Experimental Quantum Error Correction

Raymond Laﬂamme Institute for Quantum Computing

www.iqc.ca

Help from : Martin Laforest, Osama Moussa other colleagues at IQC

Plan

The need for quantum error correction and the accuracy threshold theorem Characterising noise Characterising and demonstrating control • Implementing error correcting codes, noiseless subspaces and subsystems • Implementing encoded gates Extracting entropy, algorithmic cooling Conclusion The death of QComputers˚(1995)˚ Threshold theorem

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

16 Ingredients for FTQEC

Knowledge of the noise Good quantum control Ability to extract entropy Parallel operations Characterising noise in q. systems Process tomography:

X † X ρf = AkρiAk = χklPkρiPl k kl For one qubit, 12 parameters are required as de- scribed by the evolution of the Bloch sphere:

For n qubits, we need to provide 42n−4n numbers to do so.

6 Coarse graining

• We are not interested in all the elements that describe the full noise superopeartor but only a coarse graining of them.

• If we are interested in implementing quantum error corrrection, we can ask what is the probbaility to get one, or two, or k qubit error, independent of the location and independent of teh type of error σx,y,z. The question is can we do this eﬃciently?

• Coarse graining is equivalent to implement a sym- metry.

7 Coarse graining

1) to coarse error type average over SU(2)⊗n Z X † † † ρf = dµ(U)U AkUρiU AkU k This is an example of a 2-design, and the integral can be replaced by a sum X X † † † ρf = CαAkCαρiCαAkCα k α where Cα belongs to the Cliﬀord group ∼ SP with −iπX −iπY −iπZ P = {1l,X,Y,Z}, S = {e 4 , e 4 , e 4 }

2) coarse grain the position by symmetrising using permutation πs

8 Coarse graining

Λi (n) ρ † † out m πs Ci Λ Ci πs σm,i,s

Λi,s If we implement all the elements in the Clifford group and permutation, we would have an exponential number of terms , but the sum can be estimated by sam- pling and using the Chernoff bound. (see Emerson et al. arXiv:0707.0685 ) In practice, implementing the symmetrisation can be done by starting with the state |000 ... i and measure the Hamming size (i.e. the number of 1) in the final state.

9 NMR implementation: 2 qubits Experimental results

10 Checking for noise independence

If the noise is independent ω cω = (c1) Randomized Benchmarking of Quantum Gates • Wish to characterize the computationally relevant errors per gate in a general context • Process tomography gives complete information about a particular instance of a particular gate and is limited by errors in preparation and measurement • Furthermore, we need a scalable scheme to benchmark a large-scale quantm information processor efficiently • Scheme is to apply a sequence of random gates and measure the fidelity decay as a function of the number of gates

E. Knill, et al. arXiv:0707.0963 Depolarizing the noise

State Measure Preparation P G P G ..... P R P • Random Pauli gates (P) arrise naturally in fault-tolerant constructions • Random compuational gates (G) are Clifford group gates which can be efficiently tracked classicaly • Recovery gate (R) returns state to one which gives known answer with certainty • Averaging over many random sequences is equivalent to “twirling” the noise turning the map into a depolarizing map with one parameter d: 1 Λ (ρ) = A ρA† Λ¯ = (1 d) ρ + d 11 k k → − 2n k

R.F. Selection

• Can select subset of spins that see the same r.f. field • Sequence selects spins within +/- 2% of ideal r.f. field strength Pulses with r.f. Selection 1.02 • Exponential de- Simple Pulses 1.01 Fit cay suggests gate Composite Pulses 1 Fit error does not de- 0.99

0.98 pend on where in se- 0.97 quence it lies 0.96

Fidelity • Suggests an er- 0.95

0.94 ror per randomized 0.93 computational gate 0.92 of 5x10-4/3x10-4 for 0.91 simple/composite 0.9 0 10 20 30 40 50 60 70 80 90 100 Number of Computational Gates pulses 2 qubit phase damping code Q. Error Correction for Phase

Error Encoding Decoherence Decoding Correction α|0i+ β|1iY90 Y-90 α|0i +β|1 i

|0> Y90 Y-90

|0> Y90 Y-90 Toffoli gate

α|0i + β|Ø1i i j° Errors: + − j i j°i α| + ++i +Øβ| − −−i − + ° j i g α| − ++i + β| + −−i j i g αj | + −+i + β| − +−i g αj | + +−i + β| − −+i (α|0i + β|1i)|00i j (α|1i + β|0i)|11i Control-Not (α|0i + β|1i)|01i 00 00 → 00 (α|0ii+ β|j1i)|10j i i j i 01 → 01 11 = j i (α|0i + β|1i) ⊗ j i 10 → 11 j 01 { 1 2 j i 11 → 10 j i j ∼ 1 − 3γ { 10 |±i = √ (|0i±|1i) 2 j i

Version:Version: January January 17,2 17,2001; 2001; Typeset Typeset on June on June 12, 12,2002,14:58 2002,15:23 2 2 12 Phase QEC NMR circuit NMR implementation of the decoding and error correction:

Toﬀoli gate:

and the full decoding and Toﬀoli, including some optimization

35 3˚bit˚code˚in˚an˚ion˚trap 5 bit quantum error correcting code

Encoding Decoding M Y:90 ZZ:90 Implementation of the 5 bit code with C1 Z:270 -X:90 -Y:90 Y:90 the stabilizer Z2Y 3Y 4X5, Z1Y 2Y 3X4, C2 Y:90 -Y:90 Y:90 -Y:90 2 3 4 5 1 2 3 4 C X:90 Z:270 -X:90 -Y:90 Y Z Z Z and X Z X Z , including de- 3 90-y rotation C4 -X:90 -Y:90 coding and error correction for a basis of 1 Correcting Z errors M qubit errors. ZZ:45 C1 z C2 C3 C4 ZZ:-45 Correcting X errors M -y z y S swap C1 S S C2 12 C3 c0-not Fidelity C4 8 Correcting Y errors M 90x rotation S C1 x z -x x 4 C2 45 rotation C3 z C4 0.5 0.6 0.7 0.8 0.9

1.0

zC1 0.5 xM yC4 zC4 yM xC3 yC1 xC3 zM xC1 zC2 yC2 no xC2 zC3 xC4

Relative polarization 0.0 -100 -50 0 50 100 Hz

25 Decoherence˚free˚subspaces Encoded˚gates Noiselesssubsystem Algorithmic cooling with heat bath

12 Multiple Rounds of Algorithmic Cooling

Hm1/2

Hbath

Refresh Thermal Contact Register Operation • By using heat-bath able to surpass 9 x 10 10 Shannon/Soresnsen bound of 1.5X

9 heat-bath polarization

7 Polarization Boost

) 1.69X Boost 6 w.r.t. heat-bath Compression C2 C C 5 Step 1 m 4 1 1.39 0.47 0.49 Intensity (a.u. 3 1.56 2 2 0.68 0.71

1 3 1.64 0.76 0.79

0 4 1.69 0.79 0.84 −10 C1 −5 C2 0 5 Cm 10 Frequency (kHz) Conclusion

In order to error correct, we need to have Knowledge of the noise Good quantum control Ability to extract entropy Parallel operations Recent experiments have demonstrated these elements individually but we need to pull them together.

It is only the beginning of experimental QEC. Thanks