Real-time adaptive estimation of decoherence timescales for a single qubit Muhammad Junaid Arshad1Christiaan Bekker1Ben Haylock1Krzysztof Skrzypczak1Daniel White1Benjamin Griffiths2Joe Gore3Gavin W. Morley3Patrick Salter4Jason Smith2

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Real-time adaptive estimation of decoherence timescales for a single qubit

Muhammad Junaid Arshad,1Christiaan Bekker,1Ben Haylock,1Krzysztof Skrzypczak,1Daniel

White,1Benjamin Griﬃths,2Joe Gore,3Gavin W. Morley,3Patrick Salter,4Jason Smith,2

Inbar Zohar,5Amit Finkler,5Yoann Altmann,6Erik M. Gauger,1and Cristian Bonato1, ∗

1SUPA, Institute of Photonics and Quantum Sciences,

School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK

2Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK

3Department of Physics, University of Warwick, Coventry CV4 7AL, UK

4Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK

5Department of Chemical and Biological Physics Weizmann Institute of Science, Rehovot 7610001, Israel

6Institute of Signals, Sensors and Systems, School of Engineering and Physical Sciences,

Heriot-Watt University, Edinburgh EH14 4AS, UK

Characterising the time over which quantum coherence survives is critical for any implemen-

tation of quantum bits, memories and sensors. The usual method for determining a quantum

system’s decoherence rate involves a suite of experiments probing the entire expected range of this

parameter, and extracting the resulting estimation in post-processing. Here we present an adaptive

multi-parameter Bayesian approach, based on a simple analytical update rule, to estimate the key

decoherence timescales (T1,T∗

2and T2) and the corresponding decay exponent of a quantum system

in real time, using information gained in preceding experiments. This approach reduces the time

required to reach a given uncertainty by a factor up to an order of magnitude, depending on the

speciﬁc experiment, compared to the standard protocol of curve ﬁtting. A further speed-up of a

factor ∼2 can be realised by performing our optimisation with respect to sensitivity as opposed to

variance.

I. INTRODUCTION

Decoherence, resulting from the interaction of a quan-

tum system with its environment, is a key performance

indicator for qubits in quantum technologies [1] includ-

ing quantum communication, computation and sens-

ing. Decoherence timescales determine the storage time

for quantum memories and quantum repeaters, a cru-

cial metric for quantum communication networks [2–5].

Rapid benchmarking of decoherence timescales in plat-

forms such as superconducting qubits [6, 7] or silicon

spin qubits [8, 9], is a critical validation and quality as-

surance step for the development of large-scale quantum

computing architectures, and has the potential to im-

prove error correction protocols eﬃciently close to fault-

tolerance thresholds. In quantum sensing, the role of de-

coherence is two-fold. On one side, decoherence sets the

ultimate performance limit of the sensors [10]. On the

other hand, decoherence itself can be the quantity mea-

sured by a quantum sensor, as it provides information

about the environment. An example of this is relaxom-

etry, where the rate at which a polarised quantum sen-

sor reaches the thermal equilibrium conﬁguration gives

information about diﬀerent physical processes in the en-

vironment [11–14].

Decoherence rates can be measured by preparing the

system into a known quantum state and probing it at

varying time delays to determine the probability of decay

from its initial state. The standard protocol for decoher-

ence estimation involves a series of measurements with

∗c.bonato@hw.ac.uk

time delays set over a pre-determined range, reﬂecting

the expected value of the decoherence rate, and ﬁtting of

the result to a decay function. As the range of time de-

lays is determined in advance, some of the measurements

will provide little information on the decoherence of the

system, since the time delay is either much shorter than

the true decoherence rate, resulting in no decay, or much

longer, resulting in complete decay.

Here we introduce an real-time adaptive protocol to

measure decoherence timescales T1,T∗

2and T2for a sin-

gle qubit [15], respectively corresponding to relaxation,

dephasing, and echo decay time [1], together with the

coherence decay exponent β. While the proposed algo-

rithms are very general and can be applied to any quan-

tum architecture, our experiments are implemented on

a single spin qubit associated with a nitrogen-vacancy

(NV) centre in diamond.

Adaptive techniques have been shown to be central to

progress across a broad range of quantum technologies

[16]. Early work in this ﬁeld involved the implementa-

tion of adaptive quantum phase estimation algorithms on

photonic systems [17], later extended to frequency esti-

mation with applications to (static) DC magnetometry

with single electron spins [18–20]. Alternative adaptive

protocols for the estimation of static magnetic ﬁelds are

based on sequential Bayesian experiment design [21, 22]

and ad-hoc heuristics [23], later applied to the charac-

terisation of a single nuclear spin [24]. Real-time adap-

tation of experimental settings has also been shown to

be advantageous when measuring spin relaxation [25] or

tracking the magnetic resonance of a single electron spin

in real-time [26–29]. Furthermore, adaptive techniques

have been investigated to enhance photonic quantum sen-

arXiv:2210.06103v4 [quant-ph] 24 Jan 2024

sors [30–32], as a control tool for quantum state prepa-

ration [33] and to extend quantum coherence of a qubit

by manipulating the environment [34–36].

Despite these pioneering experiments, several impor-

tant methodological questions still remain open. A prior-

ity concern is that adaptive protocols introduce an over-

head, given by the time required to compute settings on

the ﬂy for the next iteration. It is crucial to minimise this

computation time, since it can slow the protocol down to

the point that the overhead can reverse the gain in mea-

surement speed compared to a simple parameter sweep.

This has not been considered in many cases, in particu-

lar where algorithms were investigated through computer

simulations [21, 22, 27, 37] or as oﬀ-line processing of

pre-existing experimental data [23]. While the optimi-

sation of complex utility functions can possibly deliver

the best theoretical results, this could be practically less

advantageous than near-optimal approaches with very

fast update rules in minimising total measurement du-

ration. A second issue is related to the fact that, for

multi-parameter Hamiltonian estimation, standard ap-

proaches such as the maximisation of Fisher information

can fail, as the Fisher information matrix becomes sin-

gular when controlling the evolution time [38]. This has

stimulated researchers to ﬁnd ad-hoc heuristics, for ex-

ample, the particle guess heuristic [23, 24, 38] for the esti-

mation of Hamiltonian terms; these heuristics, however,

do not necessarily work beyond Hamiltonian estimation.

A third question is related to what quantity should be

optimised. Previous work has targeted the minimisa-

tion of the variance of the probability distribution for the

quantity of interest [24, 39]. While this is clear when all

measurements feature the same duration, the answer is

less straightforward when adapting the probing time. If

two measurements with diﬀerent probing times result in

a similar variance, the protocol should prefer the shorter

one, minimising the overall sensing time.

Here we address these open questions, presenting theo-

retical and experimental data about the adaptive estima-

tion of decoherence for a single qubit, using NV centres

as a case study. Compared to other recent investigations

of adaptive protocols [23–25], our experiments utilize a

very simple analytical update rule based on the concept

of Fisher information and the Cram´er-Rao bound. By

exploiting state-of-the-art fast electronics, we experimen-

tally perform the real-time processing in than 50µs, an

order of magnitude shorter than previous real-time ex-

periments [24], negligible compared to the duration of

each measurement. Such a short timescale makes our

approach useful for qubits where fast single-shot read-

out is available such as trapped ions [40], superconduct-

ing qubits [41] and several types of spin qubits [42–45],

and could be further shortened in future work by imple-

menting the protocols on ﬁeld-programmable gate array

(FPGA) hardware.

In the case of multi-parameter estimation, previous

work on Hamiltonian estimation had pointed out that the

Cram´er-Rao bound cannot be used in the optimisation

as the Fisher information matrix is singular and cannot

be inverted [38]. Here we address this issue by utilising

multiple probing times, showing that the Fisher informa-

tion matrix can be inverted and that the corresponding

adaptive scheme provides better performance than non-

adaptive approaches. Finally, we discuss what quantity

needs to be targeted to achieve the best sensor perfor-

mance, experimentally demonstrating the superiority of

optimizing sensitivity, deﬁned as variance multiplied by

time, over optimizing variance. As a ﬁgure of merit, sen-

sitivity encourages faster measurements.

Our work tackles these general questions using the

characterisation of decoherence as a test case. While

adaptive approaches have been investigated in the case

of phase and frequency estimation [17–24], also in rela-

tion to Hamiltonian learning [23], the case of decoherence

is much less explored, with only one work targeting the

estimation of the relaxation timescale T1[25]. Here we

provide the ﬁrst complete characterisation of the three

decoherence timescales typically used in experiments (T1,

T∗

2and T2), together with the decoherence decay expo-

nent β.

II. THEORY

Decoherence and relaxation are processes induced by

the interaction of a qubit with its environment, leading

to random transitions between states or random phase

accumulation during the evolution of the qubit. These

processes are typically estimated by preparing a quan-

tum state and tracking the probability of still measuring

the initial state over time, which can be captured by a

functional form [10]

p(t)∝1

21−e−χ(t).(1)

Although the noise processes induced by interaction

with the environment can be complex, χ(t) can often be

approximated by a simple power law:

χ(t)∝t

Tχβ

,(2)

where Tχand βdepend on the speciﬁc noise process [1].

For white noise, the decay is exponential with β= 1.

For a generic 1/f qdecay, relevant for example for super-

conducting qubits, with a noise spectral density as ∝ωq,

χ(t) scales as χ(t)∝(t/Tχ)1+q[46].

In the case of a single electronic spin dipolarly coupled

to a diluted bath of nuclear spins, the decay exponents

have been thoroughly investigated, with analytical solu-

tions available for diﬀerent parameter regimes [47]. If the

intra-bath coupling can be neglected, the free induction

decay of a single spin is approximately Gaussian (β= 2)

[48, 49]. The Hahn echo decay exponent T2can vary, typ-

ically between β∼1.5−4 depending on the speciﬁc bath

parameters and applied static magnetic ﬁeld [47, 50].

(a)

(b)

(c)

𝜋

2 𝜋

532 nm

AWG

T𝜒_est

(

real-time Bayesian inference

time time time

Microcontroller

APD

pr(T𝜒 ) ∼ p0(T𝜒 )p(r|T𝜒)

r = (r1,r2..., rn)

Bayesian update

outcome r

feedback

r clicks in

R trails

real-time Bayesian inference for averaged readout

example of experimental adaptive estimation

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

time (μs)

epoch #

0.25

0.15

0.20

0.10

0.05

0.30

time (μs)

20 30 4010

p0( )

T𝜒Experiment( )

𝜏

opt ≈ ξ· 𝜒

𝜏

pu( )

T𝜒

p( )

T𝜒

p( )

T𝜒

p( )

T𝜒

Figure 1. Real-time adaptive feedback. (a) Schematic of the real-time adaptive protocol demonstrated in this work, using

the electronic spin associated with a nitrogen-vacancy (NV) centre in diamond. An Arbitrary Waveform Generator (AWG) is

used to generate pulses for manipulation of the spin qubit. The spin state is then optically measured and the detected photon

count rate is used by the microcontroller to estimate the value of the decoherence timescale Tχ(and the decay exponent β)

using Bayesian inference. The microcontroller also computes the optimal probing time τfor the next measurement, passing

the value to the AWG, which builds the next estimation sequence accordingly. (b) In our experiment, a single interrogation of

the qubit does not provide suﬃcient information to discriminate qubit state. Hence Rmeasurements are performed, and the

resulting number rof detected photons are used to update p(Tχ) through Bayes’ rule and to compute τfor the next experiment.

2) shown for each measurement epoch (here the decay

exponent is known a-priori as β= 2). The probability p(T∗

2), initially uniform, converges towards a narrow peak as more

measurement outcomes are accumulated. The experimental adaptively-optimised values for τare shown on the bottom plot.

In Sec. II A, we assume the decay exponent βto be

known, and we only focus on estimating the decay time

Tχ. This is a practically-relevant situation in cases where

the nature of the bath is well-understood and the decay

exponent βis known, at least approximately, a priori.

We then extend our analysis to the simultaneous estima-

tion of Tχand βin Sec. II B.

Fig. 1(a) sketches the operation of the real-time adap-

tive sensing system developed in our study. We have

utilised the electronic spin associated with a nitrogen-

vacancy (NV) center in diamond as the qubit, which is

initialised and readout by optical pulses. The qubit state

is manipulated by microwave (MW) pulses, created in

real-time by an Arbitrary Waveform Generator (AWG)

based on an external digital input. After the application

of a pulse sequence, the qubit is optically readout, with

the spin state information enconded in the number photo-

luminescence photon counts during optical illumination.

The core of our adaptive system is a real-time micro-

controller, which uses the detected photon count rate to

estimate the values of the decoherence timescale (Tχ) and

the decay exponent (β) via Bayesian inference. As shown

in the inset, the probability distribution starts out as uni-

formly ﬂat, but begins to converge around the true value

after a few iterations. Based on the estimated value in

the current iteration, the microcontroller computes the

optimal probing time (τ) for the subsequent measure-

ment and communicates this value to the AWG, which

then constructs the next estimation sequence accordingly.

This cycle repeats for several iterations until a desired

level of error in the estimation of the target quantity is

reached. Fig. 1(b) shows the ﬂow of the experimental

estimation sequence. For our experiments, a single mea-

surement of the qubit lacks the information required for

discriminating its state eﬀectively. Therefore, we conduct

R measurements, to obtain r detected photons, enough to

discriminate the spin state. Such counts are then utilised

to update the probability distribution p(Tχ) using Bayes’

rule. After the Bayesian update, updated probability

distribution is used to compute the optimal settings and

provide feedback for the subsequent measurements. Fig.

1(c) shows an example of experimental estimation of T∗

performed by an adaptive Ramsey experiment, plotted as

the evolution of p(T∗

2) for increasing estimation epochs.

In the beginning, p(T∗

2) is a uniform distribution in the

range 0-8 µs, which then converges to a singly-peaked

distribution after more and more measurement outcomes

are processed. In the case of an NV centre in a high-

purity diamond, the decay is expected to be Gaussian

(β= 2) [49]. As described later in Sec. II A, the optimal

adaptive rule for this case is to choose the probing time

as τopt ∼0.89 ·ˆ

T∗

2(see Eq. 20, where ˆ

T∗

2is the current

estimate of T∗

2computed from the probability distribu-

tion p(T∗

2). The chosen values for τare shown on the

bottom plot, illustrating how they converge very fast to

the optimal value τopt ∼0.89 ·(T∗

2)true ∼2.23 µs.

A. Adaptive Bayesian estimation

We utilize Bayesian inference, exploiting Bayes’ the-

orem to update knowledge about the decoherence time

Tχand decay exponent βin the light of a set of new

measurement outcomes denoted by ⃗m ={m1,m2, . . . }.

Thanks to its ﬂexibility in accounting for experimental

imperfections and for integrating real-time adaptation of

the experimental setting while remaining easy to inter-

pret mathematically, the Bayesian framework [16, 51] has

been widely applied in quantum technology, from sens-

ing [19, 23–25], to the tuning of quantum circuits [52, 53],

and model learning [54]. In this section, we will restrict

the discussion to the characterisation of the decoherence

time Tχ; the extension to a multi-parameter case, with

the simultaneous estimation of Tχand β, will be pre-

sented in Sec. II B.

For each binary measurement outcome mn(mn=

0,1), the probability distribution of Tχ, which represents

our knowledge about Tχ, is updated as

P(Tχ|m1:n)∝P(mn|Tχ)P(Tχ|m1:(n−1)),(3)

where m1:n={m1, . . . , mn}. Here, P(Tχ|m1:(n−1)) is the

posterior probability after (n−1)-th update and proceeds

to serve as prior distribution at the n-th iteration, and

P(mn|Tχ) is the likelihood function

P(m|Tχ) = 1 + eimπe−(τ /Tχ)β

2.(4)

Note that this likelihood depends on τ(which we will

adjust later) but this dependency is omitted in P(m|Tχ)

to simplify the notation. Our approach to adaptive es-

timation is to derive a simple expression for the optimal

parameter settings, that can be computed in real-time by

an analytical formula without adding much extra compu-

tation time to the sensing process.

A conventional approach to updating τadaptively

would be to use the information gain as criterion [21,

52, 55]. However, this involves integrals (with respect

to Tχ) requiring numerical evaluation and an associated

signiﬁcant computational overhead. Here, we instead em-

ploy an approximation of the Bayesian information ma-

trix (BIM) (a 1 ×1 matrix, in the case of a single param-

eter) [56] which links to the classical Fisher information

[57]. While computing the BIM also requires the com-

putation of an integral (more precisely an expectation)

with respect to Tχ, this can now be easily approximated

as explained in Appendix A.

The Cram´er-Rao lower bound (CRLB) of Tχrepresents

the minimum reachable variance for any (unbiased) esti-

mator of Tχ, and is inversely proportional to the Fisher

information F. Thus, maximizing Fwith respect to the

control parameter τis expected to improve our estimate

of Tχ.

As shown in Appendix A, we examine a Bayesian form

of the CRLB, computing the corresponding Fisher infor-

mation FBcan be computed as:

FB(τ)≈β2(τ/ ˆ

Tχ)2β

χhe(2τ/ ˆ

Tχ)β

−1i(5)

where ˆ

Tχis a point estimate of Tχbefore each measure-

ment.

While we are unable to maximize ˆ

FB(τ) analytically,

approximate solutions exist. We found that the heuristic

τopt ≈ξ·ˆ

Tχ(6)

leads to satisfactory results, where ξis a parameter that

depends on β. Some numerically-computed values for ξ

are listed in Table I, for some common values of β.

The Fisher information Fas a function of the ground

truth value for Tχ=T∗

2and the probing time τ, from

Eq. (17) is plotted in Fig. 2(a). F(τ, T ∗

2) is normalised

by its maximum with respect to τ, for each value of T∗

The plot shows clearly that the maximum of F(τ) has a

linear dependence on T∗

2, following Eq. (17) (shown as

the red dashed line).

B. Multi-parameter estimation

In many practical situations, it is important to learn

both the decoherence timescale, Tχ, and the noise expo-

nent, β, as the latter provides useful information about

the nature of the qubit environment.

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Real-timeadaptiveestimationofdecoherencetimescalesforasinglequbitMuhammadJunaidArshad,1ChristiaanBekker,1BenHaylock,1KrzysztofSkrzypczak,1DanielWhite,1BenjaminGriffiths,2JoeGore,3GavinW.Morley,3PatrickSalter,4JasonSmith,2InbarZohar,5AmitFinkler,5YoannAltmann,6ErikM.Gauger,1andCristianBonato1,∗1SUPA,...

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Real-time adaptive estimation of decoherence timescales for a single qubit Muhammad Junaid Arshad1Christiaan Bekker1Ben Haylock1Krzysztof Skrzypczak1Daniel White1Benjamin Griffiths2Joe Gore3Gavin W. Morley3Patrick Salter4Jason Smith2

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