The XZZX surface code

Benjamin J. Brown

arXiv:2009.07851

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With thanks to my coauthors:J. Pablo Bonilla Ataides,David K. Tuckett,Stephen D. Bartlettand Steven T. Flammia

Scaling quantum computersTo scale a quantum computer we must:

Control a large number of qubits below threshold as they performrepeated stabilizer measurements1,2.

Furthermore, there is a huge overhead cost to performfault-tolerant quantum logic operations.3

1M. Takita et al., Phys. Rev. Lett. 119, 180501 (2017)2C. Kraglund Andersen et al, Nat. Phys. 16, 875 (2020)3C. Gidney and A. Fowler, Quantum 3, 135 (2019)

Scaling quantum computersCat qubits 4, 5 (that experience highly biased noise) have beenproposed as high-quality qubits (with bias preserving gates 6, 7) forsurface code implementations8.

=

= DC flux-bias for the SNAIL

= out-of-plane interconnectthrough which all the microwave drives (such as the two-photon drives, coupling drives for CX gates and drives for single-qubit gates) are applied

= resonator for qubit readout

SNAIL

4 μm

d

a

2 cm

Time

c

b

e f

4A. Grimm et al. Nature 584, 205 (2020)5R. Lescanne et al., Nat. Phys. 16, 509 (2020)6J. Guillaud and M. Mirrahimi, Phys. Rev. X 9, 041053 (2019)7S. Puri et al. Sci. Adv. 6, eaay5901 (2020)8C. Chamberland et al. arXiv:2012.04108 (2020)

The XZZX surface code

The XZZX surface code9 is locally equivalent to the standardsurface code10

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9X.-G. Wen, Phys. Rev. Lett. 90 16803 (2003)10A. Kitaev, Ann. Phys. 303, 2 (2003)

The XZZX surface code

. . . except every other qubit is rotated by a hadamard gate.(the green qubits are rotated)

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The XZZX surface codeThe XZZX code demonstrates:

I Thresholds that match the hashing bound for all uniformsingle-qubit Pauli noise channels.

I Exceptionally high fault-tolerant thresholds for biased noisemodels

I Reduced overheads (by a factor O(1/ηd/4

)) at low error rates

and high bias η

I We also argue that we can maintain these advantages whileperforming computation

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Error correction with the surface code

We use stabilizers to measure (Pauli) errors E .

SE |ψ〉 = (−1)E |ψ〉

On the surface code, errors appear as strings, and defects appearin pairs at their endpoints11.

This can be viewed as a defect parity conservation law12,13.11E. Dennis et al., J. Math. Phys. 43, 4452 (2002)12A. Kitaev, Ann. Phys. 303, 2 (2003)13BJB and D. J. Williamson, Phys. Rev. Research 2, 013303 (2020)

Symmetries and conservation lawsMWPM is possible because the surface code respects a defectconservation symmetry14,15

Mathematically, the symmetry is apparent in the stabilizer group∏v

Av = 1 ⇒∏v

av = 1

(av = ±1 eigenvalues of stabilizers Av . )

⇒ #v with av = −1 is even ≡ defect conservation (mod2)

14E. Dennis et al., J. Math. Phys. 43, 4452 (2002)15BJB and D. J. Williamson, Phys. Rev. Research 2, 013303 (2020)

A symmetry with respect to Pauli-Z errors16, 17

We find a richer space of symmetries if we restrict our error model.

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Pauli-Z errors commute with the product of stabilizers along thediagonal and therefore respect a 1D symmetry on the XZZX code.

Decoding Z errors is therefore equivalent to decoding therepetition code (50% threshold).

16BJB and D. J. Williamson, Phys. Rev. Research 2, 013303 (2020)17D. K. Tuckett, Phys. Rev. Lett. 124, 130501 (2020)

Symmetries for all Pauli errors18, 19

The XZZX code has many symmetries for other error models.

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The XZZX code has symmetries with respect to Pauli-X errorsalong perpendicular diagonals to those for Pauli-Z errors.

Pauli-Y errors have the same symmetry as the CSS surface code(we exploited these symmetries in earlier work)

18BJB and D. J. Williamson, Phys. Rev. Research 2, 013303 (2020)19D. K. Tuckett, Phys. Rev. Lett. 124, 130501 (2020)

High threshold error ratesThe XZZX code has a threshold that matches the zero-ratehashing bound for all uniform single-qubit Pauli channels.

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Threshold

errorrate:pc

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Threshold

errorrate:pc

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pc

XZZX CSS

R ≡ k/n ≥ 1− H(~p), (1)

H(~p) - Shannon entropy.

~p = (pX , pY , pZ ) with pO the probability that Pauli error O occurs.

R - rate with R = 0 the zero-rate hashing bound.

Numerics exceeding hashing

At high bias, our numerics demonstrate a threshold above thezero-rate hashing bound

100 101 102 1030.1

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Bias: η

Threshold

errorrate:pc

XZZX

CSS

Hashing

∞13,17,21,25

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29,33,37,41

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Code distances used in pc estimationpc−

ph.b

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η = 30

η = 100

η = 300

η = 1000

See also other work on exceeding hashing for instance20,21,22

20D. DiVincenzo et al., Phys. Rev. A 57, 830 (1998)21G. Smith and J. Smolin, Phys. Rev. Lett. 98, 030501 (2007)22J. Bausch and F. Leditzky, arXiv:1910.00471 (2019)

Exceeding hashing

A zero-rate code with a threshold above hashing implies we cansend information at non-zero rate with p above hashing.

We need the following:

I A finite rate code Rout = Kout/Nout > 0

I An inner code of constant size Nin with threshold pth. > ph.b.

Concatenating gives a family of codes with rate:

R ′ = Rout/Nin > 0. (2)

using qubits with ph.b. < p < pth. for some constant Nin.

Numerics exceeding hashing

We also exceed hashing with a suboptimal matching decoder

100 101 102 1030.1

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Bias: η

Thresholderrorrate:pc

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CSS

Hashing

∞

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dZ

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(b)

(c)

We achieved this using XZZX codes with different boundaryconditions.

The high-speed matching decoder allows us to probe very largesystem sizes

Fault-tolerant thresholds with biased noise

Matching decoders generalise readily to the fault-tolerantsetting.23, 24, 25

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Standard decoder

∞

We find exceptional fault-tolerant thresholds for the XZZX codeunder phenomenological biased noise.

23E. Dennis et al., J. Math. Phys. 43, 4452 (2002)24BJB and D. J. Williamson, Phys. Rev. Research 2, 013303 (2020)25D. K. Tuckett, Phys. Rev. Lett. 124, 130501 (2020)

Logical failure rates below thresholdWe need a low logical failure rate Pwith a small number of qubits n.

At low error rate p we can generically achieve

P ∼ pd/2 (3)

To leading order, this is the probability of d/2 errors occurring.

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The surface code uses n = O(d2) qubits.

Logical failure rates below threshold

We can choose an XZZX code with a weight n Pauli-Z logical.

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At infinite bias26, we can expect a logical failure rate like

P ∼ pn/2. (4)

. . . but what happens at finite bias?

26D. Tuckett et al. Phys. Rev. X 9, 041031 (2019)

Logical failure rates below threshold

In practice we find errors ofd/4 low rate errorsand d/4 high rate errors.

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P ∼ pd/4︸︷︷︸d/4 highrate errors

×(p

η

)d/4

︸ ︷︷ ︸d/4 low

rate errors

=

(1

η

)d/4

︸ ︷︷ ︸factorof

improvement

pd/2. (5)

Logical failure rates below thresholdWe test the ansatz at low p using the splitting method27,28

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Physical error rate: p

Logicalerrorrate:P

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d = 15

P ∼ pd/4︸︷︷︸d/4 highrate errors

×(p

η

)d/4

︸ ︷︷ ︸d/4 low

rate errors

=

(1

η

)d/4

︸ ︷︷ ︸factorof

improvement

pd/2. (6)

27C. Bennett, J. Comput. Phys. 22, 245 (1976)28Bravyi and Vargo, Phys. Rev. A 88,062308 (2013)

Logical failure rates below thresholdFor small n and high bias we identify signatures of pd

2logical

failure rate scaling

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Code distance: d

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rate:P

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−0.2

log(p/(1− p))

LineGradients

We observe this when

pd2 �

(1

η

)d/4

pd/2 (7)

(in other words, at moderate p and small-ish n)

Logical failure rates below thresholdIt is favourable to find codes with open boundary conditions.

Changing the lattice geometry means we have logical failure rateslike

PX ∼(p

η

)dX /2

and PZ ∼ pdZ/2, (8)

at low p.

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We can therefore save resources by choosing small dXwithout compromising PZ .

Fault-tolerant quantum computation

We can perform computations using code deformations29,30, e.g.,

Braiding twists31,32

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Lattice surgery33

29R. Raussendorf et al. Phys. Rev. Lett. 98, 190504 (2007)30H. Bombin and M. A. Martin-Delgado, J. Phys. A 42, 095302 (2009)31H. Bombin, Phys. Rev. Lett. 105, 030403 (2010)32BJB et al., Phys. Rev. X 7, 021029 (2017)33C. Horsman et al., New J. Phys. 14, 123011 (2012)

Fault-tolerant quantum computation

We can maintain these advantages undergoing fault-tolerantquantum computation34.

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The one-dimensional symmetries we need for high thresholds aremaintained under initialisation and XZZX codes with twists.

34D. Litinski, Quantum 3, 128 (2019)

Forthcoming work:XZZX thresholds with cat qubit circuits

1.61.41.21.00.8 10-4

4.03.53.02.52.0 10-44.5

1-photon cat

6.25-photon cat

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pro

bab

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Total CX error probability

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Total CX error probability

Logic

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Ordinary CX

work with A. Darmawan, A. Grimsmo, S. Puri and D. Tuckett.

Outlook

We have shown the XZZX code:

I has threshold error rates that match hashing

I has exceptionally high fault-tolerant thresholds

I significantly reduced resource costs below threshold

I can maintain its performance during computational operations

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Outlook

Our work raises questions in several areas, such as:

I Have we really exceeded the hashing bound?

I How do we specialise codes for a given noise model in general?

I What is the potential for non-CSS codes?

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Backup slide: Different geometriesPrevious work35 has shown negligible difference in logical errorrates against bit-flip noise for modest p as a function of number ofqubits n.

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B.V. method

Low

est-order

approxim

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Generalised path-counting model

0.05 0.06 0.07 0.08 0.09 0.10p

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log10P

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0.9951.0001.0051.0101.0151.0201.025

p/%P

◆/P

■

35M. Beverland et al. J. Stat. Mech.:Theo. Exp. 2019, 073404 (2019)