Self-consistent Noise Characterization of Quantum Devices Won Kyu Calvin Sun1 2and Paola Cappellaro1 2 3 1Research Laboratory of Electronics Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA

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Self-consistent Noise Characterization of Quantum Devices

Won Kyu Calvin Sun1, 2 and Paola Cappellaro1, 2, 3, ∗

1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

2Department of Nuclear Science and Engineering,

Massachusetts Institute of Technology, Cambridge, MA 02139, USA

3Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

(Dated: October 19, 2022)

Characterizing and understanding the environment aﬀecting quantum systems is critical to elu-

cidate its physical properties and engineer better quantum devices. We develop an approach to

reduce the quantum environment causing single-qubit dephasing to a simple yet predictive noise

model. Our approach, inspired by quantum noise spectroscopy, is to deﬁne a ‘self-consistent’ classi-

cal noise spectrum, that is, compatible with all observed decoherence under various qubit dynamics.

We demonstrate the power and limits of our approach by characterizing, with nanoscale spatial

resolution, the noise experienced by two electronic spins in diamond that, despite their proximity,

surprisingly reveal the presence of a complex quantum spin environment, both classically-reducible

and not. Our results overcome the limitations of existing noise spectroscopy methods, and highlight

the importance of ﬁnding predictive models to accurately characterize the underlying environment.

Extending our work to multiqubit systems would enable spatially-resolved quantum sensing of com-

plex environments and quantum device characterization, notably to identify correlated noise between

qubits, which is crucial for practical realization of quantum error correction.

I. INTRODUCTION

The performance of quantum devices is often limited

by the eﬀects of their environment, even if the environ-

ment could be tamed or even turned into a resource if

it could be properly characterized [1–8]. Unfortunately,

a full characterization of the environment is usually not

possible and one has to rely on a simpliﬁed model of

the noise sources. For simpler quantum systems such

as qubits and qutrits, it is in principle always possible

to reduce a complex quantum environment to a classi-

cal noise (spectrum) model, at least for a ﬁxed dynamics

of the total system [9–11]. However, this noise model

is not guaranteed to be predictive when the system (or

bath) dynamics is changed by control, as is the case for

quantum devices. Obtaining a classical noise spectrum

that can describe the system dynamics under a broad set

of controls and predict its performance would be highly

desirable, not only to enable practical characterization

of unknown complex many-body environments (e.g., for

applications in quantum sensing or quantum device char-

acterization), but also to engineer more robust quantum

devices and control sequences tailored to the noise.

In this paper, we demonstrate an approach to build a

practical yet predictive noise model of qubit decoherence.

Our approach is to form a ‘self-consistent’ classical noise

model — that is, consistent with all observed decoherence

under various qubit dynamics — by reconciling comple-

mentary approaches to noise spectroscopy. Crucially, by

reconciling limitations of existing methods, we demon-

strate that it succeeds even when the existing methods

fail to yield the correct noise model, and is further able

∗pcappell@mit.edu

Predicted

Qubit Dynamics =Measured

Qubit Dynamics

-th Qubit Dynamics

, … ,

{|˜

fk=1(ω)|2|˜

fk=N(ω)|2}

Decoherence

{χk=1(T),..,χk=N(T)}

S(ω)

Classical Noise ModelQuantum Environment

NRI

FIG. 1. Reducing a quantum environment to a self-

consistent classical noise model. To model a quantum

environment, we attempt to develop a classical noise model

S(ω) that is consistent with the set of all observed decoher-

ence under various controlled dynamics. When such a ‘self-

consistent’ noise model is possible, as demonstrated in this

paper experimentally for an NV electronic spin in diamond

but not a nearby interacting electronic spin X several nanome-

ters away, we further verify that the self-consistent model has

predictive power even under new dynamics, conﬁrming that

it accurately models the underlying quantum bath.

to predict the system dynamics under additional con-

trol sequences. If such a self-consistent noise model is

possible, this indicates that the underlying (quantum)

bath can be eﬀectively reduced to a classical Gaussian

noise process, enabling practical characterization of the

bath with predictive power. We demonstrate this ex-

perimentally, by building a self-consistent noise model

of the electronic spin of a nitrogen-vacancy (NV) cen-

ter in diamond and subsequently verify that it is pre-

dictive even under new qubit dynamics. On the other

hand, if a self-consistent model is not possible, this in-

dicates that the underlying bath is suﬃciently complex,

either of quantum or of non-Gaussian nature. We verify

this experimentally with another electronic spin near the

NV — and indeed with further investigation verify the

quantum nature of its local environment. Finally, hav-

arXiv:2210.09370v1 [quant-ph] 17 Oct 2022

ing characterized the bath of two nearby electronic spins

in diamond, we are able to probe, with nanoscale spatial

resolution, the dominant source of noise common to both

qubits arising from the quasistatic many-body electronic

spin bath. The noise model reveals the local spin density

and timescale of spin bath dynamics with nanoscale vari-

ations, information which is inaccessible by conventional

nuclear magnetic resonance (NMR) or ensemble-sensor

techniques.

II. QUANTUM NOISE SPECTROSCOPY

Several protocols for noise spectroscopy have been de-

veloped thus far, ranging from simple sequences [12–

14] to more complex continuous [15–17] and pulsed [18–

21] control. They have successfully elucidated noise

sources (from local ﬂuctuators [18, 22–25] to spin en-

vironments [12–14, 19, 20, 26]), and their accuracy to

reproduce a given classical noise has been evaluated [27].

However, much less attention has been paid to analyze

their predictive power especially when the reconstructed

noise spectrum is only an approximation to the real noise,

i.e., whether because it arises from a quantum system [28]

or a complex classical source [29–31]—or more simply due

to experimental limitations. Here, to achieve a predictive

noise model, we propose to build a self-consistent noise

spectrum by combining complementary approaches.

The simplest approach, which we call R-E-noise spec-

troscopy, utilizes only decoherence under the free evolu-

tion [Ramsey, (R)] and spin echo (E) experiments. The

knowledge of their decay functionals and decay times

T∗

2(R) and T2(E) may be suﬃcient to fully character-

ize a noise model S(ω|~p) with unknown model param-

eters ~p [32]. While minimal in experimental cost, this

method requires a noise model that is already known

and suﬃciently simple to uniquely identify ~p [12–14].

Furthermore, it can only investigate low-frequency noise

(ω < T −1

2).

A more general approach based on dynamical-

decoupling sequences with equidistant πpulses [Carr-

Purcell-Meiboom-Gill (CPMG) pulse sequences] can in

principle reconstruct the full noise spectrum. Under

the ﬁlter-function formalism, each CPMG experiment of

inter-pulse length 2τmforms a ﬁlter |˜

fT(ω)|2that approx-

imates a delta function δ(ω−ωm), ωm= (2π)(4τm)−1.

This allows direct measurement of S(ωm) from the

simple-exponential decay χm(T) under CPMG pulse se-

quences, where

χm(T) = 1

2ZS(ω)|˜

fT(ω)|2dω

2π≈4

π2S(ωm)T. (1)

While this method can characterize arbitrary, unknown

noise spectra with high-resolution, it comes at increased

experimental cost, as one CPMG experiment is needed

per frequency. Furthermore, the bandwidth, while much

broader, is still bounded by the coherence time T2and

Rabi frequency Ω0,T−1

2< ωmΩ0[20]. In particular,

low frequencies are harder to reach in the presence of

strong noise.

III. SELF-CONSISTENT NOISE

CHARACTERIZATION

Combining these techniques, we demonstrate how to

obtain a self-consistent classical model. We start with a

minimal noise model, consistent with initial experimental

data, and incrementally reﬁne it as necessary to be con-

sistent with additional experiments. While other strate-

gies are possible, this minimizes the experimental cost.

We ﬁrst demonstrate the protocol in the concrete case of

an NV center in diamond (Fig. 1).

A. NV electronic spin qubit

The ﬁrst step is to measure the NV Ramsey dynamics.

We used the ms={0,−1}states of the NV electronic

spin (electronic spin S= 1) in an external static magnetic

ﬁeld of strength B0≈350 G aligned approximately along

the NV axis. The control was achieved with a single-tone,

resonant microwave of ΩNV

0≈6.9 MHz amplitude to drive

both 15NV hyperﬁne transitions (Azz ≈3.2 MHz).

Observing a Gaussian decay under Ramsey control

[Fig. 2(b)], we assume as our minimal model an Ornstein-

Uhlenbeck (OU) process

S(ω|b, τc) = b2(2τc)

1+(ωτc)2,(2)

characterized by two parameters (b, τc). Indeed, a qua-

sistatic or “slow” OU noise, (bsτs)1, predicts a Gaus-

sian decay, χR(T) = (bsT)2/2≡(T /T ∗

2)2. More gen-

erally, the slow-OU noise has successfully modeled noise

from a slowly ﬂuctuating spin bath [13, 14, 26] and is

expected [6] to be the dominant noise in our system [33].

Then, ﬁtting for T∗

2we identify one of two unknown pa-

rameters, bs= 0.56(2) MHz.

Given a working model S0=Ssconsistent with Ram-

sey dynamics, we can ask whether it is already pre-

dictive of echo dynamics. Unfortunately, we ﬁnd that

it is not, as while S0predicts a stretched-exponential

χE(T)≈(b2

sT3)/(12τs)≡(T/T2)3, the NV echo is domi-

nantly simple exponential [Fig. 2(c)]. Note that similarly

we could have started with the knowledge of NV echo

decay to ﬁrst search for a minimal (single-termed) noise

model consistent with echo dynamics and test whether

it is predictive of Ramsey dynamics. In such a case, we

would arrive at either a fast-OU noise Sf(τfT) or

white-noise Sw, which both yield an exponential decay.

However, neither are consistent with NV Ramsey dynam-

ics.

This suggests that the environment around the NV is

suﬃciently complex so as not to be reduced to a sin-

gle independent noise process. We thus introduce min-

imal complexity to the working model by considering

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Self-consistentNoiseCharacterizationofQuantumDevicesWonKyuCalvinSun1,2andPaolaCappellaro1,2,3,1ResearchLaboratoryofElectronics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139,USA2DepartmentofNuclearScienceandEngineering,MassachusettsInstituteofTechnology,Cambridge,MA02139,USA3Depart...

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Self-consistent Noise Characterization of Quantum Devices Won Kyu Calvin Sun1 2and Paola Cappellaro1 2 3 1Research Laboratory of Electronics Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA

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