Real-time frequency estimation of a qubit without single-shot-readout Inbar Zohar1Ben Haylock2 Yoav Romach3 Muhammad Junaid Arshad2 Nir Halay3 Niv Drucker3 Rainer St ohr4 Andrej Denisenko4 Yonatan Cohen3 Cristian Bonato2 and Amit Finkler1

2025-04-24 0 0 789.55KB 10 页 10玖币

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Real-time frequency estimation of a qubit without single-shot-readout

Inbar Zohar1,∗Ben Haylock2, Yoav Romach3, Muhammad Junaid Arshad2, Nir Halay3, Niv

Drucker3, Rainer St¨ohr4, Andrej Denisenko4, Yonatan Cohen3, Cristian Bonato2, and Amit Finkler1

1Department of Chemical and Biological Physics,

Weizmann Institute of Science, Rehovot 7610001, Israel

2Institute of Photonics and Quantum Sciences, SUPA,

Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom

3Quantum Machines, Tel-Aviv 6744332, Israel and

4Third Institute of Physics, University of Stuttgart, Stuttgart 70569, Germany

(Dated: May 12, 2023)

Quantum sensors can potentially achieve the Heisenberg limit of sensitivity over a large dynamic

range using quantum algorithms. The adaptive phase estimation algorithm (PEA) is one example

that was proven to achieve such high sensitivities with single-shot readout (SSR) sensors. However,

using the adaptive PEA on a non-SSR sensor is not trivial due to the low contrast nature of the

measurement. The standard approach to account for the averaged nature of the measurement in

this PEA algorithm is to use a method based on ‘majority voting’. Although it is easy to implement,

this method is more prone to mistakes due to noise in the measurement. To reduce these mistakes,

a binomial distribution technique from a batch selection was recently shown theoretically to be

superior, as all ranges of outcomes from an averaged measurement are considered. Here we apply,

for the ﬁrst time, real-time non-adaptive PEA on a non-SSR sensor with the binomial distribution

approach. We compare the mean square error of the binomial distribution method to the majority-

voting approach using the nitrogen-vacancy center in diamond at ambient conditions as a non-SSR

sensor. Our results suggest that the binomial distribution approach achieves better accuracy with

the same sensing times. To further shorten the sensing time, we propose an adaptive algorithm

that controls the readout phase and, therefore, the measurement basis set. We show by numerical

simulation that adding the adaptive protocol can further improve the accuracy in a future real-time

experiment.

I. INTRODUCTION

Quantum sensing is a promising technology with many

possible applications in ﬁelds such as renewable en-

ergy [1], condensed matter physics [2–5], biology [6–8],

and chemistry [9, 10]. Diﬀerent quantum systems are

studied as quantum sensors [11], and depending on the

systems’ interactions with the environment it can be used

to sense diﬀerent physical quantities such as magnetic

ﬁelds [5], electric ﬁelds [8], temperature [12], strain [13],

or pressure [14]. One of the advantages of these sensors

is the possibility of achieving high sensitivity while over-

coming the standard quantum limit (SQL) and reaching

the Heisenberg limit (HL) [15].

Recent studies have pushed the sensitivity to the HL

using entanglement [16], or quantum algorithms [17].

One algorithm, widely studied, is the phase estimation

algorithm (PEA), suggested by Kitaev [18]. This algo-

rithm aims to estimate a phase (φ) that a quantum sensor

is accumulating due to some interaction with frequency

fwith an unknown parameter in the environment. The

sensor accumulates the phase at K+ 1 diﬀerent sensing

times (τ) that grow exponentially, τ= 2kτ0, where kis

an index going from 0 to K. The shortest sensing time

∗inbar.aharon@weizmann.ac.il

τ0limits the dynamic range (DR) of the sensor to

DR = [−1

2τ0

] (1)

The longest sensing time is bounded from above by the

dephasing time, T∗

2, of the sensor 2Kτ0< T ∗

2[19].

The full algorithm is based on a quantum system with

multiple quantum bits that carry the process of estimat-

ing the phase simultaneously using the quantum Fourier

transform [17]. Such multi-qubit systems are still chal-

lenging and sometimes not available for every sensing

environment. Moreover, a single qubit sensor such as

a spin-1/2 oﬀers the ultimate spatial resolution, and any

additional gain from entangling it with additional spins

is canceled by the increase in sensor size. In these cases,

therefore, a system of a single qubit that is incorporated

with an adaptive PEA, and making use of quantum-

classical interfaces [20, 21], can be of beneﬁt. Here we

experimentally demonstrate, in real-time, a non-adaptive

PEA scheme in non-ideal but very realistic sensing con-

ditions, and show numerically the advantage of moving

this method to an adaptive one.

A. Adaptive phase estimation algorithm

The general scheme for applying adaptive PEA

(Fig. 1a) consists of a cyclic process of four steps. The

ﬁrst is a pulse sequence applied on the sensor depending

arXiv:2210.05542v2 [quant-ph] 11 May 2023

on the target frequency, f, in question and its interac-

tion with the sensor as expressed in the Hamiltonian,

H(f). This pulse sequence will use the same exponen-

tially growing sensing time as in the quantum PEA only

in sequential order, from the shortest to the longest, and

not simultaneously, similar to the Kitaev’s iterative PEA

[22]. After each pulse sequence with one sensing time,

the sensor is measured, and the outcome (u) can be one

of the two states of the sensor - zero or one. This out-

come is used in the second step to update the probability

function, Pm(u|f), to measure the sensor state, |0ior |1i,

given that there is interaction with the unknown param-

eter with frequency f. The nature of the sensor’s inter-

action with the target parameter in the pulse sequence is

encoded in the probability function.

The critical step of the algorithm is in step 3 , where

one applies a Bayesian update to estimate the unknown

parameter [23–25]

Pposterior(f|u)∝Pm(u|f)Pprior(f|u) (2)

where P(f|u) is the probability function of the mea-

surement outcome given the target parameter, subscript

posterior is the new probability after each Bayesian

update and prior is the old one from the last update.

P(u|f) is the probability function of the target parame-

ter given the outcome of the measurement is u, the sub-

script mdenotes a single outcome. Since the adaptive

uniform distribuon

Measurement

Calculate

Adapt variables

Bayesian update

(a)

Threshold

Single shot readout

number of posive results

Threshold

number of posive results

non single-shot readout

(b) (c)

FIG. 1. (a) Graphical illustration of the adaptive phase esti-

mation algorithm comprising four steps: (1) A pulse sequence

suitable for the estimation of the unknown parameter, given

the nature of the interaction between the sensor and the pa-

rameter. This pulse sequence will be applied with exponen-

tially growing sensing times. The state of the sensor is mea-

sured after every sensing time. (2) Calculating the proba-

bility function for the state of the sensor given the unknown

parameter. (3) Using Bayes’ Theorem to update the proba-

bility function for the parameter. (4) Calculating the optimal

variables for extracting maximal information from the next

iteration. After Mkiterations for each sensing time, the ﬁnal

distribution will be the estimate of the unknown parameter.

(b-c) Schematic illustration of the measurement outcome of a

single-shot (b) or averaged (c) sensor.

PEA applies the sensing scheme with diﬀerent sensing

times sequentially, each measurement holds less infor-

mation about the phase than the quantum PEA. The

penalty in the full scheme is that each sensing time is

measured multiple times by changing one of the sensing

variables, as is illustrated in step 4 . The number of it-

erations Mk=G+ (K−k)Ffor each sensing time grows

as the sensing time gets shorter, where Gand Fare op-

timized parameters, and kis the index of the sensing

time [26]. The adaptive character of the scheme is es-

tablished in step 4 . In this step, the optimal variables

for gaining maximal information are calculated based on

the last probability function and then transferred to the

pulse sequence of the next iteration.

Adaptive PEA has been studied extensively. Theoret-

ical works suggested controlling the sensing phase or the

sensing time [27] to enhance sensitivity. Others used nu-

merical simulations [28, 29], and several did experimental

studies with diﬀerent sensors [23, 30] to prove the feasi-

bility and beneﬁts of this protocol. All of these studies

were performed with a single-shot readout (SSR) sen-

sor, where the state of the sensor can be measured after

one measurement with high ﬁdelity (Fig. 1b). Neverthe-

less, in some cases, non-SSR sensors are the only possible

sensing approach, for instance, for imaging nanoscale bi-

ological samples with high special resolution and in am-

bient conditions. These sensors are characterized by the

high ratio of classical noise added in the measurement,

for example, low photon collection eﬃciency in optically

read-out systems, compared to the quantum projection

noise of the system [11]. This causes the histogram of

the measurement outcomes to mix ‘0’ and ‘1’. Therefore,

assigning the measurement outcome to one state of the

sensor with high ﬁdelity, i.e., in one shot, is impossible

(Fig. 1c).

For a non-SSR sensor, the pulse-sequence and the mea-

surement should be applied for many repetitions to assess

the sensor state, still with a non-negligible error. This sit-

uation requires adjusting the probability function P(u|f)

used in the Bayesian update to the averaged measure-

ment result. The most common and simple solution is

to use a threshold that is calculated based on the prob-

ability to measure a positive outcome from the sensor at

each of the states, which can be a collection of photons

for an optically measurable sensor (See Appendix V E).

In this method the measurement is repeated for Rtimes

and the number of positive outcomes ris assigned to a

state of the sensor, u, based on the calculated threshold;

we call this method “majority voting”. This approach

results in a binary outcome from a large batch of size R

repetitions of the measurement. This method’s beneﬁt

is the possibility of using the probability function and

Bayesian update as in the SSR sensor scheme [31]. How-

ever, it disregards most of the possible outcomes from

the Rrepetitions by using only a binary span of results.

Therefore, it is also more prone to noise, where a noisy

measurement can be assigned to the wrong binary option

[32].

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Real-timefrequencyestimationofaqubitwithoutsingle-shot-readoutInbarZohar1,BenHaylock2,YoavRomach3,MuhammadJunaidArshad2,NirHalay3,NivDrucker3,RainerStohr4,AndrejDenisenko4,YonatanCohen3,CristianBonato2,andAmitFinkler11DepartmentofChemicalandBiologicalPhysics,WeizmannInstituteofScience,Rehovot76100...

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Real-time frequency estimation of a qubit without single-shot-readout Inbar Zohar1Ben Haylock2 Yoav Romach3 Muhammad Junaid Arshad2 Nir Halay3 Niv Drucker3 Rainer St ohr4 Andrej Denisenko4 Yonatan Cohen3 Cristian Bonato2 and Amit Finkler1

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