The XZZX surface code Resilience of tailored codes to biased noise

arXiv:2009.07851 Steven Flammia AWS Center for Quantum Computing

Joint work with J. Pablo Bonilla-Ataides, David K. Tuckett, Stephen D. Bartlett, Benjamin J. Brown University of Sydney David Poulin

1 Concepts: Surface code (CSS)1,2,3

• Qubits on vertices • X-type (Z-type) stabilizers around dark (light) faces • for code states • Error induces defect at face where • Logical operators as shown

1 Kitaev, Ann. Phys. 303, 2 (2003). 2 Bravyi and Kitaev, arXiv:quant-ph/9811052 (1998). 2 3 Bombin and Martin-Delgado, Phys. Rev. A 76, 012305 (2007). Concepts: Decoding

• Minimum-weight matching • Maximum-likelihood (ML) (MWM)1 • Identify most probable logical • Pair X (Z) defects minimizing coset of equivalent errors distance over graph • Optimal by definition • Near-optimal for bit-flip noise • Efficient approximation using • Sub-optimal for tensor networks2,3,4 depolarizing • Efficient • Efficient fault-tolerant fault-tolerant extension not extension known

1 Dennis, Kitaev, Landahl, and Preskill, J. Math. Phys. 43, 4452 (2002). 2 Bravyi, Suchara, and Vargo, Phys. Rev. A 90, 032326 (2014); 3 Darmawan and Poulin, Phys. Rev. E 97, 051302 (2018). 3 4 Tuckett, Darmawan, Chubb, Bravyi, Bartlett, and Flammia, Phys. Rev. X 9, 041031 (2019). Concepts: Noise channels

• Bit-flip

• Phase-flip

• Depolarizing

4 Concepts: Noise channels

• Biased-noise, let

assume and define bias (e.g. Y-bias),

• Limits is depolarizing noise is pure Y noise • Similarly defined for X-biased and Z-biased noise

5 Concepts: Hashing bound

• Theorem (Hashing bound)1: • There exists a stabilizer quantum error-correcting code that achieves the hashing bound for a Pauli channel of the following form:

where and is the entropy of this probability vector. • Proof invokes random codes • Lower bound on possible threshold error rate for Pauli channel • Set (surface code), parametrize in terms of single-qubit error rate , and solve for • Depolarizing , Bit-flip / phase-flip • Open question: tight bounds on quantum capacity2,3

1 Wilde, Quantum Information Theory, Cambridge University Press (2013). 2 DiVincenzo, Shor, and Smolin, Phys. Rev. A 57, 830 (1998). 6 3 Kianvash, Fanizza and Giovannetti, arXiv:2008.02461 (2020). Surface code resilience Through the ages (2004 – 2019)

7 Surface code thresholds: Optimal decoding

• Bit-flip / phase-flip threshold (2004)1: ~11%

• Depolarizing threshold (2012)2: ~19%

• Biased noise (2018-19): next slide

1 Dennis, Kitaev, Landahl, and Preskill, J. Math. Phys. 43, 4452 (2002). 2 Bombin, Andrist, Ohzeki, Katzgraber, and Martin-Delgado Phys. Rev. X 2, 021004 (2012). 8 Surface code thresholds with biased noise1,2

• Hashing bound (gray line) • Bit-flip / Phase-flip ~11% • Depolarizing ~19% • Pure Y 50% • Y-biased thresholds track hashing bound

1 Tuckett, Bartlett, and Flammia, Phys. Rev. Lett. 120, 050505 (2018). 2 Tuckett, Darmawan, Chubb, Bravyi, Bartlett, and Flammia, Phys. Rev. X 9, 041031 (2019). 9 Y such high thresholds?1

Pure X logical Pure Y logical Ways to fail • Pure X noise:

• Pure Y noise:

Deform with Zero Y-stabilizers xxxxxxxxx X-stabilizers

1 Tuckett, Darmawan, Chubb, Bravyi, Bartlett, and Flammia, Phys. Rev. X 9, 041031 (2019). 10 Tailoring to dephasing noise1,2

• Dephasing (Z-biased) noise is dominant in many physical architectures

• Tailor surface code for Z-biased noise by simply exchanging Y ↔ Z

• Bias-preserving syndrome extraction is possible3,4

1 Tuckett, Bartlett, and Flammia, Phys. Rev. Lett. 120, 050505 (2018). 2 Tuckett, Darmawan, Chubb, Bravyi, Bartlett, and Flammia, Phys. Rev. X 9, 041031 (2019). 11 3 Puri, St-Jean, Gross et al, Science Adv. 6, eaay5901 (2020). 4 Guillaud and Mirrahimi Phys. Rev. X 9, 041053 (2019). The XZZX code (These are the new results!)

12 Concept: Single-qubit Pauli noise

• General single-qubit Pauli noise channel

where is a stochastic vector, i.e.

13 Concept: Single-qubit Pauli noise

• Visualizing threshold error rates Hashing bound

14 CSS surface code thresholds: Optimal decoding

• Achieves hashing bound for Y-biased noise (also )

• Well below hashing bound when X or Z errors dominate

• 211 thresholds (111 estimates)

15 Concepts: Surface code (XZZX)1

• Qubits on vertices

• XZZX stabilizers around all faces

• Logical operators as shown

1 X.-G. Wen, Phys. Rev. Lett. 90, 016803 (2003). 16 XZZX surface code thresholds: Optimal decoding

• Achieves hashing bound for all single-qubit Pauli noise channels

• Code with local stabilizers and efficient decoder (unlike random codes)

• Exceeds hashing bound in some cases?

17 Surface code thresholds – Hashing bound

CSS XZZX

18 XZZX surface code X- or Z-biased thresholds

• For high bias , we observe threshold estimates exceeding hashing bound

• Threshold is an asymptotic property • We investigate using sets of larger code distance (and converged ML decoder)

19 One surface code for every Pauli channel

20 Efficient biased-noise decoding Symmetry matching decoders

21 Concepts: Code symmetry MWM decoding1

• CSS code (toric for simplicity) • Symmetry of code: • For any error : • Parity conserved: errors induce even number of defects (-1 eigenvalues) • Pair defects and fuse to find correction • Treats X and Z errors separately • Does not handle Y errors well

1 Dennis, Kitaev, Landahl, and Preskill, J. Math. Phys. 43, 4452 (2002). 22 System symmetry MWM decoding1: CSS code

• CSS code with pure Y noise • Symmetry w.r.t. error model

• For any Y-error :

• Parity conserved: Y-errors induce even number of defects along rows (cols.) • Pair defects in rows (cols.) to construct error bounding cluster and fuse within cluster to find correction

1 Tuckett, Bartlett, Flammia, and Brown, Phys. Rev. Lett. 124, 130501 (2020). 23 System symmetry MWM decoding1: CSS code

• Finite bias • Low-rate X or Z errors weakly violate the system symmetry • Decoder weakly matches between rows (cols.)

• Fault-tolerance • Measurement errors violate the system symmetry • Symmetry is restored by repeating measurements and checking parity of time adjacent measurements2 1 Tuckett, Bartlett, Flammia, and Brown, Phys. Rev. Lett. 124, 130501 (2020). 24 2 Dennis, Kitaev, Landahl, and Preskill, J. Math. Phys. 43, 4452 (2002). System symmetry MWM decoding: XZZX code

• XZZX code with pure Z noise • Symmetry w.r.t. error model

• For any Z-error :

• Parity conserved: Z-errors induce even number of defects along diagonals • Pair and fuse defects along diagonals to find correction • Finite bias and FT tricks work as before

25 Efficient biased-noise decoding Exceptional fault-tolerant thresholds

26 Symmetry matching thresholds: q = 0

• Ideal measurements: • CSS code, Y-biased decoder1 • (tailored CSS code, Z-biased) • XZZX Y-biased similar • XZZX code, Z-biased decoder • Near hashing-bound for intermediate bias • Simpler, performs better

1 Tuckett, Bartlett, Flammia, and Brown, Phys. Rev. Lett. 124, 130501 (2020). 27 Symmetry matching thresholds: q = ph.r + pl.r.

• Phenomenological noise model • Faulty measurements:

• CSS code, decoding X / Z separately • Solid: optimal 3.3% for bit-flip1 • Dashed: MWM 2.93% for bit-flip2 • CSS code, Y-biased decoder3 • (tailored CSS code, Z-biased) • XZZX code, Z-biased decoder • For depolarizing, it is MWM • For pure Z, exceeds 9%

1 Ohno, Arakawa, Ichinose, and Matsui, Nucl. Phys. B697, 462 (2004). 2 Wang, Harrington, and Preskill, Ann. Phys. 303, 31 (2003). 28 3 Tuckett, Bartlett, Flammia, and Brown, Phys. Rev. Lett. 124, 130501 (2020). Exceptional fault-tolerant thresholds

CX gate infidelity CX gate infidelity 0.015 0.025 0.035 0.045 0.05 0.10 0.15 0.20

10-1

10-2

10-3

10-4 Logical error probability -4 0.6 0.8 1.0 1.2 1.4 1.6 2.0 3.0 4.0 5.0 6.0 7.0 8.0 10

29 Bias-preserving fault-tolerant computation

30 Concept: Lattice surgery1,2,3

• Figures adapted from Game of Surface Codes - Litinkski (2019)4 (CC BY 4.0)

• FT computation • FT Initialization • FT Pauli product measurements (via lattice surgery) • Magic state distillation 1 Horsman, Fowler, Devitt, and Meter, New J. Phys. 14, 123011 (2012). 2 Yoder and Kim, Quantum 1, 2 (2017). 31 3 Litinski and von Oppen, Quantum 2, 62 (2018). 4 Litinski, Quantum 3, 128 (2019). FT initialization: XZZX code

Preparing 2-qubit patch in • Initialize red (blue) qubits in X (Z) basis • System is in +1 eigenstate of shaded faces and logical Z • We can correct Z (X) errors on red (blue) qubits; X (Z) errors act trivially • Measure stabilizers to project onto eigenstate of unshaded faces

32 FT Pauli product measurement: XZZX code

• Measuring of logical qubit and ancilla • Initialize blue qubits in Z basis (+1 eigenstate of boundary faces) • Product of dark faces yields measurement • Twist introduces a branch in the symmetry requiring minor decoder adaptation

33 Bias-preserving fault-tolerant computation

34 The XZZX surface code is awesome! arXiv:2009.07851

There are still qualitative improvements to the surface code waiting to be discovered!

35

XZZX surface code X- or Z-biased thresholds

Maximum Likelihood decoder convergence analysis 37 Sub-threshold scaling

• With XZZX code, only logical operator of Pauli-Z has weight • At infinite bias, failures due to high-rate errors1 • At finite bias, this effect persists for high bias, large and small codes • Plot • Solid lines: fit to • Dashed lines: fit to • Although a small-size effect, this regime may be relevant to early surface code experiments

1 Tuckett, Darmawan, Chubb, Bravyi, Bartlett, and Flammia, Phys. Rev. X 9, 041031 (2019). 38 Sub-threshold scaling

• At small and modest , or larger codes, we expect most failures due to O(d) mix of low and high-rate errors • Ansatz

where is an entropy term with1 • Plot • Solid lines: ansatz with • Decoder corrects low-rate errors in presence of more high-rate errors • Loosely-speaking, effective physical error rate is

1 Beverland, Brown, Kastoryano, and Marolleau, J. Stat. Mech. 2019, 073404 (2019). 39 Symmetry matching thresholds: XZZX code with circuit noise

Plots courtesy of Shruti Puri, see her FTQT talk for full details • Generic syndrome extraction noise model Idle qubit bias=50 • CX, CZ, initialization, 10-1 measurement 10-2 10-1 • Ordinary CX gates 10-3 • Reduce bias, mapping high- rate errors to low-rate errors -4 10 10-2 Bias preserving CX Ordinary CX • Bias-preserving CX gates Logical error probability 0.014 0.016 0.018 0.020 0.022 0.014 0.016 0.018 0.020 0.022 • Higher thresholds and lower CX, CZ gate infidelity CX, CZ gate infidelity logical failure rates Measurement error=Gate infidelity/2 • Such gates are possible with certain bosonic cat qubits1,2

1 Puri, St-Jean, Gross et al, arXiv:1905.00450 (2019). 2 Guillaud and Mirrahimi Phys. Rev. X 9, 041053 (2019). 40 Symmetry matching thresholds: XZZX code with circuit noise

Plots courtesy of Shruti Puri, see her FTQT talk for full details • Noise model for cat qubit in driven transmon CX gate infidelity CX gate infidelity • Photon number 0.015 0.025 0.035 0.045 0.05 0.10 0.15 0.20 • (left) • (right) -1 10 • Single photon loss rate 10-2 • Lower x-axis 10-3 • tolerates 6% CX error 10-4 • e.g. transmon system with Logical error probability -4 nonlinearity of 10 MHz 1.0 1.4 2.0 3.0 4.0 5.0 6.0 7.0 8.0 10 0.6 0.8 1.2 1.6 would need bare transmon lifetime of ~64 µs • Proof-of-concept Kerr-cat experiment1 achieved ~15 µs 1Grimm, Frattini, Puri et al, Nature 584, 205–209 (2020). 41