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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
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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
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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.
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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
State tomography of code qubits
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State tomography of code qubits
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.