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