Quantum Firmware: Engineering error resistance at the physical level for quantum computation Support Michael J. Biercuk Quantum Control Laboratory Centre for Engineered Quantum Systems School of Physics, The University of Sydney
Outline • Motivation (quickly) • Quantum Firmware for error suppression at the physical level • Noise-filtering: An Experiment-Friendly analytical approach • Experiments – quantum firmware in the lab • Noise filtering in nontrivial gates
Decoherence due to environmental noise: A Fundamental Limit Dephasing Relaxation
How do we deal with this error? • Closed-loop feedback control ⇒ Quantum Error Correction • Open-loop control ⇒ Dynamical Error Suppression Aim: Improve QEC performance by driving down physical error rates
Spin Echo to Dynamical Decoupling Hahn 1950, NMR Simple application of control pulses suppresses errors… Significant flexibility in pulse number, pulse timing, pulse types… The art is in efficient sequencing 95, 180501 (2005). PRA 58 58, 2733 (1998). PRA 59, 59, 4178 (1999). PRL 95
Vision: “Quantum Firmware” • Efficient physical-layer error evasion strategy • Simple implementation • Absorbed into machine-language abstraction: invisible to programmer • Useful for *any* quantum technology • Similar to DRAM refreshes
What are our interests: • Making dynamical error suppression accessible – i.e. Please don’t speak group theory to me • Calculating error rates in real environments instead of assuming some p • Considering realistic constraints imposed by hardware • Designing quantum control approaches compatible with these constraints
Noise Filtering
Calculating the effect of fluctuating noise β (t) represents dephasing noise is the error Net Phase is convolution of noise and toggling frame
Capturing the average behavior Time Domain Convolution Fourier Domain Product Uhrig Uhrig et al. et al. , , PRL PRL 98, 100504 (2007); 98, 100504 (2007); Cywinski Cywinski et al., et al., PRB PRB 77, 174509 (2008). 77, 174509 (2008). Biercuk Biercuk et al et al ., ., Nature Nature 458, 996 (2009). Biercuk et al., , 996 (2009). Biercuk et al., J. Phys. B J. Phys. B 44, 154002 (2011). , 154002 (2011).
Capturing the average behavior Time Domain Convolution Fourier Domain Product Coherence: Coherence preserved if F small for dominant frequencies Uhrig Uhrig et al. et al. , , PRL PRL 98, 100504 (2007); 98, 100504 (2007); Cywinski Cywinski et al., et al., PRB PRB 77, 174509 (2008). 77, 174509 (2008). Biercuk Biercuk et al et al ., ., Nature Nature 458, 996 (2009). Biercuk et al., , 996 (2009). Biercuk et al., J. Phys. B J. Phys. B 44, 154002 (2011). , 154002 (2011).
Fully characterize any sequence by F Stopband Passband Rolloff ~ Filter Order Filter Function Normalized Frequency [ �� ] Biercuk et al., Biercuk et al., J. Phys. B J. Phys. B 44, 154002 (2011). , 154002 (2011).
Fully characterize any sequence by F Plug-In High Pass Filter PHP-1000+ Typical Performance Curves Insertion Loss 0 10 20 30 Insertion Loss (dB) 40 50 60 70 80 90 100 1 10 100 1000 10000 Frequency (MHz) Biercuk et al., Biercuk et al., J. Phys. B J. Phys. B 44, 154002 (2011). , 154002 (2011).
Timing Constraints Reduce Performance (Daniel didn’t tell the whole story…there are other prices to pay) Filter Function Dimensionless Angular Frequency ( 2 � Hz � ) 10 GHz clocking in 1 ms experiment reduces performance by 80 dB @ ωτ = 1 Biercuk, Doherty, Uys , J. Phys. B 44 44, 154002 (2011).
Digital modulation for large systems The Walsh Functions Walsh Index, n Time Bin Normalized Pulse Location Other benefits: ease of sequencing, unified framework, long-time storage… For more see K. Khodjasteh’s talk, Thursday 2:40 PM Hayes, Khodjasteh, Viola, Biercuk, arXiv: 1109.6002, to appear in PRA
Quantum Firmware and Noise Filtering in the lab
Experimental Quantum System Crystal of Beryllium ions
Ions in a Penning Trap B=4.5 T B=4.5 T 9 Be + at 4.5T Fluorescence Cooling Field Sensitive 9 Be + F=1 124 GHz F=2 ν c ~ 7.6 MHz 7.6 MHz ν z ~ 600 kHz 600 kHz 20 kHz ν m ~ 20 kHz 458, 996 (2009). Biercuk et al ., Quant. Info. Comp . 9, 920 (2009). Biercuk et al ,. Nature 458
DD suppresses decoherence 1.0 T 2 ~2.5 ms 0.8 Population !"# ! 0.6 0.4 0.2 0.0 0 1 2 3 4 Ramsey Precession Time (ms) Biercuk et al ., Quant. Info. Comp . 9, 920 (2009).
DD suppresses decoherence 1.0 Coherence time increases ~10X (for 10 pulses) 0.8 Probability !" ! # 0.6 0.4 Free-Induction 0.2 10-pulse CPMG Decay 0.0 0 10 20 30 40 50 60 Free-Precession Time (ms) Biercuk et al ., Quant. Info. Comp . 9, 920 (2009).
The filter function works! 2.5 n =4 ! 2.0 Error Probability n =5 ! 1.5 n =6 ! 1.0 n =8 ! 0.5 n =10 ! 0.0 10 20 30 40 50 Total Free Precession Time (ms) Solid: CPMG, Open: UDD (Different Pulse Spacings) Biercuk et al ., Nature 458 458, 996 (2009). Biercuk et al ., Phys. Rev. A 79 79 062324 (2009).
The filter function works! 2.5 n =4 ! 2.0 Error Probability n =5 ! 1.5 n =6 ! 1.0 n =8 ! 0.5 n =10 ! 0.0 10 20 30 40 50 Total Free Precession Time (ms) Solid: CPMG, Open: UDD (Different Pulse Spacings) Biercuk et al ., Nature 458 458, 996 (2009). Biercuk et al ., Phys. Rev. A 79 79 062324 (2009).
Noise Injection: Modeling Other Qubits 5 10 4 10 3 10 1/ ω 2 10 1 10 0 10 -1 10 -2 10 -3 10 6 8 2 4 6 8 2 4 6 8 2 3 4 10 10 10 5 10 4 10 3 10 2 10 1 10 0 10 -1 10 -2 10 6 8 2 4 6 8 2 4 6 8 2 3 4 10 10 10 ! (2 " Hz) 79 062324 (2009). Biercuk et al ., Nature 458, 996 (2009). Biercuk et al ., Phys. Rev. A 79
UDD can outperform CPMG Red=UDD, Black=CPMG UDD Better 2.5 V N = 0.9 V 2.0 Noise Strength Error Probability Error Probability 0.7 V 1.5 0.5 V 1.0 0.3 V 0.5 CPMG Better 0.1 V 0.0 0 1 2 3 4 Free Precession Time (ms) S � ( � ) (rad/s) 4 10 1 10 -2 10 (c) -5 10 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 3 4 10 10 10 Angular Frequency (rad/s) MJB et al ., Nature 458 458, 996 (2009). MJB et al ., Phys. Rev. A 79 79 062324 (2009)., MJB et al QIC 9, 920 (2009)
Optimization by autonomous feedback 1 CPMG 4 UDD 2 Error Probability LODD 0.1 4 2 0.01 5X improvement over UDD 4 8X improvement over CPMG 2 0.001 1 2 3 4 Total Free Precession Time (ms) We have realized improved performance by tailoring the filter function Biercuk et al ., Nature 458 458, 996 (2009)
Interlude: Mike’s opinion on UDD
Dynamical decoupling studies w/ions…
Why did this work? High-fidelity control. 99.92% Fidelity/Gate 1.00 0.95 Average Fidelity 0.90 1 0.85 0.80 0 ! " =185 ! s 0.5 1.0 1.5 2.0 Pulse Length (ms) 0.75 50 100 150 200 Computational Gate Number Need to develop new Need to develop new flexible flexible high-fidelity high-fidelity control hardware, and understand gate errors control hardware, and understand gate errors 77, 012307 (2008). Biercuk et al ., Quant. Info. Comp . 9, 920 (2009). PRL 97 97 , 220407 (2006). PRA 77
Dynamical protection beyond Memory Green, Uys, & Biercuk, arXiv:1110.6686
Recommend
More recommend
![Reversing firmware using radare2 [H2HC] A. Kochkov October, 2014 Motives Implement FOSS](https://c.sambuz.com/903352/reversing-firmware-using-radare2-h2hc-s.webp)
