Master Equation Emulation and Coherence Preservation with Classical Control of a Superconducting Qubit Evangelos Vlachos1 2Haimeng Zhang3 2Vivek Maurya1 2Jeffrey

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Master Equation Emulation and Coherence Preservation with Classical Control of a

Superconducting Qubit

Evangelos Vlachos,1, 2 Haimeng Zhang,3, 2 Vivek Maurya,1, 2 Jeﬀrey

Marshall,4, 5 Tameem Albash,6, 7 and E. M. Levenson-Falk1, 2, ∗

1Department of Physics & Astronomy, Dornsife College of Letters, Arts,

& Sciences, University of Southern California, Los Angeles, CA 90089, USA

2Center for Quantum Information Science & Technology,

University of Southern California, Los Angeles, CA 90089, USA

3Department of Electrical Engineering, Viterbi School of Engineering,

University of Southern California, Los Angeles, CA 90089, USA

4Quantum Artiﬁcial Intelligence Laboratory (QuAIL),

NASA Ames Research Center, Moﬀett Field, CA, 94035, USA

5USRA Research Institute for Advanced Computer Science (RIACS), Mountain View, CA, 94043, USA

6Department of Electrical and Computer Engineering,

University of New Mexico, Albuquerque, New Mexico 87131, USA

7Department of Physics and Astronomy and Center for Quantum Information and Control,

University of New Mexico, Albuquerque, New Mexico 87131, USA

Open quantum systems are a topic of intense theoretical research. The use of master equations

to model a system’s evolution subject to an interaction with an external environment is one of the

most successful theoretical paradigms. General experimental tools to study diﬀerent open system

realizations have been limited, and so it is highly desirable to develop experimental tools which

emulate diverse master equation dynamics and give a way to test open systems theories. In this paper

we demonstrate a systematic method for engineering speciﬁc system-environment interactions and

emulating master equations of a particular form using classical stochastic noise in a superconducting

transmon qubit. We also demonstrate that non-Markovian noise can be used as a resource to extend

the coherence of a quantum system and counteract the adversarial eﬀects of Markovian environments.

I. INTRODUCTION

The study of open quantum systems remains an ac-

tive area of research at the frontier of understanding

the range of phenomena allowed by quantum mechan-

ics. Open systems are characterized by a system of in-

terest having signiﬁcant interactions with a number of

uncontrolled environmental degrees of freedom, giving

rise to decoherence in the primary system. In the case

where environmental interactions take the form of purely

Markovian (memoryless) decoherence, a master equation

of Lindblad form (ME) [1,2] can be written and solved,

in principle. However, when environmental interactions

lead to non-Markovian eﬀects, i.e. when the environ-

ment has ﬁnite-time correlations that in turn aﬀect the

system (“ﬁnite memory”), theoretical descriptions are

much more challenging. The Nakajima-Zwanzig equa-

tion [3] provides an exact physical description of such a

setup, but the equation is in general not solvable. In

fact, it is diﬃcult to even write down such an equation

as it requires a complete description of the environmental

degrees of freedom [4]. Simpler, more easily solved de-

scriptions exist [5,6], such as the post-Markovian mas-

ter equation (PMME) [7], Gaussian collapse model [8],

quantum collisional models [9–11], time-convolutionless

master equations [4], and the pseudo-Lindblad master

∗Corresponding author: elevenso@usc.edu

equation (PLME) [12]. However, these are diﬃcult to in-

terpret physically, so it remains an open question how to

write a solvable physical description of an arbitrary open

quantum system.

Despite signiﬁcant theoretical progress, experimental

tests of open quantum system theories are more lim-

ited. Progress has been made in ﬁtting MEs to mea-

sured dynamics [12,13] and simulating Markovian en-

vironments [14], and techniques exist to simulate spe-

ciﬁc non-Markovian eﬀects [15], for example by embed-

ding the system into a larger Markovian system [16–19].

However, there is still no general experimental toolkit.

Developing new capabilities to simulate non-Markovian

MEs remains highly desirable, as they would allow new

experimental tests of the validity of open system models.

In addition, many non-Markovian environments can be

used as resources for enhancing coherence of a target sys-

tem, and so this environmental engineering can be used

to improve the ﬁdelity of practical quantum processes

[20].

A particular class of non-Markovian ME, the gen-

eralized Markovian master equation (GMME) is often

exactly solvable via Laplace transforms [21–24]. This

ME describes a system undergoing Markovian dephas-

ing while coupled to a non-Markovian environment with

some ﬁnite memory. If the Lindblad operators associated

with the Markovian and non-Markovian interactions act

along orthogonal directions and the non-Markovian en-

vironmental memory is suﬃciently long, the coherence

of the system may be extended compared to the case

arXiv:2210.01388v3 [quant-ph] 20 Feb 2024

where only the Markovian background dephasing exists.

Crucially, the dynamics described by the GMME may,

in some circumstances, be emulated with noisy classi-

cal driving. The GMME is thus an ideal test case for

emulation of target ME dynamics with an experimental

system.

In this paper we demonstrate protocols for emulation

of GMME dynamics with classical control by noisily driv-

ing a single superconducting transmon qubit. Our nu-

merical simulations and experimental measurements con-

form well to the analytic solutions of the GMME in their

regimes of validity. We also extend our protocol to a

new regime, where the background dephasing itself is not

perfectly Markovian, and model this numerically with a

Bloch-Redﬁeld master equation. We explore the limits

of such regimes and describe possible extensions of this

protocol. Our results provide a basis for emulation of

more arbitrary open system dynamics and add another

experimental tool for open system engineering.

II. BACKGROUND

A. Theory

Our goal is to emulate the generalized Markovian mas-

ter equation

dtρ(t) = γiLi(ρ(t)) + LjZt

k(t−t′)ρ(t′)dt′,(1)

where Li,Ljare Lindbladians with Lindblad operators

σi, σj(i, j ∈ {x, y, z}) respectively [25], and k(t−t′) is

the memory kernel of the quantum environment. This de-

scribes a system (here, a qubit) with Hamiltonian H= 0

undergoing Markovian dephasing due to Liwhile inter-

acting with a non-Markovian environment via Lj. In the

case where k(t−t′) = δ(t−t′) the environment is fully

Markovian, and the qubit state purity decays exponen-

tially. When the environmental memory is ﬁnite, state

purity decays non-monotonically and the coherence time

may be extended [22]. Note that arbitrary choices of k

may lead to non-physical dynamics, and so care must be

taken in its selection.

To emulate this GMME we follow the recipe given

in Ref. [22] and replace the non-Markovian environ-

ment with a stochastic classical drive given by ˆ

Hd(t) =

2B(t)σj. We set this drive such that its classical autocor-

relation function is equal to the desired quantum memory

kernel, ⟨B(t)B(t′)⟩=k(t−t′), where the expectation

value is taken over many realizations of the stochastic

drive. In order for the stochastic classical drive to emu-

late Eq. (1), the axis of the drive Hamiltonian must be

orthogonal to the axis of the background Lindblad oper-

ator, i.e. i̸=j, so that the classical drive Hamiltonian

anticommutes with the background Lindblad operator.

A derivation of how this stochastic classical drive can

give rise to the GMME is given in Section VI A.

We choose to focus on two example memory kernels:

exponentially decaying memory

k(t−t′) = B2

0e|t−t′|/τk,(2)

and modulated decaying memory

k(t−t′) = 1

2B2

0e|t−t′|/τkcos(2πν(t−t′)) .(3)

We identify these as noise Type I and Type II respec-

tively. For the decaying memory of noise Type I (i.e.

decaying autocorrelation), we use a telegraph signal that

switches between ±B0in a Poisson process with mean

switching time τk[21]. For the modulated decaying

memory of noise Type II (i.e. modulated decaying au-

tocorrelation), we realize it with two methods. The

ﬁrst is to take random telegraph noise and multiply it

with cos (2πνt +ϕ), where ϕis a random phase between

[0,2π). The second method utilizes the Wiener-Khinchin

theorem, which states that a signal’s autocorrelation is

the Fourier transform of its power spectrum [26]; more

details are included in Section VI B.

B. Experimental Protocol

Our goal is to realize the noisy drive Hamiltonians

described above by subjecting our qubit to noisy con-

trol tones. We use two noise injection protocols, labeled

“XY” and “XZ” after the qubit axes that the noise is

injected along (Xbeing the non-Markovian component).

The XZ protocol is described in detail in Section III D;

here we describe the XY protocol. We ﬁrst precisely mea-

sure the qubit transition frequency ωq=ω01 using stan-

dard Ramsey interferometry with no added noise. We

perform all qubit drives at this frequency, so that in the

rotating frame of the drive the qubit Hamiltonian is 0 (as

required by Eq. 1) and the drive causes rotations about

an axis in the XY plane, with the drive phase determin-

ing the axis.

We then proceed to inject noise. The pulse sequence

is depicted in Figure 1. First we prepare the |1⟩state by

applying a πpulse. Noise along both Xand Yaxes (σx

and σyterms in the drive Hamiltonian) can dephase this

state, but as we see in our protocol, they can be engi-

neered to counteract each other. We take a white noise

signal sampled at 1.2 GS/s (previously generated in soft-

ware) and feed it into the Q port of an IQ mixer, with a

local oscillator (LO) at ωq. The output signal is a tone

at ωq, 90◦phase shifted from the LO, with its amplitude

modulated by the white noise signal. This is eﬀectively a

noisy stochastic σydrive, which causes rapid dephasing

of the |1⟩state. The goal of this is to generate a purely

Markovian environment for the qubit, emulating the ﬁrst

term in Eq. (1). The result is a monotonic exponential

decay in ﬁdelity with respect to that state, characterized

by a time constant τ0, which serves as a benchmark for

coherence preservation later. To emulate the second term

in Eq. (1), a stochastic signal with non-zero memory,

which we refer to as generalized Markovian (GM) noise,

is fed into the same mixer’s I port. This ensures a phase

diﬀerence of π/2 between the two drives, and so this drive

is eﬀectively a σxterm in the rotating frame. The eﬀec-

tive drive Hamiltonian is ˆ

Hd(t) = 1

2ΩM(t)σy+1

2ΩN(t)σx,

where ΩM,N (t) are the Markovian (white) noise and GM

noise signals, respectively. After an evolution time t, the

qubit is measured in the σzbasis, i.e. with no additional

pulses. The evolution time tis swept and each measure-

ment is repeated to build up statistics and take an expec-

tation value ⟨σz⟩at each time point. This entire sequence

is then repeated Ntimes, each time with a new instance

of white and GM noise. The resulting Ncurves are aver-

aged over the diﬀerent noise realizations and ﬁnally com-

pared to the result of the master equation solution. We

also generate simulated qubit ﬁdelity curves under the

inﬂuence of white and GM noise by numerically solving

the stochastic Schr¨odinger equation (SSE) and averaging

over many noise realizations (i.e. over many qubit trajec-

tories). These simulations treat the transmon as a true

qubit; we conﬁrmed with simulations that the transmon’s

ﬁnite anharmonicity is not expected to have a signiﬁcant

eﬀect (see Section VI E for details). We measure and sim-

ulate the eﬀects of GM noise over a broad range of noise

parameter values, i.e. the amplitude B0, mean switch-

ing time τk, and modulating frequency ν(for noise Type

II only). Prior to each parameter point, the qubit and

readout mixers are automatically calibrated to minimize

leakage at ωLO , and the π-pulse is also re-calibrated to

minimize state-preparation-related errors.

III. RESULTS

A. Background Markovian Dynamics

Before measuring the eﬀect of GM noise, we ﬁrst in-

ject only white noise into the qubit in order to emu-

late a Markovian background. We measure state ﬁdelity

F(ρ(t)) = ⟨ψ0|ρ(t)|ψ0⟩, where |ψ0⟩is the initial pure

state, as a function of time and extract the coherence (ﬁ-

delity) decay time τ0. This will later serve as a reference

value for coherence enhancement. The amplitude of the

white noise is adjusted to yield τ0≈1−2µs, down from

its bare value of ∼100 µs. This ensures that the domi-

nant dephasing process is due to our injected Markovian

noise. An example of qubit state ﬁdelity under the inﬂu-

ence of such noise is shown in Figure 2. The results show

a monotonic, exponential decrease in ﬁdelity as function

of time. We compare experimental results with the an-

alytic solution of the master equation and with ﬁdelities

obtained by numerically solving the SSE, averaged over

simulated trajectories. The results show good agreement,

indicating that the qubit is experiencing a Markovian en-

vironment to a good approximation. This measurement

is repeated immediately before measuring the eﬀects of

GM noise with a given set of parameters, and the ﬁt τ0

Generalized Markovian Noise Channel (σ

)

Markovian Noise Channel (σ

)

2σ

2Β0

V=0

State

Preparation Readout

Noise Injection

Figure 1. (a) XY noise injection protocol for noise Type I.

The qubit is prepared to the excited state by applying a π

pulse on the X axis (lower curve, in red). After that, white

noise of variance σ2is injected along the Y axis via the Q

channel of the IQ mixer (upper curve, in blue), while GM

telegraph noise of amplitude B0is injected along the X axis

via the perpendicular I channel. The amplitude of the white

noise σis adjusted to reduce the coherence time τ0to ≈1µs

when B0= 0. After a variable time t, we read out the qubit

state in the Z basis. For noise Type II, the GM noise is multi-

plied by a cosine with random phase or is generated using the

Wiener-Kinchin method. (b) Noise instances for the 3 diﬀer-

ent types of noise we inject and the corresponding memory

kernels. The ﬁrst waveform is a white noise instance, used

to emulate a Markovian background. The second waveform is

an example of random telegraph noise with τk= 2µs, used for

noise Type I. The third waveform is generated by multiplying

a random telegraph signal (τk= 2µs) by cos[2πνt +ϕ], where

ν= 2 MHz, used for noise Type II. The total length of these

waveforms is 10µs.

is used as a reference value—this accounts for any slow

drifts in τ0that may result from, e.g., mixer miscalibra-

tion.

B. Noise Type I

Next, we measure the eﬀect of GM noise Type I (im-

plemented as random telegraph noise) added on top of

the Markovian background. The results are shown in

Figure 3. We judge the eﬃcacy of our emulation proto-

col based on two criteria: the qualitative behavior of the

ﬁdelity and the quantitative modiﬁed ﬁdelity decay time

of our qubit, which we call τ. In Figure 3(c), we plot

ﬁdelity versus time averaged over many instances of GM

and white noise for GM noise generated with strength

B0= 1500 kHz and switching rate 1/τk= 0 (i.e. in-

ﬁnite environmental memory). The experimental data

show excellent quantitative agreement with the SSE sim-

ulations and the analytic GMME solution. The ﬁdelity

develops oscillations that decay with an exponential en-

velope with decay time τ≈2τ0. Our models assume a

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MasterEquationEmulationandCoherencePreservationwithClassicalControlofaSuperconductingQubitEvangelosVlachos,1,2HaimengZhang,3,2VivekMaurya,1,2JeffreyMarshall,4,5TameemAlbash,6,7andE.M.Levenson-Falk1,2,∗1DepartmentofPhysics&Astronomy,DornsifeCollegeofLetters,Arts,&Sciences,UniversityofSouthernCaliforn...

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Master Equation Emulation and Coherence Preservation with Classical Control of a Superconducting Qubit Evangelos Vlachos1 2Haimeng Zhang3 2Vivek Maurya1 2Jeffrey

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