towards practical classical processing for the surface code arxiv:1110.5133 austin g. fowler, adam...

Towards practical classical processing for the surface code

arXiv:1110.5133

Austin G. Fowler, Adam C. Whiteside, Lloyd C. L. Hollenberg

Centre for Quantum Computation and Communication TechnologySchool of Physics, The University of Melbourne, Australia

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Overview

• why study the surface code?• what is the surface code?• classical processing challenges

– correcting errors fast enough– interpreting logical measurements

• complexity optimal error processing– performance in detail– 3 second fault-tolerant distance 1000

• summary and further work

Why study the surface code?• no known parallel arbitrary

interaction quantum computer (QC) architecture

• many 2-D nearest neighbor (NN) architectures– ion traps, superconducting

circuits, quantum dots, NV centers, neutral atoms, optical lattices, electrons on helium, etc

• concatenated codes perform poorly on 2-D NN architectures– Steane, Bacon-Shor pth~2x10-5

– distance d = 9 Steane code currently requires 2304 qubits

– d = 27 requires 100k+– nq = d3.52

• surface code performs optimally on a 2-D NN QC

• surface code has– pth~10-2, so lower d required– low overhead implementations

of the entire Clifford group– flexible, long-range logical

gates– distance 9 surface code

currently requires 880 qubits– distance 10 requires 1036– distance 27 requires 4356– nq = 6d2

– lower overhead and 10x higher pth than any other known 2-D topological code

The surface code is by far the most promising and practical code currently known.

Surface code: memory

• a minimum of 13 qubits (13 cells) are required to create a single fault-tolerant surface code memory

• the nine circles store the logical qubit, the four dots are repeatedly used to detect errors on neighboring qubits

• an arbitrary single error can be corrected, multiple errors can be corrected provided they are separated in space or time

• in a full-scale surface code quantum computer, a minimum of 72 qubits are required per logical qubit

• the error correction circuitry is simple and transversely invariant

detect X error detect Z error

minimum-size logical qubit scalable logical qubit

order of interactions

Single logical qubit gates• when using error

correction, only a small set of single logical qubit gates are possible

• for the surface code, these are, in order of difficulty, X, Z, H, S and T

• X and Z are applied within the classical control software

• H can be applied transversely after cutting the logical qubit out of the lattice

• S and T involve the preparation of ancilla states and gate teleportation

• the ancilla states must be distilled to achieve sufficiently low probability p of error

• an efficient, non-destructive S circuit exists, removing the need to reinject and redistill Y states

Hadamard after cutting (stop interacting with qubits outside the green square)

state distillation p → 7p3

gate teleportation S, T, RZ()

ancilla state required for S

ancilla state required for T

non-destructive S circuit

Surface code CNOT• there are two types of

surface code logical qubit, rough and smooth

• the simplest CNOT (CX) has smooth control and rough target

• CX can be defined by its effect on certain operators

• given a state XI = , CX = CXXI = CXXICXCX = XXCX

• XI→XX, IX→IX, ZI→ZI, IZ→ZZ defines CX

• moving (braiding) the holes (defects) that define logical qubits deforms their associated logical operators, performing computation

• movement is achieved by dynamically changing which data qubits are error corrected

• more complex braid patterns are required for rough-rough CX

smooth qubit

rough qubit

smooth-rough CNOT

CNOT braid patterns

Long-range CNOT• stretching the braid

pattern enables arbitrarily long-range CNOT

• the need to propagate error correction information near each braid prevents violations of relativity

• Clifford group computations can proceed without waiting for this information

• this enables CNOTs to be effectively bent backwards in time

• note that single control, multiple target CNOT is a simple braid pattern

• arbitrary Clifford group computations can be performed in constant time

Classical processing challenges• correcting errors fast enough

– a single round of surface code QEC can involve as few as five sequential quantum gates

– depending on the underlying technology, this may take less than a microsecond

– need to be able to process a complex, infinite size graph problem in a very small constant time

• interpreting logical measurements– surface code quantum

computing is essentially a measurement based scheme

– logical gates, including the identity gate, introduce byproduct operators

– determining what these byproduct operators are is exceedingly difficult, especially after optimizing a braid pattern

5

3

7

12

4

6

Correcting errors fast enough• consider the life cycle

of a single error• in the bulk of the

lattice, a single gate error is always detected at two space-time points

• given a detailed error model for each gate, the probability that any given pair of space-time points will be connected by a single error can be calculated

• these probabilities can be represented by two lattices of cylinders

• recently completed open source tool to perform such error analysis and visualisation

detect X error detect Z error

• the primal (Z errors) and dual (X errors) lattices are independent deterministically constructed objects

• terminology: dots and lines• weight of line –ln(pline)• stochastically detected

errors are represented by vertices associated with specific dots

• edges between vertices have weight equal to the minimum weight path through the lattice

• by choosing a minimum weight matching of vertices, corrections can be applied highly likely to preserve the logical state

Correcting errors fast enough

• minimum weight perfect matching was invented by Jack Edmonds in 1965

• very well studied algorithm, however best publicly available implementations have complexity O(n3) for complete graphs

• don’t support continuous processing

• don’t support parallel processing• actually quite slow

Correcting errors fast enough

• as recently as this year [PRA 83, 020302(R) (2011)], distance ~ 10 was the largest surface code that had been studied fault-tolerantly

• renormalization techniques have been used to study large non-fault-tolerant surface codes [PRL 104, 050504 (2010), arXiv:1111.0831]

• need something much better if surface code quantum computer is to be built

• to describe our fast matching algorithm, replace complex 3-D lattice with simple uniform weight 2-D lattice (grey lines)

• vertices (error chain endpoints) are represented by black dots

• matched edges are thick black lines

• shaded regions are space-time locations the algorithm has explored

Correcting errors fast enough

Complexity optimal error correction

Complexity optimal error correction

choose a vertex

Complexity optimal error correction

explore local space-time region until other objects encountered

Complexity optimal error correction

if unmatched vertices encountered, match with one

Complexity optimal error correction

choose another vertex

Complexity optimal error correction

expand until other objects are encountered, build alternating tree

1

2

Complexity optimal error correction

it alternating tree outer space-time regions can’t be expanded, form blossom

1

Complexity optimal error correction

uniformly expand space-time region around blossom until other objects encountered

1

Complexity optimal error correction

two options in this case, unmatched vertex or boundary, choose vertex

1

Complexity optimal error correction

choose another vertex

Complexity optimal error correction

form alternating tree

1

2

Complexity optimal error correction

expand outer nodes, contract inner nodes

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2

Complexity optimal error correction

form blossom

1

Complexity optimal error correction

grow alternating tree

1 2

Complexity optimal error correction

expand outer nodes, contract inner nodes, forbidden region entered

1 2

Complexity optimal error correction

undo expand outer, contract inner

1 2

Complexity optimal error correction

undo grow alternating tree

Complexity optimal error correction

undo form blossom

Complexity optimal error correction

undo expand outer, contract inner... done until have additional data

Complexity optimal error correction• noteworthy features:

– many rules, but each rule is simple and cheap

– on average, each vertex only needs local information

– fewer vertices = faster, simpler and more local processing

– algorithm remains efficient at and above threshold

• average runtime is O(n) per round of error correction

• the runtime is independent of the amount of history, which can be discarded after a delay, finite memory required

• algorithm can be parallelized to O(1) using constant computing resources per unit area (unit width in the example)

• current implementation is single processor only

Performance in detail• we simulate surface codes

of the form shown on the right

• depolarizing noise of strength p is applied after every unitary gate, including the identity

• measurement reports the wrong eigenstate with probability p but leaves the state in the reported eigenstate

• for each d and p, our simulations continuously apply error correction until 10,000 logical errors have been observed

• the probability of logical error per round of error correction is then calculated and plotted

Performance in detail

0.1

0.5

0.2%

0.5%

Performance in detail• matching, while still efficient, is much

slower around and above the threshold error rate

• very large blossoms appear• at d = 55, p = 10-2, approximately 6

GB of memory required• at an error rate p = 10-3, it is

straightforward to simulate d = 1000• over 4 million qubits• 3 seconds per round of error

correction• approximately 100 µs per vertex pair,

at least 90% of that is memory access time

• parallelized, this would be sufficiently fast for many quantum computer architectures

• but not all...

• achieving sufficiently fast, high error rate, fault-tolerant, parallel surface code quantum error correction is extremely challenging

• without further work, classical processing speed will limit the maximum tolerable error rate

Summary and further work• complexity optimal algorithm

– O(n2) serial– O(1) parallel

• accurately handle general error models• sufficiently fast at p = 10-3 for many

architectures• next 12 months:

– parallelize the algorithm– benchmark on commercially available

parallel computing hardware– reduce memory usage, improve core speed,

raise practical p– take error correlations into account– develop algorithms capable of simulating

braided logical gates– get building!