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PRX QUANTUM 2, 040301 (2021)
Quantum Phase Estimation Algorithm with Gaussian Spin States
Luca Pezzè* and Augusto Smerzi†QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, Firenze 50125, Italy
(Received 10 October 2020; revised 21 April 2021; accepted 30 July 2021; published 1 October 2021)
Quantum phase estimation (QPE) is one of the most important subroutines in quantum computing. Ingeneral applications, current QPE algorithms either suffer an exponential time overload or require a setof—notoriously quite fragile—GHZ states. These limitations have so far prevented the demonstration ofQPE beyond proof of principles. Here we propose a new QPE algorithm based on a cascade of Gaus-sian spin states (GSSs) and a suitable adaptive measurement protocol. GSSs are renownedly resilient andhave been created experimentally in a variety of platforms, from hundreds of ions up to millions of coldand ultracold neutral atoms. We show that our protocol achieves a QPE sensitivity overcoming previousschemes, including that obtained with GHZ states, scales linearly with time, and is robust against certainsources of noise and decoherence, such as detection noise, in particular. Our work paves the way towardan efficient implementation of the QPE, as well as applications of atomic squeezed states for quantumcomputation.
DOI: 10.1103/PRXQuantum.2.040301
I. INTRODUCTION
Quantum phase estimation (QPE) [1–3] is the buildingblock of known quantum computing algorithms providingexponential speedup [4], including the computation of theeigenvalues of Hermitian operators [5], such as molecularspectra [6,7], number factoring [8–10], and quantum sam-pling [11]. All these applications require the calculation ofan eigenvalue of a unitary matrix U|u〉 = eiθ |u〉, where |u〉is the corresponding eigenstate. This problem can be castas the estimation of an unknown phase θ ∈ [−π ,π), wherethe information about θ is encoded into one or more ancillaqubits via multiple applications of a controlled-U gate [4].The QPE problem plays a key role also in the alignmentof spatial reference frames [12] and clock synchronisation[13], with further developments in atomic clocks [14], andclock networks [15].
According to the current paradigm [1,2], QPE algo-rithms are implemented iteratively, without requiring theinverse quantum Fourier transform [3,4,16,17]. IterativeQPE consists of multiple steps, each step being realizedin two possible ways. (i) Using a single ancilla qubit(|0〉 + |1〉)/√2 that interrogates 2k times the controlled-U gate in temporal sequence [18–20]. In this way, the
*[email protected]†[email protected]
Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license. Fur-ther distribution of this work must maintain attribution to theauthor(s) and the published article’s title, journal citation, andDOI.
ancilla qubit is transformed to (|0〉 + ei2kθ |1〉)/√2 and thephase information is extracted via a Hadamard gate H , fol-lowed by a projection in the computational basis. (ii) Using2k ancilla qubits in a Greenberger-Horne-Zeilinger (GHZ)state (|0〉⊗2k + |1〉⊗2k
)/√
2 that interrogate the controlled-U gate in parallel [14,15,21–23]. In this case, the GHZstate is transformed to (|0〉⊗2k + ei2kθ |1〉⊗2k
)/√
2 and thephase can be extracted by applying a collective Hadamardgate H⊗2k
followed by a parity measurement [21–23].While the above output states are characterized by aperiodicity 2π/2k in θ , an unambiguous estimate of θ ∈[−π ,π) is obtained by taking a sequence of steps usingk = 0, . . . , K and eventually repeating the procedure anoptimal and finite number of times ν at each step [1,2].Using total resources NT = ν
∑Kk=0 2k = ν(2K+1 − 1), it
is possible to estimate an unknown θ with a sensitivityreaching the Heisenberg scaling �θ ∼ 1/NT [18,23,24].
These protocols have critical drawbacks. The sequentialprotocol (i) considers 2k applications of the controlled-U gate that require exponentially large implementationtimes. This approach has been demonstrated experimen-tally [18] using multiple passes of a single photon througha phase shifter and it has been more recently applied toeigenvalue estimation [19] and magnetic field sensing [20].The parallel protocol (ii) scales linearly with the imple-mentation time but requires GHZ maximally entangledstates containing an exponentially large, 2k, number ofparticles. These states are notoriously fragile, since theloss of even a single particle completely decoheres thestate [25,26] and irremediably breaks down the quantumalgorithm. GHZ states are currently implemented with upto 20 particles [27,28]. Generally speaking, the current
2691-3399/21/2(4)/040301(21) 040301-1 Published by the American Physical Society
LUCA PEZZÈ and AUGUSTO SMERZI PRX QUANTUM 2, 040301 (2021)
grand challenge in quantum technologies is to go beyondproof-of-principle demonstrations toward implementationsovercoming “classical” bounds. In the contest of QPE, itis evident that present shortcomings call for a radicallynew approach that is both scalable with respect to tem-poral resources and exploits experimentally robust, easilyaccessible, quantum states.
Here we develop a new QPE algorithm that isimplemented with Gaussian spin-squeezed states (GSSs)[29–31] and a suitable adaptive measurement protocol.The circuit diagram of the quantum algorithm is shownin Fig. 1 and is discussed in detail below. The iterativealgorithm scales linearly with time and is broken into mul-tiple steps, each implemented with a GSS. GSSs have aGaussian particle statistical distribution and are way morerobust to noise and decoherence than GHZ states. Indeed,GSSs have been experimentally demonstrated in a varietyof platforms [30], with a few hundreds ions [32], several
thousands degenerate gases [33,34], up to millions of cold[35–38] atoms. Recent experiments have reported the gen-eration of GSSs with long lifetimes (up to 1 s) [39] anda large (up to 1011) number of particles [40]. The short-coming of GSSs—that has prevented so far their use inQPE—is that they provide high sensitivity only in a rel-atively small phase interval centred around the optimalphase value θ = 0 [30,31]. Our algorithm overcomes thislimitation by a classical adaptive phase feedback. It uses acascade of GSSs to progressively drive the unknown phaseto the optimal sensitivity points of a final highly populatedand highly squeezed state. An analytical optimization ofthe cascade, with respect to both the number of particlesand the squeezing parameter of each GSS, demonstrates aphase sensitivity
�θest = 4NT
(1)
(a)
(b)
State preparation Phase encoding Phase feedback and readout Output
FIG. 1. Circuit diagram of the quantum phase estimation algorithm described in this work. (a) The K + 1 ancilla states prepared in|0〉⊗Nk (k = 0, . . . , K) are first rotated by applying a collective Hadamard gate H⊗Nk (white box). The green box is a spin-squeezinggate, with sk being the squeezing parameter (see the text). The yellow box represents a sequence of Nk controlled-U gates, each appliedto a single ancilla qubit and providing phase encoding conditioned by the eigenstate |u〉 of the register: the sequence of controlled-Ugates is schematically represented by a single gate indicated as U⊗Nk
c . The black circle at the end of each vertical yellow line indicatesthe group of qubits to which the phase encoding is applied. Pink boxes are rotation phase gates Rz(−�j ) implementing a classicalphase feedback (pink lines) based on the estimate �j at step j < k. The readout is obtained by applying a rotation gate R⊗N
x (bluebox) followed by a measurement of the eigenstates of Jz (gray box). The output of the algorithm, after K + 1 steps, is an estimate,θest = ∑K
k=0�k, of the unknown phase θ . (b) Illustration of the kth step of the algorithm (bottom) and the corresponding Husimirepresentation of the quantum state on the Nk-qubits Bloch sphere (top) for each operation. The output signal 〈J out
z 〉k has a sinusoidaldependence on θk (see the text), from which the estimated �k is extracted and used to phase feedback the next (k + 1) step of thealgorithm.
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for the estimation of any arbitrary phase θ ∈ [−π ,π),where NT is the total number of qubits used in the full esti-mation algorithm. This result overcomes the performanceof other QPE algorithms proposed so far in the literature[18,23,24], including those using GHZ states [14,15,21–23]. Moreover, our protocol is implemented with a singlemeasurement of a collective spin observable at each stepand the phase is extracted with a practical estimationtechnique.
II. PHASE ESTIMATION ALGORITHM
The circuit representation of K + 1 steps of thealgorithm is shown in Fig. 1(a), while Fig. 1(b) illustratesa single step. The phase estimation uses NT particles thatcan access two internal or spatial modes. The ensembleof NT qubits is divided into K + 1 spin-polarized states|0〉⊗Nk , one for each step k = 0, 1, . . . , K of the algorithm,with Nk � 1 and
∑Kk=0 Nk = NT. We also introduce col-
lective spin operators Jn = ∑Nkj =1 σ
(j )n /2, where σ (j )n is the
Pauli matrix of particle j along the axis n = x, y, z in theBloch sphere. Each state |0〉⊗Nk first goes through a col-lective Hadamard gate H⊗Nk , which prepares the coherentspin state |CSS(Nk)〉 = [(|0〉 + |1〉)/√2]⊗Nk aligned alongthe x axis of the generalized Bloch sphere; see Fig. 1(b).Except at the zeroth step k = 0, the state |CSS(Nk)〉 thengoes through a spin-squeezing gate (represented in Fig. 1by the green box) that generates the GSS
|ψk(Nk, sk)〉 = 1√N
Nk/2∑
μ=−Nk/2
e−μ2/(s2kNk)|μ〉y . (2)
This concludes the state preparation. Here, s2k = 4(�Jy)
2k/
Nk is the squeezing parameter (sk < 1 for metrologicalspin-squeezed states [30,31,41]; see the discussion below),(�Jy)
2k = 〈ψk|J 2
y |ψk〉 − 〈ψk|Jy |ψk〉2 is the variance of Jy(and analogous definition for the other spin operators), the|μ〉y are eigenstates of Jy with possible eigenvalues μ =−N/2, −N/2 + 1, . . . , N/2, and N provides the normal-ization. Note that the spin-squeezing gate creates entangle-ment among the Nk ancilla qubits [29,42]. It can imple-mented in a variety of ways and experimental systems; seeRefs. [30,31] for reviews.
The phase encoding is obtained from a controlled-U gateapplied to each qubit. The gate gives a phase shift θ tothe qubit in the state |1〉, while leaving |0〉 unchanged [3–5]: more explicitly, Uc|1〉|u〉 = eiθ |1〉|u〉 and Uc|0〉|u〉 =|0〉|u〉, where |u〉 is an eigenstate of U that is stored inthe register and eiθ is the corresponding eigenvalue. InFig. 1 we follow the conventional representation of thecontrolled-U gate by a box on the register connectedto a single qubit [4]: the corresponding transformationto Nk qubits is simplified, for illustration reasons, to asingle box that we indicate as U⊗Nk
c . Overall, the U⊗Nkc
operation applied to |ψk〉|u〉 (yellow line in Fig. 1) isequivalent to a collective spin rotation of the state |ψk〉by the angle θ around the z axis, namely U⊗Nk
c |ψk〉|u〉 =e−iθ Jz |ψk〉|u〉 [43]. Note that, in Fig. 1, each horizontalthick black line follows the collective transformation of Nkqubits. In principle, there should be Nk vertical yellow linesconnecting a single-qubit controlled-U gate to each singlequbit of the state |ψk〉: for simplicity, we have representedthe transformation U⊗Nk
c as a single thick yellow line inthe figure. The algorithm requires, in total, NT single-qubitcontrolled-U gates.
The readout consists of (i) a collective spin rota-tion of the state around the z axis, implementing thephase feedback (see the discussion below); (ii) a col-lective π/2 rotation around the x axis, R⊗Nk
x = e−iπ Jx/2;and (iii) a final measurement of Jz, with possible result−Nk/2 ≤ μk ≤ Nk/2. At the kth step of the algorithm, thestate before the projective measurement is thus |ψout
k 〉 =e−iπ Jx/2e−iθk Jz |ψk〉, where θk is the overall rotation anglearound the z axis, given by the sum of θ and the phasefeedback angle. The average value of Jz on the output statedepends sinusoidally on θk: 〈J out
z 〉k ≡ 〈ψoutk |Jz|ψout
k 〉 =〈Jx〉k sin θk with 〈Jx〉k = 〈ψk|Jx|ψk〉. The phase estimationprotocol at each step of the QPE algorithm is based on thestandard method of moment [30]: replacing 〈J out
z 〉 with theresults μk of a single measurement of Jz on the output stateleads to the estimate
�k(μk) = arcsinμk
〈Jx〉k
(3)
of θk. The sensitivity of the estimator �k is quantifiedby the mean squared error, (��k)
2 = ∑Nk/2μk=−Nk/2
P(μk|θk)
[�k(μk)− θk]2, where P(μk|θk) = |〈ψoutk |μk〉|2 is the
probability of obtaining the result μk for a givenθk. Using error propagation [30], we have (��k)
2 ≈(�J out
z )2k/(d〈J outz 〉k/dθk)
2, where (�J outz )2k = 〈(J out
z )2〉k −〈J out
z 〉2k . This gives
(��k)2 = (�Jy)
2k
〈Jx〉2k
+ (�Jx)2k
〈Jx〉2k
tan2 θk, (4)
where the (�Jx,y)2k are spin variances calculated on
the state |ψk〉, and we have taken into account thefacts that 〈Jz〉k = 0 and 〈Jz Jx〉k = 0 for the GSSEq. (2). The spin moments appearing in Eq. (4) canbe calculate analytically; see Appendix A for details.For s2
kNk � 1, we find 〈Jx〉k = Nke−1/(2s2kNk)/2, 〈Jy〉k =
〈Jz〉k = 0, 〈J 2x 〉k = N 2
k (1 + e−2/(s2kNk))/8, 〈J 2
y 〉k= Nks2
k/4,
and 〈J 2z 〉k = N 2
k (1 − e−2/(s2kNk))/8. The first term in Eq.
(4) is the Wineland spin-squeezing parameter [41], ξ 2R =
s2ke1/s2
kNk , corresponding to the phase uncertainty at θk = 0.
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LUCA PEZZÈ and AUGUSTO SMERZI PRX QUANTUM 2, 040301 (2021)
For s2kNk � 1, ξ 2
R decreases with sk and reaches the min-imal value e/Nk for s2
kNk = 1. The second term in Eq.(4) is due to the bending of the squeezed states inthe Block sphere [see, e.g., Fig. 1(b)], (�Jx)
2k/〈Jx〉2
k =2 sinh2[1/(2s2
kNk)]. This term increases when decreasing
sk [in particular (�Jx)2k/〈Jx〉2
k ≈ 1/(2s4kN 2
k ) for s2kNk � 1],
and is responsible for the strong dependence of (��k)2 on
θk. This contribution prevents the use of a single (highly)squeezed state for phase estimation: a problem solved hereusing the adaptive algorithm of Fig. 1.
The key operation of the algorithm is the phase feedback(represented by pink lines and boxes in Fig. 1): after phaseencoding, the state e−iθ Jz |ψk〉 is sequentially rotated byeiJz�0 eiJz�1 · · · eiJz�k−1 ≡ ⊗k−1
j =0 Rz(−�j ), with �j beingthe estimated value at the step j < k (depending of themeasurement result μj ), Eq. (3). In particular, the overallrotation angle of state (2) is θk = θ − ∑k−1
j =0 �j for k ≥ 1,and θ0 = θ . The feedback rotation places the GSS |ψk〉close to its optimal sensitivity point, namely on the equa-tor of the generalized Block sphere; see Fig. 1(b). Theestimated value �k(μk) at the kth step is then used toimplement the adaptive phase rotation at the (k + 1)th step,and so on.
After K steps the circuit outputs are K + 1 val-ues �0, . . . ,�K : we take their sum, θest(μ0, . . . ,μK) =∑K
k=0�k(μk), as the estimate of the unknown θ . Notethat the circuit described above leads to an estimate of θwithin the inversion region [−π/2,π/2] of the arcsin func-tion, Eq. (3). The algorithm can be extended to the fullrange θ ∈ [−π ,π) by a slight modification of the zerothstep only: in this case, the state |CSS(N0)〉 is rotated bye−i(θ/2)Jz (instead of e−iθ Jz ). The estimator at the zeroth stepis now �0(μ0) = 2 arcsin(2μ0/N0) ∈ [−π ,π), which hasa sensitivity (��0)
2 = 4/N0: the factor 4 being a directconsequence of the factor 2 dividing the rotation angle.Below, we analyze the sensitivity of the QPE algorithmfor the estimation of an arbitrary θ ∈ [−π ,π).
III. PHASE SENSITIVITY
The sensitivity of the QPE algorithm after K steps isquantified by the mean squared error of θest, which weindicate as (�θest)
2. As detailed in Appendix B, (�θest)2
is given by
(�θest)2 =
∫
dθK P(θK)(��K)2 = E[(��K)
2], (5)
where P(θK) is the statistical distribution of θK (corre-sponding to the overall rotation angle in the last stepof the algorithm) and E[·] indicates statistical averaging.This equation is qualitatively understood by the fact thatθest − θ = ∑K
k=0�k − θ = �K − θK such that the sensi-tivity of the full QPE algorithm is thus given by the average
sensitivity in the estimation of θK . Indeed, it should benoted that θK depends on the stochastic values�j at all pre-vious j < K steps. In other words, θK is a random variableand, according to Eq. (5), the sensitivity of the overall QPEalgorithms is given by averaging (��K)
2 over all possi-ble values of θK . As shown in Appendix B, the quantitiesE[(��k)
2] for k = 0, . . . , K are related via the recursiverelation
E[(��k)2] = (�Jy)
2k
〈Jx〉2k
+ (�Jx)2k
〈Jx〉2k
E[(��k−1)2] (6)
with initial condition E[(��0)2] = 4/N0. Equation (6) can
be eventually generalized in the presence of noise by calcu-lating the spin moments for the noisy states, as illustratedbelow. Replacing the analytical expression for the collec-tive spin moments in Eq. (6), and assuming that s2
kNk � 1[44], we obtain
E[(��k)2] = s2
k
Nk+ E[(��k−1)
2]2s4
kN 2k
. (7)
First, we minimize Eq. (7) with respect to sk, which gives
s2kNk = {N 2
k E[(��k−1)2]}1/3. (8)
This equation is also understood as a recursive relation giv-ing the squeezing parameters s1, s2, . . . , sK as a function ofN0, N1, . . . , NK , with initial condition s0 = 1 (for the coher-ent spin state). Using the optimal values of s1, . . . , sK givenin Eq. (8), we have
mins1,...,sK
E[(��K)2] =
(32
)(3/2)(1−1/3K )
× 41/3K
N 4/3K N 4/9
K−1 · · · N 4/3K
1 N 1/3K
0
, (9)
which can now be optimized as a function of N0, N1, . . . ,NK with the constraint of a fixed total number of qubitsNT = ∑K
k=0 Nk. Replacing N0 = NT − ∑Kk=1 Nk in Eq. (9)
and taking the derivative with respect to N1, . . . , NK givesa set of K linear equations that can be recast in the matrixform (A + uu ) · x = NTu, where x = (NK , . . . , N1), u =(1, 1, . . . , 1) and
A = 14
⎛
⎜⎜⎜⎜⎜⎝
1/3K−1 0 0 · · · 00 1/3K−2 0 · · · 00 0 1/3K−3 · · · 0...
......
. . ....
0 0 0 · · · 1
⎞
⎟⎟⎟⎟⎟⎠
.
The solution of the linear set of equation is foundusing the Sherman-Morrison formula, giving x = [NT/
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QUANTUM PHASE ESTIMATION... PRX QUANTUM 2, 040301 (2021)
(2 × 3K − 1)]A−1u. This leads to the optimal number ofparticles in each state |ψk〉,
Nk = 4 × 3k−1N0, (10)
and the optimal squeezing parameter
s2k = 35/2−(3/2)(1/3k−1)−k
2(5/2)(1−1/3k−1)
1
N 1−1/3k
0
, (11)
for k = 1, . . . , K . In particular, the first step of the protocolis implemented with a GSS containing N1 = 4N0 parti-cles and, at each further step, the number of particles isincreased by a factor 3: Nk+1 = 3Nk for k = 1, . . . , K − 1.The protocol uses GSSs that are more and more squeezed(namely, with sk decreasing with k) as the number ofsteps increases, while s2
kNk saturates to the asymptoticvalue
√27/2 for k � 1. Finally, using Eqs. (9) and (10)
and N0 = NT/(2 × 3K − 1) (which is obtained by sum-ming the geometric series NT = ∑K
k=0 Nk), we obtain theanalytically optimized sensitivity
(�θest)2 = α2
K
N 2−1/3K
T
. (12)
This equation very rapidly (in the number of steps K)approaches the Heisenberg scaling [45] with respect to thetotal number of qubits and with a prefactor that convergesto αK→∞ ≈ 4. It is also worth noting that the sensitivityis exponential in the number of steps K ∼ log3 NT, while,with standard QPE protocols, K ∼ log2 NT [1,2,18,23].Already for K = 3 (which corresponds to a four-stepalgorithm using one coherent spin state and three spin-squeezed states with decreasing sk) we obtain a sensitivityO(N−53/27
T = N−1.96T ), which is very close to the Heisen-
berg scaling. Further details on the analytical minimizationleading to Eq. (12) are reported in Appendix C.
In Fig. 2(a) we show the results of numerical MonteCarlo simulations of our QPE algorithm where θ is chosenrandomly in [−π ,π) with a flat distribution. The sensi-tivity �θest approaches the Heisenberg scaling, Eq. (1), asa function of the number of qubits (or, equivalently, thenumber of steps K + 1), independently of the initial valueof N0. This is confirmed by NT ×�θest approaching theconstant value 4 for large NT and different initial N0 [col-ored symbols in Fig. 2(a)]. Here, parameters sk and Nk arechosen by a numerical optimization of E[(��k)
2], recur-sively for k = 1, . . . , K . The expected results using Eq. (5)(colored dashed lines, corresponding to dierent values ofN0) agree well with ab-initio numerical simulations of thealgorithm (symbols). Circles are obtained for N0 = 100and after K = 13 steps of the algorithm. The solid line is4/NT.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0
0.2
0.4
0.6
0.8
1.0
102 104 106 108
101
102
NT
P (| θ−θ est| ≥
ϵ )ϵNT /4
−5 0 5NT(θ − θest)/π
P(θ−θ est)
(a)
(b)
θ/π
K = 8
−5 0 5NT(θ − θest)/π
K = 13
N T×Δθ
est
0 1−1
4
8
0N T×Δθ
est
FIG. 2. Phase sensitivity of the QPE algorithm. (a) Phase sen-sitivity �θest as a function of the total number of particles NTand for different initial values of N0: N0 = 102 (red lines andsymbols), N0 = 103 (green), and N0 = 104 (blue). Symbols areresults of numerical Monte Carlo simulations where θ is cho-sen randomly in [−π ,π). The dotted lines are guides to theeye connecting results of a numerical optimization of Eq. (5)over sk and Nk for k = 1, . . . , K . The gray line is 2/
√NT while
the horizontal solid line is �θest = 4/NT. The inset shows �θestas a function of θ . Numerical data (circles) correspond to thecase K = 13 and N0 = 100; the solid line is �θest = 4/NT. (b)Error probability P(|θ − θest| ≥ ε) as a function of εNT/4. Sym-bols are Monte Carlo results obtained with N0 = 102 and K = 8(crosses), K = 10 (stars), K = 13 (circles). The solid gray lineis Eq. (13). The insets show the distributions P(θ − θest) forN0 = 100 and different values of K . Results in all panels areobtained from 104 independent repetitions of the protocol.
In Fig. 2(b) we analyze the behavior of the estimatorθest. As shown in the insets of Fig. 2(b), the distributionof θ − θest is—to a very good approximation—a Gaussiancentered in 0, namely the estimator is statistically unbiased(note that the histograms are obtained for 104 independentrepetitions of the algorithm for random θ ). Furthermore,the probability of making an error ε in the estimation of anarbitrary θ is
P(|θest − θ | ≥ ε) = 1 − erf[εNT/(4√
2)], (13)
where erf(x) is the error function; see Fig. 2(b). The errorprobability is thus exponentially small with NT.
So far, we have illustrated the protocol with quan-tum states having a fixed and known total number of
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LUCA PEZZÈ and AUGUSTO SMERZI PRX QUANTUM 2, 040301 (2021)
particles. In some experimental implementations, forinstance with ultracold atoms, it is possible to fix the num-ber of particles only in average, with a fluctuating numberof atoms in a single realization. As shown in AppendixD, our protocol is unaffected by these fluctuations. Wealso note that in some applications, as in atomic clocks[14,15,49], it is necessary to maximize the sensitivity byimplementing each round of the protocol with the maxi-mum possible number of particles available, namely Nk =N for all k [50].
IV. IMPACT OF DECOHERENCE
While GSSs are more robust to GHZ states, they are cer-tainly not immune to noise: the aim of this section is toshow how the QPE algorithm generalizes in the presenceof noise and decoherence. Interestingly, the algorithm canbe optimized numerically (and analytically, in some limits)to take into account possible experimental imperfections.As an illustration, we assume the creation of ideal GSSsthat are affected by noise before being used in the QPEalgorithm and assume noiseless phase rotations (which aretypically implemented on time scales much shorter thansqueezed-state preparations). We first consider collectivedephasing along a specific axis. In this case, the phase sen-sitivity scaling �θest = O(N−1
T ) is also recovered in thepresence of strong noise, provided that the GSSs are prop-erly aligned with respect to the dephasing axis. Second, weconsider full depolarization of the ideal GSSs. In this case,as a main effect, noise imposes a lower bound on the small-est available squeezing parameter s of the noisy state. Thismimics, in practice, the effect of noise in the creation ofGSSs in experimental systems.
Collective dephasing along an arbitrary axis n in theBloch sphere is described by the transformation
�n[|ψk〉] =∫ π
−πdφP(φ)e−iφJn |ψk〉〈ψk|eiφJn . (14)
This provides a stochastic rotation e−iφJn with an angle φdistributed with probability P(φ), where |ψk〉 is the idealGSS, Eq. (2). Without loss of generality (upon a furtherproper rotation of the state |ψk〉) we consider depolariza-tion along the n = y axis. The noise leaves unchanged themoments of Jy—in particular, it does not affect the squeez-ing along the y axis—but it decreases the length of thecollective spin 〈Jx〉 and increases the bending of the state inthe sphere, namely (�Jx)
2. These effects can be taken intoaccount by replacing, in the recursive relation Eq. (6), thespin moments calculated for the noisy state. In particular,the parameters Nk and sk can be optimized in the pres-ence of noise (see Appendix D for details). Results of thenumerical simulations (symbols) are shown in Figs. 3(a)and 3(b). These are compared with the expected behaviour[dashed lines in Fig. 3(a) and solid line in Fig. 3(b)]. For a
(a) (b)
(c) (d)
FIG. 3. Phase sensitivity of the QPE in the presence of noise.Panels (a) and (b) consider the effect of collective dephasingdescribed by Eq. (14) with P(φ) ∝ eγ cosφ . Panel (a) reports�θestas a function of the total number of particles NT and for differ-ent values of γ : γ = 10 (red lines and symbols), 4 (green), 2.5(blue), and 2 (black). Symbols are results of Monte Carlo simu-lations, while dotted lines are expected results (see Appendix D).Panel (b) shows the prefactor αdeph
∞ of the Heisenberg scaling inEq. (15) as a function of γ . Circles are results of Monte Carlosimulations (obtained asymptotically in the number of steps); thesolid line is the expected result—see Appendix D. In panels (c)and (d) we consider full depolarization. Panel (c) shows �θestas a function of NT with the symbols representing the results ofsimulations for different values of D: D = 10−4 (blue circles),10−6 (green), and 10−8 (red). Lines are the analytical results (seethe text). The black dashed line is the noiseless case. Asymp-totically in NT, the sensitivity follows Eq. (17) with prefactorβD shown in panel (d) as a function of D. Here the blue lineis βD = 1.75 × D1/4, obtained from a fit. Red circles are resultsof numerical simulations.
sufficiently large number of steps, the protocol reaches theHeisenberg scaling (for K � 1)
�θest = αdeph∞NT
, (15)
where the prefactor αdeph∞ is determined by the coeffi-
cients∫ π−π dφ cos(2λφ)P(φ), with λ = 0, 1; see Appendix
D. Note that the Heisenberg scaling (15) is recoveredeven when the width of P(φ) is of the order of 2π ,highlighting the robustness of the QPE algorithm to thissource of noise. This is in contrast with QPE protocolsimplemented with GHZ states, where there is no pre-ferred rotation axis that guarantees robustness to dephasingnoise [51]. On the other hand, if the spin-squeezed state isnot properly aligned with respect to the noise axis, noisewill impose a lower bound to the squeezing parameter sand the overall effect associated with dephasing will be
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QUANTUM PHASE ESTIMATION... PRX QUANTUM 2, 040301 (2021)
similar to that of full depolarization, which is discussedbelow.
Full depolarization (within the symmetric subspace ofdimension N + 1) is described by the transformation
�D[|ψk〉] = (1 − D)|ψk〉〈ψk| + D
N + 11, (16)
where 0 ≤ D ≤ 1 is the depolarization parameter. Thisnoise affects the spin moments along all directions and,in particular, it imposes a lower bound to the smallestachievable squeezing parameter s2 ≤ s2
min = DN/3; seeAppendix D. Simulations of the QPE in the presence of fulldepolarization are shown in Figs. 3(c) and 3(d). In general,any noise in the state preparation that limits the prepara-tion of the highly squeezed state, e.g., particle losses (seeAppendix D), would have an effect similar to that of fulldepolarization. The QPE protocol in the presence of noisefollows the ideal (noiseless) phase estimation sensitivityup to a number of steps k � k (corresponding to a numberof particles Nk ∼ 4 × 3kN0) for which sk ≥ smin. The QPEprotocol is robust in the sense that, if D � 1 such thatk � 1, we recover the Heisenberg scaling (1) in a finiterange of the total number of particles. When k ≥ k, the sen-sitivity of the QPE algorithm saturates to the asymptoticvalue �θest ∼= √
D/3; see the dashed lines in Fig. 3(c).This saturation can be avoided by modifying the algorithm:the phase estimation is now repeated with multiple copiesof the squeezed state |ψk〉 of Nk particles and squeezingparameter sk and without phase feedback between differ-ent steps. Asymptotically in NT the algorithm reaches asensitivity (see Appendix D)
�θest = βD√NT
(17)
with a prefactor βD ∝ D1/4 that is smaller that 1 (whilefor separable qubit states, we would have�θest = 1/
√NT).
The asymptotic behavior Eq. (17) is in qualitative agree-ment with the no-go theorem of Refs. [25,26] that preventsreaching the Heisenberg limit for an asymptotically largenumber of particles and for the noise model of Eq. (16).Symbols in Fig. 3(c) show results of numerical MonteCarlo simulations, in excellent agreement with analyticalcalculations. In panel (d) we show βD as a function of Dwhere the results of simulations (circles) are compared tothe expected scaling with D (solid line).
V. NONLINEAR READOUT AGAINSTDETECTION NOISE
Using a nonlinear readout protocol [52–56], the QPEalgorithm can be made robust against detection noise,which is one of the most important sources of noisein atomic experiments targeting entanglement-enhanced
Gai
n (d
B)
(b)
(c)
Nonlinear readoutState preparation Phase encoding
Gai
n (d
B)
(a)
FIG. 4. Phase sensitivity of the QPE in the presence of detec-tion noise. (a) Modified kth step of the QPE algorithm (for k ≥ 1)including the OAT gate (indicated as τk, orange) and its inverse(−τk); see the text. Panels (b) and (c) show the sensitivity gainin the presence of detection noise. In panel (b), we consider aconstant σDN at all steps (namely independently from the numberof particles). In panel (c), we consider σDN = γDN
√Nk, which
depends on the number of particles at each step. The solid blackline is a semianalytical prediction (see Appendix E); the cir-cles are results of numerical simulations for the estimation of arandom θ ∈ [−π ,π) repeated 104 times. The horizontal dashedline corresponds to �θest = 4/NT. The vertical axis in panels (b)and (c) shows the gain in decibels with respect to the standardquantum limit, namely −10 log10[(�θest)
2NT].
sensitivities. Nonlinear readout has been demonstratedexperimentally for atomic squeezed states in Ref. [55]:it consists of a proper nonlinear evolution applied beforeparticle-number detection. In this case, each step of theprotocol of Fig. 1 is modified as shown in Fig. 4(a). Thespin-squeezing gate is generated by applying a one-axis-twisting (OAT) evolution e−iτk J 2
z [57] [represented by theorange box indicated as −τk in Fig. 4(a)] to an initialcoherent spin state, where τk is an effective nonlinear-evolution time. The readout here consists of four trans-formations: (i) the phase feedback discussed previously(pink box); (ii) an inverse OAT transformation, e+iτk J 2
z
(orange box indicated as −τk), (iii) a rotation of π/2around the x axis, R⊗Nk
x = e−iπ Jx/2 (bluebox), and, finally,(v) a measurement of Jz. Notice that, here, we considerphase encoding and feedback implemented as rotationsaround the y axis, corresponding to the overall transfor-mation e−iθk Jy . This can be implemented as the transfor-mation e−iθk Jz , as discussed in Fig. 1, combined with addi-tional rotations around the x axis [not explicitly shown inFig. 4(a)] : e−iθk Jy = ei(π/2)Jx e−iθk Jz e−i(π/2)Jx . Precisely, the
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LUCA PEZZÈ and AUGUSTO SMERZI PRX QUANTUM 2, 040301 (2021)
output state before the final projective measurement is nowgiven by
|φoutk (Nk, τk, θk)〉 = e−i(π/2)Jx e+iτk J 2
z e−iθk Jy |φk(Nk, τk)〉,(18)
where |φk(Nk, τk)〉 = e−iτk J 2z |CSS(Nk)〉 is the state after the
OAT gate. The rotation angle θk is estimated by invert-ing the relation 〈J out
z 〉k = 〈φoutk |Jz|φout
k 〉 as a function of θk,which is evaluated numerically; see Appendix E.
We have studied the QPE protocol with nonlinear read-out. It should be noted that, while the advantage of nonlin-ear readout has previously been analyzed for the estimationof a phase shift θ ≈ 0 [52–56], we extend it here for theestimation of an arbitrary phase θ ∈ [−π ,π) with sen-sitivity robust against detection noise. The protocol isoptimized numerically over Nk and τk at each step (seeAppendix E) and, in the noiseless case, it reaches perfor-mances analogous to those of the QPE protocol of Fig. 1.The advantage of the nonlinear readout is obtained whenincluding detection noise. We model the noisy detection bythe convolution P(μk|θk) = ∑Nk/2
ν=−Nk/2PDN(μk|ν)P(ν|θk),
where P(ν|θk) = |z〈ν|φoutk (Nk, τk, θk)|2 is the probability
that a measurement of Jz gives results ν, and PDN(μk|ν) ≈e−(ν−μ)2/(2σ 2
DN) is a Gaussian convolution function, whereσDN is the detection noise parameter. In Figs. 4(b) and 4(c)we report the results of numerical simulations of the QPEalgorithm showing the sensitivity (�θest)
2 as a functionof σDN. In particular, in panel (b), σDN is assumed con-stant for all steps of the protocol, while in panel (c) wehave taken a detection noise that depends on the numberof particles Nk at each step, namely σDN = γDN
√Nk for
the step k. In the two panels we plot (�θest)2 for the
estimation of an arbitrary phase θ ∈ [−π ,π) using anoptimized (K = 4)-step protocol. Numerical simulations(red circles) agree very well with expected results (solidline) and show that a large gain with respect to the stan-dard quantum limit is also possible in the presence of largedetection noise (note that in current experiments σDN =5 ÷ 20 [30], approximately independent from the numberof particles).
VI. DISCUSSION AND CONCLUSIONS
To summarize, our QPE algorithm for the estimationof an arbitrary phase θ ∈ [−π ,π) uses Gaussian spinstates and an adaptive measurement protocol. It reachesa sensitivity scaling �θest = O(1/NT) with respect to thetotal number of qubits NT or, equivalently, the total num-ber of applications of a controlled-U gate. There are twomain differences with respect to the standard QPE algo-rithms [1,2] using single ancilla qubits [18,24], includingthose based on the inverse quantum Fourier transform[3,4,6]. (i) The number of controlled-U gates in the GSS
algorithm of Fig. 1 scales linearly with NT rather thanexponentially. This means that, if we take into accountthe time tU necessary to implement a single controlled-Ugate, the GSS algorithm is exponentially faster. This prop-erty is also shared by QPE algorithms using GHZ states[22,23] that are however very fragile to noise. The shorttime required by the algorithm makes it well suited for theestimation of time-varying signals. More explicitly, if weindicate with α = dθ(t)/dt the local slope of the signal,we obtain the condition α � O(tUN 2
T )−1 for the Heisen-
berg limited estimation of the time varying θ with theGSS algorithm, compared to the more stringent conditionα � O(tUNTeNT)−1 for the QPE algorithms using singleancilla qubits. (ii) Differently from Kitaev’s QPE proto-col [1,2], the phase estimation at each step of the GSSalgorithm is based on a single measurement. The knowl-edge about θ is progressively sharpened using GSSs witha higher and higher number of particles and decreasingsqueezing parameters. The key operation of the algorithmis the phase feedback that can be understood as an adap-tive measurement [58,59] able to place the GSS aroundits most sensitive point. Differently from common adap-tive strategies [58,59], our protocol uses states of unevenlydistributed qubits as a resource for phase estimation. Inparticular, the analytical optimization provided by Eqs.(10) and (11) allow to fully predetermine the number ofparticle and squeezing at each step. Therefore, the adap-tive measurement in the GSS algorithm does not requirethe numerical optimization of states, operations, and/orcontrol phases [23,24], nor the support of any classicalmemory to store the phase distribution [18]. The over-all computational time associated with the implementa-tion of the phase feedback in an algorithm of K stepsis (K + 1)× tF , where tF is the time needed for a sin-gle phase feedback, which can be safely assumed to bethe same for each step of the protocol. According to Eq.(12), the phase uncertainty becomes exponentially closeto the Heisenberg scaling by requiring a linear increaseof computational time associated with the phase feedbackimplementation.
Commonly to all QPE algorithms, our protocol usescontrolled-U gates to map a quantity of interest to a phaseto be estimated. The GSS algorithm thus shares all knownapplications of QPE [5,7,8], while using noise-resilientquantum states. The use of GSSs, which are routinelycreated in labs, can open a novel route for experimentswith cold and ultracold atoms toward applications in quan-tum computing, quantum computational chemistry, andquantum simulation.
ACKNOWLEDGMENTS
We acknowledge financial support from theEuropean Union’s Horizon 2020 research and innovationprogramme—Qombs Project, FET Flagship on QuantumTechnologies Grant No. 820419.
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QUANTUM PHASE ESTIMATION... PRX QUANTUM 2, 040301 (2021)
APPENDIX A: ANALYTICAL CALCULATION OFTHE SPIN MOMENTS
We consider a Gaussian state of N particles
|ψ(N , s)〉 = 1√N
N/2∑
μ=−N/2
e−μ2/(s2N )|μ〉y , (A1)
where s is a squeezing parameter (s < 1 for metrologicalspin-squeezed states; see below), the |μ〉y are eigenstatesof Jy (Jy |μ〉y = μ|μ〉y), and N = ∑N/2
μ=−N/2 e−2μ2/(s2N ) isthe normalization. We calculate mean values and variancesof the collective spin operators Jx,y,z as a function of Nand s.
1. First moments
We have
〈Jx〉 =N/2∑
μ=−N/2
e−μ2/(s2N )e−(μ+1)2/(s2N )
N
×√
N2
(N2
+ 1)
− μ(μ+ 1). (A2)
For s2N � 1, we can replace the sum over μ with an inte-gral in a continuous variable m. For s � √
N , we canneglect boundary effects and extend the integration from−∞ to +∞. We write
〈Jx〉 ≈√
N2
(N2
+ 1)√
2πs2N
∫ +∞
−∞dm e−m2/(s2N )
× e−(m+1)2/(s2N )
√
1 − m(m + 1)(N/2)(N/2 + 1)
.
Since the Gaussian wave function has a width of approxi-mately equal to s2N , it constraints the variable m to values|m| ∼ s
√N � N : we can thus expand in Taylor the square
roots in the above equation to obtain
〈Jx〉 ≈√
N2
(N2
+ 1)√
2πs2N
∫ ∞
−∞dm
× e−m2/(s2N )e−(m+1)2/(s2N )
×[
1 − 12
m(m + 1)(N/2)(N/2 + 1)
+ O(
m(m + 1)(N/2)(N/2 + 1)
)2]
≈√
N2
(N2
+ 1)[
1 − 1(N/2)(N/2 + 1)
s2N − 18
]
× e−1/(2s2N ).
In particular, in the case s = 1 we recover the coher-ent spin state result 〈Jx〉 = N/2 up to corrections ofthe order of 1/N . When s2N ≈ 1, the discretenessof μ becomes important and the continuous-variableapproximation breaks down. For s2N � 1, the overlapbetween Gaussians in Eq. (A2) is well approximatedby e−μ2/(s2N )e−(μ+1)2/(s2N ) ≈ e−1/(s2N )(δμ,0 + δμ,−1), andreplacing is into Eq. (A2), we obtain 〈Jx〉 ≈ 2e−1/(s2N )√(N/2)[(N/2)+ 1]. Overall, keeping only the leading
term in N , we have
〈Jx〉 ≈⎧⎨
⎩
N2
e−1/(2s2N ) for s2N � 1,
Ne−1/(s2N ) for s2N � 1.(A3)
The other spin moments are equal to zero due to thesymmetry properties of the Gaussian state (A1):
〈Jy〉 = 0 and 〈Jz〉 = 0. (A4)
In Fig. 5(a) we plot 〈Jx〉 as a function of s. Dots areexact values, the solid line is the analytical approxima-tion for s2N � 1, while the dashed line is the analyticalapproximation for s2N � 1.
2. Second moments
For the second moment of the Jx spin operator, we have
〈J 2x 〉 = N
4+
(N 2
8− s2N
8
)
+ 12
N/2∑
μ=−N/2
e−μ2/(s2N )e−(μ+2)2/(s2N )
N
√N2
+ μ+ 2
×√
N2
+ μ+ 1
√N2
− μ
√N2
− μ− 1. (A5)
For s2N � 1, a continuous-variable approximation and aTaylor expansion yield
〈J 2x 〉 ≈ N
4
(N2
+ 1)
(1 + e−2/(s2N ))− s2N8
.
We recover the coherent spin state result 〈J 2x 〉 = N 2/4
for s = 1 up to O(1/N ). In the opposite limit s2N �1, we consider e−μ2/(s2N )e−(μ+2)2/(s2N ) ≈ e−4/(s2N )δμ,−2 +e−2/(s2N )δμ,1 + e−4/(s2N )δμ,0 in Eq. (A5) and get
〈J 2x 〉 ≈ N 2
8(1 + e−2/(s2N ))+ N
4
(
1 − s2
2
)
.
Overall, keeping only the leading term in N , we find that
〈J 2x 〉 ≈ N 2
8(1 + e−2/(s2N )) (A6)
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LUCA PEZZÈ and AUGUSTO SMERZI PRX QUANTUM 2, 040301 (2021)
1.0(a) (b) (c)
(d)(e) (f)
FIG. 5. Spin moments as a function of the squeezing parameter s. Dots are exact values and lines are analytical expressions: solidlines are for s2N � 1 while dashed lines are for s2N � 1. The insets in panels (c) and (e) are enlargements around s2N = 1. The spinsqueezing parameter in panel (f) is ξ 2 = N (�Jy)
2/〈Jx〉2. In all panels N = 105 and the vertical black dotted line is s2N = 1.
for both s2N � 1 and s2N � 1. In Fig. 5(b) we plot 〈J 2x 〉 as
a function of s (dots), compared to the analytical approx-imation (solid line). The calculation of 〈J 2
y 〉 is straightfor-ward,
〈J 2y 〉 = s2N
4. (A7)
Finally, the calculation of 〈J 2z 〉 is similar to the calculation
of 〈J 2x 〉 and we have
〈J 2z 〉 = N
4+
(N 2
8− s2N
8
)
− 12
N/2∑
μ=−N/2
e−μ2/(s2N )e−(μ+2)2/(s2N )
N
√N2
+ μ+ 2
×√
N2
+ μ+ 1
√N2
− μ
√N2
− μ− 1,
giving
〈J 2z 〉 ≈ N 2
8(1 − e−2/(s2N )) (A8)
for both s2N � 1 and s2N � 1, to the leading order in N .
3. Variances and covariances
Combining Eqs. (A3) and (A6) we have
(�Jx)2 ≈
⎧⎪⎪⎨
⎪⎪⎩
N 2
8(1 − e−1/(s2N ))2 for s2N � 1,
N 2
8(1 + e−2/(s2N ))− N 2e−2/(s2N ) for s2N � 1,
(A9)
while combining Eqs. (A4) and (A7),
(�Jy)2 = s2N
4, (A10)
and from Eqs. (A4) and (A8),
(�Jz)2 ≈ N 2
8(1 − e−2/(s2N )) (A11)
for both s2N � 1 and s2N � 1. In Figs. 5(c) and 5(d), weplot (�Jx)
2 and (�Jz)2, respectively, as a function of s,
while in Fig. 5(e) we plot the ratio (�Jx)2/〈Jx〉2
. Finally,the covariances are identically zero,
〈{Jx, Jy}〉 = 0, 〈{Jx, Jz}〉 = 0, and 〈{Jy , Jz}〉 = 0,(A12)
due to the symmetry of the state (A1).
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QUANTUM PHASE ESTIMATION... PRX QUANTUM 2, 040301 (2021)
4. Spin-squeezing parameter
The metrological spin-squeezing parameter is obtainedby taking the ratio between Eqs. (A10) and (A3):
ξ 2 = N (�Jy)2
〈Jx〉2 ={
s2e1/(s2N ) for s2N � 1,s2
4 e2/(s2N ) for s2N � 1.(A13)
The function s2e1/(s2N ) reaches a minimum at s2N = 1.Therefore, for 1/N � s2 < 1, the spin-squeezing param-eter is always decreasing with s, reaching a minimumvalue ξ 2
min = e/N . For s2N � 1, the exponential term inEq. (A13) dominates and the spin-squeezing parameterincreases with s: in other words, the decrease of 〈Jx〉 withs dominates over the squeezing of (�Jy)
2. In Fig. 5(f)we compare the above approximate expressions with theexact calculation of ξ 2, as a function of the squeezingparameter s.
APPENDIX B: DERIVATION OF EQS. (5) AND (6)
The sensitivity of the estimator θest(μ0, . . . ,μK) =∑Kk=0�k(μk) is formally given by
(�θest)2 =
∑
μ0,...,μK
P(μ0, . . . ,μK |θ)[θest(μ0, . . . ,μK)− θ ]2,
(B1)
where μk is the measurement result at the kth step of thealgorithm (with k = 0, 1, . . . , K) and the sum runs overall possible measurement results at all steps. The differ-ent measurement events are independent and, taking intoaccount the frequency feedback, we can write
P(μ0, . . . ,μK |θ) = P(μ0, . . . ,μK−1|θ)× P[μK |θK(μ0, . . . ,μK−1)],
where θK(μ0, . . . ,μK−1) = θ − ∑K−1k=0 �k(μk). As
noted in the main text, θest(μ0, . . . ,μK)− θ = �K
(μK) − [θ − ∑K−1k=0 �k(μk)] = �K(μK) − θK (μ0, . . . ,
μK−1). Equation (B1) can thus be rewritten as
(�θest)2 =
∑
μ0,...,μK−1
P(μ0, . . . ,μK−1|θ)
×∑
μK
P[μK |θK(μ0, . . . ,μK−1)]
× [�K(μK)− θK(μ0, . . . ,μK−1)]2. (B2)
As a consequence of the stochastic nature of the mea-surement events, the phase θK(μ0, . . . ,μK−1) is also astochastic variable with distribution
P(θK) =∑
μ0,...,μK−1
P(μ0, . . . ,μK−1|θ)
× δ[θK(μ0, . . . ,μK−1)− θK ], (B3)
where δ is the Dirac delta function. Using Eq. (B3), we canthus rewrite Eq. (B2) as
(�θest)2 =
∫
dθK P(θK)∑
μK
P(μK |θK)[�K(μK)− θK ]2
=∫
dθK P(θK)(��K)2
≡ E[(��K)2], (B4)
which recovers Eq. (5) of the main text. Next, we substi-tute Eq. (4) into Eq. (B4) and note that, since we expectθK � 1, we can approximate tan2 θK ≈ θ2
K . We thus have
E[(��K)2] = (�Jy)
2K
〈Jx〉2K
+ (�Jx)2K
〈Jx〉2K
∫
dθK P(θK)θ2K . (B5)
Note that θk(μ0, . . . ,μK−1) = θ − ∑K−1k=0 �k(μk) = θK−1
(μ0, . . . ,μK−2)−�K−1(μK−1) and, thus, using Eq. (B3),
∫
dθK P(θK)θ2K =
∑
μ0,...,μK−1
P(μ0, . . . ,μK−1|θ)[θK−1(μ0, . . . ,μK−2)−�K−1(μK−1)]2
=∑
μ0,...,μK−2
P(μ0, . . . ,μK−2|θ)∑
μK−1
P[μK−1|θK−1(μ0, . . . ,μK−2)]
× [θK−1(μ0, . . . ,μK−2)−�K−1(μK−1)]2.
Analogously as above, we can understand θK−1(μ0, . . . ,μK−2) as a stochastic variable with distribution
P(θK−1) =∑
μ0,...,μK−2
P(μ0, . . . ,μK−2|θ)δ[θK(μ0, . . . ,μK−2)− θK−1],
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LUCA PEZZÈ and AUGUSTO SMERZI PRX QUANTUM 2, 040301 (2021)
and write∫
dθK P(θK)θ2K
=∫
dθK−1P(θK−1)
×∑
μK−1
P(μK−1|θK−1)[θK−1 −�K−1(μK−1)]2
=∫
dθK−1P(θK−1)(��K−1)2
≡ E[(��K−1)2]. (B6)
Combining Eqs. (B5) and (B6), we have thus demonstratedthat
E[(��K)2] = (�Jy)
2K
〈Jx〉2K
+ (�Jx)2K
〈Jx〉2K
E[(��K−1)2].
Furthermore, we can write
E[(��K−1)2] = (�Jy)
2K−1
〈Jx〉2K−1
+ (�Jx)2K−1
〈Jx〉2K−1
×∫
dθK−1P(θK−1)θ2K−1, (B7)
and repeat the above reasoning to obtain the recursive for-mula (6). It should be noted that in the zeroth step of theprotocol, the phase is fixed (and not random), equal to θ :E[(��0)
2] = (��0)2 = 4/N0, which provides the initial
condition for the recursive formula (6).
APPENDIX C: DETAILS ON THE NUMERICALAND ANALYTICAL MINIMIZATIONS
When substituting the analytical expressions for theaverage spin operators and variances discussed above, Eqs.(A3), (A9) and (A10), for s2N � 1, into Eq. (6), we obtain
E[(��k)2] = 2s2
k + Nk(1 − e−1/(s2kNk))2E[(��k−1)
2]
2Nke−1/(s2kNk)
.
(C1)
We recover Eq. (7) when taking the leading order ins2
kNk � 1. The optimization of Eq. (C1) over sk leads to
s4kN 2
k − s2kNk + N 2
k
(e−2/(s2
kNk) − 12
)
E[(��k−1)2] = 0,
(C2)
which can be solved analytically in the limit s2kNk � 1,
giving Eq. (8). By replacing Eq. (8) into Eq. (7) gives
minsk
E[(��k)2] = 3
2
(E[(��k−1)
2]N 4
k
)1/3
. (C3)
Recursively to K steps of the algorithm, we obtain
mins1,...,sK
E[(��K)2] =
(32
)1+1/3+1/9+···+1/3K−1
× N−4/3K N−4/9
K−1 · · · N−4/3K
1
(N0
4
)−1/3K
.
(C4)
The above equation is optimized with respect tosk for k = 1, . . . , K (with s0 the initial condition).We now optimize it with respect to Nk for k =0, . . . , K , with the constraint of a total NT = ∑K
k=0 Nkparticles. Imposing (∂/∂Nk){1/mins1,...,sK E[(��k)
2]} = 0(with k = 1, . . . , K) provides the following set of K cou-pled linear equations:
(1
4 × 3K−1 + 1)
NK + NK−1 + NK−2 + · · · + N1 = NT,
NK +(
14 × 3K−2 + 1
)
Nn−1 + NK−2 + · · · + N1 = NT,
NK + NK−1 +(
14 × 3K−3 + 1
)
NK−2 + · · · + N1 = NT,
...
NK + NK−1 + NK−2 + · · · +(
1 + 14
)
N1 = NT.
These can be recast in matrix form as
(A + uu ) · x = NTu,
where x = (NK , . . . , N1), u = (1, 1, . . . , 1), and A is givenby
A = 14
⎛
⎜⎜⎜⎜⎜⎝
1/3K−1 0 0 · · · 00 1/3K−2 0 · · · 00 0 1/3K−3 · · · 0...
......
. . ....
0 0 0 · · · 1
⎞
⎟⎟⎟⎟⎟⎠
. (C5)
The matrix A + uu can be inverted using the Sherman-Morrison formula,
(A + uu )−1 = A−1 − A−1uu A−1
1 + u A−1u,
where we note that
u A−1u = Tr[A−1] = 4 × 3KK∑
k=1
13k = 2(3K − 1).
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QUANTUM PHASE ESTIMATION... PRX QUANTUM 2, 040301 (2021)
Therefore,
(A + uu )−1u = A−1u1 + Tr[A−1]
.
Finally,
x = NT
2 × 3K − 1A−1u,
that is,
NK = 4 × 3K−1
2 × 3K − 1NT = 4
3× NT
2 − 1/3K ,
NK−1 = 4 × 3K−2
2 × 3K − 1NT = 4
9× NT
2 − 1/3K ,
...
Nk = 4 × 3k−1
2 × 3K − 1NT = 3k−1 × 4
3K × NT
2 − 1/3K ,
...
N1 = 42 × 3K − 1
NT = 43K × NT
2 − 1/3K ,
and
N0 = NT −K∑
k=1
Nk = 13K × NT
2 − 1/3K .
These results are summarized by Eq. (10). In particular,we have N1 = 4N0 and Nk+1 = 3Nk for k = 1, . . . , K − 1.Finally, by substituting these values into Eq. (C3), weobtain Eq. (12), where
αK = 3(9/4)(1−1/3K )−K/(2×3K )
2(11/4)(1−1/3K )−1/3K
(
2 − 13K
)1−1/(2×3K )
. (C6)
Equation (12) reaches a scaling of sensitivity with NT thatis arbitrarily close to the Heisenberg scaling when increas-ing the number of steps of the protocol. In particular,limK→∞ αK = 39/4/27/4 ≈ 3.52.
1. Pseudocode of the algorithm
For further clarity, we report below the pseudocode ofthe algorithm, according to the analytical optimization dis-cussed above. Note that, as initial conditions, we can eitherfix the total number of qubits, NT, or the number of qubitsin the zeroth-step state, N0. In the former case, the codestarts with N0 = NT/(2 × 3K − 1), whereas in the lattercase, it uses a total of NT = (2 × 3K − 1)N0 qubits. Wealways assume that N0 � 1. Given NT qubits, one can fur-ther optimize the algorithm over the number of steps K(see below).
The analytical optimization over Nk and sk, Eqs. (10)and (11), respectively, relies on the condition s2
kNk � 1:when this condition is not met, we perform a numer-ical optimization of Eq. (C1) over Nk and sk, recur-sively. Numerical Monte Carlo simulations of the QPEalgorithm require the probability distribution P(μk|θk) inorder to simulate stochastic measurement events. It canbe calculated exactly as P(μk|θk) = |z〈μ|e−i(π/2)Jx e−iθk Jz
|ψk〉|2, up to Nk ≈ 104. For larger values of Nk, we approx-imate P(μk|θk) as a Gaussian distribution centered in〈J out
z 〉k with width (�J outz )2k , where J out
z can be expressedas a function of the spin moments of the states after statepreparation as J out
z = Jx sin θk + Jz cos θk. We confirm thatthe two approaches give equivalent results for small valuesof Nk.
2. Results for a fixed number of steps
In Fig. 6 we show the sensitivity as a function of NTfor the estimation of a random phase in [−π ,π ] and afixed total number of steps: K = 1 (black circles and lines),K = 2 (blue), K = 3 (green), and K = 4 (red). Circles areresults of numerical Monte Carlo simulations. The solidlines are solutions of a numerical optimization over sk andNk for k = 0, . . . , K (with s0 = 1). The dashed line is Eq.(12). The agreement between the numerical results and theanalytical prediction relies on the condition s2
kNk � 1 forall k = 0, . . . , K . Using Eq. (12), we obtain
s2K NK ∼ N 1/3K
T . (C7)
We thus obtain s2kNk � 1 for large NT : when increasing
K , increasingly larger values of NT are required in order tofulfill the condition s2
kNk � 1. This expected behavior issupported by the numerical calculations shown in Fig. (6).
040301-13
LUCA PEZZÈ and AUGUSTO SMERZI PRX QUANTUM 2, 040301 (2021)
102 103 104 105 106 107 108101
102
103
104
NT
SQLK = 1K = 2K = 3K = 4
N2 T×(Δ
θ est)2
FIG. 6. Phase sensitivity for a fixed number of steps. Plot of�2θest multiplied by N 2
T as a function of the total number ofparticles NT for K-step protocols: K = 1 (black), K = 2 (blue),K = 3 (green), and K = 4 (red). Dots are the results of MonteCarlo simulations taking θ as a random phase in [−π ,π ]. Thesolid lines are expected sensitivities (obtained from numericaloptimization), while the dashed lines are the analytical predic-tions, Eq. (12), which hold in the large-NT limit. Dotted lines areguides to the eye. The solid gray line is the standard quantumlimit (SQL) (�θ)2SQL = 1/NT.
This figure also shows that, for a given NT, there is an opti-mal number of steps K that optimizes the sensitivity. Forinstance, at NT = 1000, the case K = 2 overcomes boththe cases K = 1 and K = 3. The two-step protocol reachesan optimal point at NT ≈ 1000, where NT�
2θest,K = 4.64.
3. Heisenberg scaling
Given a fixed N0 (which here plays the role of an ini-tial condition) and asymptotically large NT and K , Eq. (12)predicts a sensitivity
�θest = 3.52NT
(C8)
in the estimation of any θ ∈ [0, 2π ]. This result relies onthe condition s2
kNk � 1 for all k = 0, . . . , K . The quantitys2
kNk can be written as a function of N0 as
s2kNk = 3(3/2)(1−1/3k−1)
2(5/2)(1−1/3k−1)−2N 1/3k
0 , (C9)
where we have used Eqs. (8), (10) and (12). The value ofN0 is negligible in Eq. (C7) for large k. Asymptotically ink, we find that
limk→∞
s2kNk =
√33
2≈ 3.67. (C10)
We thus conclude that the condition s2kNk � 1 is not satis-
fied for large k. We can improve the analytical predictionof Eq. (C6) is different ways [see Fig. 7(a) for a comparisonbetween the different methods].
(a) Heuristically (without a rigorous justification), wecan take into account the exponential in the denom-inator of Eq. (C1) by writing
�θest = αK
N 1−1/(2×3K )T
, (C11)
where αK = αK e1/(2s2K NK ) and s2
K NK is given inEq. (C7). This predicts, for large NT and K , the
102 104 106 108
5
10
15
2025
105 1010 1015100
101
102
103
104
105
105 1010 1015100
102
104
106
108
1010
NT
N T×Δθ
est
(a)
NT NT
N T×Δθ
est
s2 kN k
(b) (c)N0 = 108N0 = 107N0 = 106N0 = 105N0 = 104N0 = 103N0 = 102
FIG. 7. (a) Sensitivity �2θest as a function of the total number of particles NT for a given initial N0 (here N0 = 100). The green dotsare calculated from Eq. (12), reaching Eq. (C8) asymptotically in NT. The red dots are calculated from Eq. (C11), reaching Eq. (C12)asymptotically in NT. The black dots are obtained from the numerical optimization of sk at each step, while using the analytical optimalsequence N1 = 4N0 and Nk+1 = 3Nk for k ≥ 1. The blue dots are obtained from the numerical optimization of sk and Nk at each step.Lines are guides to the eye. (b) Sensitivity as a function of the total number of particles NT for different values of N0 (different colors).Points are obtained from a full optimization over sk and Nk at each step (k = 1, . . . , K and s0 = 1); dashed lines are guides to the eye.The gray line is �θ = 2/
√NT. The horizontal black line is Eq. (C14). (c) Quantity s2
kNk as a function of NT for different values of N0[same as in panel (b)]. The gray line is the case s0 = 1, while the horizontal black line is Eq. (C15).
040301-14
QUANTUM PHASE ESTIMATION... PRX QUANTUM 2, 040301 (2021)
sensitivity
�θest = 4.034NT
, (C12)
where the prefactor is α∞e1/√
2×33 . This value iscloser than Eq. (C6) to the optimal numerical result(see below).
(b) We can also improve the prediction of Eq. (C6)by keeping the sequence of Nk particles as dis-cussed above, namely N1 = 4N0 and Nk+1 = 3Nkfor 1 ≤ k ≤ K − 1, and choose the optimal squeez-ing parameter sk at step k by numerically solving theEq. (C2), recursively, with E[(��0)
2] = 4/N0 asinitial condition. In this case, we obtain a sensitivity
�θest = 4.01NT
. (C13)
(c) Finally, we have performed a full numerical opti-mization over Nk and sk (for k ≥ 1). For large N0,the optimal sequence is N1 ≈ 2.2N0 and Nk+1 =3.5Nk for k ≥ 1, which is only slightly different fromthe optimal sequence found analytically. The fullnumerical optimization provides
�θest = 3.97NT
, (C14)
independently from the initial N0; see Fig. 7(b). Theasymptotic value of s2
kNk for k → ∞ is
s2∞N∞ = 4.009 (C15)
[see Fig. 7(c)], which is slightly larger than theanalytical prediction (C8).
In conclusion, the analytical study and the numericalanalysis fully support our claim that the phase estimationprotocol reaches a sensitivity at the Heisenberg scaling4/NT, as discussed in the main text.
APPENDIX D: IMPACT OF NOISE AND OTHERPOSSIBLE IMPERFECTIONS
1. Number of particle fluctuations
When implementing our quantum phase estimation pro-tocol with atomic spin-squeezed states, it important toconsider the impact of fluctuations of the atom number. Wetake into account this effect by replacing state (A1) with the
statistical mixture
ρ(N , s, σN ) =+∞∑
N=0
e− (N−N )2
2σ2N
N
N/2∑
μ,ν=−N/2
e−(μ2+ν2)/(s2N )|N ,μ〉y〈N , ν|, (D1)
where N provides the normalization (Tr[ρ] = 1), N =Tr[ρN ] is the average number of particles, and the |N ,μ〉y
are common eigenstates of Jy with eigenvalue μ and ofthe Casimir invariant of J 2
z + J 2y + J 2
z with eigenvalue(N/2)(N/2 + 1). This state reduces to Eq. (A1) in the limitσN → 0. The calculation of the mean value and variancesof average spin moments gives
〈Jx〉ρ = N2
e+1/(2s2N ), (D2)
(�Jx)2ρ = σ 2
N + N 2
8(1 + e−2/(s2N ))2 − N 2
4e−1/(s2N )
≈ N 2
8(1 − e−1/(s2N ))2, (D3)
for s2N � 1 and σ 2N � N 2 (note that, typically, we can
consider σ 2N = N and the condition σ 2
N � N 2 is fulfilled
102 104 106 108
5
10
15
20
25
NT
NT×Δθ
est
FIG. 8. Phase estimation with states of a fluctuating number ofparticles, Eq. (D1). The figure shows the phase uncertainty �θestas a function of the total average number of particles NT. Redcircles are results of numerical Monte Carlo simulations taking θas a random phase in [−π ,π ]. Dots are expected semianalyticalpredictions with the dashed line being a guide to the eye. Thesolid black line is 4/NT and the protocol starts with N0 = 100particles.
040301-15
LUCA PEZZÈ and AUGUSTO SMERZI PRX QUANTUM 2, 040301 (2021)
for N � 1), and
(�Jy)2p = s2N
4. (D4)
These equations are analogous to Eqs. (A3), (A9) and(A10), respectively, with the replacement N → N . Theprotocol discussed above, in the case of a fixed N, canthus be straightforwardly generalized to this case, provid-ing the sensitivity 4/NT with NT = ∑K
k=1 Nk being the totalaverage number of particles.
In Fig. 8 we show results of Monte Carlo simulations ofthe protocol where the total number of particles is chosenrandomly at each step k = 0, 1, . . . , K within a Gaussiandistribution of variance σ 2
Nk= Nk and the average number
of particles is taken in the sequence N1 = 4N0 and Nk+1 =3Nk for 1 ≤ k ≤ K − 1.
2. Collective dephasing
We now study the effect of dephasing in the statepreparation, while neglecting noise during the phase shiftoperation, assuming that these take a negligible time withrespect to the decoherence time scale. Upon an additionalrotation of the probe states |ψ(N , s)〉, we take the y axisas the dephasing axis. We describe collective dephasingby the transformation ρ(N , s) = �y[|ψ(N , s)〉], giving thedensity matrix
ρ(N , s) =∫ π
−πdφP(φ)e−iφJy
× |ψ(N , s)〉〈ψ(N , s)|eiφJy ,
where the states |ψ(N , s)〉 are given in Eq. (A1), and e−iφJy
has the physical meaning of a stochastic phase rotationaround the y axis with a probability distribution P(φ) [with∫ π−π dφP(φ) = 1]. We have
〈Jx〉ρ = 〈Jx〉|ψ〉C1, (D5)
(�Jx)2ρ = 〈J 2
x 〉|ψ〉C2 − 〈Jx〉2|ψ〉C
21 + 〈J 2
z 〉|ψ〉S2, (D6)
(�Jy)2ρ = (�Jy)
2|ψ〉 = s2N/4, (D7)
where
C1 =∫ π
−πdφP(φ) cosφ, C2 =
∫ π
−πdφP(φ) cos2 φ,
S2 =∫ π
−πdφP(φ) sin2 φ.
Here the pendix |ψ〉 refers to the mean values and vari-ances of collective spin operators calculates for the state|ψ(N , s)〉. The y depolarization leaves unchanged themoments of Jy—in particular, it does not change thesqueezing along the y axis—but it decreases the lengthof the collective spin, 〈Jx〉, and increases the bending ofthe state in the sphere, namely 〈J 2
x 〉. Substituting values(D5)–(D7) into Eq. (6) and following the protocol dis-cussed above, we obtain the recursive relation for step k[we assume that the same dephasing noise P(φ) affects allstates in the phase estimation protocol],
E[(��k)2] = 2s2
k + Nk[1 + (C2 − S2)e−2/(s2kNk) − 2C2
1 e−1/(s2kNk)]E[(��k−1)
2]
2Nke−1/(s2kNk)C2
1
, (D8)
which assumes that E[(��k−1)2] � 1 in order to neglect
higher-order expansion terms in the tangent function, andrecovers Eq. (C1) in the case C1,2 = 1 and S2 = 0. Theoptimization over sk leads to the equation
s4kN 2
k − s2kNk + N 2
k
((C2 − S2)e−2/(s2
kNk) − 12
)
× E[(��k−1)2] = 0. (D9)
Figure 9 shows the results of a numerical optimization overboth sk and Nk, and for different values of N0. The noise
distribution is conveniently chosen as
P(φ) = eγ cosφ
2π I0(γ ), (D10)
where I0(x) is the modified Bessel function and γ isthe dephasing rate. In the case γ � 1, P(φ) can bewell approximated as a Gaussian with width σ 2
φ = 1/γ .Overall, we see that the protocol is very robust to z-depolarization noise: for large K , it reaches a Heisenbergscaling
�θest = αdephK
NT, (D11)
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QUANTUM PHASE ESTIMATION... PRX QUANTUM 2, 040301 (2021)
105 1010 1015100
101
102
103
104
105
N0 = 108
N0 = 107N0 = 106
N0 = 105
N0 = 104N0 = 103
N0 = 102
NT
N T×Δθ
est
FIG. 9. Phase estimation in the presence of collective dephas-ing. Phase sensitivity �θest as a function of NT, for y-dephasingnoise parameter γ = 4 [see Eq. (D10)] and for different values ofN0. Circles are results of the numerical optimization of the recur-sive relation (D8) over sk and Nk for k = 1, . . . , K . Dashed linesare guides to the eye. The solid line is �θ = 2/
√NT.
independently of N0, where the prefactor αdephK depends on
C1, C2, and S2. In Fig. 9 we show the expected phase sen-sitivity as a function of NT for γ = 4 and different valuesof N0. Symbols are obtained by numerically optimizingEq. (D8) over Nk and sk at each step. Furthermore, inFig. 3(a) of the main text we show a comparison betweenthe numerical optimization and Monte Carlo simulationsof the protocol. In Fig. 3(b) of the main text we plot theprefactor αdeph
K as a function of γ .
3. Full depolarization
We consider full depolarization within the symmetricsubspace of dimension N + 1, ρ(N , s) = �D[|ψ(N , s)〉],with
ρ(N , s) = (1 − D)|ψ(N , s)〉〈ψ(N , s)|
+ D
N + 11, (D12)
where 0 ≤ D ≤ 1. The mean value of any collective oper-ator J αn becomes
〈J αn 〉ρ = (1 − D)〈J αn 〉|ψ〉 + D
N + 1Tr[J αn ], (D13)
for any direction n and any power α. Taking into accountthe facts that
Tr[Jn] = 0 and Tr[J 2n ] =
N/2∑
μ=−N/2
μ2
= N (N + 1)(N + 2)12
, (D14)
and using Eqs. (A3) and (A9) in Eq. (A10), for s2N � 1we have
〈Jx〉ρ = (1 − D)N2
e−1/(2s2N ), (D15)
(�Jx)2ρ = (1 − D)N 2
8[1 + e−2/(s2N ) − 2(1 − D)e−1/(s2N )]
+ DN (N + 2)12
, (D16)
(�Jy)2ρ = (1 − D)s2N
4+ DN (N + 2)
12. (D17)
It should be noted that, according to Eq. (D17), thevariance (�Jy)
2ρ
cannot be smaller than DN (N + 2)/12.Identifying this value as s2
minN/4, we obtain a smallestachievable squeezing parameter s2
min � DN/3, meaningthat, when a state (A1) with squeezing parameter s �smin is depolarized according to Eq. (D12), it does notlead to a decrease of variance (�Jy)
2ρ
below DN (N +2)/12. Substituting the above spin moments into Eq. (6),we obtain the recursive relation for the K th step of thealgorithm
E[(��k)2] = 3(1 − D)s2
k + D(Nk + 2)
3(1 − D)2Nke−1/(s2kNk)
+ 3(1 − D)Nk[1 + e−2/(s2kNk) − 2(1 − D)e−1/(s2
kNk)] + 2D(Nk + 2)
6(1 − D)2Nke−1/(s2kNk)
E[(��k−1)2]. (D18)
In the following we restrict our study to the case D � 1. In Fig. 11 we plot the numerical optimization of recur-sive relation (D18) for D = 10−8, taking N0 = 100 and s0 = 1 as initial conditions. The value s2
min = DN/3 identifies a
040301-17
LUCA PEZZÈ and AUGUSTO SMERZI PRX QUANTUM 2, 040301 (2021)
102 103 104 105 106 10710–6
10–5
10–4
10–3
10–2
10–1
NT
Δθest
FIG. 10. Phase estimation in the presence of full depolariza-tion. Phase sensitivity �θest as a function of NT for full depo-larization noise with parameter D = 10−8 and N0 = 100. Opencircles are the results of the numerical optimization over sk andNk of recursive relation (D18). The black dashed line is the idealnoiseless case (D = 0) that follows the Heisenberg scaling (C14)for large NT. The black horizontal line is Eq. (D19), which pre-dicts a saturation of �θest at a value fully identified by D. Thevertical dotted line corresponds to the critical NT, Eq. (D20),which identifies the transition from a noiseless regime to a noise-dominated one. The dots are obtained by repeating the estimationprotocol with a depolarized squeezed state of Nk particles andsqueezing parameter k. The thin solid red line is Eq. (D21), whilethe thick solid red line is a fit �θest = βD/
√NT for large NT.
critical step k (smin = sk). For k � k, the protocol essen-tially follows the ideal protocol and depolarization noisedoes not play any major role. For k � k, the sensitivityquickly saturates to the value
�2θest,k�k = D
3(1 − D)2. (D19)
The critical value k can be obtained by using the rela-tion s2
kNk = 4 [see Eq. (C14)], assuming that k � 1
(which is valid for D � 1). Using s2ks2 = DNk/3, we
obtain N 2k
= 12/D and, using Eq. (10), we have k =log3[3
√3/(2N0
√E)]. This value corresponds to a total
critical number of particles
NT =k∑
k=0
Nk = 3
√3D
, (D20)
which is obtained by noting that NT = 3Nk/2 and is high-lighted by the vertical dotted line in Fig. 11. The corre-sponding sensitivity reached by the protocol at this point is
approximatively �θest,k ≈ 4/NT, according to Eq. (C13).For k � k, it is convenient to modify the algorithm byavoiding phase feedback between the different steps ofthe protocol and repeating the estimation using multiplecopies of |ψ(Nk, sk)〉 [which are depolarized according toEq. (D12)]. Using m copies of the state gives a statisticalgain factor 1/
√m in sensitivity, such that
�θest ≈ 4
NT√
m= 8
3
(D
12
)1/4 1√NT
. (D21)
To derive this equation we have used NT = NT + mNk ≈mNk for m � 1 since NT = 3Nk/2. Equation (D21), shownby the thin solid red line in Fig. 11, slightly underesti-mates the prefactor βD of the asymptotic βD/
√NT scaling
of sensitivity. Nevertheless, Eq. (D21) accurately catchesthe behavior βD ∝ D1/4, as shown in Fig. 3(d) of the maintext.
4. Particle losses
To show the robustness of our QPE protocol againstlosses, we consider a simple model. We focus on bosonicmodes, a and b, and model the loss of particles by cou-pling to additional modes, A and B, respectively, that areinitially in the vacuum state: aη = √
1 − ηa + √ηA and
102 104 106 108 1010
10–15
10–10
10–5
100
NT
Δθest
FIG. 11. Phase estimation in the presence of losses. Phasesensitivity �θest as a function of NT in the presence of losses.Symbols are obtained by optimizing recursive relation (D25)over sK and NK for k = 1; . . . , K : blue dots are obtained forη = 10−4 and green squares for η = 10−6. The respective solidlines are calculated from Eq. (D26). The black dashed line isthe ideal noiseless case (η = 0) that follows the Heisenberg scal-ing (C13) for large NT. Vertical dotted lines are NT = 6/(η − η2)
(see the text). The initial number of particles is N0 = 100.
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QUANTUM PHASE ESTIMATION... PRX QUANTUM 2, 040301 (2021)
bη = √1 − ηb + √
ηB, where η is a loss coefficient rang-ing from 0 (no loss) to 1 (full particle loss from a and bmodes). Considering Jx = (a†
ηbη + b†ηaη)/2, Jy = (a†
ηbη −b†ηaη)/2i, Jz = (a†
ηaη − b†ηbη)/2, we have
〈Jx〉η = (1 − η)N2
e−1/(2s2N ), (D22)
(�Jx)2η = (1 − η)2N 2
8(1 − e−1/(s2N ))2 + η(1 − η)N
4,
(D23)
(�Jy)2η = (1 − η)2s2N
4+ η(1 − η)N
4. (D24)
Similar to the case of full depolarization, there isa lower bound to (�Jy)
2η: it cannot be smaller than
Nη(1 − η)/4. From the relation Ns2min/4 = Nη(1 − η)/4
we identify a smallest squeezing parameter s2min =
η(1 − η). For η � 1, we can thus identify criticalalgorithm step k, such that, for k � k, the noisyalgorithm follows the ideal noiseless one. Using s2
kNk =
4, sk = smin and NT = 3Nk/2, we obtain NT = 6/(η − η2).
We substitute the above spin moments into Eq. (6)and obtain the recursive relation for the kth step of thealgorithm
E[(��k)2] = 2(1 − η)s2
k + 2η + [(1 − η)Nk(1 − e−1/(s2kNk))2 + 2η]E[(��k−1)
2]
2(1 − η)Nke−1/(s2Nk). (D25)
For NT � NT, we thus achieve the Heisenberg scaling�θest = 4/NT. For NT � NT, the noise associated with par-ticle losses dominates the algorithm. It is more convenientto repeat the phase estimation using m copies of the state|ψ(Nk, sk)〉. The sensitivity thus follows
�θest =√
η
(1 − η)
1√NT
. (D26)
APPENDIX E: NONLINEAR READOUT
As discussed in the main text, the output state of the kthstep of the protocol is now
|φoutk (Nk, τk, θk)〉 = e−i(π/2)Jx e+iτk J 2
z e−iθk Jy e−iτk J 2z |CSS(Nk)〉.
(E1)
The rotation angle θk is estimated by inverting the relation〈J out
z 〉 = 〈φoutk |Jz|φout
k 〉 as a function of θk. In Figs. 12(a)and 12(b) we plot 〈J out
z 〉 and (�J outz )2 as a function of θ for
N = 1000 and τ = 0.01. We see that the function 〈J outz 〉 is
monotonic (and thus invertible) around θ = 0 in a regionθ = [−�, �], where � depends on τ and N . In Fig. 12(c) weplot � as a function of τ . As we see, � = π/4 in the limitτ = 0, while � quickly decreases by increasing τ .
The statistically averaged sensitivity at the kth step ofthe algorithm, in the presence of detection noise, is
E[(��k)2] =
∫ �k
−�k
dθk
((�J out
z )2k + σ 2DN
(d〈J outz 〉k/dθk)2
)
P(θk), (E2)
10–4 10–3 10–2 10–10.00
0.05
0.10
0.15
0.20
0.25
0.0
0.5
1.0
1.5
2.0
2.5
3.0104
–0.50 –0.25 0.00 0.25 0.50–0.50
–0.25
0.00
0.25
0.50
⟨ Jout z⟩/N
(Δ Jout z)2
θ /π–0.50 –0.25 0.00 0.25 0.50
θ /π τ
ℓ
(a) (b) (c)
FIG. 12. Plots of (a) 〈J outz 〉 and (b) (�J out
z )2 as a function of θ for N = 1000 and τ = 0.01. The shaded region indicates the inversionregion θ ∈ [−�, �] around θ = 0 where 〈J out
z 〉 is monotonic. Panel (c) shows � as a function of τ .
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LUCA PEZZÈ and AUGUSTO SMERZI PRX QUANTUM 2, 040301 (2021)
TABLE I. Optimization parameters Nk and τk used in Fig. 4 ofthe main text in the case σ 2
DN = 0.
Step N τ
0 600 01 500 6.6 × 10−3
2 2400 5.8 × 10−3
3 6600 5.5 × 10−3
where σ 2DN is the detection noise parameter, and P(θk) is
taken as a Gaussian distribution centered at θk = 0 and ofvariance E[(��k−1)
2]. The optimal values of τk and Nk arefound by minimizing E[(��k)
2] × ∑kj =0 Nj , with initial
conditions E[(��0)2] = 4/N0. In particular, we find that,
for the optimal parameters, the quantity 2∫ �k
0 dθkP(θk) isnegligible. In Table I we list the values of τk and Nk usedin the numerical simulations giving Fig. 4 of the main textin the case σ 2
DN = 0. For σ 2DN �= 0, we have optimized only
with respect to τk, obtaining values similar to the noiselesscase.
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