1. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. Numerical Analysis of Robust Phase Estimation Kenneth Rudinger (SNL), Shelby Kimmel (QuICS)

2. Main question  You want to characterize single-qubit operations: G x (α), G y (ε), θ=arccos(X  Y)  Desire self-consistently, efficiently (Heisenberg-limited accuracy)).  There are (at least!) two good options:  Robust Phase Estimation (RPE)  Gate Set Tomography (GST) When should you use RPE and when should you use GST? 2

3. RPE and GST Key concept: Amplifying errors through structured, repeated sequences (germs). Tomographic probing achieved with short sequences after preparation or before measurement (fiducials). 3   G1G1 G1G1 G2G2 G2G2 FiFi FiFi FjFj FjFj G1G1 G1G1 G2G2 G2G2 FiFi FiFi FjFj FjFj Repetitions logarithmically spaced (k=1,2,4,…,1024,…) Only weak assumptions about gate set (ρ, E, {G i }) required (unlike standard process tomography). Recorded frequencies ( /( + )) constitute data set; processed to produce desired estimates. Heisenberg scaling achieved in accuracy Error ~ 1/k max      

4. GST and RPE GSTRPE Estimated quantitiesSPAM and process matrices- ~30 parameters (Also, e.g., diamond norms) (Also also non-Markovianity scores) Rotation angles, rotation axis angle- (3 paramaters). Number of fiducial pairs (n f ) 362 Number of germs (n g )83 Sequences per generation (n f *n g ) 2886 Analysis coreNon-linear least-squares likelihood maximization (expensive!) Inverse trignometric functions, “simple” minimization 4

5. 5 N=32 Mean of 100 trials All angle errors = 0.01 radians Heisenberg scaling achieved in accuracy* (including error bars)! *Cutoff for θ in RPE. RPE in blue. G x rotation error (α) G x -G y axis-angle error (θ)G y rotation error (ε) RPE vs GST: Simulations

6. 6 N=32 Mean of 100 trials All angle errors = 0.01 radians Heisenberg scaling achieved in accuracy* (including error bars)! *Cutoff for θ in RPE. RPE in blue. GST in green. G x rotation error (α) G x -G y axis-angle error (θ)G y rotation error (ε) RPE vs GST: Simulations

7. Resource costs  How do experimental resource costs compare?  Estimate error of 10 -4  GST- 2*10 3 sequences; 6*10 4 clicks. 23 s analysis runtime.  RPE- 60 sequences; 1.92*10 3 clicks. 0.34 s analysis runtime. 7 G x rotation error (α)

8.  Trapped 171 Yb + on a chip; microwave gates.  N=370, k=1024 GSTRPE π/2-α(-6.4±1.6)*10 -5 (-1.7±1.1)*10 -4 π/2-ε(-2.7±1.1)*10 -5 (-1.3±1.1)*10 -4 θ (6.4±1.7)*10 -5 (2.1±0.06)*10 -4 Experimental results! G x rotation (α) G x -G y axis-angle (θ)G y rotation (ε) 8

9.  Trapped 171 Yb + on a chip; microwave gates.  N=370, k=1024 GSTRPE π/2-α(-6.4±1.6)*10 -5 (-1.7±1.1)*10 -4 π/2-ε(-2.7±1.1)*10 -5 (-1.3±1.1)*10 -4 θ (6.4±1.7)*10 -5 (2.1±0.06)*10 -4 Experimental results! G x rotation (α) G x -G y axis-angle (θ)G y rotation (ε) ΣN GST =8.7*10 5 ΣN RPE =2.4*10 4 ΣN GST =8.7*10 5 ΣN RPE =2.4*10 4 RPE and GST (mostly) agree! 9

10. Conclusions  First numerical and experimental demonstration of robust phase estimation.  RPE can efficiently and consistently estimate gate rotation angles and axes.  Experimental RPE results are consistent with gate set tomography; RPE costs significantly less than GST.  …But RPE gives no additional information about gate sets (process matrices, fidelities, diamond norms, Markovianity scores).  Want a first look at gate performance or a quick tuneup? Use RPE! Want more? Use GST! 10