Detecting arbitrary single-qubit errors in a planar …...© 2015 IBM Corporation Detecting arbitrary single-qubit errors in a planar sublattice of the surface code Easwar Magesan

© 2015 IBM Corporation

Detecting arbitrary single-qubit errors in a planar sublattice of the surface code

Easwar Magesan

Quantum Cybernetics and Control Workshop

Nottingham, UK

January 22, 2015

arXiv: 1410.6419

IBM


Baleegh Abdo, David Abraham, Lev Bishop, Nick Bronn, Jerry M Chow, AntonioCorcoles, Andrew Cross, Oliver Dial, Stefan Filipp, Kent Fung, Jay M Gambetta, JaredHertzberg, George Keefe, Mark Ketchen, Chris Lirakis, Nick Masluk, Doug McClure,Hanhee Paik, John Rohrs, Mary Beth Rothwell, Jim Rozen, Martin Sandberg, WilliamShanks, Sarah Sheldon, John A Smolin, Srikanth Srinivasan, Matthias Steffen , MaikaTakita

IBM QUANTUM COMPUTING GROUP

TJ Watson Research CenterYorktown Heights, NY

IBM


Motivation

Implement surface code quantum computing with

superconducting qubits.

Why superconducting qubits? Improving coherence

times, circuit QED for control and measurement,

engineering and design of qubit properties.

Why surface code? High threshold for fault-tolerant

quantum computation, nearest-neighbor

interactions, one and two-qubit gates.

IBM

© 2015 IBM Corporation4

Beginning to realize larger networks of

qubits:

Motivation

Demonstrate important surface code operations:→

Square lattice – move into second dimension.

Arbitrary errors – not just bit, phase etc.

Here we implement a complete single-qubit

error detection experiment that incorporates

various key ingredients for implementing the

surface code.

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Outline

• Superconducting qubits – the transmon,

• Control, characterization and measurement,

• The surface code,

• 4-qubit device,

• [[2,0,2]] error-detection protocol,

• Experimental results.

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01

12= 01

23=01

01

12= 01

23=01

SC

SC

Barrier/weak-link

“Josephson Phase” Josephson RelationsNon-linear inductor

Superconducting qubits: anharmonicity

• Josephson energy: characteristic energy stored in inductor-

• charging energy: characteristic energy stored in capacitor -

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• No SQUID loop: simpler systems to characterize, avoid flux-noise.

• However no frequency tunability and more stringent on junction fabrication.

-Location of transition frequencies and anharmonic levels govern gate speeds

Physical qubit: transmon

Increasing EJ/EC ratio

Transition from cooper

pair box1 to transmon2.

[1] V. Bouchiat et al., Physica Scripta. T76 (1998) [2] J. Koch et al., Phys. Rev. A 76, 04319 (2007)

Single junction

Entanglement: leverage all-microwave schemes (i.e. cross-resonance).

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Control: single-qubit gates

X

Y

Z

X YI quadrature Q quadrature

[2] F. Motzoi et al., Phys. Rev. Lett. 103, 110501 (2009)

Phase errors due to leakage terms are

reduced by DRAG.

DRAG pulse2

pulse

amplitu

dederivative scaling

zero offset

bus

pad

qubit

-Use a circuit-QED architecture1.

[1] A. Blais et al., Phys. Rev. A 69, 062320 (2004)

Typical gate times: 20-60 ns.

Gaussian pulse shapes,

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Q1 Q2

control target

Two-qubit gates: Cross-resonance

Quantum Bus1

[1] J. Majer et al., Nature, 449 (7161) (2009)

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Two-qubit gates: Cross-resonance

Q1 Q2

control target

Quantum Bus1

Drive Q1 at

Q2 frequency:

Two qubit gate contribution:

Single qubit gate contribution:

m represents classical cross-talk

Leakage to higher levels is the main error when

want to eliminate

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Naive scheme:

CR at target Q freq

on control Q

Decoupling scheme1:

1.split CR in half

2.Echo control

3.flip sign of CR

4.Echo control

+CR -CR

+CR -CR

[1] A. D. Córcoles et al., PRA 87, 030301(R) (2013)

Harnessing cross-resonance: the ZX90 gate

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State

PreparationMeasurement

•Dimension grows exponentially with

number of qubits.

• State Preparation And Measurement

errors become main limitation for

reasonably good gates.

• Generate random sequences of gates from elements of the Clifford group

• Add an undo gate at the end of each sequence to return to the ground state.

• Measure the projection of the final state onto the ground state.

• Repeat for different sequence lengths, average over random realizations of sequences and

fit to model.

Measurement

Process Tomography Vs Randomized Benchmarking

QPT

RB1,2

Process

Efficient and estimates gate fidelity independent of SPAM errors!

[1] E. Knill et al, PRA 77, 012307 (2008) [2] E.M. et al, PRL 106, 180504 (2011)

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Measurement

with

for one qubit 24 Cliffords; 1.875 pulses per Clifford on average

Randomized Benchmarking: Single-qubit gates

The Clifford group is the normalizer of the Pauli group

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Ingredients:

• Single qubit gates

• Two-qubit entangling gate

Deterministically creating a set of Clifford operations spanning the two-qubit Clifford group

Can produce the two-qubit Cliffords in the following manner:

Single

qubit

gates

Single

qubit

gatesCNOT

Single

qubit

gates

2CNOTs

+

+Single

qubit

gates

3CNOTs+

576

576

5184

518411520

TWO

QUBIT

CLIFFORDS

Randomized Benchmarking: Two-qubit gates

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[1] J.M. Gambetta et al. PRA, 77, 012112 (2008)

Qubit QND measurements

Transmon-resonator Hamiltonian:

-Drive resonator and emitted trajectory1 corresponds to qubit measurement.

-State-dependent shift of resonator frequency.

-Characterize measurements by assignment fidelity

or full measurement tomography3 (takes back-action

into account).

[3] J.M. Chow et al. Nat. Comm. 5, 4015 (2014)

-Classification of measurement trajectories based on

machine learning algorithms2

[2] E.M. et al. arXiv:1411.4994 (2014)

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Surface code-Example of a “topological” quantum error-correcting (stabilizer) code1

[1] A. Kitaev, Russian Mathematical Surveys 52 (6) (1997)

N code qubits on the edges. Vertex stabilizer Plaquette stabilizer

X

X

X

XZ

ZZ

Z

Logical qubit - lattice defect (ignore stab.)

Logical Z - loop of Z operators around defect.

Logical X - chain of X operators connecting

defect to boundary.

Logical CNOT – braiding defects (topological).

Error-detection and correction – measure stabilizers and use weight

matching algorithms2 (classical post-processing) to determine error locations.

[2] W. Cook and A. Rohe, INFORMS J. Comput. 11, 138 (1999).

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• One logical qubit (defect) corresponds to 4 code plus 9 syndrome qubits.

• Repetitive measurements on syndrome qubits provides information for

error-correction.

MU

Surface code

A threshold exists:

-Introduce syndrome qubits to measure stabilizers in QND manner.

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Surface code: mapping the lattice

Want to reduce the

number of couplers

(bus resonators) per

qubit.

Two bus resonators per qubit yields required connectivity

Have a desired layout and a set of quantum components (qubits,

resonators etc). How do we map the layout onto an SC qubit architecture?

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Z X

• [[2,0,2]] error-detection experiment

-Stepping stone to full plaquette of 8 qubits

-Demonstrate projection onto either XX or ZZstabilizer of the same code qubits

Chow et al. Nat. Comm. 5, 4015(2014)

Surface code: mapping a 2x2 lattice

2 code and 2 syndrome qubits

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• 4 qubits, 4 bus resonators, 4 independent readouts

• 6.5-6.8 GHz readout frequencies

• 7.5-8 GHz bus frequencies

• anharmonicities ~ 330 MHz

Cavity (GHz) Qubit (GHz) T1 (ms) T2-echo(ms)

Q1 6.50 5.303 33 17

Q2 6.70 5.101 36 16

Q3 6.50 5.291 31 18

Q4 6.70 5.415 29 22

CR Pulse length (ns) Gate Fidelity

317 0.9390 CR12

300 0.9370 CR23

367 0.9410 CR34

158 0.9650 CR41

• Gate fidelities characterized via randomized benchmarking (RB)1.

• Single-qubit gate fidelities above 0.997.

Four qubit ring: device parameters

[1] E.M. et al, PRL 109, 080505 (2012)

2*Chi/(2*Pi) (MHz) Kappa/(2*Pi) (kHz)

Q1 -3.0 615

Q2 -2.0 440

Q3 -2.5 287

Q4 -2.8 1210

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Experimental schematic

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[[2,0,2]] error-detection protocol

• Contains many of the pieces required to demonstrate surface code operations in the full plaquette:

-Operations are performed in a planar array of qubits.

-QND stabilizer measurements via high-fidelity syndrome measurements.

[[n,k,d]] stabilizer code:

n- #physical qubits, k- #encoded qubits,

d- distance.

• [[2,0,2]] – detect arbitrary single qubit errors on a fixed code state

• Stabilizers: XX and ZZ .

• Measuring stabilizers project onto a Bell basis state – outcomes detect an error occurred.

In addition to demonstrating error-detection…

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ZZ Stabilizer XX Stabilizer

Stabilizers

-Measures bit parity

-Projects onto even and odd bit

parity subspaces

Even:

Odd:

-Measures phase parity

-Projects onto even and odd phase

parity subspaces

Even:

Odd:

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Stabilizers

Syndrome Code state

Orthonormal Bell basis!

-Syndromes map to

orthogonal states

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X Error

Error detection via Stabilizers

Z Error

Y Error

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Q1

Q2

Q3

Q4

(C1) (C2)

(S1)

(S2)

Protocol Implementation

Constructing CNOTs

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X 22

X 5

Single-qubit gates

Two-qubit gates

Refocus gates X 20XX Stabilizer

Protocol ImplementationZZ Stabilizer

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1. Perform state tomography of code state for various

errors (conditional on syndrome measurement results).

2. Track rotation errors as a continuous function of rotation

angle – demonstrate sinusoidal behavior of probabilities.

3. Apply more general unitary errors and observe

probabilities of syndrome measurement outcomes.

Basic outline of experiment

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−1.5 −1 −0.5 0 0.5 1 1.50

0.1

0.2Measurement Histograms

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

−1.5 −1 −0.5 0 0.5 1 1.50

0.1

0.2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.1

0.2

Integrated Value

-Bin shots of syndrome qubits

according to threshold values.

Reconstruct conditional states by

state tomography.

-Correlate shots1 of code qubits to

create a conditional measurement

vector y

-Want Pauli basis representation x

of the state:

-Constrain state to be positive semidefinite and solving a semidefinite program gives

a physical state.

-Simple linear inversion to obtain x: potentially unphysical state (not positive semidefinite).

~95.9%

~94.7%

~94.1%

~96.5%

M1

M2

M3

M4

ground excited

1. C. Ryan et al., arXiv 1310.6448 (2013)

Characterizing readout and state tomography

y and x related via:

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Q2 excited

Q2 ground

Q4 ground Q4 excited

M4 (Arbitrary Voltage)

M2 (

Arb

itra

ry V

oltage)

-1 0 1

-1

0

1

2

-2

Correlated histograms of syndromes

Example: No error

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0001

1011

0001

1011

-1

0

1

Real parts

0001

1011

0001

1011

-1

0

1

Imaginary parts

0001

1011

0001

1011

-1

0

1

Real parts

0001

1011

0001

1011

-1

0

1

Imaginary parts

State Fidelity

0.8491

State Fidelity

0.8046

State tomography of code qubits

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0001

1011

0001

1011

-1

0

1

Real parts

0001

1011

0001

1011

-1

0

1

Imaginary parts

0001

1011

0001

1011

-1

0

1

Real parts

0001

1011

0001

1011

-1

0

1

Imaginary parts

State Fidelity

0.8195

State Fidelity

0.8148


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Important point: A standard circuit simulation using

experimental parameters predicts state fidelities of ~0.75-

0.76.

State fidelities are ~0.81. Why?

-Errors in preparing the initial Bell state on the code

qubits show up as incorrect assignment of syndrome

measurement outcomes.

-Assuming the number of shots used to perform

tomography is large, the conditional state of the code

qubits is insensitive to state-preparation errors.

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Tracking Y-Errors

{0,-} and {1,+}

curves have this

form because ZZ

parity check

implemented first

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Tracking X-Errors

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Tracking Z-Errors

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Take into account assignment errors by normalizing by baseline probability

increased variance in data.

Decoherence during quantum process affects phase-flip errors the most.

Detecting arbitrary errors

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Moving forward

-Continue to design and build 2D networks of qubits to

realize larger subsections of the surface code.

Realize logical qubits and operations.

-Improve gate times, gate fidelities, and readout fidelities.

-Use both spatial and temporal information of errors for

minimum weight-matching algorithms.

Documents

Detecting arbitrary single-qubit errors in a planar …...© 2015 IBM Corporation Detecting arbitrary single-qubit errors in a planar sublattice of the surface code Easwar Magesan