Matt Reed

Yale University

Boston, MA - February 28, 2012

Three-qubit quantum error correction with superconducting circuits

Leo DiCarlo

Simon Nigg

Luyan Sun

Luigi Frunzio

Steven Girvin

Robert Schoelkopf

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)

z

y

x

Quantum bit

Why do we need to correct?

Control signal

Sta

te v

alue

“0”

“1”

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

Classical bit

Small control fluctuations do cause a change in the system state!

p ~ 10-2 - 10-5

Classical repetition code

Probability p of having a bit flipped

“Binary symmetric channel”

0 0001 111

Repetition code: send each bit three times, then vote

Reduces classical error rate to 3p2 – 2p3

1-p0

1 1

0

1-p

p

pSen

tR

eceived

• No cloning theorem• Measurements project qubits• Errors are continuous

Can we do this for quantum computing? Some reasons to think no:

Quadratic!

GHZ-like states

“I don’t know where they are pointing, but I know they’re pointing in the same direction”

But we can make

All ZiZj correlations are +1, independent of and

both 0Qubits 1 and 2 are either: both 1or

Z1Z2 = = +1=

It is not possible to go from

Flipping GHZs

Flipped State Z1Z2 Z2Z3

None +1 +1

Q1 -1 +1

Q2 -1 -1

Q3 +1 -1

Each error has a different observable! - The basis for the bit flip code

What happens when we flip one of the qubits in a GHZ-like state?

Z1Z2 = +1Z2Z3 = +1

Z1Z2 = -1Z2Z3 = +1

Independent of and

Four errors = two classical bits

Circuit quantum electrodynamicsOur system: superconducting qubits coupled to a microwave resonator

In analogy to cavity QED:

Transmon qubits

Transmission-line resonator bus• 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, 173601 (2010)

Four-qubit cQED device• Four transmon qubits coupled to single 2D microwave resonator

• 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 467 574 (2010)

cavity

Q2

Q3

Flux bias on Qubit 1 (a.u.)

Q1

Fre

quen

cy (

GH

z)

Adiabatic multiqubit phase gates

Interactions on two excitation manifold give entangling two-qubit conditional phases

A two qubit phase gate can be written:

Top qubit flux bias (a.u.)

Entanglement!

DiCarlo, et al. Nature 460, 240 (2009)

Interactions on two excitation manifold give entangling two-qubit conditional phases

Can give a universal “Conditional Phase Gate”

A two qubit phase gate can be written:

Entanglement!

Top qubit flux bias (a.u.)

Adiabatic multiqubit phase gates

DiCarlo, et al. Nature 460, 240 (2009)

Sudden multiqubit phase gates

Suddenly move into resonance with

Or transfer to in 6 ns!

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, 167005 (2003)

Entangled states on demand

0

0

0 /2yR

/2yR /2

yR

01

Sta

teTo

mog

rap

hy T

1

20 10 10

94%F T T

DiCarlo, et al. Nature 467 574 (2010)

GHZ states on demand

0

0

0

Sta

teTo

mog

rap

hy

/2yR

/2yR

/2yR

/2yR

/2yR

01

10

88%F GHZ GHZ

Can simply change the preparation of Q2 to encode any stateDiCarlo, et al. Nature 467 574 (2010)

error

Error correction with GHZ states

encode

X

X

or

X

or0

0

0 1

diagnose fix

0 1

0

0X

Logic

X

X

GHZ state for

Measurements force finite rotations to full flips

Nielsen & Chuang

nose

Ions: Chiaverini et al. Nature 432, 602 (2004)NMR: Cory et al. PRL 81, 2152 (1998)

Works for any single error

Measurement-free QECToffoli implements classical logic • only acts on flipped subspace

Toffoli(CCNot)

gate

encode

X

0

0

0 1

diagnose fix

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)

Feed-forward measurement hard in this first expt- Measurement-free version of the code

0 1

0

0

Reset(potentially)

Toffoli gate with noncomputational statesTwo-qubit gate requires two excitations

Three-qubit interaction: third excited state The essence!

Adiabatic interaction:

Three-qubit phase here!

This interaction is small, so use intermediary

Sudden transfer:

Identical for:

111101011001110100010000

111101011001110100010000

Classical input state

Classical output state

Classical truth table

Classically, a phase gate does nothing. So we dress it up to make it a CCNOT

F = 86%

How do we prove the gate works? First, measure classical action

Optics: Lanyon Nat. Phys. 5, 134 (2009)Ions: Monz PRL 102, 040501 (2009)SCQs: Mariantoni Science 334, 61 (2011) Fedorov Nature 481, 170 (2012)

(>50% the time of an equivalent construction)

Quantum process tomography of CCPhase

Theory

Input operatorOutput operator

0.0

0.6

0.3

Input operatorOutput operator

Experiment

F = 78%

4032 Pauli correlation measurements (90 minutes)

Want to know the action on superpositions:

(but now with 64 basis states)

Invert to find

Protection from single qubit bit-flip errorsPrepare

Encode inthree-qubit

state

“Error” rotation by some angle

Decode syndromes

Correct subspace with error

Ideally, there should be no dependence of fidelity on the error rotation angle

Measure single-qubit state fidelity to

Correction fidelity vs. bit-flip error rotation

Encode, single known error, decode, fix, and measure resulting state fidelity

Error on Q3

No correction

Error on Q1

Error on Q2

(No-correction curve has finite fidelity because its duration is the same as the corrected curves)

Error syndromes

Look at two-qubit density matrices of ancillas after a full flip

Is the algorithm really doing what we think?

IIZIIZZZ

IIZIIZZZ

1

0

-1

No error

Top flip

Bottom flip

X

Protected flip

X

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

( )zR

( )zR

( )zR

More realistic error model: Simultaneous flips on each qubit happen with probability

2sin ( / 2)p

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

No correction

Simultaneous phase-flip errorsTo measure the effect of the code on any state, test with four one-qubit basis states

Corrected

Depends only quadratically on error probability!

For better coherence, see 3D Cavity session L39 (room 109B)

Summary

• Realized the simplest version of gate-based QEC• Both bit- and phase-flip correction

• Not fault-tolerant (gate based, un-encoded)

Reed, et al. Nature 482, 382 (2012)

• 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

Questions?

Reed, et al. Nature 482, 382 (2012)

CCNot gate pulse sequence

More than two times faster than equivalent two-qubit gate sequence

Example: extract

I Z I ZI I Z I Z Z ZZ I I Z Z I ZI Z Z

4 ZZZ

no pre-rotation:

on Q1 and Q2:

on Q1 and Q3:

on Q2 and Q3:

0,xR

0,xR

0,xR

Joint Readout

000 000

M

( )xR ( )xR ( )xR

ZZZ

I Z I ZI I Z I Z Z ZZ I I Z Z I ZI Z Z I Z I ZI I Z I Z Z ZZ I I Z Z I ZI Z Z I Z I ZI I Z I Z Z ZZ I I Z Z I ZI Z Z

000 000M

I Z I ZI I Z I Z ZI Z ZZ I I Z Z I Z Z

Three qubit state tomography

DiCarlo, et al. Nature 467 574 (2010)