66].

The paper is organized as follows: Section II sketches

the control setup for optimizing quantum experiments

and describes the conventional method for estimating the

transfer function and its inclusion in the optimization. In

Sec. III, we detail our non-linear estimation method us-

ing non-linear kernels. We also describe how to derive the

transformation matrix and its gradient. The non-linear

eﬀects on quantum operations are shown with a numeri-

cal example of Rydberg atom excitations in Sec. IV. We

apply the estimation methodology to our numerical Ry-

dberg example in Sec. V and discuss requirements on the

available measurement data. Finally, we consider diﬀer-

ent numerical optimization methods in combination with

our estimation method in Sec. VI (see also Appendix A)

and conclude in Sec. VII. The raw data ﬁles from the

simulations performed for this work are provided in [67].

II. TIME-DEPENDENT CONTROL PROBLEMS

We aim to eﬃciently transferring the population from

an initial quantum state to a ﬁnal target state. The

evolving state of a quantum system is described by its

density operator ρ(t) and the corresponding equation of

motion is written for coherent dynamics as

˙ρ=−i[H(t), ρ] + L(ρ).(1)

The form of the Lindblad term L(ρ) is discussed in

Sec. IV while the Hamiltonian can be expressed as

H(t) = Hd+X

ui(t)Hi.(2)

The free-evolution or drift component is given by Hd,

while Hidenotes the control Hamiltonians which are mul-

tiplied with time-dependent control pulses ui(t). More

precisely, our goal is to transfer a quantum system from

a given initial pure state with density operator ρito a

0.00 0.25 0.50

Time (µs)

100

200

300

Amplitude (MHz)

246

A B C D E F

Distortion type

(a)

10−3

10−9

Mean absolute scaled error

(b)

Distorted pulse Linear Quadratic

Figure 1. Quadratic estimation of distorted pulses [Eq. (7)] is

preferable to linear estimation [Eq. (4)]: (a) A pulse is numer-

ically distorted (solid line); later the distortion is estimated

up to linear (dashed line) and quadratic terms (dotted line).

The quadratic estimation better matches the actual distorted

pulse when compared to the linear estimation. (b) Numeri-

cally computed errors for diﬀerent types of distortions [includ-

ing the distortion C plotted in (a)] generated by Eqs. (20)–

(21) are plotted for both the linear and the quadratic estima-

tion. The error is deﬁned in Eq. (22) and describes the diﬀer-

ence between the actual distorted and the estimated pulse.

target pure-state density operator ρtin time Tby vary-

ing the control pulses ui(t) while minimizing the cost

function

C= 1 − |hρt|ρ(T)i|2= 1 − |Tr[ρ†

tρ(T)]|2,(3)

where Tr(M) denotes the trace of a matrix M. This cost

function measures the diﬀerence between the target-state

density operator ρtand the ﬁnal-state density operator

ρ(T). In this work, we employ gradient-based optimiza-

tion methods, which are described and discussed in Sec-

tion VI and Appendix A.

The experimental realization of control pulses ui(t) re-

lies on several devices, which might introduce systematic

distortions and reduce the overall control eﬃciency. It is

our objective to determine these systematic distortions in

order to adapt the control pulses during the optimization

and counteract any adverse eﬀects. For a linear distor-

tion, we can calculate its transfer function

T(ω) = Y(ω)

X(ω)(4)

in the Fourier domain as the ratio of the Fourier trans-

form of the input and output pulses x(t) and y(t), i.e.

before and after the distortion has taken place. Alterna-

tively, we can calculate the impulse response I(t) of the

system which relates the input and output pulse in the

time domain using the convolution

y(t)=(x∗ I)(t) = Z∞

−∞

x(τ)I(t−τ)dτ. (5)

Figure 1 highlights that a linear model might not be suf-

ﬁcient for estimating experimental distortions as it can-

not account for non-linear eﬀects. Non-linear eﬀects are

demonstrated in Fig. 1(a) by passing one estimated ex-

ample pulse through a numerically generated distortion

[see Eqs. (20)–(21)]. When estimating the distortion co-

eﬃcients using a linear model, the resulting distorted

pulse does not match in Fig. 1(a) with the actual dis-

torted pulse. However, the quadratic estimation with

a non-linear model (as described in Sec. III) precisely

recovers the actual distorted pulse. Non-linear models

are, e.g., preferable for Rydberg excitations which are de-

tailed with realistic experimental parameters in Sec. IV.

III. NON-LINEAR ESTIMATION METHOD

We provide now a general approach for estimating non-

linear distortions in a controlled quantum system and ex-

plain how this estimation approach can be incorporated

into the synthesis of robust optimal control pulses.

A. Truncated Volterra series method

We characterize non-linear distortions using the trun-

cated Volterra series method [68]. The Volterra series is

a mathematical description of non-linear behaviors for a

wide range of systems [69]. In analogy to Eq. (5), we can

write the general form of the Volterra series as

y(t) = h(0) +

n=1 Zb

. . .Zb

h(n)(τ1, . . . , τn)

j=1

x(t−τj)dτj

(6)

where x(t) is assumed to be zero for t < 0 as we con-

sider general, non-periodic signals. The output function

y(t) can be expressed as a sum of the higher-order func-

tionals of the input function x(t) weighted by the cor-

responding Volterra kernels h(n). These kernels can be

regarded as higher-order impulse responses of the system.

The Volterra series in Eq. (6) is truncated to the order

P < ∞and it is called doubly ﬁnite if aand bare also

ﬁnite. For a causal system, the output y(t) can only de-

pend on the input x(t−τj) for earlier times (i.e. t≥τj)

which results in a≥0; recall that x(t−τj) = 0 for τj> t.

The Volterra series can therefore also model memory ef-

fects (which are assumed to be of ﬁnite length) and it is

not restricted to instantaneous eﬀects.

The discretized form of the Volterra series truncated

to second order (i.e. P= 2) is given by ([68, Eq. 2.25])

yn=h(0) +

R−1

j=0

h(1)

jxn−j+

R−1

k,`=0

h(2)

k` xn−kxn−`,(7)

The discrete output entries ynhave Ntime steps with

n∈ {0, . . . , N−1}which are obtained from Ldiscrete

input entries xqwhere xq= 0 for q < 0. Note that

N=L+R−1≥L, where R≥1 denotes the as-

sumed memory length of the distortion. The memory

length Rquantiﬁes how the response at the current time

step depends on the input of previous time steps, i.e., R

bounds the number of previous time steps that can af-

fect the current one. Volterra kernel coeﬃcients of the

zeroth, ﬁrst, and second order are represented by h(0),

h(1)

j, and h(2)

k` . The matrix given by h(2)

k` is symmetric.

We are characterizing the transfer function by estimating

the kernel coeﬃcients in Eq. (7). The number Mof the

to-be-estimated coeﬃcients scales quadratically with the

memory length R(in general, the number of coeﬃcients

scales with RP). Although the Volterra estimation can

be extended to any higher order P > 2, we will focus in

this work on the quadratic case.

For the estimation process, we assume that we are pro-

vided with a training data set consisting of input-output

pulse pairs (x(t), y(t)) from an experimental device (or a

sequence of devices) which causes the distortion. Next,

we discuss how given the training data, we can estimate

the kernel coeﬃcients in Eq. (7) by minimizing some er-

ror measures (such as the mean square error) between

the modeled output and the measured output.

B. Truncated Volterra series via least squares

We can choose from diﬀerent methods to estimate the

Volterra series. The most widely used ones are the cross-

correlation method of Lee and Schetzen [70] and the ex-

act orthogonal method of Korenberg [71]. We choose the

latter due to its simplicity and as it does not require an

inﬁnite-length input. We can write Eq. (7) as

yn=

M−1

m=0

unm km(8)

or equivalently as the matrix equation Y=UK or







yN−1





=





u00 u01 ··· u0,M−1

u10 u11 ··· u1,M−1

uN−1,0uN−1,1··· uN−1,M−1













kM−1





,

(9)

where Kis deﬁned in Eq. (11) below. We follow the

convention that the entries of a given matrix (or vector)

Dare represented by dij (or di). Here, n∈ {0, . . . , N−1}

and m∈ {0, . . . , M−1}where

M= 1+R+R(R+1)/2 (10)

denotes the number of coeﬃcients that need to be esti-

mated to describe the quadratic Volterra series. In par-

ticular, unm are obtained from the input pulses via (recall

again xq= 0 for q < 0)

unm =









1 for m= 0,

xn−m+1 for m∈ {1, . . . , R},

xn−axn−bfor m∈ {R+1, . . . , M−1},

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Compensatingfornon-lineardistortionsincontrolledquantumsystemsJuhiSingh,1,2,RobertZeier,1,yTommasoCalarco,1,2andFelixMotzoi11ForschungszentrumJulichGmbH,PeterGrunbergInstitute,QuantumControl(PGI-8),52425Julich,Germany2InstituteforTheoreticalPhysics,UniversityofCologne,50937Koln,Germany(Dated:Ma...

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Compensating for non-linear distortions in controlled quantum systems Juhi Singh 1 2Robert Zeier

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