Experimental Quantum Error Correction Raymond Laflamme Institute - PowerPoint PPT Presentation

Experimental Quantum Error Correction Raymond Laflamme Institute for Quantum Computing, Waterloo laflamme@iqc.ca www.iqc.ca QEC11, December 2011

Plan � Introduction � Benchmarking and certifying gates � Implementations of QEC � Conclusion

Threshold theorem Threshold theorem See Andrew Landahl ‘s talk A quantum computation can be as long as required with any desired accuracy as long as the noise level is below a threshold value -6 ,-5,-4,...,-1? P < 10 Knill et al.; Science, 279, 342, 1998 Kitaev, Russ. Math Survey 1997 Aharonov & Ben Or, ACM press Significance: ... Preskill, PRSL, 454, 257, 1998 -imperfections and imprecisions are not fundamental objections to quantum computation -it gives criteria for scalability -its requirements are a guide for experimentalists -it is a benchmark to compare different technologies

Threshold theorem Threshold theorem Accuracy threshold Theorem proved... what is left? See Andrew Landahl ‘s talk -what is the value of the threshold? -what is the operational cost? A quantum computation can be as long as required with any desired accuracy as long as the noise level is below a threshold value -6 ,-5,-4,...,-1? P < 10 Knill et al.; Science, 279, 342, 1998 Kitaev, Russ. Math Survey 1997 Aharonov & Ben Or, ACM press Significance: ... Preskill, PRSL, 454, 257, 1998 -imperfections and imprecisions are not fundamental objections to quantum computation -it gives criteria for scalability -its requirements are a guide for experimentalists -it is a benchmark to compare different technologies

Ingredients for FTQEC � Parallel operations � Good quantum control � Ability to extract entropy � Knowledge of the noise • No lost of qubits • Independent or quasi independent errors • Depolarising model • Memory and gate errors • . . .

Ingredients for FTQEC � Parallel operations � Good quantum control � Ability to extract entropy � Knowledge of the noise • No lost of qubits • Independent or quasi independent errors • Depolarising model • Memory and gate errors • . . . and lots of qubits...

Progress in experimental QIP • # of qubits vs time 14 14 Cat State Metrology 13 13 Cat State Metrology Benchmarking 12 11 NMR Photons 10 Neutral Atoms Superconducting Circuits 9 Trapped Ions Number of Qubits 8-Qubit Quantum Dots W State 8 Benchmarking Shor Red text: Specially 7 prepared states 6 Photons Entangled • Increasing control of qubits 6 4 Photons 5-qubit Entangled QEC 5 4-Particles Entangled Shor’s 4-spin Cluster 4 Maximally State Entangled 3-qubit QFT QEC Phase QEC Grover QEC 3 To oli 3-qubit Swap QEC CNOT Grover CNOT 2 CNOT CNOT Swap Two-level Cooper Spin States Pair Triggered Single 1 Observed Box Photons from a QD 0 1985 1990 1995 2000 2005 2010 2015 Year Adapted from Michael Mandelberg Ladd, T. D., et al., Nature, 464(7285), 45–53, 2010

Progress in experimental QIP • # of qubits vs time yesterday we heard... 14 14 Cat State Metrology overhead: 10^3 is better than 10 ^11 ... 13 13 Cat State Metrology Benchmarking 12 11 NMR Photons 10 Neutral Atoms Superconducting Circuits 9 Trapped Ions Number of Qubits 8-Qubit Quantum Dots W State 8 Benchmarking Shor Red text: Specially 7 prepared states 6 Photons Entangled • Increasing control of qubits 6 4 Photons 5-qubit Entangled QEC 5 4-Particles Entangled Shor’s 4-spin Cluster 4 Maximally State Entangled 3-qubit QFT QEC Phase QEC Grover QEC 3 To oli 3-qubit Swap QEC CNOT Grover CNOT 2 CNOT CNOT Swap Two-level Cooper Spin States Pair Triggered Single 1 Observed Box Photons from a QD 0 1985 1990 1995 2000 2005 2010 2015 Year Adapted from Michael Mandelberg Ladd, T. D., et al., Nature, 464(7285), 45–53, 2010

e − i π 2 X 2 Y R → e − i π ( θ ) 0  0  Benchmarking gate Usually we think of the circcuit model: Prepare a state, com- pute, measure | | | M 0  1  n Other possibility is to use only generators of the Clifford group (generated by Hadamard, Phase gate and CNOT), with state preparation and measuremen in the computational basis: | | | M 0  1  and include the preparation of | π/ 8 � , or ρ = 1 l + 1 21 √ ( X + Y + Z ) 3

Benchmarking gates Knill et. al. PRA, 77, 012307, (2008)

Benchmarking gates

Benchmarking gates Technology 1 Qubit Gate Error Date 2 . 0(2) × 10 − 5 Single Trapped Ion 2011 1 . 3(1) × 10 − 4 Liquid State NMR 2009 1 . 4(1) × 10 − 4 Atoms in Optical Lattice 2010 1 . 4(2) × 10 − 4 ESR 2011 8((1) × 10 − 4 Trapped Ion Crystal 2009 4 . 8(2) × 10 − 3 Single Trapped Ion 2008 5(2) × 10 − 3 Solid State NMR 2011 7(5) × 10 − 3 Superconducting Transmon 2010

Randomized Benchmarking Reference Randomized Benchmarking Reference 1 1 0.99 0.99 0.98 0.98 Fidelity Fidelity 0.97 0.97 0.96 0.96 0.95 0.95 0 50 100 150 200 0 50 100 150 200 Number of Computational Gates Number of Computational Gates (a) (b) Randomized Benchmarking Reference Benchmarking Simulated Echo 1 1 Benchmark Absolute Fit 0.8 Imaginary 0.99 Real 0.6 Signal (arbitrary) 0.98 Fidelity 0.4 0.2 0.97 0 f(x) = a*exp(b*x) 0.96 Coefficients (with 95% confidence bounds): a = 0.9998 (0.9996, 1) −0.2 b = −0.0002517 (−0.0002545, −0.000249) 0.95 −0.4 0 50 100 150 200 0 1000 2000 3000 4000 Number of Computational Gates Time (ns) (c) (d)

Benchmarking gates Multi-qubit Comparison Summary Table System Error/Fidelty Reference NJP 11 013034 liquid-state NMR 0.0047 (2009) Nat. Phys. 4 463 ion-trap (single) 99.3% (2008) Nature 460 240 superconducting 91% (2009) Science 320 1326 NV centre 89% (2008) PRL 93 080502 Linear Optics 90% (2004) arXiv:0907.5552 Neutral Atoms 73% (2009) Nature 455 1085 ESR 95% (2008) A generalisation of the 1 qubit benchmarking can be found in E. Magesan, J. M. Gambetta, and J. Emerson, Phys. Rev. Lett. 106, 180504 (2011).

Characterising noise in q. systems Process tomography: A k ρ i A † � � ρ f = χ kl P k ρ i P l k = k kl For one quibt, 12 parameters are required as described by the evolution of the Bloch sphere: ρ f χ 1 1 χ 1 1 , X χ 1 1 , Y χ 1 ρ i       1 , 1 1 , Z 1 1 1 1 ρ f χ X , 1 1 χ X , X χ X , Y χ X , Z ρ i       X X ρ f χ Y , 1 1 χ Y , X χ Y , Y χ Y , Z ρ i  =            Y Y ρ f χ Z , 1 1 χ Z , X χ Z , Y χ Z , Z ρ i Z Z For n qubits, we need to provide 4 2 n − 4 n numbers to do so.

Coarse graining Emerson, Silva, Moussa, Ryan, Laforest, • We are not interested Baugh, Cory, Laflamme, Science 317, 1893, 2007 in all the elements that describe the full noise superopeartor but only a coarse graining of them. • If we are interested in implementing quantum er- ror corrrection, we can ask what is the probability to get one, or two, or k qubit error, independent of the location and independent of the type of error σ x,y,z . The question is can we do this efficiently? • Coarse graining is equiv- alent to implement a sym- metry.

Coarse graining 1) we don’t want to know which qubit is affected, coarse grain the position by symmetrising using permutation π s 2) turn the noise into a depolarizing one for each qubit, coarse grain error type average over SU (2) ⊗ n � dµ ( U ) U † P k Uρ i U † P † � ρ f = χ kl l U kl This is an example of a 2-design, and the integral can be replaced by a sum α P † � � C † α P k C α ρ i C † ρ f = χ kl l C α α kl where C α belongs to the Clifford group ∼ SP with 4 Y , e − i π l , X, Y, Z } , S = { e − i π 4 X , e − i π 4 Z } P = { 1

Coarse graining To estimate the noise, start with the state | 000 . . . � , implement the symmetrisation group and the Clifford group and count how many bits have been flipped. Λ i ( n ) C † σ out ρ m π s π † C i Λ |000...> |010...> m,i,s s i Λ i,s If we implement all the elements in the Clifford and permutation group, we would have an exponential number of terms , but the sum can be estimated by sampling and using the Chernoff bound. (see Emerson et al. Science 317, 1893, 2007)

Errors in Clifford gates Adapt the idea for Clifford gates Practical experimental certification of com- putational quantum gates via twirling O. Moussa, M.P. da Silva, C.A. Ryan and R. Laflamme

Errors in Clifford gates Use malonic acid in solid state One qubit can be benchmarked using the Knill procedure: and Clifford gates using the new procedure Note: the difference between b) and c) is improving the pulse (“fixing”)