Bayesian Quantum Multiphase Estimation Algorithm

Valentin Gebhart,1, 2 Augusto Smerzi,1 and Luca Pezze1

1QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, 50125 Firenze, Italy2Universita degli Studi di Napoli ”Federico II”, Via Cinthia 21, 80126 Napoli, Italy

Quantum phase estimation (QPE) is the key subroutine of several quantum computing algorithmsas well as a central ingredient in quantum computational chemistry and quantum simulation. WhileQPE strategies have focused on the estimation of a single phase, applications to the simultaneousestimation of several phases may bring substantial advantages; for instance, in the presence of spatialor temporal constraints. In this work, we study a Bayesian algorithm for the parallel (simultaneous)estimation of multiple arbitrary phases. The protocol gives access to correlations in the Bayesianmulti-phase distribution resulting in covariance matrix elements scaling as O(N−2

T ), with respect tothe total number of quantum resources NT . The parallel estimation allows to surpass the sensitivityof sequential single-phase estimation strategies for optimal linear combinations of phases. Further-more, the algorithm proves a certain noise resilience and can be implemented using single photonsand standard optical elements in currently accessible experiments.

I. INTRODUCTION

Quantum phase estimation (QPE) provides a link be-tween quantum computation, simulation, and metrology.In fact, several quantum technology tasks can be cast asthe estimation of an unknown eigenphase θ ∈ [0, 2π] ofa unitary matrix U , U |u〉 = eiθ|u〉, where |u〉 is the cor-responding eigenstate. The problem can be efficientlysolved using QPE algorithms [1–3]. QPE is the keysubroutine of quantum algorithms providing exponentialspeedup [4], including number factoring [5–7], compu-tation of molecular spectra [8–11], and quantum sam-pling [12]. It also finds applications in the alignment ofspatial reference frames [13], clock synchronization [14],frequency estimation [15, 16], and clock networks [17].In quantum metrology [18–20], ideas developed for QPEalgorithms have been applied to the estimation of an in-terferometric phase θ with Heisenberg-limited sensitivity,both locally [21] and in the full θ ∈ [0, 2π] domain [22–27]. Experimental QPE has been implemented with sin-gle photons [22, 23], including the simulation of Hamil-tonian spectra on an integrated photonic device [28, 29],and NV centers [30].

So far, QPE algorithms have mainly focused on theestimation of a single eigenphase of U . Besides tailoredpostprocessing techniques [31, 32], existing multiphaseestimation algorithms follow a sequential scheme wherean ancilla qubit encodes a single parameter at a time andis then measured. However, there is an urgent demand forparallel-multiphase-estimation techniques [33–42], wheredifferent phases are estimated simultaneously [43–45]. In-deed, several problems of interest involve the joint esti-mation of multiple parameters and/or their linear com-binations. Important examples include imaging [46, 47],sensing [48–50], and biological probing [51, 52].

In this paper, we extend Kitaev’s iterative QPE algo-rithm [1, 2, 22, 24, 53] to the parallel (simultaneous) es-timation of d ≥ 1 eigenphases θ = (θ1, ..., θd) of U , usinga finite number of resources NT . Here, NT is quanti-fied as the total number of applications of a controlled-U

gate [22, 24, 54]. In general, parallel multiphase estima-tion requires us to access phase correlations in the out-put state (which are optimally captured by a Bayesianapproach) while avoiding the proliferation of marginal-probability tails in the d-dimensional posterior phase dis-tribution. Our protocol achieves a sensitivity scalingO(N−1

T ) for the simultaneous estimation of arbitrary vec-tor parameters θ ∈ [0, 2π]d, as well as for arbitrary lin-ear combinations of the d phases. The algorithm inheritsimportant properties from Bayesian approaches to single-parameter estimation [21, 28, 30, 53, 55, 56], such as noiseresilience and the optimal use of measurement data. Theexploitation of correlations in the d-dimensional Bayesiandistribution allows us to overcome the sensitivity achiev-able with the sequential scheme for the estimation of cer-tain linear combinations of the phases, for fixed NT . Ow-ing to the primary relevance of QPE in quantum tech-nologies, our methods may be applied to a variety of rele-vant problems and algorithms related to joint estimationof multiple (eigen)phases.

Our protocol goes beyond the usual approach to mul-tiparameter estimation [33–39, 41] based of the analysisof the quantum Fisher information matrix (QFIM) [57–59]. First, the optimal measurement strategy to saturatethe QFIM generally depends the specific θ, while our al-gorithm uses a single approach for the estimation of anyθ ∈ [0, 2π]d. Furthermore, the (frequentist) sensitivitypredicted by the QFIM is only achieved asymptoticallyin the number of repeated measurements, which gener-ally hinders the scaling with respect to the total numberof resources. In particular, fixing NT allows us to clarifyunambiguously under which conditions parallel strategiesovercome sequential ones, thus settling a long-standingdebate in the literature [33–38].

The paper is structured as follows. In Sec. II,we present the quantum circuit and the correspondingBayesian algorithm for the parallel estimation of d ≥ 1eigenphases of a generic unitary operator U . In Sec. III,we discuss the performance of the algorithm in both thepresence and the absence of noise. In Sec. IV, we proposea multiround protocol for the parallel estimation of d op-

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Figure 1. Quantum circuits for parallel (a) and sequential(b) multiphase estimation. The corresponding quantum algo-

rithm is detailed in the text. Hd+1 and Zφd+1 (H and Zφ2 ) aregeneralized (single-qubit) Hadamard and phase gates, respec-tively. |0〉 denotes a (d+ 1)-dimensional ancilla qudit, |0〉 anancilla qubit and Uc is a controlled-U gate.

tical phases that can be implemented with single photonsand standard optical elements. Finally, we conclude inSec. V.

II. BAYESIAN ALGORITHM FOR PARALLELQUANTUM MULTIPHASE ESTIMATION

A. Quantum circuit of the algorithm

Our iterative multiphase estimation algorithm is basedon the quantum circuit shown in Fig. 1(a). As is com-mon in QPE algorithms [1, 3, 4, 8, 9], we assume (i) theexistence of an oracle having access to d eigenstates of Uand (ii) the possibility of generating controlled-UM gates,with M being an integer that we specify below. The cir-cuit consists of a register of d eigenstates |u1〉, ..., |ud〉of U and a (d + 1)-dimensional ancilla qudit. The reg-

ister is prepared in the product state⊗d

j=1 |uj〉. The

ancilla is initialized in |0〉 (bold for the ancilla states)and then transformed by a (d + 1)-dimensional gener-alized Hadamard gate Hd+1 defined by 〈m|Hd+1|n〉 =ei[2πmn/(d+1)]/

√d+ 1, where m,n ∈ {0, . . . , d} label

the computational basis states. This gate prepares theancilla in the superposition (|0〉 + · · · + |d〉)/

√d+ 1.

Next, d + 1 adjustable (known) controlled phases φ =(φ0, ..., φd) are imprinted on the ancilla via Zd+1(φ) givenby 〈m|Zd+1(φ)|n〉 = δmne

iφm . The key operation of thequantum circuit is the generalized controlled-U gate

Uc = |0〉〈0| ⊗ 1⊗ · · · ⊗ 1+ |1〉〈1| ⊗ U ⊗ 1⊗ · · · ⊗ 1+ · · ·+ |d〉〈d| ⊗ 1⊗ · · · ⊗ 1⊗ U, (1)

where 1 is the identity applied to the respective regis-ter state. The gate Uc is applied M ∈ N times. If theancilla is in the state |0〉, the register states are acted

upon by the identity. Otherwise, if the ancilla is in thestate |j 6= 0〉, the unitary U acts on the jth registerstate |uj〉, efficiently resulting in a phase eiMθj imprintedon the ancilla state. Therefore, after this step, the an-

cilla state is given by 1√d+1

∑dn=0 e

i(Mθn+φn)|n〉, where

we defined θ0 = 0, while the state of the register qu-dits does not change since they are prepared eigenstatesof U . A second Hadamard gate finally maps the ancilla

to the state 1d+1

∑dm,n=1 e

i(Mθn+φn+2πmnd+1 )|m〉, which is

then measured in the computational basis. The probabil-ity of measuring the ancilla in one of the d+ 1 outcomeso = 0, . . . , d is

P (o|θ,φ,M) =1

(d+ 1)2

{d+ 1 + 2

∑n

cos[Mθn + βn]

+ 2∑n<m

cos[M(θn − θm) + γnm]}, (2)

where βn = φn − φ0 + 2πno(d+1) and γnm = φn − φm +

2π(n−m)o(d+1) .

B. Bayesian processing

The algorithm consists of multiple rounds where, inthe kth round, the basic quantum circuit of Fig. 1(a)using M = 2k (k = 0, 1, . . . ) is repeatedly applied. Aftersetting the control phases φ, Bayes’ theorem states thatthe posterior probability distribution corresponding to ameasurement with result o is

P (ϑ|o,φ,M) =P (o|ϑ,φ,M)P (ϑ)∫

ddϑ P (o|ϑ,φ,M)P (ϑ), (3)

up to a normalization constant, where ϑ is a contin-uous d-dimensional variable, P (o|ϑ,φ,M) is given inEq. (2) and P (ϑ) is the Bayesian prior, which quanti-fies our knowledge about θ before the measurement. Asinitial condition (before the first measurement), we usea uniform prior P (ϑ) = 1/(2π)d. The posterior distri-bution P (ϑ|o,φ,M) expresses the degree of belief thatθ = ϑ, where θ is the true and unknown value of thevector parameter that we want to estimate, which is as-sumed constant while running the phase estimation al-gorithm. Given the posterior distribution, we (i) updatethe Bayesian prior according to P (ϑ|o,φ,M) 7→ P (ϑ);(ii) calculate θj = arg

∫ddϑ eiϑjP (ϑ|o,φ,M) which pro-

vides an estimate of θj ; and (iii) calculate the Bayesianprobability, Phalf =

∫C

ddP (ϑ), of finding the true phase

in the hypercube C =∏j [θj − π/2k+1, θj + π/2k+1] cen-

tered around θ. If Phalf ≤ 1 − ε, for some small ε thatwe indicate as the decision parameter, we set new controlphases φ and repeat the measurement, keeping the sameM = 2k as in the previous measurement. If Phalf > 1− ε,we resrict the phase domain to the domain C, renormal-ize the Bayesian probability such that

∫C

ddP (ϑ) = 1,

3

and proceed to the next (k + 1)th round of the algo-rithm, namely, setting M = 2k+1. In Table I, we showthe pseudo code of the Bayesian algorithm. A detaileddiscussion of how many measurements are performed ineach round is given in Appendix A.

We observe that the use of random control phasesφ ∈ [0, 2π]d, together with repeated measurements atfixed values of M , efficiently avoids the proliferation oftails in the Bayesian distribution, an effect that decreasesthe performance of the algorithm (see Appendix B). Inthe case of single-phase estimation, there are optimaladaptive methods to choose the control phase [22, 24]that, however, cannot straightforwardly be generalizedfor the estimation of d phases. Furthermore, the use ofrandom φ avoids a (classical) computational slowdownassociated with the calculation of optimal control phases.Alternatives to random control phases such as quasiran-dom sequences are discussed in Appendix B.

A second important aspect is the cutting of the phasedomain before increasing M . Cutting is necessary toguarantee a finite memory for storing the posterior distri-bution when running the algorithm with large M . With-out cutting, there is an upper bound on M imposed bythe computer memory due to an increasing grid size.Whenever M is increased, our protocol instead allowsto restrict the distribution to the hypercube C of highprobability, avoiding the need to increase the number ofgrid points. Numerically, we find that a small finite gridsize is advantageous because of small memory require-ments and because it automatically performs a coarsegraining (smoothening) of the Bayesian posterior. Thedrawback is that each cutting of the phase domain pro-

duces an error probability P(d)err (ε) of the true value θ not

being in the cut-out region C, thus resulting in a biasedestimation. Reduction of the error probability requiresincreasing the number of measurements in each round,and thus the total resources. In practice, from the nu-merical simulations reported in the following, we observe

Input: {ε, kmax}P (ϑ) = 1/(2π)d

for k = 0, . . . , kmax − 1:M = 2k; Phalf = 0while Phalf < 1− ε:φ = generate random()o = measurement(M,φ)P (ϑ) = Bay update(P (ϑ), o,M,φ)Phalf = compute(P (ϑ))

P (ϑ) = cut grid(P (ϑ))P (ϑ) = normalize(P (ϑ))

return: P (ϑ)

Table I. The pseudocode of the Bayesian multiphase estima-tion algorithm. The initially flat phase distribution P (ϑ) isupdated during kmax rounds of the algorithm. In the kthround, we set M = 2k and perform measurements with ran-dom control phases φ until we find Phalf > 1−ε, i.e., until theprobability distribution is localized enough to cut the grid.

that P(d)err (ε) ∼ ε, with a prefactor depending slightly on

d. The decreasing of the error probability P(d)err (ε) requires

increasing the number of measurements at each round asO(log(1/ε)) (see Appendix A).

III. RESULTS

A. Figure of merit and resource counting

We quantify the performance of the algorithm by thed× d Bayesian covariance matrix

Vij = 4

∫ddϑ sin

ϑi − θi2

sinϑj − θj

2P (ϑ|o,φ,M). (4)

The diagonal entries of V correspond to the Holevo vari-ance for small Vjj [60] while the off-diagonal elementsquantify correlations in the Bayesian distribution. Equa-tion (4) gives the variance in the estimation of any lin-ear combination of parameters n · θ according to Vn·θ =∑di,j=1 niVijnj , where n is a d-dimensional real vector.As for resource counting, we follow the standard quan-

tification of single parameter-estimation [22, 24, 54],namely, we count the number of applications of thecontrolled-U gate, while assuming the existence of an ora-cle having access to d eigenstates of U and the possibilityof generating controlled-UM gates. We want to empha-size why the generalized controlled gate Uc correspondsa single application of the gate U : as one can see fromEq. (1), the application of the gate Uc can only imprintthe eigenphases θj (and not d × θj) on any initial state.This is in contrast to the application of d unitaries U(say, acting on the register states) that are able to im-print the phase d × θj by properly choosing the initialstate. Furthermore, considering the equivalence of the(multipass) Bayesian multiphase estimation algorithm toa protocol making use of N00N states [c.f. Sec. IV andEq. (12], the M -fold application of Uc corresponds to asingle application of the multiparameter phase shift ona generalized N00N state with M particles, counting asM resources according to traditional resource countingin quantum metrology.

Consequently, the total resources used up to the Kth

round of the algorithm are NT =∑K−1k=0 mk2k, where

mk is the number of measurements performed in thekth round, while the total number of measurements is

Nmeas =∑K−1k=0 mk [53].

B. Sensitivity of the Bayesian algorithm in thenoiseless case

We perform extensive numerical simulations of theBayesian multiphase estimation algorithm presentedabove. In Fig. 2, we show exemplary d-dimensionalBayesian posterior distributions for (a) d = 2 and (b)d = 3 parameters, after simulating the measurements up

4

(a) (b)

Figure 2. Examples of the Bayesian probability distribution for (a) d = 2 and (b) d = 3, obtained after five rounds of the

algorithm. The red point indicates θ. The phase domains are restricted to a fraction of the hypercube C =∏dj=1[θj − c, θj + c]

with c = π/2k and k = 4 in both cases. The color scale shows the Bayesian posterior P (ϑ) (a) and marginal Bayesiandistributions (b) normalized to the maximum, maxϑP (ϑ).

to the fifth round of the algorithm (k = 4, M = 16). TheBayesian distributions are highly localized in a fractionof the hypercube C and, as we discuss below, with a veryhigh probability around the true θ. They are also clearlyanisotropic – a direct consequence of the cross terms inEq. (2) – showing correlations between the different esti-mators θj .

In Fig. 3, we show the results of the numerical sim-ulation of the Bayesian covariance matrix, Eq. 4. Uni-formly generated θ ∈ [0, 2π]d are repeatedly estimatedusing NT ≈ 109, corresponding to 25 rounds, or about500 measurements. Fig. 3 shows the Bayesian variancesVjj and covariances Vij for i 6= j for (a) d = 1, (b)d = 2 and (c) d = 3. The dots indicate numerical re-sults obtained after each round of the algorithm (noticethat there is no post-selection of optimal results). Theyspread around an average value (solid line) mainly due tofluctuations of the number mk of repeated measurementsat each round. The horizontal dashed line is a numericalfit for NT � 1, giving

V =C

(1)H (ε)

N2T

, for d = 1, (5)

V =C

(2)H (ε)

N2T

(1 0.47

0.47 1

), for d = 2, (6)

and

=C

(3)H (ε)

N2T

1 0.45 0.450.45 1 0.450.45 0.45 1

, for d = 3, (7)

where the fitting parameter C(d)H (ε) depends on the de-

cision parameter ε. A fit of C(d)H as a function of ε

(c.f. Fig. 4, blue dots), gives C(1)H = 3.13 + 2.50 ln 1/ε

for d = 1; C(2)H = 10.8 + 13.8 ln 1/ε for d = 2; and

C(3)H = 40.1 + 26.2 ln 1/ε for d = 3. In particular, the

results of Fig. 3, C(2)H ≈ 138 and C

(3)H ≈ 281, are ob-

tained for ε = 10−4. We thus have a scaling O(N−2T ) of

both diagonal and off-diagonal elements of the covariancematrix. Also, the Bayesian covariance matrix scales ex-ponentially with the number of measurements or, equiv-alently, with the number of ancilla qudits, as we show inAppendix C.

Furthermore, we analyze the error rate P(d)err as a func-

tion of the decision parameter ε. More precisely, we cal-

culate P(d)err as the probability – per round – that θ /∈ C,

namely, that the true value of the parameter lies outsidethe cut-out region C, resulting in a biased estimation of

θ. P(d)err is estimated as follows. We observe numerically

that, after a larger error probability in the first round, theerror probability during the algorithm remains roughlyconstant (see Appendix D). Therefore, by assuming aconstant error probability per round, we slightly overes-timate the asymptotic error probability per round. Wesimulate Nsim runs of the algorithm for different valuesof the decision parameter ε, resulting in Nerr errors (wecount an error whenever the true value of θ lies outsidethe hypercube C, θ /∈ C). After running k rounds, theprobability of no error, 1− Perr,T, is related to the (con-

stant) error probability per round, P(d)err , by

1− P (d)err,T = (1− P (d)

err )k. (8)

Using this relation and estimating Perr,T = Nerr/Nsim,

we find an estimate for P(d)err (see Fig. 4, orange squares).

The error bars are obtained by error propagation andthe use of ∆Perr,T =

√Nerr/Nsim, assuming a Poisson

distribution of error counts.A fit of P

(d)err (ε) with the model c × ε yields the error

5

(a) (b) (c)

Figure 3. Bayesian variances Vij (i = j) and covariances Vij (i 6= j) multiplied by N2T , for (a) d = 1, (b) d = 2 and (c) d = 3.

Dots are results of numerical simulations and correspond to values calculated at the end of each round of the algorithm. Solidlines indicate averages converging to horizontal lines is the average. It converges to a horizontal lines Vij ∼ N−2

T for large NT

(dashed, obtained from a fit) that provide the coefficients C(d)H in Eqs. (5-7). Here, we report results of 100 simulations of the

full algorithm up to k = 25, estimating phases randomly chosen in [0, 2π]d with uniform distribution. The decision parameteris set to ε = 10−4.

rate scalings P(1)err = 0.94ε, P

(2)err = 0.78ε, and P

(3)err =

0.58ε, see Fig. 4. Note that a more accurate fit withthe model c1 × εc2 yields the parameters c1 = 0.57 andc2 = 0.91 for d = 1, c1 = 0.94 and c2 = 1.04 for d = 2,and c1 = 0.64 and c2 = 1.02 for d = 3. For a proper

determination of the scaling of P(d)err (ε), one should also

perform more simulations. However, for our qualitativescaling comparison of parallel and sequential protocols,this rough estimation is sufficient.

In the case d = 1, we can directly compare our re-sults with existing single-phase estimation algorithms.References [22, 24] have studied “backward” algorithmsthat, in analogy to the semiclassical implementation ofthe inverse quantum Fourier transform [61], are madeof phase estimation rounds in a decreasing sequenceM = 2K , 2K−1, ..., 1. Exploiting an optimized adap-tive protocol, the algorithm of Ref. [22] reaches V =Cs/N

2T with Cs = 23. The advantage of these ap-

proaches is that the error probability is negligible. Thedrawback is that increasing the sensitivity by increas-ing K requires to restart us the algorithm. In con-trast, our algorithm belongs to the class of “forward”algorithms [23, 27, 28, 31, 53], where the phase sensitiv-ity increases progressively as the protocol proceeds with,ideally, no upper bound in the number of rounds. TheBayesian forward protocol of Ref. [53] approximates, ateach round, the Bayesian distribution with a Gaussianfunction and reaches Cs = 22 (with a finite probabilityof biased estimation). As shown in Fig. 3(a), we overcome

the previous results, namely, we reach C(1)H = 22, at the

price of an error probability of P(1)err = 5 × 10−4. Fur-

thermore, our algorithm does not suffer from breakdownproblems or an uncontrolled increase of error probabil-ity [53] due to the proliferation of tails in the Bayesiandistribution. The nonadaptive protocol of Ref. [23] showsa negligible error probability at the price of a larger pref-actor Cs = 40.5.

C. Comparison between sequential and parallelmultiphase estimation

The estimation of d eigenphases can be performedeither following a parallel protocol using an (d + 1)-dimensional ancilla qudit, as illustrated above, or follow-ing a sequential protocol where each phase is estimatedseparately using ancilla qubits [see Fig. 1(b)]. In thelatter case, any linear combination of d eigenphases isestimated with sensitivity

Vn·θseq =

d∑j=1

n2jVj =

C(1)H d2

∑dj=1 n

2j

N2T

, (9)

where Vj is the sensitivity of estimating θj in a single-parameter estimation algorithm and NT is the total num-ber of resources used in the multiphase estimation. Weassume here that all phases are estimated using the sameresources (note that an optimization of resources, de-pending on the values of nj , is also possible).

Compared to the sequential protocol, the parallel pro-tocol offers two key advantages. (1) The estimation ofmultiple phases simultaneously might be necessary, ormight prove particularly efficient, in systems and appli-cations where spatial or temporal constraints (e.g., fortime-varying signals) prevent the estimations of singlephases independently. (2) By exploiting correlations inthe Bayesian multiphase distribution, certain combina-tions of the eigenphases can be estimated with highersensitivity in the parallel case compared to the sequentialcase, for a fixed error rate Perr in both cases. Explicitly,in the case d = 2, the sequential strategy estimates θ1 andθ2 in a single-phase estimation and then infers θ1−θ2 with

sensitivity Vθ1−θ2 = 8C(1)H (Perr)/N

2T with C

(1)H (Perr) =

3.13 + 2.50 ln 0.94/Perr. On the other hand, the parallel

estimation uses Vθ1−θ2 = 2(1 − 0.47)C(2)H (Perr)/N

2T with

C(2)H (Perr) = 10.8+13.8 ln 0.78/Perr. Fixing, for instance,

Perr = 10−3, the sequential protocol yields Vθ1−θ2 =162/N2

T , while the parallel protocol scales as Vθ1−θ2 =109/N2

T . Analogously, for d = 3 and Perr = 10−3, the

6

(a) (b) (c)

Figure 4. The fitting parameter C(d)H (ε) of Eqs. (5)-(7) (blue dots) and the error rate P

(d)err (ε) (orange squares) as a function

of the decision parameter ε, for (a) d = 1, (b) d = 2, and (c) d = 3. The solid gray lines are fits (see the text), error barsare estimated standard deviations of the error counts (see the text). Averages are obtained by simulating the algorithm up toM = 224 with 105 repetitions for d = 1 and d = 2 and 5× 104 repetitions for d = 3, respectively.

sequential scheme measures all phase differences withVθj−θk = 364/N2

T , compared to Vθj−θk = 227/N2T for

the parallel protocol.

D. Sensitivity of the Bayesian algorithm inpresence of noise

In the following, we extend the Bayesian multiphase es-timation algorithm in the presence of noise. In particular,we assume that each phase imprinting is accompanied bynoise characterized by a decoherence rate Γj for the jthphase. In particular, we generalize the qubit phase damp-ing channel [4] to the qudit case. Formally, if the ancillastate is given by the density matrix ρ, the phase damping

is described by the channel E(ρ) =∑dj=0KjρKj , where

the Kraus operators are defined as (K0)mn = e−ΓmMδmn(Γ0 = 0) and (Kj)mn =

√1− e−2ΓjMδmjδnj for j =

1, . . . , d. In the case d = 1, this model results in a decayof the measurement probabilities to white noise, a modelcommonly considered in single-phase estimation [53]. Inthe optical implementation (cf. Sec. IV), the noise modelcan be understood as a mode-dependent dephasing dur-ing each phase imprinting.

The algorithm in the presence of noise follows thenoiseless one with setting M = minj [2

k, 1/Γj ] in the kthround. In other words, we stop increasing M as soon asthe noise becomes dominant. After this point, the covari-ance matrix V follows a shot-noise scaling (Vij ∼ N−1

T ).The pseudocode of the Bayesian algorithm in the pres-ence of noise is shown in Table II.

In Fig. 5, we show results of numerical simulations,plotting NTVij as a function of NT , for d = 2 phaseswith Γ1 = 0.02 and Γ2 = 0.01. The change of scaling ofV around NT = 104 is evident, approaching a constantvalue for sufficiently large NT . In this regime, the esti-mation is still overcoming the shot-noise limit Vθj = N−1

T(dashed line in Fig. 5). Note that since the strong cor-relation in the noiseless case stems from the cross termsof Eq. 2, the correlations now decay with a larger rateΓ1 + Γ2. We also mention that an asymmetric noisecan be balanced in the estimation algorithm by perform-ing each measurement with an M -dependent unbalanced

Input: {ε, kmax,Γj}P (ϑ) = 1/(2π)d

for k = 0, . . . , kmax − 1:M = minj [2

k, 1/Γj ]; Phalf = 0while Phalf < 1− ε:φ = generate random()o = measurement(M,φ)P (ϑ) = Bay update(P (ϑ), o,M,φ)Phalf = compute(P (ϑ))

P (ϑ) = cut grid(P (ϑ))P (ϑ) = normalize(P (ϑ))

return: P (ϑ)

Table II. The pseudocode of the Bayesian multiphase estima-tion algorithm in the presence of noise. The measurementstrategy is similar to the one in the noiseless case (cf. Ta-ble I). Here, in the kth round, we choose M = minj [2

k, 1/Γj ]dependent on the different dephasing rates Γj .

101 102 103 104 105 106

NT

10−2

10−1

100

101

VijNT

V11NT

V22NT

V12NT

Figure 5. The scaling of the Bayesian variances Vij (i = j)and covariances Vij (i 6= j) with the total number of resourcesNT in the presence of dephasing. The dots indicate resultsof numerical simulations after each round and the lines cor-respond to average values. The dashed line is the shot-noiselimit N−1

T . Here, d = 2, Γ1 = 0.02, Γ2 = 0.01 and ε = 10−4.

Hadamard gate Hd+1 instead of the first Hadamard gate.

7

Figure 6. The optical implementation of the parallel multi-phase estimation algorithm. A single photon enters a mul-tiarm interferometer composed of two (d + 1)-port beam-splitters (BSd+1) that enclose a control (φ0, ..., φd) and anunknown (θ1, ..., θd) optical phase shift in each mode. Thephase shifts θj are defined with respect to a reference phaseθ0. The photon is finally measured in the outcoming modes.

IV. OPTICAL IMPLEMENTATION

Above, we focus on the parallel estimation of d eigen-phases of a unitary matrix: a task that can find possi-ble applications in a wide variety of quantum computingtasks based on QPE. In addition, here we show that ourmethods can be straightforwardly applied to multiparam-eter estimation problems in quantum optical interferom-etry. In particular, we consider the parallel estimation ofd optical phase shifts using a generalized Mach-Zehnderinterferometer setup.

The protocol can be realized with single photons andstandard optical elements and is thus accessible by cur-rent state-of-the-art technology [43, 62, 63] (see Fig. 6).The extended Hadamard gate is realized by a multiportbeam splitter BSd+1 [62, 64], such as a tritter for d = 2or a quarter for d = 3, recently implemented with inte-grated optical circuits [43, 45, 62, 63], that can be alsorealized with a cascade of 50:50 beam splitters [44]. Forinstance, in the case d = 2, the unitary transformationof a tritter is given by [62, 65]

UBS,3 =1√3

1 1 11 ei2π/3 ei4π/3

1 ei4π/3 ei2π/3

(10)

which corresponds precisely to H3 in Sec. II. In the cased = 3, the corresponding optical element is called a quar-ter and is parametrized by a implementation-dependentparameter γ [64],

UBS,4(γ) =1

2

1 1 1 11 eiγ −1 −eiγ1 −1 1 −11 −eiγ −1 eiγ

. (11)

For γ = π/2, this corresponds exactly to H4 in Sec. II,

while for other γ, the final probability has to bemarginally adjusted.

The different phases θ1, ..., θd and the (random) con-trol phases φ0, ..., φd are imprinted on the different opti-cal modes. The phase shift on mode j is implemented bythe transformation exp[i(Mθj + φj)nj ], where nj is thenumber operator for the jth mode. The phase “ampli-fication” M × θj is obtained by M passes of the phaseshift θj , as demonstrated experimentally for single-phaseestimation in Ref. [22]. Note that all d phases are definedwith respect to an unknown reference phase θ0 in the ze-roth arm. Finally, after a second multiport beam splitter,the photon is measured in one of the d+1 output modes,resulting in statistics described by Eq. (2). As discussedabove, the optical scheme of Fig. 6 would reach a multi-phase estimation variance with optimal scaling O(N−2

T )with respect to the total resources NT (quantified hereas the total number of applications of the phase shift θ),and exponentially with the number nphot of single pho-tons used (cf. Appendix C).

Finally, we mention that, in principle, the protocol canbe equivalently implemented with multimode N00N-likestates [33, 36–38, 66]

|ψ〉 =|M, 0, ..., 0〉+ |0,M, 0..., 0〉+ |0, ..., 0,M〉√

d+ 1. (12)

The different phases in each mode are imprinted by asingle application of the phase shifts, namely, exp[i(θj +φj)nj ]. Projecting the final state on the set of states

|ψo〉 =1√d+ 1

(|M, 0, ..., 0〉+ e

i2πod+1 |0,M, ..., 0〉

+ · · ·+ ei2πdod+1 |0, 0, ...,M〉

), (13)

where o = 0, . . . , d labels the measurement outcome, re-sults in the probabilities of Eq. (2) (redefining the randomcontrol phases φj → φj/M). In this scheme, the phasesM×θj are imprinted by a single application of the phaseshift θj but using M photons. Regarding resources, thetwo optical schemes are equivalent [22, 54, 67]. Althoughproposed in Ref. [33] and largely studied as a benchmarkin the multiphase estimation literature [36–38], the gen-eralized NOON state of Eq. (12) has not been realizedexperimentally so far.

V. CONCLUSIONS

To summarize, we have presented a Bayesian quantumalgorithm for the simultaneous estimation of d arbitraryphases θ ∈ [0, 2π]d with covariance matrix elements scal-ing as O(N−2

T ) with respect to the total number of re-sources NT . The running of the algorithm requires thesupport of a classical memory to store the Bayesian dis-tribution. This is necessary to calculate variances, confi-dence intervals, and estimators, and to monitor the pro-cess of the algorithm. Our algorithm is well suited for

8

the estimation of a few parameters, realizable with anoptical device [43, 62, 63]. For larger values of d, othercutting strategies that do not scale with d can be consid-ered, such as Gaussian approximations of the Bayesiandistribution [53] or adaptive updates of the grid pointsusing Monte Carlo methods [45, 68]. However, the im-pact of these methods on the probability of obtainingbiased estimations has to be carefully analyzed.

In the context of multiparameter quantum metrology,our protocol provides a viable strategy to estimate ar-bitrary phases θj and their linear combinations θ · nwith ∆2(θ · n) ∼ 1/N2

T , for fixed resources NT � 1.Furthermore, our analysis highlights that correlations inthe Bayesian distribution (only accessible by the paral-lel scheme) can be exploited for the estimation of certaincombinations of phases with a sensitivity overcoming thatachievable by sequential (single-parameter) estimationstrategies. Finally, an optical version of the algorithmusing single photons and linear optical elements can beimplemented in state-of-the-art experiments [43, 62, 63].Equivalently, the algorithm can be implemented usinggeneralized N00N-states [33, 36–38]. Recently, a multi-round protocol using entangled qubits to measure a spe-cific linear combination of multiple phases (at an optimalpoint) has been realized [69, 70]. In contrast, our proto-col is able to measure all linear combinations of phasessimultaneously.

ACKNOWLEDGMENTS

We acknowledge financial support from the Euro-pean Union’s Horizon 2020 research and innovation pro-gramme - Qombs Project, FET Flagship on QuantumTechnologies Grant No. 820419.

APPENDIX

Appendix A: Number of measurements per round

We now discuss the distribution of the number of mea-surements in the different rounds of the phase estimation

(a) (b) (c)

Figure 7. (a) The average number of measurements perround, nmeas, used in the different rounds of the algorithmsimulated for 1000 runs. (b-c) The distribtuion of the numberof measurements nmeas used in (b) the fifth round and in (c)the 20th round of the algorithm. d = 2, ε = 10−4.

102 103 104 105

1/ε

10

12

14

16

18

20

nm

eas

Figure 8. The scaling of the average number of measure-ments per round, nmeas, (excluding the first round) used inthe algorithm depending on the decision parameter ε. For tendifferent values of ε, we simulate 100 runs of the algorithm andaverage the results. d = 2.

algorithm. Here, nmeas(k) is the number of measure-ments used in the kth round when averaged over manyruns of the algorithm. It is related to the average totalnumber of measurements after K rounds, 〈Nmeas(K)〉,as 〈Nmeas(K)〉 =

∑Kk=0 nmeas(k). In Fig. 7(a), we show

nmeas depending on the round for 1000 simulations ofthe algorithm until M = 224 for d = 2 and ε = 10−4.We see that after the first round, a constant number ofmeasurements is performed on average per round. In thefirst round, more measurements are performed becausethe initial flat Bayesian distribution has to become suf-ficiently localized before the algorithm switches to thesecond round.

In Figs. 7 (b) and 7 (c), we show the distribution of thenumber of measurements nmeas in (b) the fifth round andin (c) the 20th round for the simulation of 1000 runs usedfor Fig. 7 (a) (indicated in various colors). We observethat the distribution of nmeas is similar for the differentrounds and reason that, due to the tail of the distribution,the use of a constant number of measurements per roundmight lead to an increased error if chosen too small or to

a large constant C(d)H of the Heisenberg scaling if chosen

too large.

Having checked that after the first measurement, thenumber of measurements per round is constant on av-erage, we now check the scaling of this average numbernmeas (excluding the first round) depending on the deci-sion parameter ε. In Fig. 8, we show nmeas for differentvalues of ε averaged over the different rounds of 100 runsof the algorithm for each ε and d = 2. We see that nmeas

scales as O(ln 1/ε) (or, equivalently, O(ln 1/P(d)err )), a scal-

ing that is also observed in the Kitaev phase estimationalgorithm [1]. In Ref. [1], the unknown phase is esti-mated (with high probability) up to an additive error εusing O(ln 1/ε) ancilla qubits.

9

Appendix B: Choice of the control phases during thealgorithm

Here, we will discuss the choice of the random controlphases during the algorithm. First, we discuss the cased = 1 for simplicity. If we do not vary the control phase(without loss of generality, consider φ = 0), the Bayesianupdate results in products of cos2 θ/2 and sin2 θ/2 withmultiplicities depending on the different measurementoutcomes. This will eventually lead to a double-peakstructure in the Bayesian distribution. Measurementswith a larger M in later rounds are not sufficient to re-solve the two peaks, leading finally to an erroneous cut-ting and to a biased estimation. This behaviour is shownin Fig. 9(b). It is known that in the case of d = 1, thisobstacle can be solved by alternating the control phasebetween φ2n = 0 and φ2n+1 = π/2 for the nth measure-ment in each round (see, e.g., Ref. [23]). The same effectis reached by using random (but known) control phasesin each measurement.

In the case d = 2, an alternative to using random con-trol phases is to set the control phases φ in the nthmeasurement to (0, 0) if n mod 4 = 1, (π/2, 0) if nmod 4 = 2, (0, π/2) if n mod 4 = 3 and (π/2, π/2) if

n mod 4 = 0. For ε = 10−3, this leads to C(2)H = 113

and P(2)err = 7.1 × 10−4, compared to C

(2)H = 105 and

P(2)err = 7.7× 10−4 for random phases.A different approach is to use so-called quasi-random

sequences [71]. Quasi-random sequences are determinis-tic sequences of tuples xn that densely cover [0, 2π]2 forn→∞ and appear random but, at the same time, cover[0, 2π]2 more evenly than random numbers during theconvergence. Using, for example, the Halton sequence in

two dimensions as control phases, we observe C(2)H = 101

and P(2)err = 8.5× 10−4 for ε = 10−3. We see that the al-

ternatives of random control phases discussed here yieldvery similar results to random ones. A minor advantagecould be achieved by using quasi-random control phases.

For d > 2, similar choices of control phases can be made.

Appendix C: Scaling of the precision with thenumber of measurements

We study the exponential scaling of the Bayesian co-variance matrix V with the number of measurements,Nmeas. In Fig. 10, we plot the scaling of the differentcomponents Vij as a function of Nmeas for the multiphaseestimation algorithm estimating d = 2 phases and a de-cision parameter ε = 10−4. Shown are results from 200realizations of the algorithm up to M = 224 (dots) andan exponential fit Vjj(Nmeas) = 0.61 exp[−0.082Nmeas](solid line). An analogous scaling can be observed ford = 1 and d = 3. In the implementation of the algorithmusing controlled-U operations, Nmeas corresponds to thetotal number of ancilla qubits that have to be preparedand measured. For the optical implementation using sin-gle photons, Nmeas is the total number of photons usedin the algorithm and for the optical implementation us-ing generalized N00N states, Nmeas is the total amountof applications of the unknown phase shifts.

Appendix D: Analysis of error probability per round

In Fig. 11, we show simulations of the error probabilityfor 106 runs of the algorithm with d = 1, using decisionparameters (a) ε = 10−2 and (b) ε = 10−3. We ob-serve that after an increased error probability in the firstround, the error probability per round remains roughlyconstant and even decreases slightly for later rounds.Therefore, the error analysis in Sec. III, which countsall errors that occur after 25 rounds of the algorithmand assumes a constant error probability in each round,represents an overestimation of the real asymptotic errorrate of the algorithm.

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