Demonstration of a Quantum Error Detection Code Using a Square Lattice of Four Superconducting Qubits (2015) [0.2Cm] A.D. Córco

Demonstration of a quantum error detection code using a square lattice of four superconducting qubits (2015)

A.D. Córcoles, Easwar Magesan, Srikanth J. Srinivasan, Andrew W. Cross, M. Steffen, Jay M. Gambetta and Jerry M. Chow

Noah H. J. Halford, Jeffrey D. Mohan and Jann H. Ungerer

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 1 Outline Quantum Error Correction Syndrome extraction Stabilizer formalism Experimental Design Surface code Transmon qubits Physical realization Implemented quantum error correction scheme Functionality of stabilizers Detection of general errors Results Single-shot histograms and state tomography Conclusions

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 2 Classical error correction

• Classically, only one possible type of error: bit ﬂip (0 ↔ 1) • Easy solution: redundant coding, e. g.,

0L = 000 , 1L = 111

• Detect and ﬁx by parity checks (or counting)

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 3 Quantum error correction

• Can still detect bit ﬂips, e. g., by measuring parity with Z1Z2 and

Z2Z3 (same bit → +1):

Z1Z2 |000i = |000i

Z1Z2 |100i = − |100i . .

• Qubits have phase: how to correct for phase ﬂips?

Z1Z2 |000i = (+1) |000i

Z1Z2(− |000i) = (+1)(− |000i) . .

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 4 Quantum error correction

• Phase ﬂips mean we need more complex encoding schemes: this is the crucial part of QEC • Example: Shor code,

1 |0i = √ (|000i + |111i)(|000i + |111i)(|000i + |111i) L 2 2

• Bit ﬂips: Zi Zi+1 , Zi+1Zi+2

• Phase ﬂips: X1 ··· X6 , X4 ··· X9

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 5 Syndrome extraction

• Another problem: measurement of quantum states is destructive • Solution: measure correlations, output on “ancilla” qubits

Example: bit ﬂip on |ψi = α |000i + β |111i

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 6 Stabilizer formalism

• Stabilizer group

S = {S1, S2, ... , Sn} • Stabilizers commute, have eigenvalues ±1

• Space of codewords is intersection of λ = +1 eigenspaces of Si (stabilizers should not affect codewords) • Allows for speciﬁc errors, whose corresponding operators

anticommute with one Si

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 7 Stabilizer formalism

• Example: 3-qubit repetition code,

|0iL = |000i , |1iL = |111i

• Stabilizer group

S = hZ1Z2, Z2Z3i = {I, Z1Z2, Z2Z3, Z1Z3}

• Bit ﬂip on qubit i can be detected

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 8 Surface code

• Code qubits (purple) and syndrome/ancilla qubits (green: bit-ﬂip and yellow: phase-ﬂip) at sites of 2D lattice • Stabilizer code, nearest-neighbor connectivity, decoupled code qubits ⇒ high fault-tolerant error thresholds, scalable geometry

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 9 Transmon qubits

• {|0i , |1i} ≈ {|Ngi , |Ng + 1i} on island (Cooper pair box) + † − • Single-qubit gates via Jaynes-Cummings: gi (aσi + a σi ) • 2-qubit gates implemented by virtual photon exchange: 1 −1 −1 + − − + Hint = 2 gi gj (∆i + ∆j )(σi σj + σi σj ) • Single-shot readout of qubit state via dispersive phase shift 2 z converted via mixer to voltage signal: ωr + (gi /∆i )σi

QIP lecture slides, Blais et al., Phys. Rev. A (2007)

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 10 Physical realization

Primitive tile of surface code: 2 code qubits (next-nearest neighbors), each coupled to ancilla qubits via superconducting coplanar waveguides

100µm

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 11 Quantum error correction scheme

[2,0,2] code • prepare code word |ψi = √1 |00i + |11i and initialize ancilla 2 qubits as |0i and |+i = √1 |0i + |1i 2 • infuse error on one code qubit • parity check stabilizers ZZ and XX implemented with CNOT gates • syndrome readout on ancilla qubits D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 12 Parity check stabilizers and error detection

stabilizers |ψi = √1 |00i + |11i is eigenstate of ZZ and XX with eigenvalue 1 2

= I ⊗ |0ih0| ⊗ I + I ⊗ |1ih1| ⊗ X · |0ih0| ⊗ I ⊗ I + |1ih1| ⊗ I ⊗ X = |0ih0| ⊗ |0ih0| + |1ih1| ⊗ |1ih1| ⊗ I + |0ih0| ⊗ |1ih1| + |1ih1| ⊗ |0ih0| ⊗ X

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 13 Parity check stabilizers and error detection

action on qubits after error

error code qubits syndrome qubits I |00i + |11i |00i X |01i + |10i |10i Z |00i − |11i |01i Y |01i − |10i |11i

⇒ single-qubit error detection possible

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 14 General error detection

θ arbitrary single-qubit error U = exp −i 2 nˆ · ~σ

error syndrome probability of measuring 2 θ I 00 cos 2 2 θ 2 X 10 sin 2 nx 2 θ 2 Z 01 sin 2 ny 2 θ 2 Y 11 sin 2 nz

• discretization of errors via projective measurements

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 15 State tomography and single-shot histograms no error X error Z error Y error

|ψi = |00i + |11i |ψi = |01i + |10i |ψi = |00i − |11i |ψi = |01i − |10i

F = 0.849 F = 0.820 F = 0.805 F = 0.815

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 16 Single-shot correlated syndrome measurements for arbitrary errors

= Xθ = Zθ = Yθ

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 17 Single-shot correlated syndrome measurements for arbitrary errors

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 18 Conclusions

• implemented a scalable system using nearest-neighbor coupled superconducting qubits • implemented high-ﬁdelity one- and two-qubit gates • achieved high single-shot assignment ﬁdelities • enabled detection of arbitrary single-qubit errors via non-demolition measurements

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 19 References

[1] Córcoles, A. D. et al. Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nature communications 6 (2015). [2] Ristè, D. et al. Detecting bit-ﬂip errors in a logical qubit using stabilizer measurements. Nature communications 6 (2015). [3] Kelly, J. et al. State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519, 66–69 (2015). [4] Nielsen, M. A. & Chuang, I. Quantum computation and quantum information (2002). [5] Wallraff, A. Quantum information processing implementations lecture notes (2017). [6] Blais, A. et al. Quantum-information processing with circuit quantum electrodynamics. Phys. Rev. A 75, 032329 (2007).

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 20 Circuit decomposition

• State preparation of next-nearest neighbor code qubits (Q1, Q3)

is mediated by bit-ﬂip syndrome qubit (Q2) • Error ε controllably introduced via single-qubit rotations about

X, Y , or Z axes (e.g. Xπ/3 or XY ) • Stabilizers implemented with ZZ (bit-ﬂip) and XX (phase-ﬂip) gates

D PHYS N. H. J. Halford, J. D. Mohan, J. H. Ungerer May 29, 2017 21