Experimental demonstration of cheap and accurate phase estimation
EExperimental demonstration of cheap and accurate phase estimation
Kenneth Rudinger, Shelby Kimmel, Daniel Lobser, and Peter Maunz Center for Computing Research, Sandia National Laboratories, Albuquerque, NM 87185, [email protected] Joint Center for Quantum Information and Computer Science (QuICS), University of Maryland, [email protected] Sandia National Laboratories, Albuquerque, NM 87185
We demonstrate experimental implementation of robust phase estimation (RPE) to learn thephases of X and Y rotations on a trapped Yb + ion qubit. We estimate these phases with uncer-tainties less than 4 · − radians using as few as 176 total experimental samples per phase, and ourestimates exhibit Heisenberg scaling. Unlike standard phase estimation protocols, RPE neither as-sumes perfect state preparation and measurement, nor requires access to ancillae. We cross-validatethe results of RPE with the more resource-intensive protocol of gate set tomography. INTRODUCTION
As quantum computers grow in size, efficient and ac-curate methods for calibrating quantum operations areincreasingly important [1–4]. Calibration involves esti-mating the values of experimentally tunable parametersof a quantum operation and, if incorrect, altering thecontrols to fix the error.When these tunable parameters are incorrectly set, itcauses the system to experience coherent errors. Coher-ent errors (versus incoherent errors) are more challeng-ing for error correcting codes to correct [5, 6], making itharder to reach fault-tolerant thresholds [7–9]. Hence itis important to correct these types of errors in order tobuild a scalable quantum computer. While recent tech-niques using randomized compiling [10] mitigate the ef-fects of coherent errors, removing as much of the coherenterrors as possible still gives the best error rates.Calibration can be challenging to perform without ac-curate state preparation and measurement (SPAM) esti-mates [11, 12]. Thus proper calibration of quantum op-erations will require robust protocols, that is, protocolsthat can accurately characterize gate parameters withouthighly accurate initial knowledge of SPAM.A new technique for calibrating the phases of gate op-erations is robust phase estimation (RPE) [13]. RPE canbe used to estimate the rotation axes and angles of single-qubit unitaries. Moreover, it is easy to implement (thesequences required are essentially Rabi/Ramsey experi-ments), simple and fast to analyze, and can obtain accu-rate estimates with surprisingly small amounts of data.RPE has advantages over standard robust characteri-zation procedures when it comes to the task of calibra-tion. RPE can estimate specific parameters of coherenterrors, whereas randomized benchmarking, while robust,can only estimate the magnitude of errors [14–18]. Whilecompressed sensing approaches can withstand SPAM er-rors [19, 20], they do not have the Heisenberg scalingRPE achieves. There is a simple analytic bound on thesize of SPAM errors that RPE can tolerate (namely lessthan 1 / √ . · − radiansin our phase estimate using only 176 total samples. Wecompare these costs to GST and find that RPE requiresorders of magnitude fewer total gates and samples toachieve similar accuracies. However, in regimes where ex-periments involving long sequences are not accessible, wefind GST potentially has better performance than RPE.Nonetheless, due to its minimal data requirements, easeof implementation and analysis, and robust estimates ofcoherent errors, RPE is a powerful tool for efficient cali-bration of quantum operations. a r X i v : . [ qu a n t - ph ] F e b PRELIMINARIES
We consider estimating the parameters α and (cid:15) fromthe single-qubit gate set [13]:ˆ X π/ α = exp [ − i (( π/ α ) /
2) ˆ σ X ] , ˆ Y π/ (cid:15) ( θ ) = exp [ − i (( π/ (cid:15) ) /
2) (cos θ ˆ σ Y + sin θ ˆ σ X )] , where ˆ σ X and ˆ σ Y are Pauli operators, α and (cid:15) are rota-tion errors in the X and Y gates, respectively, and θ isthe size of the off-axis component of the (ideally) Y gaterotation axis. There is no off-axis component to the Xgates, as we choose the X axis of the Bloch sphere to bethe rotation axis of the X gate. α and (cid:15) are parametersthat experimentalists can typically control with ease.In reality, the implemented gates will not be unitary,but instead will be completely positive trace preserving(CPTP) maps. Nonetheless, these CPTP maps will haverotation angles analogous to the angles α and (cid:15) , and inthe Supplemental Material, we show RPE can extractsuch angles. For the rest of the paper, with slight abuseof notation, we will use α and (cid:15) to refer to these moregeneral CPTP map rotations.We use both RPE and GST to extract α and (cid:15) . Fig. 1gives a schematic description of GST and RPE circuits.RPE circuits are essentially Rabi/Ramsey sequences;they consist of state preparation ρ , which for extract-ing α and (cid:15) is assumed to be not too far in trace distancefrom | (cid:105)(cid:104) | , followed by repeated applications of the X or Y gate, followed by a measurement operator M , whichis assumed to be close in trace distance to | (cid:105)(cid:104) | . (Per-forming additional, more complex “Rabi/Ramsey-like”sequences allows for RPE to extract θ as well [13]; we donot do so here.)RPE assumes all gates and SPAM are relatively closeto ideal, but tolerates errors. We use “additive error” todenote the maximum bias in the outcome probability ofany single RPE experimental sequence. This bias can bedue to SPAM errors and incoherent errors in the gates.Additive error can be tolerated as long as it is less than1 / √ ρ , followed by a gate sequence F i to simulate an alternatestate preparation. Next a gate sequence g k is appliedrepeatedly. Finally, the measurement M is preceded by agate sequence F j to simulate an alternative measurement.We refer to F i and F j as state and measurement fiducials,respectively, and g k as a germ. (For more details, see theSupplemental Material.)For both RPE and GST, running increasingly longersequences produces increasingly accurate estimates. Weuse L to parameterize the length of the sequence, as inFig. 1. We run sequences with L ∈ { , , , , . . . , L max } ,where L max is chosen based on the desired accuracy. InRPE, we repeat the gate of interest either L or L + 1times. In GST, we implement all possible combinations of state fiducials, measurement fiducials, and germs, withthe germ repeated (cid:98) L/ | g k |(cid:99) times, where | g k | is the num-ber of gates in g k and (cid:98)·(cid:99) denotes the floor function.We let N be the repetitions (samples taken) of eachsequence. We set N to be the same for all sequences ina single RPE or GST experimental run. Although thisresults in slightly non-ideal scaling in the accuracy of ourestimate [39], this is a realistic scenario for experimentalimplementation.RPE successively restricts the possible range of theestimated phase using data from sequences with largerand larger L . Inaccuracies result when the procedure re-stricts to the wrong range. For larger values of L max ,there are more rounds of restricting the range, and thusmore opportunities for failure. By increasing N when L max increases, we can limit this probability of failure.Likewise, a large additive error makes it easier to incor-rectly restrict the range, but again, taking larger N canincrease the probability of success. The interaction be-tween accuracy, N , L max , and additive errors is shown inFig. 2. This graph was created by adapting the analysisof [13] to the case of fixed N over the course of an RPEexperimental run [45]. Fig. 2 shows that, given an ad-ditive error δ , there exist good choices for N and L max ,provided that δ < / √ /L max . This is a good proxy (upto log factors) for Heisenberg scaling.In practice, experimentalists care less about Heisen-berg scaling, and more about the resources required toachieve a desired accuracy in their estimate. Thereforewe are additionally interested in how large N and L max should be to attain a desired precision. Assuming time isthe key resource, if experimental reset time is long com-pared to gate time, N becomes the dominant cost factor.On the other hand, if gate time is long compared to ex-perimental reset time, L max is the dominant factor. EXPERIMENTAL RESULTS
Here we will give estimates of α . Results for (cid:15) aresimilar and can be found in the Supplemental Material.We implement GST and RPE on a single Yb + ion in a linear surface ion trap. The qubit levels arethe hyperfine clock states of the S / ground state: | (cid:105) = | F = 0 , m F = 0 (cid:105) , | (cid:105) = | F = 1 , m F = 0 (cid:105) . Weinitialize the qubit close to the | (cid:105) state via Dopplercooling and optical pumping; we measure in the com-putational basis (approximately) via fluorescence statedetection [40]. The desired operations are X π/ and Y π/ . ⇢ F i g k j L | g k | k F j M ⇢ MX ⇡ ↵ L or L + 1 ⇢ M L or L + 1 Y ⇡ ✏ ( a ) { ( b ) FIG. 1: (Color online) (a) RPE and (b) GSTexperimental sequences. Each sequence starts with thestate ρ and ends with the two-outcome measurement M . (a) An RPE sequence consists of repeating the gatein question either L or L + 1 times. (b) In GST, a gatesequence F i is applied to simulate a state preparationpotentially different from ρ . This is followed by (cid:98) L/ | g k |(cid:99) applications of a germ —a short gate sequence g k oflength | g k | . Finally, a sequence F j is applied to simulatea measurement potentially different from M . (N,L max )(4,16) (4,1024)(16,16) (16,1024) (370,16)(370,1024) π /(2·16) π /(2·1024) R M S e rr o r uppe r bound − − − δ FIG. 2: (Color online) Analytic upper bounds on theRMSE of the RPE phase estimate. Because RPE ispotentially biased, the RMSE does not go to zero in thelimit of infinite N , but instead, approaches a floor of π/ (2 L max ) . Larger additive error δ produces a largerbias, and thus require larger N and larger L max toachieve a small RMSE. For example, N = 16 is notlarge enough to reach the floor for L max = 1024 , butincreasing N to 370 we easily saturate the bound formost values of δ .See [23] for experimental details. For the numerical anal-ysis in this work, we have used the open-source GSTsoftware pyGSTi, and have extended its capabilities toinclude RPE functionality [41].We take 370 samples of each GST and RPE sequence.(For details, see Gate Sequences in Supplemental Mate-rial.) We use L ∈ { , , , . . . , } . The GST datasetcomprises 2,347 unique sequences and 868,390 total sam- ples, while the RPE dataset comprises 44 sequences and16,280 samples. The RPE dataset further disaggregateinto disjoint sets of 22 unique sequences and 8,140 sam-ples per phase.Looking at Fig. 2, we see that N = 370 is larger thannecessary for RPE with L max = 1024 for additive errorless than ∼ .
25. To simulate experiments with fewerthan 370 samples per sequence, we randomly sample(without replacement) from the experimental dataset, sothat the new, subsampled dataset has
N <
370 samplesper sequence.We use several methods to characterize the experimen-tal accuracy of RPE. First, we apply the analytic boundson RMSE of Fig. 2. We also compare our subsampledRPE estimates to the GST estimate. Unlike RPE, GSTis an unbiased estimator [42], so going to large N (ata large cost in resources) gives standard quantum limitscaling. Using the N = 370 dataset for GST, we esti-mate α − π/ . ± . · − ; the error bars denotea 95% confidence interval derived using a Hessian-basedprocedure (see [23] for details). On the other hand, usingall RPE data we estimate α − π/ . · − , with anRMSE upper bound of π/ (2 · L max ) ≈ . × − (wherethis bound comes from Fig 2 with N = 370, assumingour additive error is less than 0 .
25; this assumption isborne out in the next section).While the RPE estimate is consistent with the GST re-sult, the accuracy is significantly lower, and we thus take α , the full data estimate from GST, to be the “true”value of α for the purposes of benchmarking RPE. Inparticular, throughout this paper, we calculate experi-mental RMSE by comparing the mean estimate from 100subsampled datasets to α . Heisenberg Scaling from RPE
To look for Heisenberg scaling in RPE estimates,we perform RPE on 100 subsampled datasets for L max ∈ { , , , . . . , } with N ∈ { , } . Wesee Heisenberg-like scaling in the experimental RMSEin Fig. 3. We also plot π/ (2 L max ), which is the ana-lytic upper bound if sufficient samples are taken to com-pensate for additive error. We see that in practice, theanalytic bounds can be pessimistic. Moreover, we seethat while the experimental RPE accuracy is sensitive to N , increasing N to 256 from 16 does not dramaticallyimprove the RMSE, improving the scaling to . /L max from . /L max . Instead, as expected, large increasesin accuracy are obtained by moving to larger L max . ThisHeisenberg-like scaling is especially important for regimeswhere the time to implement the gate sequence is longrelative to SPAM time.We believe our experimentally derived bounds are sig-nificantly better than our analytic bounds in part becauseour system is well calibrated. The analytic bounds give aworst-case analysis that accounts for bias caused by ad-versarial additive error, but RPE is effectively unbiasedfor our system, up to the accuracy we achieve. Comparison to GST
Because RPE can be biased, increasing N cannot im-prove the RMSE below π/ (2 L max ) in the worst case (seeFig. 2 and [13]). However since GST is unbiased, it al-ways benefits from increasing N. We investigate this effect in Fig. 4. We plot theRMSE for experiments with fixed L max = 1024, but N ∈ { , , . . . , } . Analytic bounds for RPE are de-rived using the same method as in Fig 2. Experimentalbounds for GST and RPE are derived from comparingthe estimates of 100 subsampled datasets to α . While the analytic RPE bounds do not improve withincreasing N , the subsampled RPE and GST datasetsshow standard quantum limit scaling. We expect this forGST, because GST is unbiased. In the case of RPE ourexperimental system happens to have very small additiveerror, and so is only very slightly biased. In this case, weexpect to see improving estimates with increasing N untilour accuracy is about the same size as our bias. Fig. 4tells us that for systems with relatively large additiveerror, where large N is feasible but large L max is not,GST can provide more accurate results.However, we see in Fig. 4 that GST pays a substantialcost relative to RPE in required number of total sam-ples (i.e., number of samples per sequence N times to-tal number of sequences). In Fig. 5, we compare thenumber of gates and samples which RPE and GST eachrequire to achieve a desired accuracy, by analyzing 100subsampled datasets with fixed N = 16 and varying L max ∈ { , , , . . . , } . We see that RPE can achievesimilar accuracy to GST while using at least an order ofmagnitude fewer total samples.For our system, acquiring the entire RPE and GSTdatasets took 10.8 minutes and 12.1 hours, respectively,and total experimental time scales linearly with N . Thuswe note that had our actual data acquisition rate been N = 16, it would have taken 28 s to acquire thatRPE dataset and about 31 minutes to acquire the GSTdataset. As for analysis time, a single RPE dataset canbe analyzed in about 0.05 s on a modern laptop. GSTanalysis takes about 20 s [46]. All datasets and analysisnotebooks are available online [43]. CONCLUSIONS
We show that robust phase estimation successfully es-timates the phases of single-qubit gates, yielding resultsthat are consistent with the full tomographic reconstruc-tion of gate set tomography and also exhibits Heisenberg-
N=16 N=256 π /(2L max ).223/L max .078/L max R M S E rr o r − − − − max FIG. 3: (Color online) RSME versus L max for RPEestimates of α from 100 subsampled datasets of size N = 16 and N = 256. While analytic bounds are atbest π/ (2 L max ), we see this can be pessimistic. Whenthe additive errors, which can bias the RPE estimate,are sufficiently small, increasing N improves RMSaccuracy. GST RMSERPE RMSE-ExperimentalRPE RMSE-Analytic- δ =0.1RPE RMSE-Analytic- δ =0.01 ⇡ · / / p S R M S E rr o r − − − − − FIG. 4: (Color online) Scaling of RMSE of estimates of α as a function of total samples (S), with L max = 1024,and N ∈ { , , . . . , } . The data point furthest leftin each sequence corresponds to N = 8, and the furthestright to N = 256 . Analytic bounds are derived using thetechniques of Fig 2. Experimental data points take theRMSE of 100 subsampled datasets for both RPE andGST. While the analytic bounds converge to π/ ∝ / √ S ) of RPE experimental estimates. As discussedin the text, this is because our experimental device hasvery low additive error, and thus the RPE estimates areessentially unbiased, and can achieve greater accuracywith increasing number of samples. GST estimates alsoexhibit standard quantum limit scaling.like scaling in accuracy. In particular, an individualphase may be estimated with a root mean squared er-ror of 3 . · − with as few as 176 total samples.Hence, RPE is a strong choice for diagnosing and cal- RPE GST R M S E rr o r − − − − Total samples10 FIG. 5: (Color online) RMSE for RPE and GSTestimates of α versus total number of samples, using100 subsampled datasets with N = 16. Each sequentialdata point corresponds to setting L max ∈ { , , , . . . , } . RPE achieves the same levelof accuracy as GST using far fewer resources.ibrating single-qubit operations. It would be interestingto investigate whether the techniques of RPE can be ap-plied to assessing other errors in single-qubit gate oper-ations in a fast and accurate manner. ACKNOWLEDGEMENTS
Sandia National Laboratories is a multi-program lab-oratory managed and operated by Sandia Corporation,a wholly owned subsidiary of Lockheed Martin Corpora-tion, for the U.S. Department of Energy’s National Nu-clear Security Administration under contract DE-AC04-94AL85000. SK is funded by the Department of Defense.The authors thank Robin Blume-Kohout and NathanWiebe for helpful conversations, and Erik Nielsen for ex-tensive software support. This research was funded, inpart, by the Office of the Director of National Intelli-gence (ODNI), Intelligence Advanced Research ProjectsActivity (IARPA). All statements of fact, opinion or con-clusions contained herein are those of the authors andshould not be construed as representing the official viewsor policies of IARPA, the ODNI, or the U.S. Government.
SUPPLEMENTAL MATERIALRobust phase estimation on CPTP maps
In RPE, [13], the gates to be analyzed are assumed tobe close to some unitaries. Then RPE allows estimationof the error parameters of those unitaries (see Eq. 1).However, there is ambiguity in this formulation, becausegiven a full description of completely positive and tracepreserving (CPTP) map E , there is not a unique unitary associated to this map. This might make it difficult tocompare RPE and GST, since GST produces an estimatefor a complete CPTP map. We now show that given aCPTP map E on a single qubit, RPE can extract thephase of the imaginary eigenvalues of that map.We will use the Pauli-Liouville representation of CPTPmaps, states and measurements (e.g. [44]). Let P i = σ i (the single-qubit Pauli matrices) and let P be the 2-by-2identity matrix. Then for a single-qubit CPTP map E ,the Pauli-Liouville representation E PL is given by E PL = (cid:88) i,j =0 tr ( E ( P i ) P j )2 | i (cid:105)(cid:104) j | . (1)In the Pauli-Liouville representation, a single qubitdensity matrix ρ is given by the vector | ρ (cid:105)(cid:105) where | ρ (cid:105)(cid:105) = (cid:88) i =0 √ ρP i ) | i (cid:105) , (2)and a positive measurement operator M is given by (cid:104)(cid:104) M | where (cid:104)(cid:104) M | = (cid:88) i =0 √ M P i ) (cid:104) i | . (3)As a consequence of these definitions, we have thattr( M E ( ρ )) = (cid:104)(cid:104) M |E PL | ρ (cid:105)(cid:105) . Thus, as in GST, using aninvertible 4 × S , we can transform all states | ρ (cid:105)(cid:105) , maps E PL , and measurements (cid:104)(cid:104) M | as E PL → S − E PL S | ρ (cid:105)(cid:105) → S − | ρ (cid:105)(cid:105)(cid:104)(cid:104) M | →(cid:104)(cid:104) M | S, (4)and not impact any observables.For single-qubit CPTP maps, E PL is a real 4 × re ± iφ be the phases of the complexeigenvalues of a map E PL . Using a similarity transfor-mation S E (in particular, the matrix whose columns arethe right eigenvectors of E PL ), we can transform E PL to E PL (cid:48) = S − E E PL S E , where E PL (cid:48) has the form E PL (cid:48) = re iφ re − iφ
00 0 0 d . (5)Now suppose we can prepare the state ρ x ≈ | + (cid:105)(cid:104) + | ,and make measurements M x and M y (measurements inthe σ x and σ y bases, respectively). By construction, weassert that, under the same similarity transformation S E ,we have S − E | ρ x (cid:105)(cid:105) = (1 / √ , / , / , T + | δ ρ x (cid:105)(cid:105)(cid:104)(cid:104) M x | S E = (1 / √ , / , / ,
0) + (cid:104)(cid:104) δ M x |(cid:104)(cid:104) M y | S E = (1 / √ , − i/ , i/ ,
0) + (cid:104)(cid:104) δ M y | . (6)We may assert the above because any errors introducedby S E get absorbed into the δ terms. (Physically | δ ρ x (cid:105)(cid:105) , (cid:104)(cid:104) δ M x | and (cid:104)(cid:104) δ M y | correspond to additive errors presentin the state preparation and measurement operations.)Then we have (cid:104)(cid:104) M x | ( E PL ) k | ρ x (cid:105)(cid:105) = 12 (1 + cos( kφ )) + δ kx (cid:104)(cid:104) M y | ( E PL ) k | ρ x (cid:105)(cid:105) = 12 (1 + sin( kφ )) + δ ky (7)where ( E PL ) k signifies acting with E repeatedly k times,and δ kx and δ ky depend on r as well as | δ ρ x (cid:105)(cid:105) , (cid:104)(cid:104) δ M x | , and (cid:104)(cid:104) δ M y | .Comparing Eq. 6 with Eq. (I.1)-(I.2) of [13], we seethat given these two types of measurements, RPE can beused to learn φ , assuming δ kx and δ ky are not too large.Thus we can directly compare estimates of rotation an-gles obtained by GST or by RPE. Gate sequences
Detailed explanations for the choice of gate sequencesused for RPE and GST are given in [13] and [23], respec-tively. Here we simply provide complete descriptions ofthe gate sequences used.Before proceeding further, we describe two notationalconventions: We denote the X π/ gate as G x , and Y π/ gate as G y . Additionally, sequences are listed in oper-ation order, not matrix multiplication order, so the se-quence G x G y means “apply the X π/ gate, and then ap-ply the Y π/ gate”.Both RPE and GST rely on gate sequences that havea well-defined structure. For GST, each sequence is ofthe following form:1. Prepare a fixed input state.2. Apply a short gate sequence (called a fiducial prepa-ration , denoted F i ) to simulate a particular statepreparation.3. Apply a short gate sequence (called a germ , de-noted g k ) (cid:98) L/ | g k |(cid:99) times, where | g k | is the numberof gates in the germ, and L ∈ Z + is the sequencelength.4. Apply a short gate sequence (called a fiducial mea-surement , denoted F j ) to simulate a particularmeasurement operation. 5. Perform and record the outcome of a fixed mea-surement.RPE uses fiducial sequences and germs as well. How-ever, the fiducial sequences are not independent of thegerm under consideration, as we will describe in moredetail when we discuss the specific RPE gate sequences.We divide experiments into generations , labeled by m ∈ { , , . . . } . Sequences in generation m have sequencelength L = 2 m . For example, for the m = 3 gen-eration, the underlying sequence (modulo fiducials) forthe germ G x is simply G x ; for the germ G x G y , it is( G x G y ) , and for the germ G y G x G y G x G x G x , it is just G y G x G y G x G x G x .In GST, for each generation and each germ, sequencesare run with every possible pairing of fiducial state prepa-ration and measurement. That is, if there are f p and f m unique fiducial preparations and measurements re-spectively, then there are f p · f m unique sequences for aparticular germ for a particular generation.In our experiments, there are 11 generations in total(ranging from m = 0 to m = 10). Additionally, our tar-get preparation operation is always | (cid:105)(cid:104) | , and our targetmeasurement operation is always σ z . GST fiducials
The preparation and measurement fiducials thatwe use for GST are, conveniently, identical. Theycorrespond to mapping both the state preparation andmeasurement vectors to the six antipodal points on theBloch sphere that intersect with the X, Y, and Z axes.Therefore, each germ at each generation generates 36different sequences. The fiducials are:1. {} (The null sequence; do nothing for no time.)2. G x G y G x G x G x G x G x G y G y G y GST germs
The germs we use for GST in this Letter are:1. G x G y G x G y G y G y G y G x G y G x G y G x G x G x G y G x G y G y G x G x G y G y G y G x G y G x G x G x G y G x G y G y RPE germs and fiducials
The fiducials and germs used in an RPE sequence willdepend on both the quantity being estimated, and thenative fixed input and fixed measurement. In partic-ular, for our experimental system, we believe that thefixed input state is close to | (cid:105)(cid:104) | and the fixed measure-ment is close σ z . Then for α (the amount of over- orunder-rotation in G x ), the germ is G x , state preparationis always the empty fiducial {} , and there are two mea-surement fiducials, the empty fiducial {} and the gate G x ; for (cid:15) (the amount of over- or under-rotation in G y ),the germ is G y , state preparation is always the emptyfiducial {} , and there are two measurement fiducials, theempty fiducial {} and the gate G y ;Therefore, every RPE sequence we apply has one ofthe following forms:1. G x m G x m +1 G y m G y m +1 for m ∈ { , · · · , } . RPE Algorithm
We use the following is the algorithm that takes rawdata counts from a robust phase estimation experiment,and returns an estimate of the phase. An open-sourceimplementation of this protocol is available online [41].
Algorithm 1
Input: (cid:126)M ∈ [ Z + ] n , a vector whose i th element M i is the numberof repetitions (samples) of the i th experiment. (cid:126)x ∈ [ Z + ] n , a vector whose i th input x i is sampled froma binomial distribution B (cos(2 i − φ ) / / δ i,x , M i ),with δ i,x ∈ [0 ,
1] for all i . (cid:126)y ∈ [ Z + ] n , a vector whose i th input y i is sampled froma binomial distribution B (sin(2 i − φ ) / / δ i,y , M i ),with δ i,y ∈ [0 ,
1] for all i . Output:
Estimate ˆ φ ∈ [ − π, π ] of φ function RobustPhaseEstimation ( (cid:126)M , (cid:126)x , (cid:126)y ) Estimate = 0 (cid:46)
Initial estimate could be any value in[ − π, π ]; algorithm would be unaffected. for i = 1 to n do L = 2 i − CurrentPhase = arctan 2(( x i − M i / /M i , − ( y i − M i / /M i )) /L (cid:46) Calculate the remainder of the estimatemod 1 /L . while CurrentPhase < (Estimate − π/L ) do CurrentPhase = CurrentPhase + 2 π/L (cid:46)
If smaller than allowed principle range, increase until inrange. end while while
CurrentPhase > (Estimate + π/L ) do CurrentPhase = CurrentPhase − π/L (cid:46) If larger than allowed principle range, decrease until inrange. end while
Estimate = CurrentPhase end for return
Estimate end function Results for (cid:15)
We now present our experimental results for the rota-tion angle (cid:15) , corresponding to Figs. 3 and 5. We seethat the (cid:15) estimate and error bar behaviors are bothqualitatively and quantitatively similar to the α behav-ior. In particular, we observe 1 /L max scaling in theRPE estimates for (cid:15) at N as low as 16, and the ob-served RMSE scaling constant is below that guaran-teed by RPE theory. Additionally, we find that, usingthe full N = 370 dataset, GST returns the estimate (cid:15) − π/ . · − ± . · − . RPE provides a con-sistent estimate of (cid:15) − π/ . · − , with an RMSEupper bound of π/ (2 · L max ) ≈ . × − . [1] D. J. Egger and F. K. Wilhelm, Phys. Rev. Lett. ,240503 (2014).[2] C. Ferrie and O. Moussa, Phys. Rev. A , 052306 (2015).[3] K. R. Brown, A. C. Wilson, Y. Colombe, C. Ospelkaus,A. M. Meier, E. Knill, D. Leibfried, and D. J. Wineland,Phys. Rev. A , 030303 (2011). N=16 N=256 π /(2L max ).254/L max .071/L max R M S E rr o r − − − − max FIG. 6: (Color online) Root mean squared error vs L max for RPE estimates of the angle (cid:15) from subsampleddatasets with N = 16 and N = 256. 100 subsampleddatasets are used for each value of N . This plot isanalogous to the plot shown in Fig. 3; for furtherdetails, see caption of that figure. RPE GST R M S E rr o r − − − − Total samples10 FIG. 7: (Color online) Root mean squared error vstotal number of samples for RPE and GST estimates ofthe angle (cid:15) from the subsampled datasets with N = 16.This plot is analogous to the plot given in Fig. 5; forfurther details, see caption of that figure. [4] Z. Hou, H.-S. Zhong, Y. Tian, D. Dong, B. Qi, L. Li,Y. Wang, F. Nori, G.-Y. Xiang, C.-F. Li, and G.-C.Guo, New Journal of Physics , 083036 (2016).[5] Y. R. Sanders, J. J. Wallman, and B. C. Sanders, NewJournal of Physics , 012002 (2016).[6] M. Guti´errez, C. Smith, L. Lulushi, S. Janardan, andK. R. Brown, Phys. Rev. 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