1. Matt Reed Yale University Boston, MA - February 28, 2012
Three-qubit quantum error correction
with superconducting circuits
Matt Reed
Yale University
Boston, MA - February 28, 2012
Leo DiCarlo
Simon Nigg
Luyan Sun
Luigi Frunzio
Steven Girvin
Robert Schoelkopf

2. Outline Why is QEC necessary? Repetition codes Our architecture: cQED
Adiabatic and sudden two-qubit phase gates
GHZ states
Efficient Toffoli gate using third-excited state
Bit- and phase-flip error correction
Reed, et al. Nature 482, 382 (2012)

3. Why do we need to correct?
Classical bit z y x
Quantum bit
Small control fluctuations do cause a change in the system state!
p ~
State value
“1”
“0”
Control signal
Small control fluctuations do not change the system state – compressed phase space
Error probability ~ 10-15
To get p~10-15 would need T1 ~ 1 year

4. Classical repetition code1 p
Sent
Received
Probability p of having
a bit flipped
“Binary symmetric
channel”
Repetition code: send each bit three times, then vote
Reduces classical error rate to 3p2 – 2p3
Quadratic!
Can we do this for quantum computing? Some reasons to think no:
No cloning theorem
Measurements project qubits
Errors are continuous

5. GHZ-like states Z1Z2 = = +1 = It is not possible to go from
But we can make
All ZiZj correlations are +1, independent of and
both 0
Qubits 1 and 2 are either:
both 1 or
Z1Z2 =
= +1 =
“I don’t know where they are pointing, but I know they’re pointing in the same direction”

6. Flipping GHZs Z1Z2 = +1 Z1Z2 = -1 Z2Z3 = +1 Z2Z3 = +1
What happens when we flip one of the qubits in a GHZ-like state?
Z1Z2 = +1
Z2Z3 = +1
Z1Z2 = -1
Z2Z3 = +1
Independent of and
Flipped
State
Z1Z2
Z2Z3
None +1 Q1 -1 Q2 Q3
Each error has a different observable! - The basis for the bit flip code
Four errors = two classical bits

7. Circuit quantum electrodynamics
Our system: superconducting qubits coupled to a microwave resonator
Transmon qubits
Transmission-line resonator bus
In analogy to cavity QED:
Protection from spontaneous emission
Qubit readout
Multiplexed qubit drives (single-qubit gates)
Mediate qubit coupling (multi-qubit gates)
cQED: Wallraff Nature 431, 162 (2004)
Bus: Majer Nature 449, 443 (2007)
Readout: Reed PRL 105, (2010)

8. Four-qubit cQED device
Four transmon qubits coupled to single 2D microwave resonator
cavity Q2 Q3
Flux bias on Qubit 1 (a.u.) Q1
Frequency (GHz)
Three qubits biased at 6, 7, and ~8 GHz
Fourth qubit above cavity and unused
T1 ~ 1 μs, T2 ~ 0.5 μs
Flux bias lines to control frequency
Nanosecond speed - two qubit gates
DiCarlo, et al. Nature (2010)

9. Adiabatic multiqubit phase gates
A two qubit phase gate can be written:
Entanglement!
Interactions on two excitation manifold give entangling two-qubit conditional phases
Top qubit flux bias (a.u.)
DiCarlo, et al. Nature 460, 240 (2009)

10. Adiabatic multiqubit phase gates
A two qubit phase gate can be written:
Entanglement!
Interactions on two excitation manifold give entangling two-qubit conditional phases
Can give a universal “Conditional Phase Gate”
Top qubit flux bias (a.u.)
DiCarlo, et al. Nature 460, 240 (2009)

11. Sudden multiqubit phase gates
Suddenly move into resonance with
Crossing measurement:
• Jump to a flux
• Wait some time
• Jump back
• Measure if in 11 (black)
or 02 (white)
Previously proposed:
Strauch et al., PRL 91, (2003)
Or transfer to in 6 ns!

12. Entangled states on demand
Tomography
State 01
DiCarlo, et al. Nature (2010)

13. Can simply change the preparation of Q2 to encode any state
GHZ states on demand 01 10
Tomography
State
Can simply change the preparation of Q2 to encode any state
DiCarlo, et al. Nature (2010)

14. Error correction with GHZ states
Measurements force finite rotations to full flips
encode
error
diagnose
nose
fix or
GHZ state for
Logic
Works for any single error
Nielsen & Chuang
NMR: Cory et al. PRL 81, 2152 (1998)
Ions: Chiaverini et al. Nature 432, 602 (2004)

15. Measurement-free QEC encode diagnose fix
Feed-forward measurement hard in this first expt
- Measurement-free version of the code
Toffoli implements classical logic
• only acts on flipped subspace
Toffoli
(CCNot)
gate
encode
diagnose
fix
Reset
(potentially)
Toffoli can be constructed with five two-qubit gates, but that’s expensive
How can we do better?
Nielsen & Chuang Cambridge Univ. Press
Ions: P. Schindler et al. Science 332, 1059 (2011)

16. Toffoli gate with noncomputational states
Two-qubit gate requires two excitations
Three-qubit interaction: third excited state
The essence!
This interaction is small, so use intermediary
Adiabatic interaction:
Sudden transfer:
Identical for:
Three-qubit phase here!

17. How do we prove the gate works? First, measure classical action
Classical truth table
How do we prove the gate works? First, measure classical action
Classically, a phase gate does nothing. So we dress it up to make it a CCNOT
111
101
011
001
110
100
010
000
Classical input state
Classical output state
F = 86%
(>50% the time of an equivalent construction)
Optics: Lanyon Nat. Phys. 5, 134 (2009)
Ions: Monz PRL 102, (2009)
SCQs: Mariantoni Science 334, 61 (2011)
Fedorov Nature 481, 170 (2012)

18. Quantum process tomography of CCPhase
Want to know the action on superpositions:
(but now with 64 basis states)
Invert to find
Theory
Input operator
Output operator
0.0
0.6
0.3
Input operator
Output operator
Experiment
F = 78%
4032 Pauli correlation measurements (90 minutes)

19. Protection from single qubit bit-flip errors
Prepare
“Error” rotation by some angle
Correct subspace with error
Encode in
three-qubit state
Decode syndromes
Measure single-qubit state fidelity to
Ideally, there should be no dependence of fidelity on the error rotation angle

20. Correction fidelity vs. bit-flip error rotation
Encode, single known error, decode, fix, and measure resulting state fidelity
No correction
Error on Q2
Error on Q1
Error on Q3
(No-correction curve has finite fidelity because its duration is the same as the corrected curves)

21. Error syndromes Is the algorithm really doing what we think?
Look at two-qubit density matrices of ancillas after a full flip
No error II ZI IZ ZZ 1 -1
Bottom flip
Top flip
Protected flip

22. Phase-flip error correction code
Bit-flips are not common errors, but phase flips are – modify code
Differs from bit-flip code by single qubit rotations; e.g. change of coordinate system
More realistic error model:
Simultaneous flips on each qubit happen with probability
Apply errors and measure fidelity to the prepared state as a function of p
Code only works for single errors.
P(2 or 3 errors) = 3p2 – 2p3
Expect quadratic dependence on p

23. Simultaneous phase-flip errors
To measure the effect of the code on any state, test with four one-qubit basis states
No correction
Depends only quadratically
on error probability!
Corrected
For better coherence, see 3D Cavity session L39 (room 109B)

24. Summary Realized the simplest version of gate-based QEC
Both bit- and phase-flip correction
Not fault-tolerant (gate based, un-encoded)
Based on new three-qubit phase gate
Adiabatic interaction with transmon third excited state
Works for any three nearest-neighbor qubits
86% classical fidelity and 78% quantum process fidelity
Reed, et al. Nature 482, 382 (2012)

25. Questions?
Reed, et al. Nature 482, 382 (2012)

26. CCNot gate pulse sequence
More than two times faster than equivalent two-qubit gate sequence

27. Three qubit state tomography
Joint
Readout
Example: extract
no pre-rotation:
on Q1 and Q2:
on Q1 and Q3:
on Q2 and Q3:
DiCarlo, et al. Nature (2010)